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Distributed edge and contour line detection for environmental monitoring with wireless sensor networks

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Distributed edge and contour line detection for environmental monitoring with wireless sensor networks

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Content DISTRIBUTED EDGE AND CONTOUR LINE DETECTION FOR ENVIRONMENTAL MONITORING WITH WIRELESS SENSOR NETWORKS by Pei-kai Liao A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful¯llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) May 2007 Copyright 2007 Pei-kai Liao Dedication This dissertation is dedicated to my parents, Chiao-mao Liao and Yu-hua Chen. ii Acknowledgements Thanks to many people's help, I am able to overcome di±culties and make this thesis possible. It is my pleasure to thank all the people who once helped me. First, I would like to show my greatest appreciation to my parents, Chiao-mao Liao and Yu-hua Chen. Many thanks for their continual support and encouragement during my Ph.D. study days. It's my pleasure to share the honor with them at this moment. Then, I want to thank my advisor, Dr. C.-C. Jay Kuo. Without his kind guidance, I won't be able to complete my Ph.D. thesis. In addition to academic knowledge and professionalskills,healsoteachesandin°uencesmealotinotheraspectssuchasworking attitude and research spirit. He is the person I respect the most. I also want to thank my thesis committee, Dr. Antonio Ortega and Dr. Ramesh Govindan. The comments they gave really inspire me a lot and thus improve the thesis. Special thanks to my good friend, Dr. Min-kuan Chang. He was my mentor when I was a junior member of the research lab. It was his assistance which allowed me to start my ¯rst step of thesis writing. Finally, many thanks to my colleagues in Multimedia Communications Lab and all good friends I met in LA. I will always remember those days we have spent together. iii Table of Contents Dedication ii Acknowledgements iii List Of Tables vii List Of Figures viii Abstract xi Chapter 1: Introduction 1 1.1 Signi¯cance of the Research . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Statistical Edge Region Detection under Neyman-Pearson (NP) Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Contour Line Extraction with Heterogeneous Sensor Networks . . 4 1.1.3 Mesh-based Contour Lines Estimation with Homogeneous Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Contributions of the Research . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Chapter 2: Background Review 11 2.1 Review of Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 The Statistical Approach . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.2 The Filter-based Approach . . . . . . . . . . . . . . . . . . . . . . 13 2.1.3 The Classi¯er-based Approach . . . . . . . . . . . . . . . . . . . . 15 2.2 Data Acquisition and Collection in Sensor Networks . . . . . . . . . . . . 17 2.3 Review on Hypothesis Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.1 Simple Hypothesis Test and Neyman-Pearson Lemma . . . . . . . 19 2.3.2 Composite Hypotheses and Uniformly Most Powerful Tests . . . . 20 2.4 Voronoi Diagram and Delaunay Triangulation . . . . . . . . . . . . . . . . 21 2.4.1 Voronoi Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.2 Delaunay Triangulation . . . . . . . . . . . . . . . . . . . . . . . . 24 iv Chapter 3: StatisticalEdgeRegionDetectionunderNeyman-Pearson(NP) Optimality Using Wireless Sensor Networks 27 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2.1 Sensor Measurement Model . . . . . . . . . . . . . . . . . . . . . . 30 3.2.2 Edge Region De¯nition . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Proposed Method for Edge Region Detection . . . . . . . . . . . . . . . . 34 3.3.1 Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.2 Statistical Decision-Fusion Method under NP Optimality . . . . . 35 3.3.3 Extension to Noisy Scalar-¯eld Sensor Measurement Data . . . . . 55 3.4 Simulation Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 58 3.4.1 Computational Complexity . . . . . . . . . . . . . . . . . . . . . . 59 3.4.2 Sensitivity to Channel Noise Interference . . . . . . . . . . . . . . 61 3.4.3 Sensitivity to Sensor Location Error . . . . . . . . . . . . . . . . . 63 3.4.4 Simulation of Edge Region Detection . . . . . . . . . . . . . . . . . 64 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Chapter 4: DistributedContourLinesExtractionwithHeterogeneousSen- sor Networks 71 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2 System Model and Problem Formulation . . . . . . . . . . . . . . . . . . . 74 4.2.1 System Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3 Proposed Scheme to Contour Line Extraction . . . . . . . . . . . . . . . . 75 4.3.1 Extrema Localization . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3.2 Contour Points Determination . . . . . . . . . . . . . . . . . . . . 83 4.3.3 Contour Line Construction . . . . . . . . . . . . . . . . . . . . . . 87 4.4 Communication Cost Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.5 Statistical Analysis of Estimation Accuracy . . . . . . . . . . . . . . . . . 91 4.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.8 Proof I: Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . 104 4.9 Proof II: Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . 105 4.10 Proof III: Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . 107 4.11 Proof IV: Derivation of R N;lk . . . . . . . . . . . . . . . . . . . . . . . . . 110 Chapter 5: A Mesh-based Approach to Contour Lines Estimation with Cross-layer Considerations Using Homogeneous Sensor Net- works 111 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.2 System Model and Model Parameters . . . . . . . . . . . . . . . . . . . . 114 5.2.1 System Model and Problem Formulation . . . . . . . . . . . . . . . 114 5.2.2 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.3 Cross Layer Considerations in System Design . . . . . . . . . . . . . . . . 118 v 5.3.1 Physical and MAC Layers Consideration . . . . . . . . . . . . . . . 118 5.3.2 Network Layer Consideration . . . . . . . . . . . . . . . . . . . . . 122 5.3.3 Adaptive Transmission with Cross-Layer Interaction . . . . . . . . 123 5.4 2-D Triangular Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . 124 5.5 Mesh-based Contour Line Estimation . . . . . . . . . . . . . . . . . . . . . 127 5.5.1 Localized Candidate Detection of Contour Points . . . . . . . . . . 128 5.5.2 Contour Line Construction . . . . . . . . . . . . . . . . . . . . . . 135 5.6 Simulation Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 141 5.6.1 Contour Nodes Inference. . . . . . . . . . . . . . . . . . . . . . . . 141 5.6.2 Contour Line Estimation . . . . . . . . . . . . . . . . . . . . . . . 143 5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Chapter 6: Conclusion and Future Work 155 6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Bibliography 159 vi List Of Tables 3.1 ComparisonofcalculatedandsimulatedP FA F foralgorithmIwhenSNR= 30dB, T 0 L =175, BER=0:001 and R=r =0:5 . . . . . . . . . . . . . . . 49 3.2 ComparisonofcalculatedandsimulatedP FA F foralgorithmIIwhenSNR= 30dB, T 0 L =175, BER=0:001, R=0:75 and r =0:5. . . . . . . . . . . . . 55 4.1 Comparison of simulated and theoretical MSE values for the contour line representingsignalvalue0:8withlocalmaximumvalue=1,noisepower=0:01, R c =0:5 and h=0:1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.1 Comparison of the centralized scheme and the proposed approach when the number of deployed sensor nodes is M . . . . . . . . . . . . . . . . . . 149 vii List Of Figures 2.1 An edge node and its relationship to an edge line.. . . . . . . . . . . . . . 12 2.2 Illustration of the classi¯er-based approach: (a) the edge node case and (b) the non-edge node case. . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 An example of the Voronoi diagram. . . . . . . . . . . . . . . . . . . . . . 22 2.4 An example of Delaunay triangulation (solid lines) and its geometry dual (dot lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1 Illustration of the error distance between the true and the marked edge regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Illustrationofedgeandnon-edgenodeswhentheprobingrangeRislarger than the tolerance range r. . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Thetrendoftheteststatisticin(3.9)whentotalnumber N ofneighboring nodesinsidetheprobingrangeofatargetsensornodeis18,localdetection rate P D is 0:9990 and local false alarm rate P FA is 0:0012. . . . . . . . . . 41 3.4 Comparison of true and approximated values of the test statistic in (3.9) when total number N of neighboring nodes inside the probing range of a target sensor node is 18, local detection rate P D is 0:9990 and local false alarm rate P FA is 0:0012. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.5 Illustration of the target sensor node, the probing range, the tolerance range and a locally straight edge line. . . . . . . . . . . . . . . . . . . . . 53 3.6 Performance comparison of algorithm I (R=0:5) and II (R=0:75) when r =0:5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.7 PerformancecomparisonofalgorithmsIIandclassi¯er-basedmethodunder di®erent bit error rates when the tolerance radius r is 0:5. . . . . . . . . . 67 viii 3.8 PerformancecomparisonofalgorithmsIIandclassi¯er-basedmethodwhen BER= 0 under di®erent levels of average location error 0, 0:1r, 0:2r, 0:3r and 0:4r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.9 Edge region detection for binary-value signal with R = 0:75, r = 0:5, where \o" represents the detected edge nodes, \x" represents the sensor nodes detected as inside the event region, \+" represents the sensor nodes detected as outside the event region, solid lines are true edges and dash lines are the limit of the de¯ned edge region. . . . . . . . . . . . . . . . . 69 3.10 Edge region detection for real-number signal with A tr = 0:5, R = 0:75, r = 0:5, where \o" represents the detected edge nodes, \x" represents the sensor nodes detected as inside the event region, \+" represents the sensor nodes detected as outside the event region, solid lines are true edges and dash lines are the limit of the de¯ned edge region. . . . . . . . . . . . . . 70 4.1 Illustration of coarse localization of local extreme points. . . . . . . . . . . 78 4.2 Sequential search of the extremum point. . . . . . . . . . . . . . . . . . . 81 4.3 Illustration of subclusters, pseudo-beams and pseudo-grids in a cluster. . . 85 4.4 Illustration of pseudo-grids and a true contour point in one pseudo-beam. 92 4.5 Comparisonofthenormaldistributionandthenoncentralchi-squareddis- tribution with thedegree offreedom N =10 and the noncentralityparam- eter ± =40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.6 Illustration of (a) the actual signal value and (b) sensor measurements of a multi-modal physical phenomenon. . . . . . . . . . . . . . . . . . . . . . 99 4.7 Quiver plots of (a) the actual and (b) the estimated gradients. . . . . . . 100 4.8 Illustration of (a) the actual contour lines and (b) the estimated contour lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.9 The e®ect of the one-hop data collection radius R c on the system perfor- mance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.10 TheoreticalMSEasafunctionof(a)thesensordensityandthenoisepower and of (b) the sensor density when R c =0:5, h=0:1. . . . . . . . . . . . . 103 5.1 System model of the information extracting process. . . . . . . . . . . . . 115 ix 5.2 System model of the local data fusion process.. . . . . . . . . . . . . . . . 116 5.3 Illustration of a K-hop data transmission system. . . . . . . . . . . . . . . 119 5.4 Illustration of interactions between application, MAC and physical layers. 124 5.5 Comparison of two sensor node deployment schemes. . . . . . . . . . . . . 125 5.6 Illustration of the Delaunay triangulation algorithm. . . . . . . . . . . . . 127 5.7 Illustration of a contour line crossing the neighborhood of a target sensor node, where one contour node candidate is shown in the middle of the top row while the others are not. . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.8 An example of an missed contour node. . . . . . . . . . . . . . . . . . . . 132 5.9 Illustration of contour point determination. . . . . . . . . . . . . . . . . . 136 5.10 Performance comparison of contour nodes inference under di®erent bit error rates over wireless channels when sensor density is 10 nodes/unit square, tolerance range r is 0:5 and probing range R is 0:75. . . . . . . . . 142 5.11 Performance comparison of contour nodes inference with di®erent levels of sensor density when BER= 0, tolerance range r is 0:5 and probing range R is 0:75. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.12 Simulation results of contour line estimation with di®erent curvature values.151 5.13 Simulationresultsofcontourlineestimationwithdi®erentlocalslopevalues.152 5.14 Simulation results of contour line estimation when Curvature=1/3, local slope=2, BER=0.01, number of quantization bits=10. . . . . . . . . . . . 153 5.15 p Sensor Density Curvature versus local slope under di®erent bit error rates. . . . . 153 5.16 p Sensor Density Curvature versus local slope under di®erent R=r ratios when r =0:5.154 x Abstract Several signal processing problems associated with wireless sensor networks deployed for the monitoring of certain environmental physical phenomena are examined in this research. A couple of distributed algorithms are proposed to extract the essential infor- mation of the monitored physical phenomena such as edges and contour lines so that a remote base station may utilize the information for further analysis. These networks often span over a large geographical area and they cannot be adequately monitored or tracked using traditional localization techniques. Due to di®erent sensing devices of deployed sensor nodes, measured sensor data can take a binary or a scalar value. For a binary ¯eld (phenomenon observed through the sensing devices which provide binary outputs only), the edge (or the boundary region) oftenprovidestheimportantinformationofconcern. Anewstatisticalapproachtogether with a false alarm control scheme to edge region detection is proposed to increase the discrimination capability of the true edge line location. Unlike previous work in the literature, no hierarchical network topology and geographical information about sensor nodes are needed. Simulation results demonstrate that the proposed approach has better performance when the noise interference is high and the probing range is small. For a scalar ¯eld (phenomenon observed through the sensing devices which are able to provide xi real-number measurements), the two-dimensional contour lines o®er an e±cient way to depictitsdensitydistribution. Threeapproachestocontourlinesestimationareproposed toprovidedi®erentlevelsofsystemperformancefordensityobservationbasedonthecost ofsysteminstallmentanditscomputationcapability. Simulationresultsdemonstratethat the proposed approaches can extract contour lines of the density distribution well in the presence of sensing and channel noise. xii Chapter 1 Introduction 1.1 Signi¯cance of the Research Withthetechnologyadvancesinminiaturizationandintegratedcircuits,themanufacture of tiny and low-cost intelligent wireless sensor nodes becomes economically practical. A wireless network composed by hundreds or thousands of intelligent sensor nodes may en- ablenewapplicationsinareassuchasbattle¯eldsurveillance,environmentalmonitoring, industrial sensing and diagnostics and infrastructure protection. In these applications, a wireless sensor network serves as the interface between the physical world and users for various purposes. Intelligent sensor nodes are deployed over the region of interest to monitor speci¯c events, targets and/or physical phenomena. Data measured by each sensor node are collected and forwarded to a remote base station for further high-level analysis. However, if data processing and fusion are not done properly in a localized and distributed fashion, data gathering in wireless sensor networks could be a large bur- den due to limited energy, scarce bandwidth and high communication cost. To utilize limited resources e®ectively, a distributed algorithm, which processes the measured data 1 locallytoreducetheinformationamountrequiredfortransmission,isusuallyanessential component in the system design of wireless sensor networks. For environmental monitoring, wireless sensor networks can be potentially used in the monitoring of certain environmental phenomena, such as chemical gas di®usion, con- taminant spreading and so on. These phenomena often span over a large geographical area instead of a speci¯c location. Due to their large size and time-varying nature, they cannot be adequately monitored or tracked using traditional localization techniques with few isolated sensor nodes. A time-driven and distributed data fusion approach is often adopted in such a system so that the remote base station can observe the monitored phenomena periodically based on the extracted information without excessive raw data transmission. Due to di®erent sensing devices adopted by deployed sensor nodes, the signal part of sensor measurements can be a binary or a scalar value. For example, consider a sensor node that provides positive (if the signal strength is larger than a prede¯ned threshold value T) or negative (if the signal strength is smaller than T) readings only. Its measure- mentstakethebinaryvalue. Ontheotherhand, whendeployedsensornodescanprovide measurement data of higher precision with multiple bits, they can be approximated as a real number, which is called a scalar ¯eld. Here, we would like to design distributed algorithms to extract the essential informa- tion (including the edge region or contour lines) of monitored physical phenomena. The information will be sent to a remote base station for further processing without excessive 2 rawdatatransmissionfromsensornetworks. Thesigni¯canceofthisresearchisdescribed below. 1.1.1 Statistical Edge Region Detection under Neyman-Pearson (NP) Optimality For the monitoring of a physical phenomenon modeled by a binary ¯eld, the edge (or the boundary region) often provides important information of concern. Even for the physical phenomenon modeled by a scalar ¯eld, edge line still reveals a lot of information of interestsifatriggeredthresholdisde¯nedtoseparateeventregionfromnon-eventregion. Forexample,itcanbeutilizedtolocalizeaphenomenonsothatproperactioncanbetaken accordingly. There have been a few methods proposed for edge detection in the context of wireless sensor networks. Nowak and Mitra [30] proposed a boundary estimation scheme where there exists a prede¯ned hierarchical topology among sensors. For a sensor network with a hierarchical topology, this approach performs well and provide quite accurate estimation results. However, additional wireless communications and protocol processingareneededtobuildupthishierarchicalnetworktopologywithincreasedenergy consumption. In addition, the hierarchical topology is sensitive to sensor node failures and each sensor node failure may initiate another round of topology rebuilding so as to maintain it in a good shape all the time. Since a wireless sensor network has quite limitedresources, itisoftenpreferredtoprocessthemeasureddatainalocalizedmanner without layer-by-layer communications in practical applications. To achieve this goal, Chintalapudi and Govindan [5] proposed three localized approaches to distributed edge 3 node detection; namely, the statistical- , the ¯lter-, and the classi¯er-based approaches. Instead of estimating the location of an edge line, the concept of edge node detection was introduced. The detected edge nodes are used to determine the region where the edge line resides. However, there was no false alarm control scheme to provide stable and low false alarm rate in [5] as the environment varies. We found that, even with a high detection rate, it becomes di±cult to observe where the true edge line is when the false alarm rate is high as well. In Chapter 3, we propose a new statistical approach together with a false alarm control scheme to increase the performance of true edge line detection. With the de¯nition given in [5], the region determined by edge nodes did not encompass the edge line inside so that it may induce a large error distance between detected edge sensors and the true edge line when the sensor density is low. To address this problem, we adopt a slightly di®erent de¯nition, where an edge region does encompass the edge line, and the error distance can be reduced to zero when there is no noise interference. In addition, algorithms proposed in [5] demand the sensor location information. Since the location of a sensor node usually comes from localization algorithms, the performance may degrade if the estimation error of sensor node localization algorithms is signi¯cant. 1.1.2 Contour Line Extraction with Heterogeneous Sensor Networks Eventhoughseveraldistributedalgorithmsofedgedetectionwereproposedinthecontext of sensor networks, e.g. [30], [5] and [25], these algorithms only o®er a binary decision (i.e. inside or outside the phenomenon) based on measured noisy data. For a physical 4 phenomenon described by a continuous range of values, the bi-level decision output may be too coarse and lead to substantial loss of important information contained by the measured data if the information of the density distribution of the physical phenomenon is desired in some applications. A contour line is a two-dimensional curve on which the value of a function is a constant. Unlike edges or boundaries, contour lines provide more detailed information of monitored phenomena. In Chapter 4, we examine the problem of contour line extraction using wireless sensor networks. In particular, a distributed scheme to determine contour lines is proposed to monitor a noisy scalar ¯eld so that they can be used to describe the ¯eld more concisely. The underlying assumption is that the monitored physical phenomenon is a quasi-static ¯eld so that data samples measured in a local sensor during a short period of time are statistically stationary (e.g. with the constant mean and variance over time). 1.1.3 Mesh-basedContourLinesEstimationwithHomogeneousSensor Networks A distributed approach using heterogeneous sensor networks is used to estimate contour lines for a physical phenomenon with a scalar ¯eld in Chapter 4. However, this approach requiresthedeploymentofhigh-costsensornodesandahierarchicalnetworkarchitecture. To improve the system design, we propose a low-cost alternative which can work with homogeneous sensor networks as well. That is, we make use of the soft information from contour node inference and the techniques investigated in Chapter 4 and attempt to estimate the contour lines using 2-D triangular meshes. 5 A mesh-based approach to contour line estimation with cross-layer considerations us- ing homogeneous sensor networks is proposed and analyzed in Chapter 5. Although the estimated contour points may not be accurate due to noise interference (including sens- ing/thermal noise and channel noise) and the scarceness of sensor nodes, a scheme is designed to correct the detection error to a certain degree so that the integrated solu- tion can provide better results ultimately. Consequently, both °exibility and robustness of the system increase. To evaluate the e®ect of cross-layer parameters on the system performance, cross-layer issues are examined. The proper system parameter settings are investigated by computer simulation. 1.2 Contributions of the Research For the research on edge region detection, the following contributions are made. 1. Clear criteria proposed for threshold value selection. Based on statistical inference, a nearly optimal two-level decision process is devel- oped. Oneislocaldecisionprocessandtheotherisdecisionfusion(globaldecision) process. By using the NP criterion, we develop a rule to select the threshold value for the decision fusion process. 2. Performance improvement of the proposed statistical approach as compared with that given in [5] when the probing range is small and noise interference is high. 6 ² Theproposedstatisticalapproachhasalowerfalsealarmratethantheclassi¯er- based approach under a comparable detection rate and this increases discrim- ination capability of the edge line. ² Thegeographicalinformationisexploitedbytheclassi¯er-basedapproach,but not by the new statistical approach. Thus, the performance of the classi¯er- basedapproachcanbea®ectedbytheaccuracyofthegeographicalinformation as well as noisy measurements. Since the geographical information is typically obtained by the use of localization algorithms, a sensor location error may exist. ² Performance comparison is conducted to investigate the impact of the sensor location error on the performance of these two approaches. It is demonstrated that the proposed statistical approach is not sensitive to the sensor location error so that it outperforms the classi¯er-based approach if this error becomes large. For research on contour lines extraction with heterogeneous sensor networks, we have made the following contributions. 