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USC Computer Science Technical Reports, no. 773 (2002)
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USC Computer Science Technical Reports, no. 773 (2002)
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Content
Localized Edge Detection in Sensor Fields
Krishna Kant Chintalapudi, Ramesh Govindan
University of Southern California, Los Angeles,
California, USA, 90007.
Abstract
A wireless sensor network that studies relatively widespread phe-
nomena (such as a contaminant flow or a seismic disturbance) may
be called upon to provide a description of the boundary of the phe-
nomenon (either a contour or some bounding box). In such cases, it
may be necessary for each node to locally determine whether it lies
at (or near) the edge of the phenomenon. In this paper, we show that
such localized edge detection techniques are non-trivial to design in
an arbitrarily deployed sensor network. We define the notion of an
edge and develop performance metrics for evaluating localized edge
detection algorithms. We propose three different approaches for lo-
calized edge detection and present one example scheme for each. In
all our approaches, each sensor gathers information from its local
neighborhood and determines whether or not it is an edge sensor.
We evaluate the performance of each of the example schemes and
compare them with respect to the developed metrics.
1 Introduction
Several physical phenomena (for instance, contaminants and seis-
mic disturbances) can span large geographic extents. Fine-grain
sensing of these time-varying phenomena can help scientists under-
stand what factors (e.g., soil density variations) affect the spread
of these phenomena. One way to architect an energy-efficient sen-
sor network for studying these phenomena is to store the detections
of the phenomena within the network and provide a query inter-
face which enables scientists to understand the temporal and spatial
properties of these phenomena.
We anticipate that one common query will ask for the spatial extent
of the phenomenon at a given time: for example, “Which sensors
saw the primary wave before time
?”. For energy-efficiency rea-
sons, it makes more sense for to design the spatial query that returns
a boundary that captures all or most nodes that satisfy the query
predicate. A geometric representation of the boundary has the po-
tential to be more concise (and therefore more energy-efficient) than
an enumeration of all nodes. Examples of such representations in-
clude contours, hulls, or bounding boxes.
An energy-efficient boundary finding algorithm will need to care-
fully nodes in the sensor network and compute the boundary “in-
network”. A key component of such an algorithm is a localized
edge detection scheme: a technique by which each node locally de-
termines (perhaps by gathering information from other nodes within
its neighborhood) whether it lies on or near the boundary specified
by the query. If a reliable technique existed for localized edge detec-
tion, then, conceptually at least, boundary finding is simply a matter
of sequentially traversing all nodes that determine themselves to be
on the edge.
Edge detection has been widely studied in the context of digital im-
age processing. Filtering [1] is one of the most common approaches
to detecting edges in images. To determine whether an image pixel
is at an edge or not, this approach applies a filter to values of a set
of neighboring pixels. As such, these techniques can be directly
applied to localized edge detection.
However, one fundamental difference between images and a sen-
sor field is the spatial regularity of information. A digital image
is a regular grid of pixels, and information is sampled at regular
intervals. Almost all standard digital image processing techniques
(Fourier transforms, high-pass filtering) rely on having information
about the image at regular intervals. Deployment and maintainance
of thousands or even millions of sensors in a grid like regular fash-
ion over large geographical extents will require impractical amounts
of effort. It is expected that sensors will be arbitrarily placed in the
sensor field and will be prone to failures or even node displacement.
With this relaxation of regularity, it is less clear that digital image
filtering can be applied to localized edge detection.
In fact irregular node placement makes it hard to even define pre-
cisely whether a node is at an edge or not. In Section 2, we show
how to circumvent this difficulty and define some metrics for local-
ized edge detection. Because the efficacy of edge detection based
on digital image filtering is unclear, we consider two other classes
of edge detection schemes in Section 3: a statistical scheme and a
classifier-based scheme (that is suggested by the pattern recogni-
tion literature [2]). We find that, over a fairly broad range of operat-
ing conditions, the classifier scheme outperforms the other schemes.
We present our evaluation results in Section 4, and conclude in Sec-
tion 5.
2 Edge Detection
In this section we describe our model of the sensor field and discuss
definitions for an “edge”. We then develop metrics to evaluate local-
ized edge detection algorithms and discuss the trade-offs involved in
the design of localized edge detection
2.1 Assumptions, Models and Terminology
In what follows, we make fairly general assumptions about the capa-
bilities of sensor nodes and the structure of sensor networks. Sensor
1
nodes can be arbitrarily deployed, but each such node knows its lo-
cation (without loss of generality, we assume that the deployment
of sensors is in the plane, and location can be specified by ).
We use the term sensor field to both mean the geographical region
covered by the deployment, and the set of sensors within the region.
Sensors can make measurement errors, and our localized edge de-
tection schemes will need to be robust to these.