1. Distributed algorithm proposed for contour lines extraction. The proposed scheme consists of two main components that work together to form an integrated solution. They are: (i) a distributed graph algorithm to support the required data exchange among sensors; and (ii) a statistical signal processing algorithm to reduce the noise e®ect and enhance the overall system performance. 7 ² For the ¯rst component, we develop a distributed graph algorithm that ¯nds extremal points, determines isolated points of the same value, and inter- connects these isolated points into a complete contour line. As a result, the phenomenonofinterestcanbeeasilymonitoredbythebasestation. Theissue to be examined for this component is the communication cost demanded to achieve the objective. The comparison of the communication cost between a centralized scheme and the proposed scheme is conducted to demonstrate the e±ciency of the proposed distributed scheme. ² Due to sensing noise and communication noise, the measured and received data are random signals. They have to be ¯ltered to reduce the noise e®ect so as to enhance the output value. In particular, the underlying signal may be a spatially-varying function and it is not easy to separate spatially-varying signal from noise with limited resources. Thus, some e®ective statistical signal processingalgorithmhastobedevelopedtoachievethisobjective. Intuitively, if the sensor density is higher and/or the noise power is lower, we should be able to determine the position of contour lines more accurately. This is the role of the second component in the proposed scheme. 2. System design criteria investigated through performance analysis. In this research, we propose a statistical processing algorithm and derive a quanti- tative relationship between the accuracy of estimated contour lines and the sensor densityandthenoisepower. Sincethesensordensityandthenoiselevelarerelated to the build-up cost of the monitoring system and the condition of the surrounding 8 environment, respectively, our analysis sheds some light on the design of a robust sensor network to meet some performance requirements. For the research on mesh-based contour lines estimation with cross-layer considera- tions using homogeneous sensor networks, the following contributions have been made. 1. Cross-layer factors considered in system design. We are concerned with a practical and energy-e±cient approach with realistic en- vironment settings. A cross-layer design principle is used to understand the inter- action of di®erent layers and the corresponding design tradeo®. 2. Adaptive data transmission proposed to resolve the excessive retransmissions prob- lem. Data transmission is one of the most challenging issues in wireless sensor net- works [42]. It may require quite a few packet retransmissions under a poor channel condition, which could cause excessive power consumption for early energy deple- tion of a sensor node. An adaptive data fusion scheme is proposed to improve the energy e±ciency. Simulation results are given to show the e®ects of di®erent parameters on the overall system performance. 3. System °exibility is enhanced by Delaunay Triangular meshes. Forirregularsensordeployment,itisdi±cultto¯ndagoodwaytodetectrandomly located contour points, which are also sensor nodes, and connect them in a sequen- tial order for contour lines estimation. This problem can be resolved by Delaunay 9 triangulation. Triangular meshes can help locate contour lines, which make system deployment easier. 4. System robustness is guaranteed by error correction. Due to noise interference and nodes failure, contour points may be erroneously detected, whichcouldlead toseriousestimation bias. Anerrorcorrection schemeis proposed together with the detection algorithm to mitigate the estimation errors or malfunctions of several sensor nodes. This ensures system robustness against noise interference and node failure. 1.3 Outline of the Thesis The thesis is organized as follows. In Chapter 2, some research background and previous workarereviewed. Then,astatisticaldecision-fusionapproachtoedgeregiondetectionis proposed,andtheperformancecomparisonofdi®erentalgorithmsisconductedinChapter 3. Chapter 4 presents a distributed scheme to contour line extraction with heterogeneous sensor networks and analyzes its performance. In Chapter 5, a mesh-based approach to contour lines estimation with cross-layer considerations using homogeneous sensor networks is described and its performance is shown by computer simulation. Finally, conclusion remarks are given and future research directions are presented in Chapter 6. 10 Chapter 2 Background Review In this chapter, previous work and some background material are reviewed. In Sec. 2.1, previous work on the environmental phenomena monitoring is reviewed. Background knowledge on data acquisition and collection in sensor networks and hypothesis tests is described in Sec. 2.2 and Sec. 2.3, respectively. In Sec. 2.4, Voronoi diagram and Delaunay triangulation, which are two famous problems in computational geometry, are reviewed. 2.1 Review of Previous Work Instead of estimating the edge line location, the concept of edge node detection was ¯rst proposedin[5]todetermineanedgeregion. Threeapproacheswereproposedfortheedge node detection in [5]: statistical-based, ¯lter-based and classi¯er-based methods. They are brie°y reviewed below. 11 2.1.1 The Statistical Approach The major advantage of the statistical approach is its robustness to noise interference if the noise statistics is known. Besides, the algorithm does not use the geographical information in the detection of edge nodes, its computational complexity is relatively lower, and it is simple to implement. Fig. 2.1 illustrates an edge node. The dashed circle represents the radio range (or the probing neighborhood) of radius R and the solid circle indicatesthetolerancerangeofradiusr,whichdeterminesthewidthofthedetectededge region. Edge Line Inside the phenomenon R r Outside the phenomenon Figure 2.1: An edge node and its relationship to an edge line. The algorithm is described as follows. 1. Local decision. Each local sensor node makes a local decision based on its own measurements. 12 2. Decision-fusion (Global decision). The target sensor node collects local decision results, x 1 ;x 2 ;¢¢¢ ;x N , from its neighboring sensor nodes in the radio range and calculates the following parameter: ¡=1¡ jm + ¡m ¡ j m + +m ¡ ; (2.1) where m + and m ¡ are numbers of 1's and 0's in the received decision results, respectively. If ¡ ¸ ° 0 , the target sensor node is claimed to be an edge node; otherwise, it is not. The value of ° 0 is related to the required detection rate and the ratio of R and r. It can be determined by o®-line simulation. 2.1.2 The Filter-based Approach Therearewell-developededgedetectiontechniquesintheimageprocessingliterature,and this approach utilizes them to detect edge nodes. Contrary to the statistical approach, the geographical information of sensor nodes is exploited by this approach. Sensor nodes in the sensor ¯eld can be viewed as pixels in an image, and a high-pass ¯lter can be adopted to detect edge nodes in wireless sensor networks. However, unlike image pixels, sensor nodes are typically not deployed in a regular grid so the ¯lter coe±cients have to be modi¯ed. The algorithm is stated below. 1. Local decision. Each local sensor node makes a local decision based on its own measurements. 13 2. Decision fusion (Global decision). The target sensor node collects local decision results, x 1 ;x 2 ;¢¢¢ ;x N , from its neighboring sensor nodes in the radio range and calculates the the following parameter ¾ = q ¾ 2 x +¾ 2 y ; (2.2) where ¾ x = X 8S2PA S 0 W x (i S ;j S )H x (i S ;j S )V S ; ¾ y = X 8S2PA S 0 W y (i S ;j S )H y (i S ;j S )V S ; and where PA S 0 represents the set of all neighboring sensor nodes within the radio range of target sensor node S 0 and (i S 0 ;j S 0 ) and (i S ;j S ) are the coordinates of the target sensor node and the neighboring sensor node. Let (m l+ ;m l¡ ) l=4 l=1 be the number of sensor nodes with local decision 1 and 0 in the lth quadrant of the radio range around the target sensor node. Then, we have W x (i S ;j S ) = 8 > > > > < > > > > : 1 m 1+ +m 1¡ +m 4+ +m 4¡ ; if i S >i S 0 1 m 2+ +m 2¡ +m 3+ +m 3¡ ; if i S <i S 0 ; W y (i S ;j S ) = 8 > > > > < > > > > : 1 m 1+ +m 1¡ +m 2+ +m 2¡ ; if j S >j S 0 1 m 3+ +m 3¡ +m 4+ +m 4¡ ; if j S <j S 0 ; 14 and H x (i S ;j S ) = 8 > > > > < > > > > : 1; if i S >i S 0 ¡1; if i S <i S 0 ; H y (i S ;j S )= 8 > > > > < > > > > : 1; if j S >j S 0 ¡1; if j S <j S 0 ; V S = 8 > > > > < > > > > : 1; if x S =1 ¡1; if x S =0 : If the calculated value of ¾ is larger than a threshold value ¾ 0 , the target sensor node is deemed an edge node; otherwise, it is not. 2.1.3 The Classi¯er-based Approach The classi¯er-based approach is based on the pattern recognition technique. The basic idea is to divide the radio range into two sets based on local decision results of sensor nodes within the radio range, which is illustrated in Fig. 2.2. A linear classi¯er was adopted in [5] for its simplicity. Even though errors may occur in some sensor nodes, the proposed algorithm can still ¯nd the best linear classi¯er to divide the radio range into two sets. Similar to the ¯lter-based approach, the geographical information is used in this approach, too. One advantage of this approach is that we do not have to select a decision threshold value in the algorithm. However, it is sensitive to noise interference. The algorithm is described in the following. 1. Local decision. Each local sensor node makes a local decision based on its own measurements. 15 Partition Line Inside the phenomenon R r Outside the phenomenon d θ 1 1 1 0 1 1 1 1 0 1 0 0 0 0 (a) Edge node Partition Line Inside the phenomenon R r Outside the phenomenon d θ 1 1 0 1 1 0 1 1 1 0 1 0 1 1 (b) Not an edge node Figure 2.2: Illustration of the classi¯er-based approach: (a) the edge node case and (b) the non-edge node case. 2. Decision-fusion. The target sensor node collects local decision results, x 1 ;x 2 ; ¢¢¢ ;x N , from its neighboring sensor nodes in the radio range. Then, it ¯nds line L S 0 (µ opt ;d opt ) that gives the maximum value of the following parameterized line model J S 0 (µ;d) = ¯ ¯ ¯ ¯ ¯ ¯ X 8S2PA S 0 V S (x S )¢SN(w S ) ¯ ¯ ¯ ¯ ¯ ¯ ; (2.3) 16 where PA S 0 is the set of all neighboring sensors in the radio range of the target sensor S 0 , V S (x S )= 8 > > > > < > > > > : 1; if x S =1 ¡1; if x S =0 with local decision value x S , SN(w)= 8 > > > > > > > > > < > > > > > > > > > : ¡1; if w <0 0; if w =0 1; if w >0 ; and w S =j S ¡tan(µ)¢i S ¡j S 0 +tan(µ)¢i S 0 ¡ d cos(µ) ; µ2[0;¼]; d2[¡R;R]: In above, (i S 0 ;j S 0 ) and (i S ;j S ) are the coordinates of the target sensor node and the neighboring sensor node. If 0 < d opt < r, then sensor S 0 is claimed to be an edge node. 2.2 Data Acquisition and Collection in Sensor Networks There are three major data reporting schemesin wireless sensor networks; namely, event- driven,query-drivenandtime-drivenschemes[1]. Theyareusuallyapplication-dependent. Intheevent-drivenscheme,whereverthemonitoredeventoccurs,thesensornodesnearby 17 will wake up for sensing. The measured data will then be transmitted and forwarded to the remote base station via relay nodes. Contrary to the event-driven scheme, the base station actively retrieves the data it desires in the query-driven scheme. However, both schemes only involve part of sensor nodes in data gathering so that the data gather- ing problem is con¯ned to sensor nodes which reside in the routes of data transmission. For the time-driven scheme, each sensor node will wake up for sensing periodically and forward the measured data to the remote base station. Since all sensor nodes in the sen- sor ¯eld are involved in the data gathering in this scheme, the data gathering problem is moreseriousthanprevioustwoschemesandmaybecomeamajorfactorinthedecreasing lifetime of the whole network. Two approaches are often adopted to avoid the excessive burden of data gathering andextendthelifetimeoftheunderlyingnetwork. Oneisdistributeddatacompressionor data aggregation, which mitigates the communication burden by eliminating redundancy andminimizingtheamountofdatainalocalizedmannerwithlittleinformationloss. The other is distributed data fusion, which extracts the information of interest by fusing the collected data in a local fusion center for future transmission. Even though distributed data fusion can decrease the amount of data for transmission signi¯cantly, the fusion process is irreversible and the speci¯c algorithm is highly application-oriented. In this work, we focus on the distributed data fusion approach. 2.3 Review on Hypothesis Tests The background knowledge on hypothesis tests is reviewed in this section. 18 2.3.1 Simple Hypothesis Test and Neyman-Pearson Lemma Consider the following hypothesis test: H 0 :µ =µ 0 ; H 1 :µ =µ 1 ; where µ is the parameter to determine while µ 0 and µ 1 are two possible values of µ. Since the parameter space of each hypothesis contains one value only, they are both simple. A critical region is de¯ned as a subset C of the whole sample space of data samples X =(X 1 ;X 2 ;¢¢¢ ;X n ) to reject null hypothesis H 0 . We use C to denote the complement set of C. Let ® = PrfX 2 CjH 0 g, which is called the size or the signi¯cance level of the test, and ¯ =PrfX 2CjH 1 g. Then a test that maximizes 1¡¯ for a ¯xed value of alpha is a most powerful (or best) test of size ®. This can be formally stated as follows. Neyman-Pearson Lemma: [14] Suppose that each data sample of X is from a distribution f(x;µ) and we wish to test H 0 : µ = µ 0 vs. H 1 : µ = µ 1 . The best test of size ® of H 0 vs. H 1 has a critical region of the form L(µ 1 ;X) L(µ 0 ;X) ¸A; (2.4) where L(µ;X) is the likelihood function of the sample vector X and A is a non-negative constant. 19 2.3.2 Composite Hypotheses and Uniformly Most Powerful Tests If the parameter space of a hypothesis contains a set of values instead of one value, the hypothesis is called composite hypotheses. Suppose both hypotheses H 0 and H 1 are composite, and they can be represented by H 0 :µ2!; H 1 :µ2¡!; whereisthewholeparameterspaceand! isasubsetofit. Letthepowerfunction ´(µ) be value of PfX 2Cjµg and the signi¯cance level be the value of sup µ2! ´(µ). Therefore, the uniformly most powerful (UMP) test is de¯ned as follows. Uniformly Most Power Test: [14] Suppose that we have a test with a signi¯cance level ®. Then, it is uniformly most powerful if its power function ´(µ) is greater than the power function ´ ¤ (µ) of any other test with the same signi¯cance level for all µ2¡!. With this de¯nition, we have the following lemma, which shows how an UMP test can be found for a composite hypotheses test. Lemma 2.1: [14] Let H 0 be simple and H 1 be composite. Suppose that a best test is found using the Neyman-Pearson lemma for H 0 vs. H 0 1 : µ = µ 1 where µ 1 is a value of µ in ¡!. If the critical region of the best test takes the same form for all µ 1 2 ¡!, then this test is UMP. 20 The UMP test usually exists and it is often easy to ¯nd for one-sided H 1 but it does not exist for two-sided H 1 . 2.4 Voronoi Diagram and Delaunay Triangulation ConceptsoftheVoronoidiagramandDelaunaytriangulationarereviewedinthissection. 2.4.1 Voronoi Diagram The Voronoi diagram, which is also known as Dirichlet tessellation, was introduced by Lejeune Dirichlet in 1850 [11]. However, only 2-D and 3-D cases were considered in his studyofquadraticforms. In1908,aRussianmathematician,GeorgyFedoseevichVoronoi, de¯ned and studied a more general n-dimensional case in [40] and, from then on, such a subdivision diagram is named after him in memory of his contribution. The Voronoi diagram has wide and useful applications in di®erent research ¯elds such as economics, geophysics, physics and engineering. Since only the 2-D Voronoi diagram is relevant to our study, we will focus on the 2-D case in this subsection. Let P =fp 1 ;p 2 ;¢¢¢ ;p n g be a set of points, called sites in the Voronoi diagram, over a 2-D plane. The Voronoi diagram is de¯ned as the subdivision of the plane into n cells, each of which corresponds to one site in P and has the property that any point inside it has the least distance to the corresponding site than any other. The Voronoi diagram of P is denoted by VD(P) and the cell corresponding to a site p i is denoted by C(p i ). 21 Considertwopoints,pandq,intheplane. Theprependicularbisectoroflinesegment pq is de¯ned as the bisector of these two points, which divides the plane into two half- planes. LetH(p;q)bethedividedhalf-planecontainingsitepandH(q;p)betheotherone containing site q. Then, the cell containing p i in the Voronoi diagram can be represented by C(p i ) = fq2R 2 jd(q;p i )·d(q;p j );j6=ig = \ 1·j·n;j6=i H(p i ;p j ); (2.5) where d(q;p i ) is the Euclidean distance between points q and p i . An example of the Voronoi diagram is shown in Fig. 2.3, where a plane is subdivided into 20 cells according to the location of 20 sites. Figure 2.3: An example of the Voronoi diagram. 22 A complete Voronoi diagram has the following properties, which are often utilized for theanalysisofitscomplexityandthedesignofcomputingalgorithms. Theorem2.1:[8] VD(P) is connected and all of its edges are either line segments or half-lines unless the sites in P are collinear. Theorem 2.2: [8] For a Voronoi diagram of n sites, the number of vertices in VD(P) is at most 2n¡5 and the number of edges is at most 3n¡6. Therefore, the complexity of VD(P) grows linearly with respect to the number of sites. Theorem 2.3: [8] For a Voronoi diagram of n sites, the following properties hold. ² Point q is a vertex of VD(P) if and only if its largest empty circle, which contains no sites inside, has three or more sites on the circle's boundary. ² The bisector between two sites, p i and p j , de¯nes an edge of VD(P) if and only if there is a point, q, on the bisector such that its largest empty circle has both p i and p j on its boundary but no other site. By(2.5), asimplewaytocomputeaVoronoicellofsitep i isto¯ndtheintersectionof all half planes H(p i ;p j ), where j6=i. However, it leads to an algorithm of computational complexity as high as O(n 2 logn) even though the complexity of the Voronoi diagram is O(n) according to Theorem 2.2. According to [8], the lower bound of the complexity 23 is O(nlogn) and the sweepline algorithm [13], also known as Fortune's algorithm, can asymptotically achieve this bound. 2.4.2 Delaunay Triangulation Delaunay triangulation, which was ¯rst introduced by Boris Nikolaevich Delaunay in 1934 [10] is the geometry dual of the Voronoi diagram. Fig. 2.4 shows an example of Delaunay triangulation and its geometry dual. We can see from the ¯gure that Delaunay triangulationcanbeeasilyobtainedbyconnectingsiteswhichareneighborstoeachother from the Voronoi diagram. Just like the Voronoi diagram, the complexity of Delaunay triangulation is also O(N), which can be proved by the following Theorem. Figure2.4: AnexampleofDelaunaytriangulation(solidlines)anditsgeometrydual(dot lines). 24 Theorem 2.4: [9] Let P be a set of n points in the plane which are not collinear. If there are k points lying on the boundary of the convex hull of P, then any triangulation of P, T(P), has 2n¡2¡k triangles and 3n¡3¡k edges. Though Delaunay triangulation can be obtained from its geometry dual, the Voronoi diagram, it is quite time-consuming and ine±cient. The following properties of Delaunay triangulation help us understand its geometry and design its computing algorithms. Theorem 2.5: [9] Give a set of n points P =fp 1 ;p 2 ;¢¢¢ ;p n g over the plane, then the following statements hold for Delaunay Triangulation. ² 4p i p j p k isaDelaunaytriangleifandonlyifthecircumscribedcirclepassingthrough three points p i , p j and p k does not contain any other point in its interior. ² The line segment formed by p i and p j is a Delaunay edge if and only if there is a pointq suchthatacirclecenteredatq containsp i andp j onitsboundaryandthere is no other points of P in its interior. The ¯rst property in Theorem 2.5 is called the circle criterion, which is often used as a critical rule to construct Delaunay triangulation [15]. There are other well-developed algorithmsforDelaunaytriangulation[35],[21],[2],someofwhichcanachievethecompu- tational complexity lower bound O(nlogn). According to [2], a d-dimensional Delaunay triangulation problem is equivalent to a d+1-dimensional convex hulls problem, which 25 means that the 2-D Delaunay triangulation can be constructed by the convex hulls algo- rithm after projecting points in the 2-D plane onto the paraboloid in the 3-D space. 26 Chapter 3 Statistical Edge Region Detection under Neyman-Pearson (NP) Optimality Using Wireless Sensor Networks 3.1 Introduction With an emerging need in environmental monitoring, military surveillance and security protection, research on wireless sensor networks has drawn a great amount of attention recently. For environmental monitoring, wireless sensor networks can be potentially used in the monitoring of certain environmental phenomena, such as chemical gas di®usion and contaminant spreading. While being harmful, these phenomena often span over a large geographical area. Due to the large size and time-varying shape, they cannot be adequately monitored or tracked using traditional localization techniques with few isolatedsensors. Itisoftenthattheedgeorboundaryregionprovidesthemostimportant information of our concern, which can be utilized to localize the monitored phenomenon so that proper action can be taken accordingly. 27 There have been a few methods proposed for edge detection in the wireless sensor networkliteratures. NowakandMitra[30]proposedaboundaryestimationschemewhere there exists a prede¯ned hierarchical topology among sensors. For a sensor network with ahierarchicaltopology,thisapproachperformswellandprovidequiteaccurateestimation results. However, additionalwirelesscommunicationsandprotocolprocessing areneeded to build up this hierarchical network topology with increased energy consumption. In addition, the hierarchical topology is sensitive to sensor node failures and each sensor node failure may initiate another round of topology rebuilding so as to maintain it in a good shape all the time. Since a wireless sensor network has quite limited resources, it is often preferred to process the measured data in a localized manner without layer-by- layer communications in practical applications. To achieve this goal, Chintalapudi and Govindan [5] proposed three localized approaches to distributed edge sensor detection; namely, the statistical- , the ¯lter-, and the classi¯er-based approaches. Instead of trying to estimate the location of an edge line, the concept of edge node detection was proposed in their work. The detected edge nodes are used to determine the region where the edge linemayreside. Amongthesethreeapproaches,theclassi¯er-basedapproachwasclaimed to have the best performance while the statistical approach was considered to have the best resistance to noise interferences. Although the statistical approach is more robust with respect to noisy measurements, the moderate performance and the lack of a clear criterion for threshold value selection have been its main drawbacks as studied in the past. 28 Inthiswork,weproposeanewstatisticalapproach{statisticalNP(Neyman-Pearson)- baseddecision-fusionapproach, whichconsistsofafalsealarmratecontrolscheme. With the de¯nition given in [5], the region determined by edge nodes did not encompass the edge line inside and thus it may induce a large error distance between the detected edge sensors and true edge line when the sensor density is low. In this work, we adopt a slightly di®erent de¯nition, where an edge region does encompass the edge line so that the error distance can be reduced to zero when there are no noise interference. Based on statistical inference, a nearly optimal two-level decision process is developed. One is local decision process and the other is decision fusion (global decision) process. By using the NP criterion, we develop a rule to select the threshold value for the decision fusion process. Both the performance and the complexity of the proposed statistical approach has been signi¯cantly improved as compared with that given in [5]. Simulation results show that the proposed statistical approach has better stability in false alarm rate and higher discrimination capability for the edge line location even though the detection rate is slightly smaller than that of the classi¯er-based approach when the probing range is large. It is worthwhile to point out another main di®erence between the statistical and the classi¯er-based approaches. That is, the geographical information is exploited by the classi¯er-based approach, but not by the statistical approach. Consequently, the perfor- manceoftheclassi¯er-basedapproachcanbea®ectedbytheaccuracyofthegeographical information as well as noisy measurements. Since the geographical information is typ- ically obtained by the use of localization algorithms, a sensor location error may exist. 29 Performance comparison is conducted to investigate the impact of the sensor location error on the performance of these two approaches. It is demonstrated that the proposed statistical approach is not sensitive to the sensor location error and it outperforms the classi¯er-based approach if this error becomes large. The rest of this paper is organized as follows. The sensor measurement model and the edge region de¯nition are described in Sec. 3.2. The proposed statistical method to determine the edge region of a monitored phenomenon is presented in Sec. 3.3. Per- formance comparison between the proposed statistical approach and the classi¯er-based approacharecarriedoutbycomputersimulationinSec. 3.4. Finally,concludingremarks are given in Sec. 3.5. 3.2 System Model Inthissection, wewilldescribethesensormeasurementmodelandde¯netheedgeregion in an environment monitored by wireless sensor networks. 3.2.1 Sensor Measurement Model Consider a wireless sensor network deployed in a large geographical area to monitor the edge (or boundary) of a phenomenon of interest. There may be thousands of sensor nodes deployed in the ¯eld. The phenomenon of interest is assumed to be quasi-static, whose statistical properties remains unchanged during K time points. Due to the use of sensing devices which provide binary measurement results only, the signal strength of the monitored physical phenomenon can be modeled by a binary ¯eld (either A or 0). 30 To detect the edge line, each sensor node reads local data and exchanges data with its neighboring sensor nodes for further data processing. Due to the thermal noise coming from the post devices , measured data may be infected. Suppose that the noise level is independent of the true signal value. Then, the received data from the sensor node at position (x;y) and time t can be modeled as X(x;y;t)=S(x;y;t)+n(x;y;t); (3.1) whereX(x;y;t)isthemeasurement,whichisascalarvalue,S(x;y;t)isthesignalstrength and n(x;y;t) is the noise. Furthermore, it is assumed that the signal strength takes a binary level, i.e., A and 0, with the following probability P (S(x;y;t)=A)=P A ; P (S(x;y;t)=0)=1¡P A ; n(x;y;t) in (3.1) is assumed to be white Gaussian noise in both the spatial and the temporaldomains. Inthefollowingmathematicalderivation,perfectknowledgeaboutthe signal strength and noise power is assumed ¯rst. With this assumption, the optimization problem based on Bayesian detection theory can be simpli¯ed and becomes easier to obtaintheoptimalsolutionsfordecisionfusionapproach. However,itisunlikelytoobtain the information apriori in reality. To develop more realistic and general methods for the solution to the edge detection, further derivation is pursued without the aforementioned assumptions. 31 3.2.2 Edge Region De¯nition Instead of estimating the true location of an edge line, the edge region that encompasses a true edge line inside is adopted. It is de¯ned below. De¯nition 1 Edge Region An edge region is de¯ned as the area that is within distance r from the true edge line so that the width of the edge region is 2r, where r is called the tolerance range of the edge region and it can be adjusted on demand. In order to determine the edge region as de¯ned above, we can either detect those senor nodes inside the edge region or estimate the boundaries of the edge region. Following [5], the ¯rst approach is adopted here. According to the edge region given above, we can de¯ne an edge node as follows. De¯nition 2 Edge Node If the distance between the edge line and a senor node is less than tolerance range r, then the sensor node is an edge node. Weshouldpointoutthatonlyedgenodesinsidethemonitoredeventwereconsideredin[5] while edge nodes inside and outside the monitored event are all taken into consideration in our method. According to our de¯nition, the true edge line will be encompassed in the internal area of the detected edge region. Mathematically, the di®erence between these two de¯nitions are actually minor. A thinner edge region will be claimed based on the de¯nition in [5]. In a sensor network with a low sensor density, there may exist a large error distance between the 32 Sen sor Nod es Edg e Nodes Sen sor Nodes M on ito re d Dangerou s Re gio n S a f e Region T ru e E dg e E rro r Di stan ce E dg e R eg i on Figure 3.1: Illustration of the error distance between the true and the marked edge regions. true edge line and the detected edge region, which may result in a problem. For example, consider the application where sensor nodes are deployed in a public facility to detect a dangerous phenomenon. People may not receive any warning signal until they are inside the dangerous area since only edge nodes inside a dangerous region will signal alarms. This is illustrated in Fig. 3.1. One way to ¯x this problem is to change the de¯nition of edge nodes to be those outside the monitored event instead of being inside. Another way istouseourmodi¯edde¯nitionofanedgenode. Notethatthelargerwidthofadetected edgeregionisnotafundamentallimitofourmodi¯edde¯nitionsincethetolerancerange r can always be adjusted according to the system requirement. 33 3.3 Proposed Method for Edge Region Detection 3.3.1 Basic Idea With the modi¯ed edge node de¯nition, the problem of edge region detection can be simpli¯ed as a binary hypotheses test, which is de¯ned as follows. 8 > > < > > : H F0 : The target sensor node is not an edge node. H F1 : The target sensor node is an edge node. Before expressing the decision rule for this test mathematically, an intuitive argument is given ¯rst. Fig. 3.2 illustrates the di®erence between edge nodes and non-edge nodes. If the probing range R, which decides the size of the data collection range, is equal to the tolerance range r, a non-edge node has all its neighbors within the probing range either inside the monitored event or all outside the monitored event. In contrast, not all neighborsofanedgenoderesidesinthesameregion. Thispropertycanbeusedtodevelop a detection algorithm. If R > r, for a non-edge node, the majority of its neighboring sensor nodes is located either outside or inside the monitored region. This observation will be translated into a mathematical criterion to distinguish these two cases. Based on thisexpression,wecan¯ndthelikelihoodfunctionsofbothhypothesesandformulatethe decisionruleforthistestbasedonthelikelihoodratiotest(LRT).Thereasontohavetwo di®erent sizes of ranges R and r in calculation is that we could thus have both enough data for noise suppressing and thinner detected edge region for edge line determination at the same time. 34 r r r r r r T rue E dge Inside the Monitored Event Outside the Monitored Event T ol e rance Range Ed g e No de E dge N o de Non -e d g e No de Non -e d g e No de P rob ing Range R R R R Figure 3.2: Illustration of edge and non-edge nodes when the probing range R is larger than the tolerance range r. Inthefollowing,adecision-fusionmethodundertheNeyman-Pearson(NP)optimality is proposed to detect edge nodes. Due to its low communication cost (only one bit information exchanged among sensor nodes per information update), it is suitable for bandwidth- and energy-limited wireless sensor networks. 3.3.2 Statistical Decision-Fusion Method under NP Optimality Theproposedapproachiscalledthestatisticaldecision-fusionmethodsincelocaldecision results are collected from sensor nodes for decision fusion based on statistical inference. There are two levels of hypotheses test performed. One is the local test and the other is the global test. The measured data are ¯rst processed in each local sensor node where a binary decision is made based on the derived local decision rule. The binary decision 35 results are then exchanged among sensor nodes for ¯nal decision making by the global decision rule. We would like to ¯nd some optimized solution for the hypothesis tests in these two levels. Case I: R=r For the convenience of mathematical derivation, the case where probing range R is equal to tolerance range r is considered ¯rst. The two hypothesis tests are shown below. Local Hypothesis (the 1st-level test): 8 > > < > > : H L0 : The target sensor node is outside the monitored phenomenon. H L1 : The target sensor node is inside the monitored phenomenon. Global Hypothesis (the 2nd-level test): 8 > > < > > : H F0 : The target sensor node is not an edge node. H F1 : The target sensor node is an edge node. Though a similar derivation was conducted in [17], it dealt with the identical local and global hypotheses. Here, hypotheses in these two levels are di®erent. This discrepancy complicates the derivations. In order to simplify the derivation, we ¯rst consider the case with perfect information about the PDFs. Let C ij represent the cost of deciding H Fi , giventhatH Fj ispresent. TheBayesiancostfunctionafteralgebraicmanipulations becomes R B =C +C F ¢P FA ¡C D ¢P D ; (3.2) 36 where C =C 01 ¢[1¡P(H F0 )]+C 00 ¢P(H F0 ); C F =(C 10 ¡C 00 )¢P(H F0 ); C D =(C 01 ¡C 11 )¢[1¡P(H F0 )]; P FA =P(H F1 jH F0 ); P D =P(H F1 jH F1 ): Wewouldliketo¯ndapair ofoptimalglobal and local decisionrulessothat(3.2) can be minimized. Here, the person-by-person optimization is adopted. Though the equations resulting from the person-by-person optimization method represents necessary but not su±cient conditions for determining the global optimal solution, this method makes the complicated optimization problem solvable. The inequality below was proved to be the optimal global decision rule in [17]: P(Y =yjH F1 ) P(Y =yjH F0 ) P(u=1jY=y)=1 ? P(u=1jY=y)=0 C F C D ; where Y = (Y 1 ;Y 2 ;¢¢¢ ;Y N ), Y i is the local decision result of the ith neighboring sensor node with value equal to 0 or 1, y is a speci¯c value of Y and u is the global decision result of the fusion center. It is obvious that the optimal solution for the global decision rule is a likelihood ratio test with a threshold value C F C D . Consequently, we can determine the value of u if the likelihood ratio is known for all possible y. In other words, we can 37 make a global decision when the local decision result of each neighboring sensor node is known. To compute the likelihood ratio, we must know the local decision rule ¯rst. Next, we will derive the local decision rule. We can rewrite (3.2) as R B =C + X Y P(u=1jY 0 ¹ )¢[C F ¢P(Y 0 ¹ jH F0 )¡C D ¢P(Y 0 ¹ jH F1 )] + X Y P(u=1jY 1 ¹ )¢[C F ¢P(Y 1 ¹ jH F0 )¡C D ¢P(Y 1 ¹ jH F1 )]; (3.3) where Y k ¹ = (Y 1 ;Y 2 ;¢¢¢ ;Y ¹ = k;¢¢¢ ;Y N ) and k = 0 or 1. Then, P(Y k ¹ jH F0 ) and P(Y k ¹ jH F1 ) can be found via P(Y k ¹ jH F0 )= Z X ¹ P(Y ¹ =kjX ¹ )G(X ¹ ;Y ¹ )dX ¹ ; (3.4) P(Y k ¹ jH F1 )= Z X ¹ P(Y ¹ =kjX ¹ )[L(X ¹ ;Y ¹ )¡G(X ¹ ;Y ¹ )]dX ¹ ; (3.5) where Y ¹ =(Y 1 ;Y 2 ;¢¢¢ ;Y ¹¡1 ;Y ¹+1 ;¢¢¢ ;Y N ), G(X ¹ ;Y ¹ ) = [P(H L0 )] N ¢P(X ¹ jH L0 )P(Y ¹ jH L0 )+[P(H L1 )] N ¢P(X ¹ jH L1 )P(Y ¹ jH L1 ); L(X ¹ ;Y ¹ ) = [P(H L0 )P(X ¹ jH L0 )+P(H L1 )P(X ¹ jH L1 )]¢B ¹ ; B ¹ = Y i6=¹ [P(H L0 )P(Y i jX i )P(X i jH L0 )+P(H L1 )P(Y i jX i )P(X i jH L1 )]: 38 After substituting (3.4) and (3.5) back into (3.3), the Bayesian cost function can be rewritten as R B = C +C ¹ (3.6) + X Y ¹ A Y ¹ ¢ ½Z P(Y ¹ =1jX ¹ )¢[C F G(X ¹ ;Y ¹ )¡C D L(X ¹ ;Y ¹ ) + C D G(X ¹ ;Y ¹ )]dX ¹ ¾ ; where C ¹ = X Y ¹ P(¹=1jY 0 ¹ )¢[C F ¢P(Y ¹ jH F0 )¡C D ¢P(Y ¹ jH F1 )]; A Y ¹ =P(u=1jY 1 ¹ )¡P(u=1jY 0 ¹ ): By observing (3.6), it is easy to ¯nd that the optimal local decision rule is given by P(X ¹ jH L1 ) P(X ¹ jH L0 ) Y ¹ =1 ? Y¹=0 T L ; (3.7) where T L = (C F +C D )¢P 0 ¢ Q i6=¹ [P(Y i jX i )¢P(X i jH L0 )]¡C D ¢P(H L0 )¢B ¹ C D ¢P(H L1 )¢B ¹ ¡(C F +C D )¢P 1 ¢ Q i6=¹ [P(Y i jX i )¢P(X i jH L1 )] ; P 0 = [P(H L0 )] N and P 1 = [P(H L1 )] N . It is obvious from (3.7) that the optimal local decision rule is also a likelihood ratio test with a derived threshold value T L . We can 39 easily compute the probability of detection, P D , and the probability of false alarm, P FA . Then the optimal global decision rule can be written as ln(¤(Y m )) = ln ½ P(Y m jH F1 ) P(Y m jH F0 ) ¾ (3.8) = ln 8 < : [1+ P(H L0 )¢P FA P(H L1 )¢P D ] m ¢[1+ P(H L0 )¢(1¡P FA ) P(H L1 )¢(1¡P D ) ] N¡m 1+[ P(H L0 )¢P FA P(H L1 )¢P D ] m ¢[ P(H L0 )¢(1¡P FA ) P(H L1 )¢(1¡P D ) ] N¡m ¡1 9 = ; H F1 ? H F0 T F ; whereY m isthevectorY withonlym1'sandT F =ln n C F C D o . Aftersomerearrangements, (3.8) can be simpli¯ed as ln(¤ 0 (Y m )) (3.9) = ln 8 < : [1+ P(H L0 )¢P FA P(H L1 )¢P D ] m ¢[1+ P(H L0 )¢(1¡P FA ) P(H L1 )¢(1¡P D ) ] N¡m 1+[ P(H L0 )¢P FA P(H L1 )¢P D ] m ¢[ P(H L0 )¢(1¡P FA ) P(H L1 )¢(1¡P D ) ] N¡m 9 = ; H F1 ? H F0 ln[exp(T F )+1]=T 0 F : However, formula (3.9) is still too complicated for a sensor node to fuse the received data, and further simpli¯cation is needed. To do so, we observe the trend of ln(¤ 0 (Y m )). If the ratio is a monotonically increasing function of m, then K-out-of-N rule can be adopted. If it is not, we have to ¯nd another way to simplify it. Fig. 3.3 shows the trend ofln(¤ 0 (Y m ))withoneexample. Obviously,itisnotamonotonicallyincreasingfunction, and the K-out-of-N rule is not appropriate. From this ¯gure, we ¯nd that the value of 40 0 2 4 6 8 10 12 14 16 18 0 10 20 30 40 50 60 70 m Test Statistic Threshold = 20 m 1 m 2 Figure 3.3: The trend of the test statistic in (3.9) when total number N of neighboring nodes inside the probing range of a target sensor node is 18, local detection rate P D is 0:9990 and local false alarm rate P FA is 0:0012. ln(¤ 0 (Y m )) is always larger than the given threshold in some continuous interval of m. Thus, we propose the following global decision rule. 8 > > > > < > > > > : Decide H F1 ; if m 1 <m<m 2 Decide H F0 ; if m·m 1 or m¸m 2 ; (3.10) where m 1 and m 2 are real number and m 1 <m 2 . Withthissimpli¯edglobaldecisionrule,thefusioncentercanperformthecomplicated data fusion processing easily and fast by counting the number of 1's in the received local decision results only. However, we do not have the exact signal and noise power informa- tion in a real-world environment, but only a family of parameterized PDFs. Therefore, 41 it is necessary to modify the above derived local and global decision rules. To tackle this problem, it is necessary to introduce temporal information into the decision rules. Supposethatnoisepowerremainsconstantoverthewholesensor¯eldandthemonitored event status remains unchanged during continuous K time points. The local hypotheses of a sensor node can be rewritten as H L0 :X(k)=n(k) H L1 :X(k)=A+n(k); where X(k) is the measured data of a local sensor node, A6=0, n(k) is zero-mean white Gaussiannoisewithvariance¾ 2 andk isthetimeindex. ByapplyingGLRT(Generalized Likelihood Ratio Test), the local decision rule can be expressed as L G (X)= f(X; ^ A; ^ ¾ 2 1 ;H L1 ) f(X; ^ ¾ 2 0 ;H L0 ) H L1 ? H L0 T L ; where X = (X(0);¢¢¢ ;X(K¡1)), f(¢) represents the PDF of X, ^ A and ^ ¾ 2 1 are chosen to maximize the numerator and ^ ¾ 2 0 is chosen to maximize the denominator. After some algebraic manipulations, we can show that ^ A and ^ ¾ 2 1 can be calculated via P K¡1 k=0 X(k) K and P K¡1 k=0 (X(k)¡ ^ A) 2 K , respectively. Furthermore, the parameter ^ ¾ 2 0 can be computed via 1 K P K¡1 k=0 X 2 (k). It has been proved in [20] that the local decision rule can be simpli¯ed as ^ A 2 1 K P K¡1 k=0 (X(k)¡ ^ A) 2 H L1 ? H L0 T 0 L : (3.11) 42 By multiplying both the numerator and the denominator with K ¾ 2 , we can show that the new numerator is a random variable with the Â 2 (1) distribution while the new de- nominator is a random variable with the Â 2 (K¡1) distribution, given that H L0 is true. These two random variables are independent of each other. The false alarm probability can be computed via P FA = P Ã U 1 U 2 K¡1 >(K¡1)¢T 0 L jH L0 ! = P(F >(K¡1)¢T 0 L jH L0 ); where U 1 = ( P K¡1 k=0 X(k)) 2 K¾ 2 , U 2 = P K¡1 k=0 (X(k)¡ ^ A) 2 ¾ 2 and F is a random variable of Fisher's F distribution with r 1 =1 and r 2 =K¡1. Since some parameters in the derived threshold value T L are usually unknown in advance, it is impossible to calculate the corresponding value of T 0 L for the derived optimal local decision rule in (3.11). Thus, the NP criterion is adopted. Given a tolerable false alarm rate, we can ¯nd the threshold T 0 L easily based on this criterion. Then the detection rate can be calculated via P D = P Ã U 2 V >¾ 2 T 0 L K jH L1 ! = Z 1 0 Z 1 ¾ r vT 0 L K h(u;v)dudv+ Z 1 0 Z ¡¾ r vT 0 L K ¡1 h(u;v)dudv; 43 where U = P K¡1 k=0 X(k) K , V = P K¡1 k=0 (X(k)¡ ^ A) 2 ¾ 2 and h(u;v) is the joint PDF of a Gaussian random variable and a Chi-square random variable of the following form: h(u;v) = 1 q 2¼ K ¢¾¢¡( K¡1 2 )2 K¡1 2 v K¡1 2 ¡1 exp ½ ¡ (u¡A) 2 K ¾ 2 ¡ v 2 ¾ : Furthermore, to take channel condition into consideration, the probability values of P D and P FA mentioned above need to be modi¯ed according to the bit error rate (BER) of the wireless channel, which can be estimated by channel state information (CSI). P D;C = P D £(1¡ \ BER)+P FA £ \ BER; P FA;C = P D £ \ BER+P FA £(1¡ \ BER); where \ BER is the estimated bit error rate. By replacing P D and P FA with P FA;C and P D;C , the derived global decision rule in (3.9) can be applied in obtaining the fusion result. Given threshold T 0 F , the decision rule can be further simpli¯ed to (3.10). Thedeterminationoftheoptimalthresholdvaluesfortheglobaldecisionrulein(3.10) will be studied below. Since the cost as given in (3.3) is usually unknown, it is di±cult to calculate threshold T 0 F in (3.9). Without the knowledge of T 0 F , we cannot ¯nd the corresponding values m 1 and m 2 . To overcome this problem, the same approach used for the local decision rule is adopted to determine these threshold values. To apply the NP criterion, we have to determine the relationship between T 0 F and the false alarm rate of the global test. Before doing so, we ¯rst ¯nd the relationship between T 0 F , m 1 and m 2 , since it will help calculate the false alarm rate latter. The curve of ln(¤ 0 (Y m )) as 44 illustratedinFig. 3.3canbeapproximatedbytwostraightlines,whichwillbemaximally separated when the value of ln(¤ 0 (Y m )) is the largest. Therefore, we can determine the mathematical formulas for these two straight lines if the corresponding m value can be found. Let C 1 = P(H L0 )¢P FA P(H L1 )¢P D and C 2 = P(H L0 )¢(1¡P FA ) P(H L1 )¢(1¡P D ) ; where P(H L1 )=P A , P(H L0 )=1¡P A , and P D and P FA are the detection rate and false alarm rate of the local decision, respectively. Then, we have ln(¤ 0 (Y m ))=m¢ln[1+C 1 ]+(N¡m)¢ln[1+C 2 ]¡ln[1+C m 1 ¢C N¡m 2 ]; (3.12) where N is the number of sensor nodes inside the probing range of the target sensor node. To ¯nd the value of m that maximizes ln(¤ 0 (Y m )), we take the di®erentiation on the right hand side of (3.12) with respect to m and equate it with zero to get m=m max . After some manipulation, we get ln(1+C 1 )¡ln(1+C 2 ) lnC 1 ¡lnC 2 ¡ln(1+C 1 )+ln(1+C 2 ) =C m max 1 ¢C N¡m max 2 : (3.13) Let C = ln(1+C 1 )¡ln(1+C 2 ) lnC 1 ¡lnC 2 ¡ln(1+C 1 )+ln(1+C 2 ) : 45 0 2 4 6 8 10 12 14 16 18 0 10 20 30 40 50 60 70 m Test Statistic True value Approximated value Figure 3.4: Comparison of true and approximated values of the test statistic in (3.9) when total number N of neighboring nodes inside the probing range of a target sensor node is 18, local detection rate P D is 0:9990 and local false alarm rate P FA is 0:0012. Take the natural log on both sides of (3.13), we have m max = lnC¡NlnC 2 lnC 1 ¡lnC 2 : With this value, the value of ln(¤ 0 (Y m )) can be approximated by 8 > > > > < > > > > : ln[¤ 0 (Y m max )]¡ln[¤ 0 (Y 0 )] m max £m+ln[¤ 0 (Y 0 )]; if m·m max ln[¤ 0 (Y m max )]¡ln[¤ 0 (Y N )] m max ¡N £(m¡N)+ln[¤ 0 (Y N )]; if m>m max : (3.14) 46 Therefore, m 1 and m 2 can be approximated by m 1 » = T 0 F ¡ln[¤ 0 (Y 0 )] ln[¤ 0 (Y m max )]¡ln[¤ 0 (Y 0 )] £m max ; (3.15) m 2 » =N¡ ln[¤ 0 (Y N )]¡T 0 F ln[¤ 0 (Y N )]¡ln[¤ 0 (Y m max )] £(N¡m max ): (3.16) The approximation results are shown in Fig. 3.4. With the above approximation results of m 1 and m 2 , we can proceed to ¯nd the relationship between the false alarm rate of the global test, P FA F , and T 0 F . Since it is di±cult to ¯nd the relationship between P FA F and T 0 F , directly, we have to derive the relationship between P FA F and m 1 , m 2 ¯rst and then substitute them by (3.15) and (3.16). The false alarm rate of the global test, P FA F , can be calculated by P FA F = Pfdecide H F1 jH F0 is trueg (3.17) = P © ln(¤ 0 (Y m ))>T 0 F jH F0 is true ª =Pfm 1 <m<m 2 jH F0 is trueg = 8 > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > : 1 2 £P 1 + 1 2 £P 2 ; if m 1 and m 2 are not integers 1 2 £[P 10 +P 3 ]+ 1 2 £[P 11 +P 4 ]; if m 1 is an integer and m 2 is not 1 2 £[P 9 +P 5 ]+ 1 2 £[P 12 +P 6 ]; if m 1 is not an integer and m 2 is 1 2 £[P 9 +P 10 +P 7 ] + 1 2 £[P 11 +P 12 +P 8 ]; if m 1 and m 2 are both integers ; 47 where P 1 = bN¡m 1 c X k=dN¡m 2 e µ N k ¶ ¢(1¡P D ) k ¢P N¡k D ; P 2 = bm 2 c X k=dm 1 e µ N k ¶ ¢P k FA ¢(1¡P FA ) N¡k ; P 3 = bN¡m 1 ¡1c X k=dN¡m 2 e µ N k ¶ ¢(1¡P D ) k ¢P N¡k D ; P 4 = bm 2 c X k=dm 1 +1e µ N k ¶ ¢P k FA ¢(1¡P FA ) N¡k ; P 5 = bN¡m 1 c X k=dN¡m 2 +1e µ N k ¶ ¢(1¡P D ) k ¢P N¡k D ; P 6 = bm 2 ¡1c X k=dm 1 e µ N k ¶ ¢P k FA ¢(1¡P FA ) N¡k ; P 7 = bN¡m 1 ¡1c X k=dN¡m 2 +1e µ N k ¶ ¢(1¡P D ) k ¢P N¡k D ; P 8 = bm 2 ¡1c X k=dm 1 +1e µ N k ¶ ¢P k FA ¢(1¡P FA ) N¡k ; P 9 = 1 2 ¢ µ N N¡m 2 ¶ ¢(1¡P D ) N¡m 2 ¢P m 2 D ; P 10 = 1 2 ¢ µ N N¡m 1 ¶ ¢(1¡P D ) N¡m 1 ¢P m 1 D P 11 = 1 2 ¢ µ N m 1 ¶ ¢P m 1 FA ¢(1¡P FA ) N¡m 1 ; P 12 = 1 2 ¢ µ N m 2 ¶ ¢P m 2 FA ¢(1¡P FA ) N¡m 2 : Consequently, if the tolerable false alarm rate, P T , is given, then T 0 F can be calculated by the following approximation: P FA F = Pfdecide H F1 jH F0 is trueg (3.18) = Prfm 1 <m<m 2 jH F0 is trueg » =P T ; where m 1 and m 2 can be substituted by (3.15) and (3.16), respectively. Thus, we have an algorithm for edge region detection, which is described below. 48 Algorithm I: 1. Local decision: From (3.11), ^ A 2 1 K P K¡1 k=0 (X(k)¡ ^ A) 2 H L1 ? H L0 T 0 L : 2. Decision fusion (Global decision): From (3.10), 8 > > > > < > > > > : Decide H F1 ; if m 1 <m<m 2 Decide H F0 ; if m·m 1 or m¸m 2 ; where m 1 and m 2 can be determined by adjusting T 0 F to yield the best approxima- tion for (3.18). Table 3.1 gives approximation results of P FA F for di®erent values of N. We see that the values of P FA F calculated by (3.17) are close to simulated ones. Thus, the derived decision rules for both local and global tests and criteria for threshold value selection work well as expected. N T 0 F m 1 m 2 Calculated P FA F Simulated P FA F 5 1.0 0.15 4.85 0.0055 0.0062 6 1.0 0.15 5.85 0.0066 0.0068 7 1.0 0.15 6.85 0.0077 0.0074 8 1.0 0.15 7.85 0.0088 0.0090 9 1.0 0.15 8.85 0.0099 0.0096 10 1.0 0.15 9.85 0.0109 0.0115 Table 3.1: Comparison of calculated and simulated P FA F for algorithm I when SNR= 30dB, T 0 L =175, BER=0:001 and R=r =0:5 49 Case II: R>r In some situations, a thinner detected edge region is preferable so that a smaller value of r is used. If algorithm I is applied in this case, radio range R has to be reduced. The reduced radio range results in a less amount of received data for decision fusion and degrades the system performance. In view of this, we may keep radio range R the same so as to collect enough data for decision fusion while reducing tolerance range r. Under this scenario, we will modify the global decision rule only and keep the local decision rule the same. The modi¯ed global decision rule is given below. 8 > > > > < > > > > : Decide H F1 ; if m 1 <m R <m 2 DecideH F0 ; if m R ·m 1 or m R ¸m 2 ; where m R is the number of 1's in the received local decision results from the sensor nodes in the probing range of a target sensor node. The remaining problem is to ¯nd appropriate threshold values, m 1 and m 2 , to determine if the target sensor node is an edge node by the statistics m R . Again, we apply the NP criterion to ¯nd the appropriate threshold values. Similar to the derivation of algorithm I, we ¯nd the analytical form of the false alarm rate ¯rst. 50 Let N and N r be the number of sensor nodes inside the probing and tolerance ranges of a target sensor node, respectively. Without loss of generality, it is assumed that the signal strength A is equal to 1. The false alarm rate can be computed by Prfm 1 <m R <m 2 jH F0 is trueg (3.19) = Prfm 1 <m R <m 2 jE 0 is true;H F0 is trueg£PrfE 0 is truejH F0 is trueg + Prfm 1 <m R <m 2 jE 1 is true;H F0 is trueg£PrfE 1 is truejH F0 is trueg; where PrfE 0 is truejH F0 is trueg = PrfE 1 is truejH F0 is trueg = 1 2 , E 0 represents the event that all neighboring nodes inside the tolerance range of a target sensor node have signal strength equal to 0 while E 1 represents the event that all of them have signal strength equal to 1. Suppose that there are k neighboring nodes with signal strength 1 inside the probing range of a sensor node and e 1 and e 2 are the numbers of false alarms (0 ! 1) and misses (1 ! 