We model an edge as follows. Consider a phenomenon that spans
some arbitrarily shaped sub-region of the sensor field. Each sensor
can, based on locally collected measurements, determine whether it
belongs to the sub-region covered by the phenomenon or not. We
call the function that makes this decision the event predicate, and
denote the event predicate at sensor by . Taking our example in
Section 1, if the phenomenon of interest is “the geographical extent
covered by the primary seismic disturbance at time
”, the event
predicate for each sensor is, informally: “Did I see a primary seis-
mic disturbance at or before
?”.
Given an event predicate, we can then define the interior of a phe-
nomenon ( ) to be the spatial region such that, if a perfectly
calibrated error free sensor were placed in this region its predicate
function would have evaluated to 1. The exterior of the phe-
nomenon can be similarly defined. Based on these, there exists an
idealized definition of the edge of a phenomenon : the edge is
the set of all points , such that every non-empty neighborhood
of intersects with both and . We call the ideal edge,
and represents the ground truth that defines the boundary of the
phenomenon.
This idealized definition is descriptive, but does not give us much in-
sight for designing or evaluating localized edge detection schemes.
The ideal edge has no “thickness” and therefore constitutes a very
restrictive definition of an edge. Intuitively, we would like a sensor
to consider itself to be an edge sensor if it is closer to the ideal edge
than any other sensor. For this reason, we introduce the notion of
tolerance of an edge detection scheme. We define a sensor to be an
edge sensor if it a) is in the interior of the phenomenon, and b) lies
within a pre-specified distance of the ideal edge. We call the
tolerance radius, and the area around a sensor node covered by a
circle of radius the tolerance neighborhood. The tolerance radius
roughly measures the “thickness” of the edge that the designer of a
localized edge detection scheme is willing to tolerate. For a given
tolerance radius, we can define metrics that enable us to compare
the efficacy of different edge detection schemes (Section 2.2).
A stronger definition an edge might be to require continuity among
the set of edge sensors—that is, that there exists a path between
every pair of edge sensors that only traverses edge sensors. We
have not adopted this definition because it seemed to be beyond the
realm of localized techniques to ensure this property. Of course, if
we could ensure this property, it would be easy to define energy-
efficient boundary finding algorithms that simply traversed edge
nodes. As such, whatever boundary finding algorithms that we build
on top of localized edge detection will need to traverse some non-
edge sensors to determine a continuous boundary for a phenomenon.
Such algorithms are beyond the scope of this paper.
Finally, detecting whether a node lies at the edge of a phenomenon
is slightly different from detecting whether a node lies on (or near)
a contour (or iso-lines; i.e. continuous curves across the sensor field
defined by sensors having the same value of, say, temperature). In
the general case, there isn’t a well-defined notion of the interior
and exterior of a contour, as much as there is a distinction between
whether a sensor detects a phenomenon or not.
sensor node
Interior
sensor node
Edge Sensor
Not an Edge Sensor
Interior
radius of tolerance
radius of tolerance
Figure 1: If the edge passes through the radius of tolerance it is
deemed an edge sensor.
Exterior
Edge Sensors
Non Edge Sensors
Thickness
Figure 2: Edge sensors will not necessarily lie on the edge, hence
the edge in a sensor field will be thick.
2.2 Metrics
There are, broadly speaking, two classes of desirable characteristics
of localized edge detection: robustness, and performance. These
characteristics inform our choice of metrics for localized edge-
detection.
In the robustness category, we would like several desirable char-
acteristics of an edge detection algorithm. First, as we shall show
later, localized edge detection algorithms can intrinsically exhibit
2
error, failing to detect an edge when there is one, or detecting an
edge when there is none. A good algorithm has low intrinsic error.
Second, localized edge detection algorithms must be relatively ro-
bust to reasonable levels of sensor calibration error. Finally, many
localized edge detection schemes employ thresholds to decide on
the existence of an edge. We would prefer schemes which are rel-
atively insensitive to the threshold settings over a broad range of
operating conditions.
In the performance category, an obvious consideration is energy ex-
pended in communication. There exists a tradeoff between energy
and accuracy in localized edge detection; intuitively, a node can
get information from a bigger neighborhood to increase the like-
lihood of a positive detection. The second performance criterion is
the quality of the result, defined by the actual thickness of the edge.
Although our definition of an edge above includes a tolerance radius
that nominally defines an edge thickness, an actual edge detection
scheme might have a thickness that is larger or smaller than this
radius.
Based on the above discussion, in this section we use the following
metrics to evaluate the performance of localized edge detection
algorithms.
Percentage Missed Detection Errors : These are sensors which
lie within the radius of tolerance but were not marked as edge sen-
sors.
(1)
Mean thickness ratio :Let be the mean distance of all
the sensors in set
to the edge . We define, tolerance but were
not marked as edge sensors.
(2)
To avoid the effect of random outliers, we consider only the closest
95% edge sensors in the mean.