0) after taken local decision and transmitted through the wireless channel, respectively. Then, we have Pfm 1 <m R <m 2 jE 0 is true;H F0 is trueg (3.20) = Pfm 1 <e 1 ¡e 2 +k <m 2 jE 0 is true;H F0 is trueg = X k P kmax (k)£ " m 1 <e 1 ¡e 2 +k<m 2 X e 1 ;e 2 P a (e 1 )£P b (e 2 ) # ; where P a (e 1 )= µ N¡k e 1 ¶ (1¡P FA ) N¡k¡e 1 P e 1 FA ; 51 P b (e 2 )= µ k e 2 ¶ P k¡e 2 D (1¡P D ) e 2 ; and P kmax (k) is the probability that there are k neighboring nodes with signal strength 1 inside the probing range of a target sensor node. The range of e 1 is from 0 to N ¡k while e 2 is from 0 to k. Suppose that there are l neighboring nodes with signal strength 0 inside the probing range of a target sensor node and e 1 and e 2 are the numbers of false alarms and misses after transmitted through the wireless channel, respectively. Then, we can obtain Pfm 1 <m R <m 2 jE 1 is true;H F0 is trueg (3.21) = Pfm 1 <e 1 ¡e 2 +N¡l <m 2 jE 1 is true;H F0 is trueg = X l P l max (l)£ " m 1 <e 1 ¡e 2 +N¡l<m 2 X e 1 ;e 2 P c (e 1 )£P d (e 2 ) # ; where P c (e 1 )= µ l e 1 ¶ (1¡P FA ) l¡e 1 P e 1 FA ; P d (e 2 )= µ N¡l e 2 ¶ P N¡l¡e 2 D (1¡P D ) e 2 ; P l max (l) is the probability value that there are l neighboring nodes with signal strength 0 inside the probing range of a target sensor node. In (3.21), the range of e 1 is from 0 to N ¡l while e 2 is from 0 to l. Next, we determine the ranges of k, l, and the values of P kmax (k) and P lmax (l). Let the edge be a locally straight line that crosses the probing range of a target sensor node. Under this assumption, the maximum value of k (or l), given that H F0 is true, is achieved when the distance between the locally straight edge 52 R r N r Nodes N-N r Nodes Locally Straight Edge Line Target Node Figure 3.5: Illustration of the target sensor node, the probing range, the tolerance range and a locally straight edge line. line and the target sensor node is slightly larger than r as shown in Fig. 3.5. Therefore, the maximal possible value of k (or l), k max (or l max ), can be calculated by $ N£ R 2 ¢cos ¡1 (r=R)¡r¢ p R 2 ¡r 2 ¼R 2 % : Thus, the value range of k (or l) is from 0 to k max (or l max ). Suppose the sensor ¯eld is an L-by-L square. Then, P k max (k) (or P l max (l)) can be approximated by 8 > > > > < > > > > : 1¡ R¡r L=2¡r ; if k (or l)=0, R¡r k max (L=2¡r) (or R¡r l max (L=2¡r) ); if k (or l)6=0. By assuming that each nonzero value of k (or l) is equally likely, the false alarm rate canbecomputedeasilywith(3.19), (3.20)and(3.21). Furthermore, thethresholdvalues, 53 m 1 and m 2 , can be found by adjusting T 0 F as well if the tolerable false alarm rate, P T , is given. The value of P T is approximately equal to Pfm 1 < m R < m 2 jH F0 is trueg. Consequently, we derive the following algorithm. Algorithm II: 1. Local decision: the same as that in algorithm I. 2. Decision fusion (Global decision): H F0 : The target sensor node is not an edge node H F1 : The target sensor node is an edge node 8 > > > > < > > > > : Decide H F1 ; if m 1 <m R <m 2 Decide H F0 ; if m R ·m 1 or m R ¸m 2 ; where m 1 and m 2 can be calculated by Prfm 1 <m R <m 2 jH F0 is trueg » =P T : Analytical and simulated results of Prfm 1 < m R < m 2 jH F0 is trueg for di®erent values of N and N r are given in Table 3.2. Even though there exists some discrepancy due to simplifying assumptions made above, it still provides reasonable threshold values m 1 and m 2 to meet the performance requirement. With algorithm II, we have more °exibility than algorithm I in adjusting the width of the detected edge region according to the system requirement without much performance degradation. 54 N N r T 0 F m 1 m 2 Calculated P FA F Simulated P FA F 15 7 13.5 2.03 13.02 0.0094 0.0061 16 7 13.5 2.03 14.02 0.0109 0.0071 17 8 13.5 2.03 15.02 0.0125 0.0078 18 8 13.5 2.03 16.03 0.0140 0.0085 19 8 13.5 2.03 17.03 0.0156 0.0094 20 9 17.0 2.55 17.52 0.0040 0.0058 Table 3.2: Comparison of calculated and simulated P FA F for algorithm II when SNR= 30dB, T 0 L =175, BER=0:001, R=0:75 and r =0:5. 3.3.3 Extension to Noisy Scalar-¯eld Sensor Measurement Data In Sec. 3.2, it is assumed that S(x;y;t) is binary-valued, either A or 0, with probability values P (S(x;y;t)=A)=P A ; P (S(x;y;t)=0)=1¡P A : However,thisassumptiondoesnotstandiftheappliedsensingdevicesareabletoprovide real-number measurement results. In this case, the signal strength of the monitored phenomenon can be modeled as a scalar ¯eld instead of a binary ¯eld and this con°icts the underlying assumption of the derived algorithms. To tackle this problem, we modify the algorithms proposed in Sec. 3.3.2 so that they still can be applied even though the sensor measurement model changes. Though there is no explicit edge line or boundary for a scalar-¯eld physical phenomenon, a triggered threshold, which is used to separate the event region from non-event region and thus de¯nes the location of the edge line, can be de¯ned by the system according to requirements for the monitored phenomenon such as chemical gas or pollution. Althoughtheunderlyingassumptionischanged,thedevelopedalgorithmsfordecision fusion (i.e. global decision) are still valid because the received data for decision fusion at 55 a local fusion center are binary in both of cases. Thus, only the local decision rule has to be modi¯ed due to the change of the local hypothesis test, which is given below. H L0 :S(k)·A tr H L1 :S(k)>A tr ; (3.22) where S(k) is the signal part of measured data X(k) at a local sensor node with location (x;y) and time k and A tr is the triggered threshold of the monitored event. Since both hypotheses are composite, the derivation of the decision rule becomes more complicated. First, we consider the following hypothesis test: H L0 :S(k)=¹ 0 H L1 :S(k)>¹ 1 ; where ¹ 1 > ¹ 0 . By applying the Neyman-Pearson lemma [14], it can be shown that the following decision rule is the best test: P K¡1 k=0 X(k) K H L1 ? H L0 T L;scalar ; (3.23) where T L;scalar can be calculated from the desired tolerable false alarm rate. Comparing the above two hypothesis tests, we ¯nd that the second one is a special case of the ¯rst one. Let ¹ 0 be any value less or equal to A tr . The above decision rule remains the best test for this series of hypotheses tests. Therefore, (3.23) is the best test for (3.22). Since 56 H L0 in (3.22) is a composite hypothesis, there is no ¯xed false alarm rate but a series of false alarm rates unless the distribution of S(k) is known. Since it is di±cult to have this information a priori, we use the signi¯cance level (in- steadofthefalsealarmrate)to¯ndappropriatethresholdT L;scalar . Thesigni¯cancelevel isactuallythelargestfalsealarmrateinthiscasealthoughitsmathematicalde¯nition[14] is more strict. The signi¯cance level of (3.23) can be calculated by Prfdecide H L1 jS(k)=A tr g (3.24) = PrfX >T L;scalar jS(k)=A tr g=Prf X¡A tr ¾= p K > T L;scalar ¡A tr ¾= p K jS(k)=A tr g = PrfZ > T L;scalar ¡A tr ¾= p K jS(k)=A tr g; whereX = P K¡1 k=0 X(k) K ,¾ 2 isthenoisepowerandZ isastandardnormalrandomvariable. With (3.24), we can easily ¯nd threshold T L;scalar when the tolerable signi¯cance level is given. Next,theproblemisto¯ndappropriatevaluesofP FA andP D forthresholdcalculation ofthederivedglobaldecisionrulessince P FA andP D varywithlocationsofsensornodes. For the derived global decision rules over the whole sensor ¯eld, a more practical way is to use the following: P FA = PrfX >T L;scalar js(k)=A tr ¡±g; (3.25) P D = PrfX >T L;scalar js(k)=A tr +±g; 57 where ± is assigned as any nonnegative value, which may depend on how large the noise power ¾ 2 is, based on the system environment. After having local decision results and calculated P FA and P D in (3.25), we can apply the developed decision fusion processes in algorithms I and II for edge region detection directly. 3.4 Simulation Results and Discussion In this section, the proposed statistical decision-fusion method and the classi¯er-based method are compared in three di®erent aspects; namely, (1) computational complexity, (2) sensitivity to channel noise interference and (3) sensitivity to sensor location error. Sincetheoriginalclassi¯er-basedmethodasproposedin[5]onlydetectsedgenodesinside the monitored event, we have slightly modi¯ed the algorithm to allow fair comparison. The modi¯ed algorithm is stated below. The modi¯ed classi¯er-based method: 1. Local decision: the same as that in algorithm I. 2. Decision-fusion: FindlineL S 0 (µ opt ;d opt )thatgivesthemaximumvalueforJ S 0 (µ;d). J S 0 (µ;d) = ¯ ¯ ¯ ¯ ¯ ¯ X 8s2PA S 0 V S ¢SN(w S ) ¯ ¯ ¯ ¯ ¯ ¯ ; where w S =y S ¡tan(µ)¢x S ¡y S 0 +tan(µ)¢x S 0 ¡ d cos(µ) ; 58 PA S 0 is the set of all neighboring sensors within the radio range of target sensor S 0 , (x S 0 ;y S 0 ) and (x S ;y S ) are the coordinates of the target sensor node and the neighboring sensor node, µ2[0;¼], d2[¡R;R], V S = 8 > > > > < > > > > : 1; if Y S =1 ¡1; if Y S =0 ; SN(w)= 8 > > > > > > > > > < > > > > > > > > > : ¡1; if w <0 0; if w =0 1; if w >0 ; and Y S is the local decision value. If jd opt j < r, then sensor S 0 is deemed to be an edge node. In the execution of proposed statistical and classi¯er-based algorithms, sensor nodes do not have to be uniformly distributed over the sensor ¯eld. However, to facilitate the performance analysis, a uniformly distributed sensor ¯eld is assumed in the following discussion. 3.4.1 Computational Complexity First, wecomparethecomputationalcomplexityofthesetwomethods. Fortheclassi¯er- based method, a full search has to be done in the domain where µ2 [0;¼], d2 [¡R;R]. Thus, if the full search is conducted at positions with µ = (0; ¼ 10 ; 2¼ 10 ;¢¢¢ ; 9¼ 10 ) and d = (¡R;¡ 4R 5 ;¢¢¢ ;0;¢¢¢ ; 4R 5 ;R), there will be 10£11 = 110 searches for one local sensor node. We count one multiplication, one division, one addition, one subtraction or one comparison of two values as one computational unit. Since each search will cost 4N 59 multiplications and 6N¡1 additions, there are 10N¡1 operations in one local search. The total computational cost is (10N¡1)£110+109+1=1100N; where N is the number of sensor nodes inside the probing range of a target sensor node. For the proposed method, both algorithms (I and II) have one level decision process only so their computational complexities are the same. The major advantage of the pro- posed methodis thatno fullsearchis carriedout in thealgorithm and the computational complexity is low as compared with the classi¯er-based method. There are N +1 addi- tions so that the total computational cost is N +1 only each information update for the proposed algorithms if the threshold values m 1 and m 2 are already selected in advance. Since the environment does not change frequently and noise statistical properties may remain unchanged for a long time, the threshold values can be calculated and stored in each sensor node prior to the monitoring of the phenomenon. Thus, the computational complexity of threshold value calculation could be neglected. Consequently, the compu- tational complexity of the classi¯er-based method is at least 1000 times larger than that oftheproposedmethod. Thismeanstheprocessingtimeoftheproposedmethodismuch shorter than that of the classi¯er-based method. In other words, it saves the energy con- sumptionforsignalprocessingaswellasmitigatesthedelaye®ectofinformationupdate. However, this statement stands only for the case where threshold values, m 1 and m 2 , are already calculated in advance and recorded in the physical memory of a sensor node. 60 Consequently, there is a tradeo® between the computational complexity and the memory usage. 3.4.2 Sensitivity to Channel Noise Interference In this subsection, the proposed methods and the classi¯er-based method are compared under di®erent bit error rates (BER) induced by channel noise interference. The e®ect of the bit errors is shown in an average fashion. For example, 0:01£ 10 bit error is added per received data when the bit error rate (BER) is 0:01. However, to observe the worst e®ect on the performance, the error is added to the most signi¯cant bit. In our computer simulation, the sensor ¯eld is a 10£10 square area with a sensor density of 10 sensors/square. Thus, there are around 8 sensor nodes inside the range when the radio range is 0:5. Unless speci¯ed otherwise, the tolerance range of 0:5 and the probing rangeof0:75areadoptedintheexperiments. Itisassumedthatenoughquantizationbits are used to quantize the measurements before digital transmission so that the equivalent quantization noise is much smaller than sensing noise and the quantization distortion could be neglected. According to the data sheet of MICA2 mote [7], sensing noise power is usually at least 30dB smaller than the signal strength of the monitored phenomenon. In the simulation, the signal strength is set binary-valued, either 1 or 0, the sensing noise power is set 0:001, which gives 30dB SNR and K is set to be 4. The threshold value of the local decision rule is chosen to yield the lowest error probability. Thus, T 0 L is 175, P D = 0:99995 and P FA = 0:00018. The simulation is conducted using a locally straight edge line. 61 10 -4 10 -3 10 -2 10 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Bit Error Rate (BER) over Wireless Channel Probability Value P D of Algorithm I P FA of Algorithm I P D of Algorithm II P FA of Algorithm II Tolerable P FA Figure 3.6: Performance comparison of algorithm I (R = 0:5) and II (R = 0:75) when r =0:5. Fig. 3.6 shows the performance comparison of Algorithm I and II. Obviously, the performance of algorithm II is much better. In addition, we see that algorithm II is more robust to bit error rate (BER) since the performance remains high even though the BER decreases. This is because algorithm II is more °exible and it allows a larger probing range to collect more data for decision-fusion while the tolerance range is still kept the same. Note that Algorithm II can keep detection rate as high as 0:5 and suppress the false alarm rate around 0:01 at the same time even though BER is as poor as 0:1. Fig. 3.7 compares the performance of algorithm II and the classi¯er-based method at di®erent bit error rates (BER). It is assumed that the location information of each sensor node is perfectly known. From Fig. 3.7, the classi¯er-based method is sensitive to wirelesschannelconditionsincethefalsealarmratecanbeevenupto40%,whichismuch 62 higher than the tolerable false alarm rate 1% and might make the detected edge region unidenti¯able, when the BER is high. As compared with the classi¯er-based method, the proposedmethodyieldsamuchlowerfalsealarmratesincetheNeyman-Pearsoncriterion is used to control it. It provides a much lower false alarm rate and makes the system work well under a poor-channel environment if the BER information can be obtained. However, the classi¯er-based method performs better than the proposed method when the probing radius is greater than 1:5r. Thus, we may arrive at the following qualitative conclusion. Theclassi¯er-basedmethodisabetterchoiceiftheresidualenergyofasensor node is high (can sustain probing range lager than 1:5r) and/or the channel condition is good (at BER lower than 0:01). In contrast, the proposed method is a better choice when the residual energy of a sensor node is low (can sustain probing range smaller than 1:5r) and/or the channel condition is poor (at BER higher than 0:01). 3.4.3 Sensitivity to Sensor Location Error In this subsection, we investigate the e®ect of sensor location error. It is clear that the proposed statistical method only uses the measurement information to decide if the target sensor node is an edge node. In contrast, the classi¯er-based method uses both the measurement data and the geographical information of sensors. From the simulation results given in the last subsection, the classi¯er-based method has better performance when the coordinates of sensor nodes are perfectly known and the probing range is large enough (above 1:5 times of the tolerance range). However, practically speaking, this sensorlocationinformationhastobeobtainedfromtheestimationofacertainlocalization 63 algorithm. Thus, it is not realistic to evaluate the performance of the edge detection methods without considering location errors. Since there are many di®erent localization algorithms to estimate the coordinates of sensor nodes, the model of sensor location error may vary when a di®erent localization algorithm is used. For convenience, a speci¯c localization algorithm call the RSSI (Re- ceived Signal Strength Indicator) method is chosen here, where the sensor location error can be modeled as a Rayleigh random variable [38]. Let ¢x i and ¢y i be the location errors along x- and y-axis, respectively, and they are modeled by a Gaussian variable N(0;¾ 2 loc ). Thus, the average location error is equal to ¹ loc = r ¼ 2 £¾ loc : With this model, a di®erent degree of location errors can be generated by varying ¾ loc . Here, we use a certain percentage of r to represent the location error since a system with a larger tolerance range can tolerate a larger location error. In Fig. 3.8, we compare the e®ect of sensor location error on two methods when the channel condition is perfect. Since algorithm II does not utilize the geographical information, sensor location error has no e®ect on its performance. For the classi¯er-based method, it is more sensitive to location error. The false alarm rate grows as the average location error increases. 3.4.4 Simulation of Edge Region Detection In this section, we perform the simulation with the proposed edge region detection algo- rithm for binary-¯eld phenomena. Consider 1000 senor nodes randomly deployed over a 64 10£10 sensor ¯eld so that the average sensor density is 10 sensor nodes per unit square. K is equal to 4 and ± in (3.25) is set to 1:5£¾ to achieve acceptable performance. To achieve the least physical memory usage in a sensor node, we choose two ¯xed threshold values, m 1 and m 2 , for the case N = 18. However, due to the random deployment of sensor nodes, the number of neighboring sensor nodes, N, may not be the same with respect to each local target sensor node. Here, we adopt a resampling technique, which is commonly used in the bootstrap method [12], to ensure each local target sensor node to have exactly 17 data samples for decision fusion though it will degrade the performance a little bit. Although the proposed method is derived under the binary-signal-value assumption, it still works for signals with a continuous value if the local decision rule is modi¯ed by (3.23). Figs. 3.9(b) and 3.10(b) illustrate the detected edge nodes and the corresponding edge region while Figs. 3.9(a) and 3.10(a) show the sensor measurements applied to the simulation. The proposed statistical method works e®ectively in both cases even though there is channel degradation. 3.5 Conclusion BasedontheBaysianstatisticalinferenceandtheNeyman-Pearsonoptimality, astatisti- caldecision-fusionmethodwasproposedtolabelthede¯nededgeregionusingdistributed wireless sensor nodes. In comparison with the classi¯er-based method proposed before, the statistical method has a lower computational complexity. Furthermore, the classi¯er- based approach demands accurate location information of sensor nodes and a low noise 65 level in the measured data for fusion. By leveraging the NP optimality, the proposed statistical approach can achieve a signi¯cantly lower false alarm rate in the presence of a high noise level and increase the discrimination capability for the edge line location. Besides, it does not demand the location information of sensor nodes. 66 1 1.25 1.5 1.75 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R/r (Probing Radius/Tolerance Radius) Probability Value P D of A II, BER=0 P FA of A II, BER=0 P D of A II, BER=0.01 P FA of A II, BER=0.01 P D of Class, BER=0 P FA of Class, BER=0 P D of Class, BER=0.01 P FA of Class, BER=0.01 Tolerable P FA =0.01 Tolerable P FA (a) Performance comparison at lower bit error rate (BER) 0 and 0:01. 1 1.25 1.5 1.75 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R/r (Probing Radius/Tolerance Radius) Probability Value P D of A II, BER=0.05 P FA of A II, BER=0.05 P D of A II, BER=0.1 P FA of A II, BER=0.1 P D of Class, BER=0.05 P FA of Class, BER=0.05 P D of Class, BER=0.1 P FA of Class, BER=0.1 Tolerable P FA Tolerable P FA (b) Performance comparison at higher bit error rate (BER) 0:05 and 0:1. Figure 3.7: Performance comparison of algorithms II and classi¯er-based method under di®erent bit error rates when the tolerance radius r is 0:5. 67 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Average Location Error (unit is r) Probability Value P D of A II P FA of A II P D of Class P FA of Class (a) R=r =0:5 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Average Location Error (unit is r) Probability Value P D of A II P FA of A II P D of Class P FA of Class (a) R=0:75 and r =0:5 Figure 3.8: Performance comparison of algorithms II and classi¯er-based method when BER=0 under di®erent levels of average location error 0, 0:1r, 0:2r, 0:3r and 0:4r. 68 0 2 4 6 8 10 0 2 4 6 8 10 -0.5 0 0.5 1 1.5 X-axis Y-axis Measured Value (a) Sensor measurements when SNR=30dB 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 X-axis Y-axis (b) Simulation result using algorithm II when SNR=30dB and BER=0:001 Figure 3.9: Edge region detection for binary-value signal with R = 0:75, r = 0:5, where \o"representsthedetectededgenodes,\x"representsthesensornodesdetectedasinside the event region, \+" represents the sensor nodes detected as outside the event region, solid lines are true edges and dash lines are the limit of the de¯ned edge region. 69 0 2 4 6 8 10 0 5 10 -0.5 0 0.5 1 1.5 X-axis Y-axis Measured Value (a) Sensor measurements when SNR=30dB 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 X-axis Y-axis (b) Simulation result using algorithm II when SNR=30dB and BER=0:001 Figure 3.10: Edge region detection for real-number signal with A tr = 0:5, R = 0:75, r = 0:5, where \o" represents the detected edge nodes, \x" represents the sensor nodes detected as inside the event region, \+" represents the sensor nodes detected as outside the event region, solid lines are true edges and dash lines are the limit of the de¯ned edge region. 70 Chapter 4 Distributed Contour Lines Extraction with Heterogeneous Sensor Networks 4.1 Introduction Even though several distributed algorithms of edge detection were proposed in the con- text of sensor networks, e.g. [30], [5] and [25], these algorithms are developed base on binary-¯eld modeling and only o®er a binary decision (i.e. inside or outside the phe- nomenon) based on measured noisy data (signal strength together with sensing/thermal noise). For a physical phenomenon modeled by a scalar ¯eld, there does not truly exist so-called boundary or edge though it still can be di®erentiated into two subregions (the region with signal strength larger than a threshold T and the region with counterpart) using local binary decision on the measurements and boundary or edge can be detected by aforementioned algorithms. However, this kind of bi-level decision output may be too coarse and lead to substantial loss of important information contained in the measured 71 data, especially when the information of the density distribution of the physical phe- nomenon is desired in applications. A contour line is a two-dimensional curve on which the value of a function is a constant. Unlike edge or boundary, contour lines provide the density information of monitored phenomena. In this chapter, we examine the problem of contour extraction using wireless sensor networks. In particular, a distributed scheme to determine contour lines is proposed to monitor a noisy scalar ¯eld so that they can be used to describe the ¯eld more concisely. The underlying assumption is that the monitored physical phenomenon is a quasi-static ¯eld so that data samples measured in a local sensor during a short period of time are statistically stationary (e.g. with the constant mean and variance over time). The proposed scheme consists of two main components that work together to form an integrated solution. They are: (i) a distributed graph algorithm to support the required data exchange among sensors; and (ii) a statistical signal processing algorithm to reduce the noise e®ect to enhance the overall system performance. For the ¯rst component, we develop a distributed graph algorithm that ¯nds the extremal points, determines isolatedpointsofthesamevalue, andinter-connectstheseisolatedpointsintoacomplete contourline. Asaresult, thephenomenonofinterestcanbeeasilymonitoredbythebase station (BS). The issue to be examined for this component is the communication cost demanded to achieve the objective. The comparison of the communication cost between a centralized scheme and the proposed scheme is conducted to demonstrate the e±ciency of the proposed distributed scheme. 72 Due to sensing/thermal noise and communication noise, the measured and received data are random signals. They have to be ¯ltered to reduce the noise e®ect so as to enhance the output value. In particular, the underlying signal may be a spatially-varying function and it is not easy to separate spatially-varying signal from noise with limited resources. Thus,somee®ectivestatisticalsignalprocessingalgorithmhastobedeveloped toachievethisobjective. Intuitively,ifthesensordensityishigherand/orthenoisepower is lower, we should be able to determine the position of contour lines more accurately. Thisistheroleofthesecondcomponentintheproposedscheme. Inthischapter, wepro- pose a statistical processing algorithm and derive a quantitative relationship between the accuracyofestimatedcontourlinesandthesensordensityandthenoisepower. Sincethe sensordensityandthenoiselevelarerelatedtothebuild-upcostofthemonitoringsystem and the condition of the surrounding environment, respectively, our analysis sheds some light on the design of a robust sensor network to meet some performance requirements in the presence of a certain environment. Therestofthischapterisorganizedasfollows. Thesystemmodelandthecontourline estimation problem are described in Sec. 4.2. The proposed contour extraction scheme is presented in Sec. 4.3 to determine the contour lines of interest. The communication cost of the proposed scheme is analyzed in Sec. 4.4. Then, statistical analysis of estimated contour lines' accuracy is given in Sec. 4.5, where an explicit relationship between the mean-squared-error (MSE) of the estimated contour lines and the sensor density and the noise level is derived. Simulation results are reported in Sec. 4.6 to corroborate our theoretical analysis in Sec. 4.5. Finally, concluding remarks are mentioned in Sec. 4.7. 73 4.2 System Model and Problem Formulation 4.2.1 System Model Consider a heterogeneous sensor network that spans over a large geographical area in order to monitor a physical phenomenon modeled by a scalar ¯eld. All participating sensornodesarestationary. Inotherwords, thereisnomobilesensornode. Twotypesof sensor nodes are deployed in this system. The ¯rst type is called gateway nodes (or high costnodes)thathaveabundantresourcesandlong-rangecommunicationcapability. They are the ideal candidates of a cluster head or a sink. They are often deployed su±ciently and uniformly so that the completion of the assigned mission can be achieved e®ectively. Due to the long-range communication capability, gateway nodes can communicate with each other directly. The other type is known as ordinary nodes (or low cost nodes) that have limited resources, and they communicate primarily with their neighboring sensor nodes since only the short-range communication is enabled. The link between any two communicating sensor nodes is modeled as a bidirectional and symmetric one. A time- drivendatadisseminationmethodisadopted. Thatis,sensornodesaresynchronizedand measured data are disseminated periodically. Finally, we assume that sensor nodes have the knowledge of their relative location, which can be achieved by algorithms described in [4] and [33]. 4.2.2 Problem Formulation In this chapter, the physical phenomenon of interest is supposed to be a quasi-static multi-modal scalar ¯eld, which means that the underlying physical ¯eld has multiple 74 extremal points over the entire covered region. These extrema are located su±ciently far apart from one another. The ¯eld changes slowly with time so that the statistics of both the ¯eld and the noise level remain unchanged over a period of time. Due to sensing noise and communication errors, the received sensor measurement in a local fusion center could be corrupted. The received measurement from a sensor node located at position (x;y) and time t i can be modeled by X(x;y;t i )=S(x;y;t i )+n(x;y;t i ); (4.1) whereS(x;y;t i )representsthesignalstrengthoftheunderlyingphysical¯eldandn(x;y;t i ) denotesthenoisee®ect,whichisassumedtobethewhiteGaussiannoise. Thestatisticsof the ¯eld value and noise are assumed unknown a priori. To e®ectively suppress the noise interference and extract contour lines without gathering raw measured data from sensor nodes, a distributed algorithm is developed, based on very limited local information. 4.3 Proposed Scheme to Contour Line Extraction Given local noisy data samples, an energy-e±cient scheme is proposed in this section to determine the locations of contours lines parameterized by di®erent ¯eld values. The proposed scheme is composed of two main components: a distributed graph algorithm, which utilizes the limited energy of each sensor to support data exchange among sensors, and a statistical signal processing algorithm, which suppresses noise using limited local information. To make the description of the proposed scheme more lucid, these two 75 componentsaredescribedjointlyinthefollowingthreesubtasks: (1)extremalocalization; (2) contour point determination; and (3) contour line construction. 4.3.1 Extrema Localization The extrema localization algorithm is cluster-based so that the power consumption can bemanagedmoree®ectively. Itconsistsofthreesteps: clustering, coarselocalizationand ¯ne localization. Clustering The clustering algorithm presented here is coordinate-based, which means the coor- dinate information of each sensor node is needed. To initiate the clustering process, the gateway nodes exchange the information of their residual energy, and the one with the largest residual energy is elected as the coordinator. The coordinator will determine the start/end time of the clustering process and the decision-waiting time, and send these data to other gateway nodes. Whentheclusteringprocessstarts,eachgatewaynodebroadcastsarecruit-for-cluster- member message, which contains the coordinates of the transmit gateway node and the decision-waitingtime,whichisshorterthanthewholeclusteringduration. Uponreceiving this message, each sensor node calculates the distance between the gateway node and itself. Whenthedecision-waitingtimeends,eachsensornodechoosestheclosestgateway node from which it receives the request as its cluster head and sends back a join-request message, which includes its coordinates and residual energy. Upon receiving the request, the gateway node stores the coordinates of these sensor nodes in its cluster table and 76 sendsoutacon¯rmationmessageandotherrelevantinformation(e.g. thecommunication schedule, the optimal routing route to the cluster head, etc.) back to that sensor node. Due to a shorter communication range of a sensor node, it is likely that it may not reach the closest gateway node directly without the help of other intermediate sensor nodes. When this happens, a sensor node will ¯nd its neighboring sensor node which is closest to the target gateway node and ask it to forward the join-request message. This process will continue until the message reaches the target gateway node. When the end time arrives, gateway nodes (a.k.a cluster heads, hereafter) exchange their cluster tables to ensure that there is no multiple-booked sensor node. For those sensor nodes that cannot ¯nish the clustering process in time, they will switch to the sleep mode to save energy. The issue of optimizing the load among cluster heads is discussed in [16]. After clustering, a hierarchical structure is formed. Sensor nodes only talk to their neighboring sensor nodes and their cluster head while cluster heads can communicate with sensor nodes under their management and other cluster heads. After the completion of clustering, the elected coordinator will decide the start time of the next step, i.e., coarse localization, and broadcast this information to other cluster heads. Coarse Localization Without loss of generality, we use the local maximum as an example. When the starttimeofcoarselocalizationarrives, theclusterheadineachclustercollectsmeasured data from its cluster members. After collecting all data samples from sensor nodes in its realm, the cluster head compares these samples, chooses the largest one and records the 77 1.6 2 0. 99 2.0 9 0.2 2 0.1 6 0 .18 1. 11 0.7 3 1 .08 No rm al Senso rs Clu ster Hea d T he re i s o ne loc al ma x i mu m! ! 1.6 2 0. 99 2.0 9 0.2 2 0.1 6 0 .18 1. 11 0.7 3 1 .08 No rm al Senso rs Clu ster Hea d T he re i s o ne loc al ma x i mu m! ! Figure 4.1: Illustration of coarse localization of local extreme points. coordinates of that corresponding sensor. Then, the cluster head exchanges the result withitsneighboringclusterheadsforcomparison. Iftheclusterhead¯ndsthattheresult it holds is a local maximum, it will broadcast a message to other cluster heads to declare a local maximal point inside its cluster. An example is illustrated in Fig. 4.1. After this two-level comparison, those cluster heads, which declare a local maximum inside their clusters, will start a ¯ne-localization process as stated below. Fine Localization Although the distributed graph algorithm described in the ¯rst two steps can provide approximate locations of extreme points, the estimated results are sometimes too coarse 78 due to the noise e®ect. Therefore, statistical signal processing techniques are adopted in this ¯ne-localization step to further suppress noise to yield a more accurate estimate. There are two main issues to deal with in this process inside a sensor cluster. One is to use collected data samples for gradient estimation at the target sensor locally. The other is to exploit the estimated gradient to locate the local extreme point. Simic and Sastry [36] proposed an algorithm to estimate the gradient in a noise-free environment, and it converges to the true gradient in a probabilistic sense. However, the sparsely collected data samples and the noise e®ect make the error of the estimated gradient higher. Without loss of generality, our following discussion is based on an uni-modal function with one local extremum. Sequential Extremum Search Process: 1. Initialization. In the beginning of this step, the cluster head inside a cluster will inform the sensor node with the largest (or the smallest) measured value, which is called the target sensor node, to initiate the successive maximum (or minimum) search. 2. Successive Ascending (or Descending) Process. The gradient of the current target sensor node is calculated using data samples from its neighboring sensor nodes. Then, it calculates the inner product of the estimated gradient and the normalized directional vector between its neighbor (within a certain radius range) and itself. After the value of the inner product is obtained, the current target sensor node chooses the neighboring sensor node that yields the largest (or the smallest) inner product value as the next target sensor node and notify it to take over this search 79 process. The current target sensor node will add its own coordinate to the noti¯ca- tion message and forward it to the next target sensor node. Thus, the noti¯cation message includes the coordinates of all previous target sensor nodes. This list will be utilized to decide when to terminate the process. This corresponds to a suc- cessive ascending (or descending) process if the extremum is the maximum (or the minimum). 3. Termination. Once receiving the noti¯cation, the target sensor node will ¯rst de- termine if it has been visited before. If not, it will continue the search process. Otherwise, the sensor node will notify the cluster head to terminate this ¯ne-tune process and send the cluster head the list of all visited target sensor nodes. Based on this list, the cluster head will compute the estimated coordinate of the max- imum point. To avoid the convergence problem in pathological cases and utilize the limited energy more e±ciently, a maximum number of iteration is assigned in the beginning of this sequential extremum search process. In the simulation, the maximum number of iteration is set to 5. At last, it will notify all other cluster heads of its completion of ¯ne localization and query the status of other cluster heads. When all cluster heads complete this step, the coordinator will set another start time of the next stage. An example is given in Fig. 4.2 to illustrate the sequential extremum search process. The advantage of this sequential search algorithm is that it involves only a small set of sensors at each step. For sensors that are not participating can be in the sleep mode to save their power until they are wakened by one of its neighbors. In this algorithm, 80 Step 1: Initialization Step 2: Gradient Estimation Estimated Gradient Step 3: Next Sensor Point Searching Step 4: Continue Step 2 and 3 Step 5: Estimate the location of maximal point Sensor Field Figure 4.2: Sequential search of the extremum point. the gradient estimation plays a critical role and the accuracy of the estimated gradient determinesthecorrectnessofthe¯naldetectionresult. Mathematically, the2-Dgradient is de¯ned as ~ µ =rS(x;y) = · @S(x;y) @x ; @S(x;y) @y ¸ T D b S(x;y) = ~ b T ¢rS(x;y); where ~ b is an arbitrary unit vector and D b is the gradient value along vector ~ b. Since we only consider one speci¯c time interval, the time dependency in (4.1) can be ignored. Thus, the gradient of X(x;y) can be rewritten as rX(x;y)=rS(x;y)+rn(x;y): (4.2) 81 WhenthegradientvaluesD b i X(x;y)ofvectors ~ b i ,i=1;2;¢¢¢ ;N, atcoordinates(x 0 ;y 0 ) are available, we can obtain a linear system as Z(N)=H(N)rS(x 0 ;y 0 )+H(N)rn(x 0 ;y 0 ); (4.3) where Z(N)=[D b 1 X(x 0 ;y 0 );¢¢¢ ;D b N X(x 0 ;y 0 )] T and H(N)=[ ~ b 1 ; ~ b 2 ;¢¢¢ ; ~ b N ] T . However, it is di±cult to get true gradient values D b i X(x;y) of vectors ~ b i , i = 1;2;¢¢¢ ;N, from scarcely distributed data samples. Therefore, the slope values derived from these discrete data samples are used to approximate them. Let the coordinates of the current sensor and its neighboring sensors within the one-hop data collection range be ~ p and ~ p i , i = 1;2;¢¢¢ ;N, respectively. Furthermore, X(~ p) and X(~ p i ) are their corre- sponding data samples. Then, (4.3) can be rewritten as Z d (N)=H(N) ~ µ+V(N)+V ² (N); (4.4) where ~ µ is the gradient direction, Z d (N)=[ X(~ p 1 )¡X(~ p) j~ p 1 ¡~ pj ;¢¢¢ ; X( ~ p N )¡X(~ p) j ~ p N ¡~ pj ] T ; H(N)=[ ~ p 1 ¡~ p j~ p 1 ¡~ pj ; ~ p 2 ¡~ p j~ p 1 ¡~ pj ;¢¢¢ ; ~ p N ¡~ p j~ p 1 ¡~ pj ] T ; V(N) = H(N)¢rn(x 0 ;y 0 ) and V ² (N) = Z d (N)¡Z(N), which is the estimation error whentheestimatedgradient,Z d (N),isusedtoapproximatethetruegradient,Z(N). The last two terms, V(N) and V ² (N), can be combined as one unknown noise term since the 82 estimationerrorcanbeviewedasacertaintypeofnoiseaswell. Thedimensionof ~ µistwo while the number of neighboring sensors, N, is usually larger than two. Then, (4.4) is an overdetermined system. Hence, ~ µ can be obtained via b ~ µ =[H(N) t H(N)] ¡1 H(N) t Z d (N). With this estimated gradient information, the maximal point can be found by the afore- mentioned algorithm. 4.3.2 Contour Points Determination This stage consists of two steps: cluster reorganization and contour points detection. They are described below. Cluster Reorganization Cluster reorganization is performed based on the established cluster structure de- scribed in the last section. The cluster head that has no local extreme point ¯nds the closest cluster head that has a local extreme point, and their cluster tables (including the information of residual energy) are exchanged so as to merge the two into one. This is followed by the re-election of cluster heads. Among newly selected cluster heads, the one with the maximal residual energy will be elected as a sink to set up a direct link to the base station for data transfer. Subclustering is needed to further segment a large cluster to facilitate its manage- ment and provide a more organized structure for contour point detection. After querying the setting of pseudo-beams from the base station, the newly-selected cluster head starts to elect subcluster heads. Since there may not exist a gateway node in a subcluster, the sensor node which is closest to the middle point of the pseudo-beam and has larger 83 residual energy will be chosen as a subcluster head. Then, the cluster head will inform all cluster members about the information of subcluster heads such as their ID numbers and coordinates. If a sensor node is not the chosen subcluster head, it will send a join- subcluster-request message to the corresponding subcluster head based on the regions determined by polar coordinates, which can be calculated by the coordinates of the esti- mated extreme point and sensor nodes. If a sensor node is the chosen subcluster head, it will wait to receive the request message from other sensor nodes until the waiting time (decided by the cluster head) is over. As the waiting time is due, the subcluster head will send a \¯nish" message to the cluster head and sensor nodes under its management so that it can proceed to the next step. The setting of these pseudo-beams can be sent into gateway sensors ahead of time so that the communication cost between the sink and the base station can be saved. An example of the ¯nal structure inside a cluster is illustrated in Fig. 4.3. Contour Points Detection After cluster reorganization, each subcluster head asks all sensor nodes under its management to broadcast their measured data to their neighboring sensor nodes and then estimate the location of contour points inside their subclusters. Also, the cluster headqueriesthesettingofpseudo-gridsandthespeci¯cvaluesthatcontourlinesrepresent from the base station via the sink, and pass these data to subcluster heads. However, similar to the setting of pseudo-beams, the data can be saved in gateway sensors in advance as well. Base on the data, the subcluster head will ask the sensor node that is closest to pseudo-grids to calculate the value of the object function based on the received 84 Sensor Cluster Pseudo-beam Sensor Subcluster Pseudo-grids Normal Sensor Subcluster Head Cluster Head Figure 4.3: Illustration of subclusters, pseudo-beams and pseudo-grids in a cluster. data samples from its neighboring sensors. The sensor node sends back the calculated value and its coordinates to the subcluster head for comparison (the subcluster head will know which pseudo-grid it represents according to the look-up table). The subcluster head chooses the largest one from the received objective function values and sends the coordinates of the corresponding pseudo-grid to the cluster head. After this process is completed,theclusterheadsendsestimatedlocationsofcontourpointstothebasestation via the sink. 85 The remaining problem is which objective function can help determine these contour points e±ciently and correctly. Here, we adopt the average log likelihood function as the objective function, ¡ 1 2 ln(2¼)¡ln(¾ n )¡ 1 2N¾ 2 n N X i=1 f(X i ¡¹) 2 g; (4.5) where N is the number of collected data samples, ¾ 2 n is the noise power, X i , i = 1;2;¢¢¢ ;N, are the collected data samples from neighboring sensor nodes and ¹ is the signal value that the particular contour line represents. To justify the objective function given by (4.5), we have the following proposition. Proposition 1 The average log likelihood function in (4.5) converges to its mean value in probability as N goes to in¯nity. This mean value has a larger value as the mean values of these random samples, X i , i = 1;2;¢¢¢ ;N, (these mean values may be di®erent from ¹) are closer to ¹. The proof of the above proposition is given in Proof I. However, when applying (4.5) to contour point determination in practice, we en- counter a problem. That is, the noise variance ¾ 2 n in (4.5) is unknown. One way to deal with this problem is to estimate ¾ 2 n based on collected data samples. However, this will demand additional computational complexity and may result in serious performance degradation when the estimation is not consistent for each pseudo-grid. The following proposition provides a better alternative to address this issue. 86 Proposition 2 Suppose that we have the same group of random samples as given in the proof of Proposition 1. The average log likelihood function in (4.5) can be rewritten as C 1 ¡ C 2 N N X i=1 (X i ¡¹) 2 ; (4.6) where C 1 and C 2 > 0 can be assigned by an arbitrary value without a®ecting the system performance, theoretically. The proof is given in Proof II. Withthisproposition,weknowthattheestimationofthenoisevarianceisnotneeded in the proposed algorithm. We can apply (4.6) by assigning arbitrary values to C 1 and C 2 > 0 for contour point determination. This reduces computational complexity and avoids the error arising from inconsistent variance estimation for each pseudo-grid. 4.3.3 Contour Line Construction After the base station receives all the information from cluster heads, it constructs the contourlinesaccordingly. Becauseofthesparsenessofsensornodes,theresolutionofeach contour line may not be good enough. In order to tackle this problem, a simple linear interpolation technique is adopted to have smooth contour lines passing over estimated contour points. It is explained as follows. To interpolate between two contour points with Cartesian coordinates (x 1 ;y 1 ) and (x 2 ;y 2 ), we ¯rst transform them from the Cartesian coordinates to the corresponding polar coordinates, (r 1 ;µ 1 ) and (r 2 ;µ 2 ) centered at the estimated extremum. Then, the polar coordinates (r ¤ ;µ ¤ ) of interpolated points can be calculated via µ ¤ = µ 1 +·(µ 2 ¡ 87 µ 1 ); ;0· ·· 1, and r ¤ = r 1 ¡r 2 µ 1 ¡µ 2 (µ ¤ ¡µ 2 )+r 2 . With these interpolated values, we can obtain smooth contour lines passing over measured contour points. 4.4 Communication Cost Analysis Although the energy required by MCU (Microcontorller Unit) occupies a signi¯cant por- tion, the energy of wireless communication among sensor nodes is still the major source of the total consumed energy [31]. This is especially true for the sink. Thus, it is im- portant to reduce the energy consumption in wireless communication. To demonstrate the advantage of the proposed distributed scheme, communication costs of a centralized scheme and the proposed schemes are analyzed and compared in this section. To perform the communication cost analysis, we make the following assumptions. ² The sensor ¯eld is over an L£L square area. ² An ordinary sensor node consumes one unit of radio energy when disseminating a data packet to its neighboring nodes within one-hop radio range, R o . ² The energy consumption of a packet-receiving con¯rmation message, which is sent back to the transmitter for con¯rmation as the receiver receives the packet, is ne- glected since its data amount is much less than one data packet. ² A data packet has a ¯xed size, which consists of the header, one unit of measured data (one data sample) and other relevant information. ² Whenadatapacketistransmittedfromasensornodetoanotherdirectly,thewhole communication cost can be divided into three components: the energy dissipated 88 by the transmitting units, the radio power consumption and the energy dissipated by the receiving units. Let E t and E r be the amount of energy required by the transmitting and receiving units in a sensor node for transmitting and receiving one packet transmission, respectively. ² Agatewaynodehastworadiomodes: theordinarymode(withone-hopradiorange R o ) and the gateway mode (with one-hop radio range R g , which is large enough for a gateway node to communicate with its neighboring ones). When the gateway node does not serve as a cluster head or the sink, it switches to the ordinary mode to save its energy consumption. Forthecentralizedscheme,alldatasamplesmeasuredbysensornodesaretransmitted to the base station for a centralized contour extraction. The total communication cost for one contour information update can be computed by E cen = E cls +E cen;SB +E dg (4.7) whereE cls isthecommunicationcostoftheclustering,E cen;SB istheenergyconsumption of the communication between the sink and the remote base station for a centralized scheme and E dg is the energy consumption of the data gathering. The communication cost of the proposed scheme consists of ¯ve major parts. Let E el , E cr and E cpd be the communication cost for extremum localization, cluster reorganization, contour points 89 determination, respectively. Then, the communication cost of the proposed scheme for one information update can be written as E dtr =E cls +E dtr;SB +E el +E cr +E cpd ; (4.8) where E dtr;SB is the energy consumption of the communication between the sink and the remote base station. Based on (4.7) and (4.8), we can ¯nd the energy di®erence as E cen ¡E dtr =(E cen;SB ¡E dtr;SB )+(E dg ¡E el ¡E cr ¡E cpd ): (4.9) We can decompose the right-hand-side of (4.9) into two parts. The ¯rst part is the di®erence of the energy dissipated by the sink for data transmission to the remote base station between the centralized and proposed schemes. This part can be rewritten as E cen;SB ¡E dtr;SB = Mf¢[( D SB R o ) 2 +E t +E r ]¡N c N cp ¢[( D SB R o ) 2 +E t +E r ] = (Mf¡N c N cp )¢ µ ( D SB R o ) 2 +E t +E r ¶ ; (4.10) where M is the total number of sensor nodes, f <1 is the data compression rate in each cluster head, D SB is the distance between the sink and the remote base station and N c and N cp are the numbers of extracted contour lines and contour points of each contour line, respectively. From (4.10), we see that the di®erence is positive if Mf is larger than N c N cp andbecomeslargerwhenthedistancebetweenthesinkandtheremotebasestation is longer. In other words, when the number of sensor nodes is large and/or a poor data 90 compression (or aggregation) techniques is adopted, the sink will dissipate more energy in the centralized scheme than the proposed distributed scheme. The second part is the di®erence of energy consumption dissipated for inter-nodes communication inside the sensor ¯eld. The part is negative due to additional energy is dissipated for the hierarchical structure construction in the proposed scheme but this value is ¯xed if the system parameters remain the same. Consequently, the proposed distributed scheme can resolve the bottleneck of data transmission from the sink to the remote base station if the total number of sensor nodes is large and/or the distance between the sink and the remote base station is large. However, it has to consume more energytobuildupahierarchicalstructurefordatafusion,whichisjusti¯ableiftheenergy of the sink is severely constrained. 