False Detection Errors : This represents the fraction of nodes
that declared themselves to be edge sensors but should not have.
Let be the curve representing the edge (as defined in Section 2.1).
Suppose set
be the set of sensors in the sensor field which are
within a distance of from . Let
be the set of sensors marked
as edge sensors by the algorithm. We define false detection errors
as,
! (3)
Here
! is the total number of sensors in the sensor field.
We are now ready to discuss some localized edge detection schemes
that illustrate the tradeoffs involved in localized edge detection.
Interior
Exterior
r
R
edge
Figure 3: Performance can be improved by gathering information
beyond the tolerance radius. Here, the sensor gathers information in
a circle of radius " , the probing radius and is able to detect an edge
which passes through the area of tolerance more reliably.
3 Three approaches to localized edge de-
tection
In this section we propose three qualitatively different approaches
to localized edge detection, in a sensor field, a statistical approach,
an approach drawn from image processing and an approach drawn
from the pattern recognition literature. Each approach can be used
to generate a “rich” family of algorithms for edge detection.
In all these approaches, each sensor gathers information from sen-
sors in its neighborhood and independently tries to determine if an
edge passes within its tolerance radius. Specifically, the sensor gath-
ers the location and the values of the event predicate (that deter-
mines whether the sensor is in the interior or the exterior of a phe-
nomenon) from each node within the neighborhood.
One parameter that determines the performance of all algorithms, to
varying extents, is the size of this neighborhood. Arbitrary place-
ment of the sensors coupled with sensor errors can result in detec-
tion errors. In general, the performance of a scheme improves as
we collect information from more sensors (larger neighborhood).
This is because the node gets more samples from the interior and
the exterior of the phenomenon, and can make more confident es-
timates even in the presence of sensor errors. However, collecting
more information incurs more communication overhead and hence
increases the energy usage of the scheme. We have already men-
tioned this energy accuracy trade-off. We represent this parameter
by a circle of radius " centered around the sensor and call it the
probing radius. Typically, the probing radius is greater than the
tolerance radius i.e. "$# (see figure 3). Generally the greater
the % ratio, the better the performance of the algorithms in terms
of errors and thickness ratio, however the communciation overhead
increases roughly as & ’" . In the rest of the paper we shall refer to
this neighborhood as the probing neighborhood.
3
3.1 The statistical approach
A general statistical scheme would gather data from the sensors in
the probing neighborhood and perform statistical analysis to decide
whether or not the sensor is an edge sensor. The advantage in this
approach is that statistical methods can be explicitly tailored to be
robust to errors, if error characteristics are known. The general
algorithm for a statistical scheme then needs three components to
be specified.
1. The information to be collected from the neighbors,
2. A set of statistics based on the information
collected from the neighbors and,
3. A boolean decision function to decide if
the sensor is an edge sensor. The decision function usually would
involve comparing a value evaluated using against a
threshold (maybe statically or dynamically assigned).
3.1.1 An example scheme
In this paper we evaluate a specific statistical scheme that we de-
signed for edge detection. The key idea behind the scheme is the
observation that if one collects the event predicate values from sen-
sors in the neighborhood, and these values form a bimodal distri-
bution (spikes at 0 and 1) then an edge is present. Let be the
number of 1 valued event predicates and be the number of zero
valued event predicates in the neighborhood. We calculate the fol-
lowing statistic,
(4)
"!
$#&%(’
) "!
$*&%(’
(5)
Our statistical scheme is intuitively simple, and therefore forms a
baseline for comparison against other schemes. One salient feature
lacking in the statistical scheme is that it does not take the geograph-
ical locations of sensors into account when making its decisions.
For arbitrarily placed sensors, as we shall see later, this can make a
difference.
Designing the statistical scheme to be robust to sensor errors is a bit
tricky, as we now explain. If the sensors were perfectly calibrated
and error free, presence of an edge would be indicated by a non-
zero value of the statistic and any
% ’ # ) would suffice. In a more
realistic scenario, with arbitrarily placed sensors having calibration
and measurement errors, the statistic would yield non-zero values in
absence of edges because of sensor errors. Also if " # , for edges
passing in the probing beighborhood which do not lie in the area of
tolerance, 4 would give a non-zero value. Then, the choice of an
“appropriate” threshold
%+’
would determine the performance of the
scheme. In general the choice of
% ’-,
) depends % , . and the
performance requirements of the application.
3.1.2 Analysis of the scheme and choice of
% ’ To gain some intuition about the choice of
% ’ and how it relates to
the tolerance radius and the probing neighborhood, we analyze the
performance of the proposed statistical scheme. For our analysis we
assume that nodes are placed in the region at locations drawn from
a uniform density function with a density . sensors per unit area.
Also we assume that the sensors make an error in evaluating the
value of