4.5 Statistical Analysis of Estimation Accuracy Theperformanceof theproposedcontourextraction algorithm withrespecttothe sensor density and the noise power is analyzed in this section. In particular, we will study the MSE performance of the estimated contour line location. Since the estimation error of extremum localization does not a®ect the whole system performance much as long as the estimatedextremumlocationisstillinsidetheregionencompassedbytheinmostcontour line, the error propagation problem can be ignored. The analysis is conducted based on the assumptions that pseudo-grids along each pseudo-beam are uniformly distributed with spacing h and that there is a distance, denoted by d, between the true contour line locationanditsnearestpseudo-grid. Itissupposedthatthenumberofsensornodesinside 91 one-hop data collection range of each pseudo-grid is N. An illustration of the locations of pseudo-grids and a true contour point are given in Fig. 4.4. Pseudo-grids True Contour Point Data Collection Range d h Z N,1 Z N,2 Z N,k-1 Z N,k+1 Z N,k Z N,n Figure 4.4: Illustration of pseudo-grids and a true contour point in one pseudo-beam. Let Z N;i , i=1;¢¢¢ ;n, be the values calculated using the corresponding pseudo-grids via (4.6). Suppose that the mean values, ¹ Z N;1 ;¢¢¢ ;¹ Z N;n , of Z N;1 ;¢¢¢ ;Z N;n are known. For a given d, the MSE can be shown as MSE d = k¡1 X l=1 P ljk ¢(jl¡kjh+d) 2 +P kjk ¢d 2 + n X l=k+1 P ljk ¢(jl¡kjh¡d) 2 ; (4.11) whereP ljk =P(Z N;l is the largestj¹ Z N;k is the largest). Nevertheless,thevalueofdisnot known in practice. In this case, the conditional MSE is averaged over all possible values ofd to obtain an ensemble MSE. The d value is assumed to be uniformly distributed over [¡0:5h;0:5h] given that ¹ Z N;k is the largest. Then, (4.11) can be rewritten as MSE = n X l=1;l6=k P ljk ¢ · (l¡k) 2 h 2 + h 2 12 ¸ +P kjk ¢ h 2 12 ; (4.12) 92 where P ljk = P(Z N;l is the largestj¹ Z N;k is the largest). It is di±cult to compute P ljk in (4.12) since random variables Z N;i , i = 1;¢¢¢ ;n, are of the noncentral chi-squared distribution and might not be independently and identically distributed. To proceed further, we ¯rst use the following proposition to characterize the property of Z N;i . Proposition 3 Consider a set of random variables X i , i = 1;¢¢¢ ;N, each of which has the normal distribution with mean ¹ i and the same variance ¾ 2 n . Let Z N = C 1 ¡ C 2 N P N i=1 (X i ¡¹) 2 , where ¹ is a real value which can be di®erent from ¹ i . Then, random variable Z N is asymptotic normal with mean ¹ Z N =C 1 ¡C 2 ¾ 2 n ¡ C 2 N N X i=1 (¹ i ¡¹) 2 (4.13) and variance ¾ 2 Z N = 2C 2 2 ¾ 4 n N + 4C 2 2 ¾ 2 n N 2 N X i=1 (¹ i ¡¹) 2 : (4.14) This is written as Z N »AN(¹ Z N ;¾ 2 Z N ) as N is large. The proof of this proposition is given in Proof III. With this proposition, we can ap- proximate a noncentral chi-squared random variable composed by N independent Gaus- sian random variables by one Gaussian random variable so that the complexity of the problem can be largely reduced. Also, from Fig. 4.5, the PDF of the noncentral chi- squared distribution is almost identical to that of the normal distribution when N =10, which means this approximation is valid when N ¸10. 93 0 20 40 60 80 100 120 140 160 180 200 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 X PDF f(X) Noncentral chi−squared Normal distribution Figure 4.5: Comparison of the normal distribution and the noncentral chi-squared distri- bution with the degree of freedom N =10 and the noncentrality parameter ± =40. Now, we start to calculate the probability value P ljk . Since the data collection ranges overlap,randomvariablesZ N;1 ;¢¢¢ ;Z N;n calculatedvia(4.6)arenotstatisticallyindepen- dent. Instead, theyareofthecorrelatedmultivariatenoncentralchi-squareddistribution. FromProposition3, weknowthatZ N;i , i=1;¢¢¢ ;n, isasymptoticallynormal. However, we need to check if they are asymptotic correlated multivariate normal. From [34], a sequence of random vectorsfX k g is AN(¹ k ;c 2 k § k ) if and only if (X k ¡¹ k )=c k d ¡!N(0;§): 94 Thus, we can assure that Z N = [Z N;1 ;¢¢¢ ;Z N;n ] t is asymptotic correlated multivariate normal since p N(Z N ¡¹ Z N ) d ¡!N(0;R); where ¹ Z N = [¹ Z N;1 ;¢¢¢ ;¹ Z N;n ] t and R is the asymptotic form of R N . The (l;k)-entry of the covariance matrix can be computed by R N;lk = ( C 2 N ) 2 [2N lk ¾ 4 n +4¾ 2 n N lk X i=1 (¹ ki ¡¹) 2 ]; where N lk is the number of sensors inside the overlapped region between data collection rangesoftwovirtualpointswithindicesl andk. ThederivationofR N;lk isgiveninProof IV. Then, we have P ljk = Z 1 ¡1 Z Z N;l ¡1 ¢¢¢ Z Z N;l ¡1 1 p (2¼) n jdet(R N )j £expf¡ 1 2 (Z N ¡¹ Z N ) t R ¡1 N (Z N ¡¹ Z N )g dZ N;1 ¢¢¢dZ N;l¡1 dZ N;l+1 ¢¢¢dZ N;n dZ N;l : (4.15) The above integral is too complicated to get a closed form, but it can be computed by the Monte Carlo method. 4.6 Simulation Results Computer simulation was conducted under the following setting: 2000 sensor nodes are randomly deployed over a 10 by 10 square with the average sensor density of 20 sensor 95 nodes per unit square. The underlying physical phenomenon is multi-modal, which is shown in Fig. 4.6, and it is assumed to be constant in four data reporting periods so that 4 noisy measurements can be averaged to suppress the noise interference in a local sensorbeforefurtherprocessing. Thevalueofeachlocalmaximumpointisdi®erentfrom each other. The value of the global maximum is 2. The space between two neighboring pseudo-grids is 0:1 and the one-hop data collection range of a sensor node is 0:5. There existsadditivewhiteGaussiannoisewithpower0:01inthemeasureddata. Sincemostof noiseinterferencecanbesuppressedeitherby¯ltering(w.r.t. sensingnoise)orbywireless communication techniques (w.r.t. communication errors), the residual noise interference is usually small. Simulation results show that the proposed statistical signal processing method can provide a reasonable estimate in the presence of a high noise level. The quiver plots of the actual and the estimated gradient values are compared in Fig. 4.7. They are closer to each other when the sensor density becomes larger. The mean square error is about 0:5116 using the parameters described above. Actually, the information needed in the extreme point search is the gradient direction. The gradient directions of almost all quivers are fairly accurate. Thus, as long as the initial node found in coarse localization is close enough to the local extreme point of interest, the proposed sequential extremum search process can converge to the neighborhood of the extreme point even though there exists some slight di®erence between the magnitudes of the actual and the estimated gradients. The actual and the reconstructed contour lines are shown in Fig. 4.8, where contour lines with four di®erent levels of signal values are determined. The signal value of each 96 Density 15 20 25 30 35 N 12 16 20 24 28 Exp. #1 0.0271 0.0158 0.0113 0.0099 0.0098 Exp. #2 0.0244 0.0125 0.0088 0.0118 0.0080 Exp. #3 0.0175 0.0119 0.0120 0.0087 0.0073 Average 0.0230 0.0134 0.0107 0.0101 0.0084 Theoretical 0.0306 0.0122 0.0115 0.0092 0.0091 Table 4.1: Comparison of simulated and theoretical MSE values for the contour line representing signal value 0:8 with local maximum value=1, noise power=0:01, R c = 0:5 and h=0:1. contour line is given next to it. We see the proposed scheme yields good estimates of these true contour lines. Fig. 4.9 shows the e®ect of the one-hop data collection radius, R c , on the system performance. We see that there exists an optimal one-hop data collection range. A largerR c valuedoesnotguaranteebetterperformanceintheproposedschemeduetothe variation of signal values. Since signal values inside one-hop data collection range may changewhenR c becomessu±cientlylarge. Thisphenomenoncanbeexplainedby(4.14), where the second term represents the inhomogeneity of signal values. Thus, the one-hop data collection range R c has to be carefully chosen to achieve optimal communication energy utilization. In this simulation, the optimal value of R c is around 0:5¡0:6. Table 4.1 compares simulated and theoretical MSE values given by (4.12) and (4.15) for the contour line with signal value 0:8. Since MSE value varies with the slope of a physical phenomenon, we compare the largest values over the sensor ¯eld. We see that the theoretical MSE value is quite close to the simulated MSE value. Fig. 4.10 shows the MSE calculated by (4.12) and (4.15) for di®erent sensor densities and noise power values. We observe from Fig. 4.10 that the performance improvement becomes less as the sensor 97 densityincreasesandthenoisepowerdecreases. TheMSEwilleventuallyconvergetothe averaged square quantization errorEfd 2 g=h 2 =12, which is related to the space between two neighboring pseudo-grids, h, but independent of the sensor density and the noise power. Thus, Fig. 4.10 provides useful information for the cost-performance tradeo® in system design. 4.7 Conclusion A distributed scheme to extract contour lines using a wireless sensor network was pro- posed for a quasi-static scalar ¯eld. The performance analysis and computer simulation were conducted to demonstrate the e±ciency of the proposed scheme in a noisy environ- ment. We found the performance improves as the sensor density is higher and/or the noise power is lower. With this performance analysis, it is possible to design a system to achieve the required performance under di®erent environmental conditions. In addi- tion, the communication cost was analyzed. As compared with the centralized scheme, the analysis shows that the proposed scheme can e®ectively resolve the data forward- ing bottleneck from the sink to the remote base station at the expense of more energy consumption in forming a hierarchical structure. 98 0 2 4 6 8 10 0 2 4 6 8 10 −0.5 0 0.5 1 1.5 2 2.5 (a) actual signal value 0 2 4 6 8 10 0 2 4 6 8 10 −0.5 0 0.5 1 1.5 2 2.5 (b) sensor measured value Figure 4.6: Illustration of (a) the actual signal value and (b) sensor measurements of a multi-modal physical phenomenon. 99 0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 9 10 (a) True Gradient 0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 9 10 (b) Estimated Gradient Figure 4.7: Quiver plots of (a) the actual and (b) the estimated gradients. 100 2 4 6 8 1 2 3 4 5 6 7 8 9 0.8 0.8 0.4 0.4 0.8 0.8 1.2 1.6 1.2 (a) actual contour lines 0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 9 10 0.8 0.8 0.4 0.4 0.8 1.2 1.6 1.2 0.8 (b) estimated contour lines Figure4.8: Illustrationof(a)theactualcontourlinesand(b)theestimatedcontourlines. 101 0.3 0.4 0.5 0.6 0.7 0.8 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 One−hop Data Collection Radius (R c ) MSE Contour line with signal value = 0.8 Contour line with signal value = 0.4 Figure4.9: Thee®ectoftheone-hopdatacollectionradiusR c onthesystemperformance. 102 15 20 25 30 35 −3 −2.5 −2 −1.5 −1 0 0.05 0.1 0.15 0.2 Sensor Density Log10(Noise Power) Mean Square Error (a) 15 20 25 30 35 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Sensor Density (nodes/unit square) Mean Square Error 0.1 0.0316 0.01 0.00316 0.001 (b) Figure 4.10: Theoretical MSE as a function of (a) the sensor density and the noise power and of (b) the sensor density when R c =0:5, h=0:1. 103 4.8 Proof I: Proof of Proposition 1 Proof: Consider N independent random samples, X 1 ;X 2 ;¢¢¢ ;X N , and X i is generated from normal distribution N(¹ i ;¾ 2 n ). We ¯rst calculate mean value e ¹ and variance f ¾ 2 of (4.5) as e ¹ = ¡ln( p 2¼¾ n )¡ 1 2 ¡ 1 2N¾ 2 n N X i=1 (¹ i ¡¹) 2 ; (4.16) f ¾ 2 = 1 4N 2 ¾ 4 n Var ( N X i=1 (X i ¡¹) 2 ) : (4.17) Recall that, for independent random variables Z i » N(¹ i ;1), X = P n i=1 Z 2 i is of the noncentral Chi-square distribution. It has a mixture distribution X » Â 2 n+2Y , where Y » P(¸), which is a Poisson distribution with parameter ¸, and ¸ = 1 2 P n i=1 ¹ 2 i [39]. With this result, we can calculate the variance of P N i=1 (X i ¡¹) 2 and rewrite (4.17) as f ¾ 2 = 1 2N + 2 2N 2 N X i=1 µ ¹ i ¡¹ ¾ n ¶ 2 : (4.18) We conclude from above that the variance of the average log likelihood function ap- proaches zero as N goes to in¯nity. In other words, the average log likelihood function converges to its mean in probability. Also, from (4.16),e ¹ will have the larger value as ¹ i , i=1;2;¢¢¢ ;N, is closer to the signal value, ¹, of interest. 104 4.9 Proof II: Proof of Proposition 2 Proof: According to (4.16) and (4.18), the mean value and the variance of the average log likelihood function can be expressed in terms of C 1 and C 2 , respectively, as e ¹ = C 1 ¡C 2 ¾ 2 n ¡ C 2 N N X i=1 (¹ i ¡¹) 2 ; (4.19) f ¾ 2 = 2C 2 2 N Ã ¾ 4 n + 2¾ 2 n N N X i=1 (¹ i ¡¹) 2 ! ; (4.20) where C 1 =¡ 1 2 ln(2¼)¡ln(¾ n ) and C 2 = 1 2¾ 2 n . We conclude from (4.20) that C 1 has no e®ect on the variance. Thus, C 1 can be assigned any scalar value. Next,weprovethatC 2 canbeassignedbyanarbitrarypositivevalueaswell. AsC 2 is decreased,thesystemperformanceisimprovedduetothedecreasingvarianceaccordingto (4.20). However,theexpectationvaluein(4.19)changesatthesametime. Tofullyunder- standhowC 2 a®ectsthesystemperformance,weconsiderthefollowingscenario. Suppose that there are two groups of data samples, X 1 ;X 2 ;¢¢¢ ;X N and X ¤ 1 ;X ¤ 2 ;¢¢¢ ;X ¤ N ¤, where X i » N(¹ i ;¾ 2 n ) and X ¤ i » N(¹ ¤ i ;¾ 2 n ). Let Z = C 1 ¡ C 2 N P N i=1 (X i ¡¹) 2 and Z ¤ = C 1 ¡ C 2 N ¤ P N ¤ i=1 (X ¤ i ¡¹) 2 . By Chebyshev's inequality, we know Pr(jZ¡EfZgj¸ K¾ Z )· 1 K 2 andPr(jZ ¤ ¡EfZ ¤ gj¸K¾ Z ¤)· 1 K 2 . Therefore, we can assure Pr(Z >Z ¤ )¸(1¡ 1 K 2 ) 2 ifjEfZg¡EfZ ¤ gj¸K¾ Z +K¾ Z ¤ or 105 ¯ ¯ ¯ ¯ ¯ [ 1 N ¤ N ¤ X i=1 (¹ ¤ i ¡¹) 2 ¡ 1 N N X i=1 (¹ i ¡¹) 2 ] ¯ ¯ ¯ ¯ ¯ (4.21) ¸ K¾ n v u u t 2 N [¾ 2 n + 2 N N X i=1 (¹ i ¡¹) 2 ]+K¾ n v u u t 2 N ¤ [¾ 2 n + 2 N ¤ N ¤ X i=1 (¹ ¤ i ¡¹) 2 ]: It is clear that C 2 does not a®ect the inequality in (4.21). Thus, the value of C 2 has no e®ect on system performance. However, C 2 is chosen to be positive so that the objective function has the maximum point. 106 4.10 Proof III: Proof of Proposition 3 Proof: First, we would like to prove that S N = 1 N P N i=1 (X i ¡¹) 2 is asymptotic normal. We have S N = 1 N N X i=1 [(X i ¡¹ i )+(¹ i ¡¹)] 2 = ¾ 2 n V N +2¾ n Y N + 1 N N X i=1 (¹ i ¡¹) 2 ; where V N = 1 N P N i=1 ( X i ¡¹ i ¾n ) 2 , Y N = 1 N P N i=1 (¹ i ¡¹)( X i ¡¹ i ¾n ). Since V N = 1 N N X i=1 ( X i ¡¹ i ¾ n ) 2 = 1 N N X i=1 U i =U; where U i »Â 2 (1), EfU i g=1 and VarfU i g=2, we have p N(U¡1)= p 2 d ¡!N(0;1)) U¡1 q 2 N d ¡!N(0;1) by the central limiting theorem. With the asymptotic normality property, we have V N = U »AN(1; 2 N ) for large N. Furthermore, Y N = 1 N N X i=1 (¹ i ¡¹)( X i ¡¹ i ¾ n )= 1 N N X i=1 (¹ i ¡¹)W i ; where W i = X i ¡¹ i ¾n »N(0;1). Thus, we have Y N »N(0; 1 N 2 N X i=1 (¹ i ¡¹) 2 ): 107 Let 1 N = [1;1;¢¢¢ ;1] t , W = [W 1 ;W 2 ;¢¢¢ ;W N ] t , W 2 = [W 2 1 ;W 2 2 ;¢¢¢ ;W 2 N ] and ¹ = [¹ 1 ;¹ 2 ;¢¢¢ ;¹ N ] t . Then, we obtain V N = 1 N N X i=1 ( X i ¡¹ i ¾ n ) 2 = 1 N 1 t N W 2 ; Y N = 1 N N X i=1 (¹ i ¡¹)( X i ¡¹ i ¾ n )= 1 N (¹¡¹1 N ) t W; and Cov(V N ;Y N ) = 1 N 2 1 t N EfW 2 W t g(¹¡¹1 N ) = 1 N 2 1 t N 0 NxN (¹¡¹1 N )=0 Thus, [V N ;Y N ] t has mean ¹ N =[1;0] t and covariance § N = 2 6 6 4 2 N 0 0 1 N 2 P N i=1 (¹ i ¡¹) 2 3 7 7 5 : By Slutsky's theorem, we have p N( 2 6 6 4 V N Y N 3 7 7 5 ¡ 2 6 6 4 1 0 3 7 7 5 ) d ¡!N(0;§= 2 6 6 4 2 0 0 0 3 7 7 5 ): By the asymptotic multivariate normality property, we get 2 6 6 4 V N Y N 3 7 7 5 »AN(¹ N ;§ N ): 108 Since S N = ( 2 6 6 4 ¾ 2 n s¾ n 3 7 7 5 ) t 2 6 6 4 V N Y N 3 7 7 5 + 1 N N X i=1 (¹ i ¡¹) 2 ; we conclude that S N » AN(¾ 2 n + 1 N N X i=1 (¹ i ¡¹) 2 ; 2¾ 4 n N + 4¾ 2 n N 2 X i=1 N(¹ i ¡¹) 2 ): Finally, Z N is also asymptotic normal since Z N = C 1 ¡C 2 S N » AN(C 1 ¡C 2 ¾ 2 n ¡ C 2 N N X i=1 (¹ i ¡¹) 2 ; 2C 2 2 ¾ 4 n N + 4C 2 2 ¾ 2 n N 2 X i=1 N(¹ i ¡¹) 2 ): The proof is completed. 109 4.11 Proof IV: Derivation of R N;lk The derivation of (l;k)-entry of covariance matrix, R N , is given as follows. R N;lk = E © (Z N;l ¡¹ Z N;l )(Z N;k ¡¹ Z N;k ) ª = EfZ N;l Z N;k g¡¹ Z N;l ¹ Z N;k = C 1 ¡ C 1 C 2 N Ef N lk X j=1 (X kj ¡¹) 2 g ¡ C 1 C 2 N Ef N X j=N lk +1 (X kj ¡¹) 2 g¡ C 1 C 2 N Ef N lk X j=1 (X lj ¡¹) 2 g + C 2 2 N 2 Ef[ N lk X j=1 (X lj ¡¹) 2 ][ N lk X j=1 (X kj ¡¹) 2 ]g + C 2 2 N 2 Ef[ N lk X j=1 (X lj ¡¹) 2 ][ N X j=N lk +1 (X kj ¡¹) 2 ]g ¡ C 1 C 2 N Ef N X j=N lk +1 (X lj ¡¹) 2 g + C 2 2 N 2 Ef[ N X j=N lk +1 (X lj ¡¹) 2 ][ N lk X j=1 (X kj ¡¹) 2 ]g + C 2 2 N 2 Ef[ N X j=N lk +1 (X lj ¡¹) 2 ] £ [ N X j=N lk +1 (X kj ¡¹) 2 ]g¡¹ Z N;l ¹ Z N;k = ( C 2 N ) 2 2 4 2N lk ¾ 4 n +4¾ 2 n N lk X j=1 (¹ kj ¡¹) 2 3 5 ; where N lk = 2D[R 2 c arccos( h 2R c )¡ h 2 q R 2 c ¡ h 2 4 ] when l6= k and N lk = N = ¼R 2 c D when l =k, given sensor density D, R c and h. 110 Chapter 5 A Mesh-based Approach to Contour Lines Estimation with Cross-layer Considerations Using Homogeneous Sensor Networks 5.1 Introduction Environmental monitoring is one of important applications in wireless sensor networks, where sensors are deployed to observe the dynamic behavior of the natural physical phe- nomena. For those physical phenomena observed through the sensing devices which can provide real-number measurement results, the signal strength can be modeled as a scalar ¯eld. Contour lines are useful in characterizing them and contour lines estimation is the basic step of environmental monitoring before any further action is taken. A robust mesh-based approach to contour line estimation is proposed for wireless sensor networks in this work. 111 There has been research work on the monitoring of physical phenomena. Nowak and Mitra [30] proposed a boundary estimation scheme where there exists a prede¯ned hier- archical topology among sensors. For a sensor network with a hierarchical topology, this approach performs well and provide quite accurate estimation results. Chintalapudi and Govindan [5] proposed three localized approaches to detect edge nodes to depict the size and the range of the monitored physical phenomena; namely, the statistical- , the ¯lter-, and the classi¯er-based approaches. More recently, a statistical approach based on the Neyman-Pearson criterion to edge node detection was proposed in [28] to improve the performance and the sensitivity to the sensor localization error furthermore. However, these algorithms cannot show the density distribution inside a monitored physical phe- nomenon but its size and range. To resolve this problem, another localized approach was proposed in [27] and [26] to estimate the locations of contour lines with noisy measure- ments. However, the proposed approaches require the deployment of sensor nodes of a higher cost and a hierarchical network architecture. Even though it provides good esti- mation performance, the system cost is higher and its robustness and °exibility may be a®ected, too. It is desirable to develop a lower cost system that works with homogeneous sensor networks. To achieve this objective, we make use of the soft information based on contour node inference and the quantities examined in [27] and [26] to estimate the contour lines using 2-D triangular meshes. A mesh-based approach to contour line estimation with cross-layer considerations us- ingwirelesssensornetworksisproposedandanalyzedinthiswork. Themaincontribution is to use a triangular-mesh-based approach to localize true contour lines with irregular 112 (or random) sensor deployment without any prede¯ned hierarchical network topology to allow greater °exibility. Although locations of estimated contour points may be erro- neous due to various noise e®ects (including sensing and channel noises) and scarcity of sensor nodes, an error compensation scheme is developed to correct detection errors to a reasonable degree so that the ultimate estimation results are accurate enough. Both the °exibility and robustness of the system increase accordingly. In this work, we also are concerned with a practical and error-robust approach with realistic environment settings to improve the communication power e±ciency without in- creasing system complexity. With proposed error-robust algorithm, we are able to avoid excessive data packet retransmission by sacri¯cing negligible performance degradation without increasing transmission power and system complexity in PHY layer such as FEC scheme. In addition, we may also apply FEC scheme, which increases system complexity in PHY layer, to save transmission power. The trade-o® depends on the system require- ments. To evaluate the impact of cross-layer parameters on the system performance, issues of the cross-layer design are examined to determine proper system parameter set- tingsforcomputersimulation. Simulationresultsaregiventoshowthee®ectsofdi®erent system parameters on the overall system performance and reveal clues of system design rules. The rest of this paper is organized as follows. The system model and the model parameters are described in Sec. 5.2. Cross-layer considerations in system design are discussed and examined in Sec. 5.3. To provide overall concept of triangular meshes, 2-D triangular mesh generation and Delaunay triangulation are reviewed in Sec. 5.4. The 113 proposed contour line estimation approach based on the mesh structure is described in Sec. 5.5. Proper parameter settings are discussed and simulation results are provided to demonstrate the system performance in Sec. 5.6. Finally, concluding remarks are given in Sec. 5.7. 5.2 System Model and Model Parameters 5.2.1 System Model and Problem Formulation Considerawirelesssensornetworkdeployedoveralargegeographicalextenttomonitora time-varying physical phenomenon with scalar measurements. To extract useful informa- tionfromthedatameasuredbysensornodeswithoutconsumingtoomuchresourceofthe whole network, measurements are ¯rst processed or fused at each local sensor node, and processed/fusedresultsarethentransmittedtoasinkoraremotebase-stationviaadata gathering tree for post-processing. Fig. 5.1 shows the system model, which illustrates threemodulesoftheinformationextractingprocess;namely,localdataprocessing/fusion, data gathering and post-data processing. Fig. 5.2 shows the model of a local data fusion process, which is conducted at each sensor node of the network, which might be the leaves or intermediate nodes of the data gathering tree. The local signal strength together with sensing/thermal noise is measured and sent to post-processing parts of a sensor node. The measured data are then quantized into B bits (2 B levels) for digital communication, where the number of quantizationbits,B,canvaryaccordingtothechannelconditionandtheresidualbattery power. Sincethecommunicationrangeofeachsensornodeisshort,itmayrequireseveral 114 L o c a l Fu sio n Cen ter Sin k Radi o Ran g e P robi n g Ran g e Rem o te Basestat io n 1) L ocal Data Fu s i on P ro c ess 2 ) Da t a G ath ering 3 ) P o st P ro ce ssin g Figure 5.1: System model of the information extracting process. hops to reach a local fusion center if farther sensor nodes are demanded to transmit their measurements. As shown in Fig. 5.2, a K i -hop wireless channel is used to transmit data collected at the ith neighboring sensor node. Due to the fading channel and wireless channel noise, the received data at the local fusion center might be corrupted. Thus, a couple of transmissions may be needed in order to meet the data quality requirement at the receiver side. After receiving measured data, the local fusion center performs data fusion to extract the requested information. To ensure that data samples measured by sensor nodes are su±cient to describe the monitored phenomenon, itisassumed thatthe largest distancebetweentwosensor nodes over the sensor ¯eld is smaller than one half of the period of the highest frequency component of the monitored phenomenon, i.e. the Nyquist sampling criterion is met. 115 K _1-H op Wi rel ess Ch a n n el B - bi t Qu an ti zer B-bi t D ata F usio n/ P roces sin g K _N- H op Wi rel ess Ch a n n el B - bi t Qu an ti zer S e n s or S e n s or Sen si n g No i se n _1( k) Sen si n g No i se n _N(k) B i t Er ro rs B i t E rro r s Re su l t s P h y s i c a l P h e n o m e n o n S _1(k) S_N( k) X_1(k) X_N ( k) Y_1(k) Y_N(k) R_N(k) R _1( k) Loca l Fu s i on Center Sen s o r Nod es Figure 5.2: System model of the local data fusion process. According to the system model shown in Figs. 5.1 and 5.2, the received measurement at the local fusion center from the ith sensor node at time k can be written as R i (k)=Q B [X i (k)]+n c;i (k); (5.1) where Q B [¢] is a quantization function with 2 B levels, X i (k) = S i (k)+n i (k) and n c;i (k) is the equivalent noise component induced by the communication bit errors. We would like to estimate contour lines with randomly deployed sensor nodes. There are two problems to address in estimating contour lines; namely, the noise e®ect and and 116 irregularpositionsofdeployedsensornodes. Inthefollowingsections,wewilltacklethese two problems and propose some solution methods. 5.2.2 Model Parameters In this subsection, we consider a practical system so as to have a better idea of model parameters. FromthedatasheetoftheMICA2mote[7],itsbarometricpressure,humidity and temperature sensors have around§1:5%,§3:5% and§0:5 ± C sensing errors at 25 ± C, respectively. By some manipulations, we ¯nd that the SNR (signal-to-noise ratio) value with respect to the sensing noise is greater than 29 dB. Thus, the impact of the sensing noise on data quality is not a major issue. Thesimpli¯edunit-diskgraphmodelisadoptedinmostpreviouswork, wherethethe channel noise e®ect is neglected. However, the link quality can be a®ected seriously by channel attenuation in practice. This is especially true when the link distance is in the transitional region [42]. According to [6], the radio chip of the MICA2 mote, Chipcon CC1000, has a noise ¯gure of 13 dB and equivalent bandwidth of 30 KHz. Given the ambient temperature of the receiver, the channel noise power can be calculated by [32] P n =(F +1)kT 0 B; whereF isthenoise¯gure(intherealnumberscale), k istheBoltzmann'sconstant,T 0 is the ambient temperature and B is the equivalent bandwidth. If the ambient temperature is 25 ± C, the channel noise power is¡115:87 dBm while the maximal transmission power of the MICA2 mote is 5 dBm [6]. 117 The log-normal shadowing model provides high accuracy in modeling the multi-path channel between sensor nodes [41]. It can be written as PL(d)=PL(d 0 )+10nlog 10 ( d d 0 )+X ¾ ; where d is the link distance, d 0 is the reference distance, n is the path loss exponent, and X ¾ is a zero-mean Gaussian random variable with variance ¾ 2 . Consider the case with n=4, PL(d 0 )=55 dB, d=25 meters and d 0 =1 meter. Then, the average SNR at the receiver is 9:95 dB. If the link distance is 20 meters, the SNR increases to 13:83 dB . We can draw two conclusions based on the above discussion. First, the channel noise could induce large data degradation if the packet retransmission or FEC mechanism is not used in MAC or PHY layer. Second, based on the model in (5.1), a proper data fusion algorithm can be used to mitigate the data distortion e®ect and the quality of the received data can be lowered to save the communication power. These two issues will be discussed in the next section. 5.3 Cross Layer Considerations in System Design 5.3.1 Physical and MAC Layers Consideration A K-hop data transmission scheme is illustrated in Fig. 5.3. As shown in this ¯gure, each hop is modeled by a binary symmetric channel with a di®erent bit error probability. 118 P_1( E ) P _K(E) Tra nsm i tti ng Node Rece i vin g Nod e Figure 5.3: Illustration of a K-hop data transmission system. With this model, the bit error probability for a K-hop wireless data transmission can be calculated by P K¡hop (E)=1¡P K¡hop (C)=1¡ K Y i=1 (1¡P i (E)); (5.2) where P K¡hop (C) is the probability for the bit to be transmitted correctly. Since each item of the product is less than 1, the bit error probability value will become larger as K (the hop number) increases. Besides, the bit error probability is dominated by the worst link. If all single-hop links have the same error probability, then the bit error probability along the K-hop path becomes P K¡hop (E) = 1¡P K¡hop (C) (5.3) = K X i=1 0 B B @ K i 1 C C A ¢(¡1) i+1 £P(E) i : The bit error rate (BER) of each hop can be computed from the E b =N 0 -to-BER mapping,whichchangesasthewirelesschannelcondition,themodulationandtheapplied 119 FEC(ForwardErrorCorrecting)schemesvary. TherelationshipbetweenSNRandE b =N 0 can be shown as E b N 0 =SNR¡10£log 10 µ bit rate bandwidth ¶ ; (5.4) where SNR is the power ratio of communication symbol over channel noise, E b is the energy per bit and N 0 is the power density of channel noise spectrum. However, for layers above the physical layer, the packet reception rate (PRR) is of our main concern. According to [42], if each bit error is independent, the PRR of a K-hop transmission can be calculated with the knowledge of BER by PRR K¡hop =[1¡P K¡hop (E)] 8f ; (5.5) where f is the packet size. This independent bit error assumption usually holds if a proper interleaving scheme is adopted. Take non-coherent BFSK modulation without FEC mechanism over the Rayleigh fading channel as an example. The BER is around 1£ 10 ¡3 at E b =N 0 = 30 dB [37]. Then, the average number of one-hop transmissions (including the transmission at the ¯rst round and re-transmission afterwards) for an error-free receiving is as high as 2:23 whenthepacketsizeis100bytes(MAC-layerpacketsizeis9to127bytesinIEEE802.15.4 standard). In other words, it consumes around 3 more dB of average communication power for one error-free packet reception if no FEC is applied. This number grows fast when the packet size or the hop number increases. As a result, it demands a lot of communication power to transmit an error-free data packet. 120 There are two approaches to resolve this problem. One is to apply FEC with 3 dB coding gain or more in PHY layer to correct bit errors in the received packet such as convolutional codes (3,1,4) and (2,1,5) [29] though it may increase system complexity (computational complexity and memory storage); the other is to design an error-robust algorithm for application layer. If the designed algorithm could sustain low data quality of received packet without large performance degradation of data fusion, the average number of transmissions will decrease and the communication power waste can be saved. Let P AE be the acceptable bit error rate of the designed algorithm. Then, the PRR can be written as PRR K¡hop;A = 8 > > > > < > > > > : [1¡P K¡hop (E)+P AE ] 8f ; if P K¡hop (E)>P AE 1; otherwise. (5.6) If P AE = 1£10 ¡3 , the average number of transmissions can be reduced to 1, which is aboutonehalfoftheoriginalone. Thougheachofthesetwoapproachescanachieveeither goodperformanceorlowcomplexity,theycanonlyavoidadditionalcommunicationpower waste and can not further improve the power e±ciency. However,communication power e±ciency can be further improved by applying two approaches together. Though it may degradedatafusionperformanceandraisesystemcomplexity,thenetworklifetimecanbe extended due to additional communication power saving. For example, communication powercouldgainadditional3dBe±ciencywithoutlargeperformancedegradationifFEC with 3 dB coding gain is applied together with the designed error-robust algorithm. 121 5.3.2 Network Layer Consideration Due to the small radio range of a sensor node, how to route data from a transmitter to a receiver via intermediate nodes e±ciently is an important issue in wireless sensor networks. There are di®erent routing protocols proposed for power-e±ciency. However, it was shown in [24], [18] that routing protocols without the geographical information in decision calculations are not scalable. To increase the scalability and the fault tolerance of the network, a geographic-based routing protocol is preferred even though it requires theknowledgeofnodelocations. Thegeographicalinformationcanbeobtainedbysensor localization algorithms or GPS (Global Positioning System). There are several geographic routing protocols proposed based on di®erent local met- rics [19], [22], [23]. Karp and Kung [19] proposed the GPSR routing protocol that con- stitutes the main idea of latter geographic routing algorithms. It uses the hop count as its local metric for routing decisions. Nevertheless, the hop count is only an approximate metricsinceitassumesanytwosensornodesresidingwithintheradiorangeofeachother communicateperfectly. Thereactuallyexistsatransitionalregionbetweentheconnected anddisconnectedregions. Hence, Bhattacharjee et al.[22]andHelmyet al.[23]proposed more realistic local metrics to replace the hop count by considering PPR and power dis- sipation. As a result, the data can be routed along the path with the best link quality or highest power e±ciency instead of the one with the least hop count to approach the optimal performance. In our system, to maintain the circular shape of the probing range (or the data collection range) for contour nodes inference, ambient nodes have to be noti¯ed the size 122 of the probing range and the geographic information of the request node when the data request is sent out. After receiving the data request and other information, ambient nodes will decide if they have to send their measurements to the request node. If the probing range covers two or more hops neighbors of the request node, geographic routing protocols proposed in [22], [23] can ¯nd the best route to send data e±ciently. Then, the request node will utilize collected measurements for the inference. 5.3.3 Adaptive Transmission with Cross-Layer Interaction As discussed above, it is di±cult to achieve both perfect data quality and good power e±ciency at the same time even with FEC, especially when the wireless channel condi- tion is poor. A more reasonable solution is to adjust these two requirements adaptively according to the current channel condition. For example, we may sacri¯ce the data qual- ity to trade for better power e±ciency if a certain amount of data distortion (including quantization and channel distortion) is allowed. This adaptive adjustment requires the information exchange between layers. Fig 5.4 shows an adaptive data transmission scheme. As shown in the ¯gure, the applicationlayerprovidesthenetworklayertheinformationaboutthesizeoftheprobing rangebasedontheresidualpowerandtherequireddatafusionqualitysothatthenetwork layer can identify all neighboring nodes inside the probing range and route the data to the request node e±ciently. The adjusted data quality requirement, especially for the tolerable channel distortion, is sent to the MAC layer by the application layer based on theaveragetransmissionnumberatthecurrenttime. Iftheaveragetransmissionnumber 123 Ap pli cat ion Layer Netw ork Layer M AC Layer PHY La y er Siz e of t he Pro bin g Ra ng e Da ta Qua lity Req uir ement Ave ra ge Numbe r o f Tran smission s Figure 5.4: Illustration of interactions between application, MAC and physical layers. increases, the application layer will raise the tolerable channel distortion to reduce the numberoftransmissionsforcommunicationpowersaving;otherwise,thetolerablechannel distortioncanbeloweredtoenhancethedatafusionperformance. Tolerablequantization distortion will then be raised to reduce the transmission bit rate for additional receiving gain by (5.4) if maximal tolerable channel distortion is met and the average transmission number still can not be decreased e®ectively. Because of the adjustability, the system has higher °exibility to adapt to the wireless channel conditions while power e±ciency is enhanced at the same time. 5.4 2-D Triangular Mesh Generation Forsensor deploymentinasquare grid, itiseasierto locate contourlines since thesensor densityisuniformoverthewholesensor¯eldandthedistancebetweenanytwoneighbor- ing nodes is ¯xed. Well-developed techniques (edge detection/contour line extraction) in 124 (a) Random deployment (b) square-grid deployment Figure 5.5: Comparison of two sensor node deployment schemes. image processing can be applied directly. However, for random sensor deployment, both the sensor density and the distance between any two neighboring nodes are not ¯xed, which makes the problem more complicated. Fig. 5.5 illustrates the di®erence when only the closest neighboring nodes are connected in two deployments. We ¯rst de¯ne adjacent neighboring nodes below. De¯nition 3 Adjacent Neighboring Nodes Adjacent neighboring nodes are sensor nodes whose Dirichlet regions are adjacent to each other, where the Dirichlet region is de¯ned as S p i =fz :d(z;p i )·d(z;p j );j6=ig; where P = fp 1 ;p 2 ;¢¢¢ ;p M g are locations of all sensor nodes over the whole ¯eld and d(p j ;p i ) is the Euclidean distance between two sensor nodes, p j and p i . 125 With the above de¯nition, we can easily ¯nd which nodes are adjacent neighbors via the Voronoi diagram and the meshed structure can be formed by connecting these sensor nodes. A more direct way is to apply Delaunay triangulation, which is a well-known geometry dual of the Voronoi diagram. We found that formed triangles in random de- ployment, just like the squares in the grid deployment, can be used as good references to determine the location of contour lines of interest. Therearemanywell-developedalgorithmsfortheformationofDelaunaytriangulation [2], [21], [35]. Due to the computational complexity concern, each of aforementioned algorithms attempts to remove any redundant step from the triangulation process to achieve the optimal computational time. Thus, they are di±cult to explain intuitively. To help readers understand the basic idea, the incremental algorithm proposed by Green and Sibson [15] is reviewed here and presented below. Delaunay Triangulation Algorithm: [3] Let P =fp 1 ;p 2 ;¢¢¢ ;p M g be the points over 2-dimensional plane. Step 1 Initialization. Find three appropriate points, p i , p j and p k , which are in point set P to form an initial triangle. Step 2 Point Increment. Add one point p m , which is not in the point set of the current graph, to the graph and calculate if there is any existing triangle whose circum- scribed circle (the circle passing through three vertices of the triangle) contains point p m . If there is, add the triangle into region R(p m ). 126 p m p m R(p m ) New Edg es (a) Step 2 Point Increment (b) Step 3 Triangle Update Figure 5.6: Illustration of the Delaunay triangulation algorithm. Step 3 TriangleUpdate. RemoveallinneredgesofregionR(p m )andconnectthevertices ofpolygonR(p m )topointp m suchthatthereareseveralnewtrianglesformedwith p m as the common vertex. Step 4 Iteration. Iterate Steps 2 and 3 until all points in P have been added to the graph. The main steps of the algorithm are illustrated in Fig. 5.6. 5.5 Mesh-based Contour Line Estimation The estimation process consists of two stages: 1) localized candidate detection of contour points;and2)contourlineconstruction. The¯rststageisconductedinalocalizedfashion 127 whereas the second stage is carried out with a centralized scheme. They are detailed in the following two subsections. 5.5.1 Localized Candidate Detection of Contour Points In order to save limited resources of a sensor network, sensor nodes are percolated in the ¯rst stage to determine which are eligible for calculations of the second stage using derived soft information. Only the sensor nodes with high probability to be the de¯ned contour nodes are selected to conduct calculations of necessary quantities and transmit them to the sink/basestation for the second stage - contour line construction. A contour node, which is a generalized case of an edge node, is de¯ned as follows. De¯nition 4 De¯nition of a contour node: Ifthedistancebetweenacontourlineofinterestandasensornodeislessthanr (calledthe tolerance range), then the sensor node is a contour node for the contour line of interest. With the above de¯nition, we are able to derive equations to inference the probability for asensornodetobeacontournodeandcandidatesofcontourpointscanbedeterminedby utilizing this probability value. Sensor nodes with low probability values can be switched to the sleep mode for the power-saving purpose and they will not conduct any further calculation until the next synchronization cycle begins. Fig. 5.7 illustrates ¯ve possible scenarios for a contour line to cross the neighborhood of a target sensor node. The shadowed and white triangles represent sensor nodes with theirsignalvaluessmallerandlargerthanthevalueofthetargetcontourline,respectively. We see from the ¯gure that a target node can be a contour node candidate if and only 128 C ontou r Node No t C ontou r N o de No t C ontou r N o de Figure 5.7: Illustration of a contour line crossing the neighborhood of a target sensor node, where one contour node candidate is shown in the middle of the top row while the others are not. if its neighbors inside the tolerance and probing ranges are separated into two regions by the contour line of interest. This concept can be formalized mathematically below. Let E 1 be the event that the targetsensornodeisacontournodecandidate,E 0;r andE 0;R betheonesthatthecontour line of interest does not cross the tolerance range and the probing range, respectively. ObservationsY 1 (k);¢¢¢ ;Y N (k)aresortedaccordingtothedistancesbetweentheirsensing 129 nodes and the target node. Given Y 1 (k);¢¢¢ ;Y N (k), the conditional probability for a target sensor node to be a contour node can be calculated as PfE 1 jY 1 (k);¢¢¢ ;Y N (k)g (5.7) = h 1¡PfE 0;r jY 1 (k);¢¢¢ ;Y Nr (k)g i £ h 1¡PfE 0;R jY Nr+1 (k);¢¢¢ ;Y N (k)g i = · 1¡ N r Y i=1 PfS i (k)>TjY i (k)g¡ N r Y i=1 PfS i (k)<TjY i (k)g ¸ £ · 1¡ N Y i=Nr+1 PfS i (k)>TjY i (k)g¡ N Y i=Nr+1 PfS i (k)<TjY i (k)g ¸ ; where T is the signal value determined by the contour line, N r is the number of sensor nodes in the tolerance range of a target sensor node, PfS i (k)>TjY i (k)g = Z 1 T f(SjY i (k))dS = R 1 T P(YjS)f(S)dS R 1 ¡1 P(YjS)f(S)dS ¯ ¯ ¯ ¯ Y=Y i (k) ; and PfS i (k)<TjY i (k)g = Z T ¡1 f(SjY i (k))dS = R T ¡1 P(YjS)f(S)dS R 1 ¡1 P(YjS)f(S)dS ¯ ¯ ¯ ¯ Y=Y i (k) : 130 IfS i (k)isassumedtobeuniformlydistributedover[S min ;S max ],thenPfS i (k)>TjY i (k)g and PfS i (k)<TjY i (k)g can be further simpli¯ed as PfS i (k)>TjY i (k)g= R Smax T P(Y i (k)jS)dS R Smax S min P(Y i (k)jS)dS ; (5.8) and PfS i (k)<TjY i (k)g= R T S min P(Y i (k)jS)dS R S max S min P(Y i (k)jS)dS : (5.9) Next, the problem is how to ¯nd the conditional probability P(Y i (k)jS). We see from Fig. 5.2 that Y i (k) = Q B [X i (k)]. Since Q B [¢] is not a linear function, the statistical properties of Y i (k) cannot be obtained from the linear combination of X i (k) directly. Let n i (k) be Gaussian white noise. Then, P(Y i (k)jS) can be calculated by P(y =(S min + ¢ 2 )+(j¡1)£¢jS) (5.10) = PfS min +(j¡1)£¢<X ·S min +j£¢jSg = PfS min +(j¡1)£¢¡S <n ·S min +j£¢¡SjSg = Q Ã (y¡ ¢ 2 )¡S ¾ n ! ¡Q Ã (y+ ¢ 2 )¡S ¾ n ! ; where S is the signal value, X the measured data, y the quantized measurement, j = 1;¢¢¢ ;2 B the quantization level, ¢ the quantization step, n the noise and Q(¢) the tail 131 Co nto ur L ine Figure 5.8: An example of an missed contour node. probability of a standard Gaussian distribution. The ¾ n value can be estimated by long- term noise power estimation locally. If n i (k) is non-Gaussian white, P(Y i (k)jS) can be computed as P(y =(S min + ¢ 2 )+(j¡1)£¢jS) (5.11) = Z S min +j£¢¡S S min +(j¡1)£¢¡S f n (t)dt; where f n (t) is the PDF of non-Gaussian noise. By combining (5.7)-(5.10), we can de- termine the approximate probability for a target sensor node to be a contour node for given quantized measurements Y i (k). Depending on the received data quality of Y i (k), the estimation performance may degrade due to data distortion. In this situation, data diversity (i.e. more data obtained from di®erent sensor nodes) can be used to enhance the performance. When the sensor deployment is sparse, there is a possibility that a contour node may not be detected even if the contour line of interest does crossing its tolerance range as 132 shown in Fig. 5.8. To address this problem, the approximate probability in (5.7) has to be modi¯ed. In Fig. 5.8, r is the tolerance range and d r is the distance between the target sensor node and its farthest neighbor inside the tolerance range. The contour line is missed by the target node when it passes through the slashed region. Then, the probability that all sensor nodes are inside the circular region with radius d r and the contour line crosses the slashed region can be used to approximate the probability that a contour node is missed due to the sparse sensor density. The approximate probability can be calculated by P P;r (d r )= µ d r r ¶ 2Nr £ r¡d r r ; (5.12) whereN r isthenumberofsensornodesinsidethetolerancerangeandd r canbecalculated by the coordinates of the sensor nodes. Since d r < r, P P;r (d r ) becomes smaller when N is higher. In other words, it can be reduced if the sensor density is higher. Though (5.12) can not provide accurate probability value, it still gives a reasonable approximation to evaluate the scarce sensor density e®ect. The probability, P P;R (d R ), that the contour line of interest crosses the probing range but the target node is missed in detection can 133 be derived in the same way by replacing r and d r in (5.12) with R and d R ,respectively. Consequently, the probability for a sensor node to be a contour node can be modi¯ed as PfE 1 jY 1 (k);¢¢¢ ;Y N (k);Dg = h ¡ 1¡PfE 0;r jY 1 (k);¢¢¢ ;Y Nr (k)g ¢ £ ¡ 1¡P P;r (d r ) ¢ i £ h ¡ 1¡PfE 0;R jY Nr+1 (k);¢¢¢ ;Y N (k)g ¢ £ ¡ 1¡P P;R (d R ) ¢ i ; (5.13) where D represents the sensor deployment information. We see from (5.13) that PfE 1 j Y 1 (k);¢¢¢ ;Y N (k);Dg < 1 even if all the measurements are accurate. Nevertheless, it approaches 1 when the sensor density becomes higher. Then, sensor nodes with high probability values are detected as possible candidates of contour points and then conduct the gradient estimation and compute the average log likelihood value as proposed in [27]. They are reviewed below. Gradient Estimation: b ~ µ =[H(N) t H(N)] ¡1 H(N) t Z d (N); (5.14) where ~ µ is the gradient direction at the location of the target sensor node, Z d (N)=[ X(~ p 1 )¡X(~ p) j~ p 1 ¡~ pj ;¢¢¢ ; X( ~ p N )¡X(~ p) j ~ p N ¡~ pj ] T ; H(N)=[ ~ p 1 ¡~ p j~ p 1 ¡~ pj ; ~ p 2 ¡~ p j~ p 1 ¡~ pj ;¢¢¢ ; ~ p N ¡~ p j~ p 1 ¡~ pj ] T ; 134 ~ p and ~ p i , i = 1;2;¢¢¢ ;N are the coordinates of the target sensor node and its neigh- boring nodes within the probing (data collection) range and X(~ p) and X(~ p i ) are their corresponding data samples. Average Log Likelihood Values: C 1 ¡ C 2 N N X i=1 (X i ¡¹) 2 ; (5.15) whereC 1 andC 2 >0canbeassignedwithanarbitraryvaluewithouta®ectingthesystem performance. After ¯nding these two quantities, these candidates send the results to the fusion center (sink or a remote base-station) through the data gathering tree for further processing. 5.5.2 Contour Line Construction Contourlineconstructioniscarriedoutinthefusioncenteratthesecondstage. Thereare three steps: 1) Delaunay triangulation, 2) contour points determination and 3) contour line connection. First, the fusion center conducts the triangulation to form a mesh structure based on the knowledge of sensor node's coordinates. The triangulation algorithm was discussed in the last section. Instead of processing the algorithm every information update, the triangulation is conducted only at the beginning of the monitoring. The results will then be stored in the memory for later usages. After triangulation, we can identify adjacent neighboring nodes of each sensor node over the ¯eld easily. 135 E s t im ated G ra d ient Tar g et S en sor Node T arg e t S en s o r Node T rue Co nto ur L in e 2. A v erage Li kel i h o o d Val ues Co m pari so n 2. A v erage Li kel i h o o d Val ues Co m pa r i s o n 1. Sea r ch i n g A ppro pr i at e A dj a cen t N ei gh b o r i n g No des Figure 5.9: Illustration of contour point determination. Next,weevaluatewhichsensornodes(thosedetectedascandidatesofcontourpoints) are appropriate contour points for contour line connection based on two quantities calcu- latedinthepreviousstage-gradientandaveragelikelihoodvalue. Inordertohavebetter estimation of contour line location, the chosen contour points have to be close to the true contour line as much as possible. The question is how to ¯nd these candidate points. We observe that these points all have the same characteristics; namely, they possess the largest average likelihood value as compared with their adjacent neighboring nodes along the direction and the reverse direction of the gradient at a local position. Thus, as illustrated in Fig. 5.9, the algorithm consists of two sub-steps, which are described below. 136 Algorithm of Contour Point Determination 1. We ¯nd two adjacent neighboring nodes located at the direction and the reverse directionoftheestimatedgradient. WiththeknowledgeoftheDelaunaytriangula- tion structure, it is easy to determine which sensor nodes are adjacent neighboring nodes of a target sensor node. Suppose that ¡ ! p t is the coordinate of the target sen- sor node and ¡ ¡ ! p a;1 ;¢¢¢ ; ¡ ¡ ! p a;L are the coordinates of its adjacent neighboring nodes. Then, we calculate the inner product of the estimated unit gradient (we only need the direction information) and unit vectors ¡ ¡ ! p a;i ¡ ¡ ! pt j ¡ ¡ ! p a;i ¡ ¡ ! p t j , where i=1;¢¢¢ ;L and choose thelargestandleastonesastheestimationoftwoadjacentneighboringnodesalong the direction and the reverse direction of the gradient, respectively. 2. After ¯nding these two adjacent neighboring nodes, we compare their average like- lihood values with that of the target sensor node. If the target sensor node possess the largest value, it is the desired contour point. In the last step, we try to connect the detected contour points with lines. Due to the noise e®ect (sensing and channel noises) and the scarcity of sensor nodes, detection results might be erroneous. Thus, an error correction scheme is adopted in the last step to lessen the e®ects induced by detection errors. Algorithm of Contour Line Connection 1. Initialization. Choose a detected contour point with the largest average likelihood value as the head point of the current contour line segment estimation from the list of unprocessed detected contour points. 137 2. Connection point search. We check if there is any adjacent neighboring node which is in the list of unprocessed detected contour points. Case 1: If there is none, we check if there is any two-hop neighboring node that is in the list of unprocessed detected contour points. A. If there is none, save the current point as the end point of the current contour line segment estimation, remove the current point from the list of unprocessed detected contour points and go to sub-step 3. If the current point is the head point of the current contour line segment estimation, it is kept in the list of unprocessed detected contour points so that other estimated contour line segment still can be connected to it without dis- connection. B. Ifthereisoneormorepoints,checkifthereisanyintermediatenode,which is one of adjacent neighboring nodes, to the chosen two-hop neighboring nodes with the most adequate direction (perpendicular to the estimated local gradient). i) If there is none, save the current point as the end point of the current contour line segment estimation, remove the current point from the list of unprocessed detected contour points and go to sub-step 3. If thecurrentpointistheheadpointofthecurrentcontourlinesegment estimation,itiskeptinthelistofunprocesseddetectedcontourpoints. 138 ii) If there is one, save the current point and the intermediate node as points for the connection of the current contour line segment estima- tion, remove the current point from the list of unprocessed detected contour points, take the chosen node as the current processing point and go to sub-step 2. If the current point is the head point of the current contour line segment estimation, it is kept in the list of unpro- cessed detected contour points. Case 2: Ifthereisone,checkiftheconnectiondirectionisadequate(perpendicular to the estimated local gradient). A. If yes, save the current point as a point for the connection of the current contour line segment estimation, remove the current point from the list of unprocessed detected contour points, take the chosen node as the current processing point and repeat sub-step 2. If the current point is the head point of the current contour line segment estimation, it is kept in the list of unprocessed detected contour points. B. If no, save the current point as the end point of the current contour line segment estimation, remove the current point from the list of unprocessed detected contour points and go to sub-step 3. If the current point is the headpointofthecurrentcontourlinesegmentestimation, itiskeptinthe list of unprocessed detected contour points. Case 3: Iftherearetwoormore,checkifthereisanyadequateconnectiondirection (perpendicular to the estimated local gradient). 139 A. If yes, choose the best one, save the current point as a point for the con- nectionofthecurrentcontourlinesegmentestimation,removethecurrent pointfromthelistofunprocesseddetectedcontourpoints,takethechosen node as the current processing point and repeat sub-step 2. If the current point is the head point of the current contour line segment estimation, it is kept in the list of unprocessed detected contour points. B. If no, save the current point as the end point of the current contour line segment estimation, remove the current point from the list of unprocessed detected contour points and go to sub-step 3. If the current point is the headpointofthecurrentcontourlinesegmentestimation, itiskeptinthe list of unprocessed detected contour points. 3. List checking. Check if there is any unprocessed detected contour point. If yes, go to sub-step 1; otherwise, go to sub-step 4. 4. Termination. Connect all of estimated contour line segments and terminate. Withtheabovealgorithm,two-hopmissings(disconnections)andone-hopfalsealarms (improper branches) can be corrected if the detected contour points do not contain two successive errors and the estimation error of local gradients is small. The proposed algorithm can also estimate two or more contour lines if they are separated at least three hops away from each other. 140 5.6 Simulation Results and Discussion 5.6.1 Contour Nodes Inference Inthissubsection,thee®ectsofdi®erentsystemparametersontheperformanceofcontour nodes inference proposed in 5.5.1 are investigated. In the simulation, the sensing noise power ¾ 2 n is set to be 30dB smaller than the power of signal strength 5, and it can be estimated accurately in a long-term manner if the environment remains steady for a long time. K is assumed to be 1. The tolerance range, r, and the probing range, R, are set to 0:5 and 0:75, respectively. The signal is chosen to be a linear function from S min = 0 to S max = 10 along the x-axis with a constant slope 1. We attempt to detect the contour line with the signal value T =5. The number of quantization bits for each measurement is 10 by default, i.e. there are 2 10 quantization levels. The BER E®ect Theperformancecomparisonfordi®erentbiterrorratesoverwirelesschannelsisgiven in Fig. 5.10, where the x-axis denotes the distance from the target sensor node to the contour line of interest. The negative and positive values mean the target node is located at the left- and right-hand side of the contour line. The dot line shows the performance of an ideal case, where all measurements are perfect and sensor density is in¯nitely large. Thee®ectofthebiterrorsisshowninanaveragefashion. Forexample,0:01£10biterror is added per received data when the bit error rate (BER) is 0:01. However, to observe the worst e®ect on the performance, the error is added to the most signi¯cant bit. We see from the ¯gure that the shape of the curve deviates from the ideal one, especially for 141 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 Distance to Contour Line Probability Ideal BER=0 BER=0.001 BER=0.005 BER=0.01 Figure 5.10: Performance comparison of contour nodes inference under di®erent bit error rates over wireless channels when sensor density is 10 nodes/unit square, tolerance range r is 0:5 and probing range R is 0:75. the slope at the absolute distance= 0:5, when the bit error rate increases. It is found that the curve still has sharp slope at the absolute distance= 0:5 when BER reaches 1£10 ¡3 . Thus, we can choose the maximal acceptable BER to be 1£10 ¡3 . Then, the data quality requirement for the MAC layer can be lowered to 1£10 ¡3 when the number of quantization bit for one measurement is 10. The Sensor Density E®ect Fig. 5.11comparestheperformancewithrespecttodi®erentlevelsofsensordensity. It is clear that a higher sensor density gives better performance because it has larger slope at the absolute distance= 0:5. This is due to the increasing number of the necessary data, which provide more useful information to the local fusion center to mitigate noise 142 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 Distance to Contour Line Probability Ideal 5 nodes/unit square 10 nodes/unit square 15 nodes/unit square Figure 5.11: Performance comparison of contour nodes inference with di®erent levels of sensor density when BER=0, tolerance range r is 0:5 and probing range R is 0:75. interferenceonprobabilitycalculationsoastoincreasethediscriminabilityofthede¯ned contour node. Thus, it helps the selection of possible candidates of contour points. 5.6.2 Contour Line Estimation A 4£ 4 sensor ¯eld with sensor density 10 nodes/square is used to demonstrate the performance of the proposed approach by computer simulation. Following the previous subsection, we set the tolerance range, r, and the probing range, R, to 0:5 and 0:75, respectively, if not speci¯ed otherwise. A cone-shaped function, de¯ned by its slope and base radius over a two-dimensional space, is used to model the monitored physical phenomenon. The relationship between sensor density, local slope of the monitored phe- nomenonandcurvatureofthecontourline,whichisthereciprocalofthebaseradius, will 143 be investigated to determine clear design rules. We attempt to localize the contour line with signal value T = 5. To evaluate how the cross-layer parameters a®ect the perfor- mance, di®erent parameter settings such as BER and quantization levels are compared and discussed based on simulation results or mathematical equations. The number of quantization bits for each measurement is 10 by default, i.e. there are 2 10 quantization levels. Based on the discussion in 5.2, the local sensing noise level is set to be 30 dB smaller than the signal level T =5 in power. The Curvature E®ect The performance comparison for di®erent curvature values is shown in Fig. 5.12, where circles stand for deployed sensor nodes, squares are estimated contour points, bold solid lines are estimated contour lines, the black bold dashed line is the true contour line and other two dashed lines are the limits of the true contour line's tolerance range. We see from the ¯gures that the proposed approach provides good performance as the curvature is 1=3 but it reachesthe performance limit as the curvaturegoes to 1=2:7. This is because the current sensor density is not high enough to provide good contour line estimation when the curvature is too large. The big circle in Fig. 5.12(b) illustrates the limit of the correction scheme. Only two-hop or smaller errors can be corrected adequately. To give a more explicit rule for the system design, we adopt a ratio to describe the system limit as p Sensor Density Curvature : (5.16) 144 The ratio is around 8 when the sensor density is 10 and the curvature is 1=2:7. Thus, it requires the ratio to be larger than 8:54 to provide acceptable estimation. For example, if we want to provide good performance for a contour line with curvature 0:7, it requires to increase the sensor density from 10 to at least 36 nodes/square. The E®ect of Local Slope Simulation results with two di®erent local slope values are compared in Fig. 5.13. We see that good performance can be achieved when the local slope is 1. The correction scheme fails when the local slope goes below 0:5. This is because, for a ¯xed noise level, a larger local slope value makes the characteristics of those true contour points more prominent, thus reducing the missing rate and the false alarm rates. It can be explained by the mean value of the average likelihood value, which is equal to C 1 ¡C 2 ¾ 2 n ¡ C 2 N N X i=1 (¹ i ¡T) 2 ; (5.17) where T is the signal value of the contour line of interest and ¹ i , i = 1;¢¢¢ ;N, are the mean values of received measurements. According to (5.17), the sensor node that gets closer to the desired contour line has a larger mean value of its average likelihood value and a larger local slope enhances the gap. Thus, a larger local slope gives better estima- tion performance. To keep the same performance for the cases with a smaller local slope, the sensor density has to increase so as to mitigate the noise and increase the accuracy of the calculated average likelihood value. 145 The E®ect of BER and Probing Range Size Fig. 5.14 shows the performance when bit errors occur in the data for processing. We see that the proposed approach can sustain BER=0:001 in an average fashion while providing good estimation performance. With this property, we may adopt the adaptive transmission scheme described in 5.3 to increase the power e±ciency of communication amongsensornodes,thusextendingthenetworklifetime. FromFigs. 5.12-5.14,wefound thatthevalueof(5.16)isnotaconstantevenwiththesamecurvatureandsensordensity. It varies with di®erent values of local slope and bit error rate. To clarify the design rules under di®erent conditions, the relationship between (5.16) and local slope with di®erent levelsofbiterrorrateand R r ratiowasinvestigatedandshowninFigs. 5.15and5.16. The region above the curves is the e®ective area for the proposed approach, which means the estimation performance is acceptable (at most one missing connection and one erroneous branch are observed) if the value of (5.16) falls in the e®ective area. In Fig. 5.15, the curve with BER=0:01 ends at local slope 1:0 because the BER-induced noise is so large that the proposed algorithm can not estimate contour lines accurately as the local slope is below 1:0. Two conclusions can be drawn from Figs. 5.15 and 5.16. First, the value of (5.16)willeventuallyconvergeevenwithincreasinglocalslopeandtheconvergedvalueis related to other system parameters such as BER and R=r. Second, larger probing range does not guarantee better performance and there exists an optimal R=r ratio. This can be explained by the variance of the average likelihood value, which is equal to [26] 2C 2 2 ¾ 4 n N + 4C 2 2 ¾ 2 n N 2 N X i=1 (¹ i ¡T) 2 : (5.18) 146 From (5.18), it is found that larger inhomogeneity of the signal strength (the term P N i=1 (¹ i ¡ T) 2 ) induces larger variance of the average likelihood ratio value and thus system performance degrades as probing range increases. The Assignment of Quantization Bits From Fig. 5.2, there are three sources of data distortion - sensing, quantization and channeldegradation. Thoughquantizationdegradationcanbeimprovedbyraisingquan- tization levels, more quantization bitscould induce larger channel degradation dueto the increase of transmission bit rate according to (5.4) and thus may make channel degrada- tion the major source of data distortion. In order to achieve the best data transmission performancewithoutlargequantizationdegradation, thenumberofquantizationbitshas to be chosen as small as possible and still provides the data quality with least distortion. Since sensing degradation is the ¯rst distortion source encountered through the whole data fusion process, the induced data distortion can not be removed or mitigated even with¯nerquantizationlevelsandperfectdatatransmission. Therefore,theinducedquan- tization noise should be smaller than sensing noise so as to make total data distortion approach to the sensing degradation as close as possible. Based on the above discussion, we have the following equation for the quantization bits assignment if linear quantizer is applied. ¢ 2 12 <¾ 2 n ; (5.19) where ¢ is the quantization di®erence and ¾ 2 n is the sensing noise power. For example, a linear quantization scheme over [0;10] for measured data is adopted in simulation. 147 Supposesignalstrengthis5andsensingnoisepoweris0:025, whichconstitutesSNR=30 dB. The number of quantization bits should be at least 5 bits. Compared with 10 bits, 5-bitquantizerprovidesadditional3dBperformancegainfordatatransmissionandthus the communication power can be further saved. Besides, nonlinear quantization schemes could be used to improve quantization e±ciency if the statistical properties of the signal strength is known. Comparison of A Centralized Scheme and the Proposed Approach For wireless sensor networks, power consumption, computational complexity and memory storage are three major concerns to evaluate the feasibility and e±ciency of a data fusion algorithm. To examine whether the proposed approach is feasible and e±- cient, it is compared with a centralized scheme in these three aspects. In the centralized scheme, all sensor measurements are transmitted to the sink or a remote base-station for contour lines estimation through spatial interpolation using received data without any local data fusion. Table 5.1 shows the comparison of the centralized scheme and the pro- posed approach. For the proposed approach, the communication cost and computational complexity increase with O( p M), where M is the number of deployed sensor nodes, be- causeonlymeasurementsfromcandidatesofcontourpointsinsteadofallaretransmitted to the sink or a remote base-station for the calculation of the second stage. Though it requires computational complexity O(M£log(M)) [3] for triangular mesh construction, triangulationisconductedatthebeginningofthemonitoringonly. Sinceadditionalmem- ory is required for the storage of the triangulation topology, memory storage increases with O(M) instead of O( p M). However, for a centralized scheme, communication cost 148 Schemes Performance Com. Cost Complexity Memory Centralized Scheme Good O(M) at least O(M) at least O(M) Proposed Approach Medium O( p M) O( p M) O(M) Table 5.1: Comparison of the centralized scheme and the proposed approach when the number of deployed sensor nodes is M increases with O(M) while computational complexity and memory storage increase with at least O(M) or higher, which depends on the algorithm used for interpolation and con- tour lines estimation. Consequently, the proposed approach has better e±ciency though there is performance degradation. 5.7 Conclusion A mesh-based approach to contour line estimation with cross-layer considerations using wirelesssensornetworkswasproposed. Thisalgorithmtogetherwithacorrectionscheme can correct two-hop errors and provide good estimation performance even though the estimation of contour points is not accurate. Simulation results demonstrated the per- formance limits and shed light on issues of system design. The ratio of the square root of the sensor density over the curvature value was proposed to determine the conditions, under which the proposed algorithm works well. The relationship between this value and localslopewasexaminedunderdi®erentlevelsofsensordensity, biterrorratesandprob- ing range sizes to clarify the system design rules. The assignment of quantization bits was also investigated to remove all redundant bits. It was found that the quantization will not a®ect estimation performance if the induced quantization noise is smaller than sensing noise. With this property, the amount of data could be further suppressed and 149 thus the communication power can be saved to extend the network lifetime. In addition, the proposed approach was shown to be more e±cient than a centralized scheme though it sacri¯ces the estimation performance. 150 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 E s t i m a t e d Co n t o ur Li n e T rue Co n t o ur L i n e (a) Curvature=1/3, local slope=2, BER=0, number of quantization bits=10 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 E s t i m a t e d Co n t o ur Li n e Tr ue Co n t o ur L i n e (b) Curvature=1/2.7, local slope=2, BER=0, number of quantization bits=10 Figure 5.12: Simulation results of contour line estimation with di®erent curvature values. 151 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 E s t i m a t e d Co n t o ur Li n e T rue Co n t o ur L i n e (a) Curvature=1/3, local slope=1, BER=0, number of quantization bits=10 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 E s t i m a t e d Co n t o ur Li n e T rue Co n t o ur L i n e (b) Curvature=1/3, local slope=0.5, BER=0, number of quantization bits=10 Figure5.13: Simulationresultsofcontourlineestimationwithdi®erentlocalslopevalues. 152 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 E s t i m a t e d Co n t o ur Li n e T rue Co n t o ur L i n e Figure 5.14: Simulation results of contour line estimation when Curvature=1/3, local slope=2, BER=0.01, number of quantization bits=10. 0.5 1 1.5 2 2.5 3 8 9 10 11 12 13 14 Local Slope (Sensor Density) 0.5 /Curvature BER=0 BER=0.001 BER=0.01 Figure 5.15: p Sensor Density Curvature versus local slope under di®erent bit error rates. 153 0.5 1 1.5 2 2.5 3 8 10 12 14 16 18 20 22 Local Slope (Sensor Density) 0.5 /Curvature R/r=1.5 R/r=1.0 R/r=2.0 Figure 5.16: p Sensor Density Curvature versus local slope under di®erent R=r ratios when r =0:5. 154 Chapter 6 Conclusion and Future Work 6.1 Conclusion To monitor the environmental phenomena such as poisonous gas di®usion, ¯re outbreak and containment spreading through di®erent sensing devices, algorithms were proposed for edge detection and contour lines estimation with wireless sensor networks in this thesis. Dependingondi®erentapplicationrequirementsandtheunderlyingsystemdesign, various sensing devices may be applied and the monitored physical phenomenon can be represented by a binary or a scalar ¯eld due to their measuring precision. Based on the sensed values, di®erent characteristics of the monitored phenomenon can be extracted by local preprocessing and sent to a remote base station for further processing without excessive raw data transmission. When the sensed data take binary values, the extracted edge/boundary information helps us identify the size and location of the event region. Even for scalar-value sensed data, edge/boundary still reveals a lot of information of interests if a triggered thresh- old is de¯ned to separate event region from non-event region. Instead of estimating 155 the location of edge lines, an edge region that encompasses the edge line inside was de- ¯ned, and a distributed algorithm for edge region detection was proposed in Chapter 3. A statistical decision-fusion method was proposed to label the de¯ned edge region using distributed wireless sensor nodes based on the Baysian statistical inference and the Neyman-Pearson (NP) optimality. As compared with the traditional classi¯er-based approach, our new statistical approach can achieve a much lower false alarm rate in the presence of a high noise level without the sensor location information by leveraging the NP optimality. In other words, the proposed statistical decision-fusion method has bet- ter edge region performance and is more robust against a harsh environment. Simulation results with cross-layer considerations were shown to demonstrate these advantages. When the sensed data take scalar values, we developed two algorithms for environ- ment monitoring in Chapters 4 and 5, respectively. With the deployment of high-cost sensor nodes, we are able to build up a hierarchical network topology for contour lines estimation. Speci¯cally,adistributedapproachtocontourlinesextractionwithheteroge- neoussensornetworksisproposedinChapter4. Toincreasesystemrobustnessandlower deployment cost, a mesh-based approach to contour lines estimation with homogeneous sensornetworkswasproposedinChapter5,whichutilizessomequantitiesandtechniques proposed in Chapter 4. For comparison, the algorithm proposed in Chapter 4 has the best contour lines detectionperformancebutdemandsthehighestinstallmentcostandcommunicationcost due to the establishment of a hierarchical network. The algorithm in Chapter 5 has the detection performance in the middle but the advantages in higher robustness and lower 156 cost. Furthermore, compared with a centralized scheme, the proposed algorithm requires less communication cost, computational complexity and possible memory storage when the network size is large though the estimation performance degrades. Depending on system and application requirements, these two proposed algorithms can be adopted to obtain di®erent levels of system performance. 6.2 Future Work Solutions to physical phenomena monitoring using the space correlation of monitored eventswere consideredin this research. Forthe dynamictacking of time-varyingphysical phenomena, we may apply developed algorithms periodically to take newly measured data into account. However, the tracking performance could be improved if the time correlation of sensed data is considered adequately. Along this research direction, several issues are worthy of further investigation as described below. ² System modeling and estimation with Markov random ¯eld Duetothetime-varyingnatureofthephysicalphenomenon,themeasured2Dsensor data over time is actually a random ¯eld. It seems feasible to develop a Markov random ¯eld to describe the dynamic behavior of the whole system to exploit the correlation in both space and time. There are model parameters to be estimated based on observed data. Depending on properties of the monitored phenomenon, its state transition can be modeled by the Markov chain, the hidden Markov model or others. With an adequate model, a more e®ective tracking algorithms could be developed. 157 ² Dynamic edge/contour lines tracking It is interesting to develop an e±cient scheme for dynamic tracking of edge regions and contour lines. For a dynamic tracking problem, a timely information update is required. Two techniques can be used together to meet the requirement. The ¯rst one is to improve the processing time of the algorithm so that the delay time can be reduced. The second one is to forecast the locations of edge or contour lines using current and past measurements so that the remote base station may have the timely information even though there is a delay. ² Delay modeling and analysis Distributedalgorithmsforedgeandcontourlinesestimationwereconsideredinthis thesis. 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Abstract (if available)

Abstract Several signal processing problems associated with wireless sensor networks deployed for the monitoring of certain environmental physical phenomena are examined in this research. A couple of distributed algorithms are proposed to extract the essential information of the monitored physical phenomena such as edges and contour lines so that a remote base station may utilize the information for further analysis. These networks often span over a large geographical area and they cannot be adequately monitored or tracked using traditional localization techniques.

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Asset Metadata

Creator Liao, Pei-kai (author)

Core Title Distributed edge and contour line detection for environmental monitoring with wireless sensor networks

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 04/17/2007

Defense Date 02/08/2007

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag contour line,data fusion,distributed algorithm,edge detection,OAI-PMH Harvest,wireless sensor networks

Language English

Advisor Kuo, C.-C. Jay (committee chair), Govindan, Ramesh (committee member), Ortega, Antonio (committee member)

Creator Email pliao@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m404

Unique identifier UC160106

Identifier etd-Liao-20070417 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-481166 (legacy record id),usctheses-m404 (legacy record id)

Legacy Identifier etd-Liao-20070417.pdf

Dmrecord 481166

Document Type Dissertation

Rights Liao, Pei-kai

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu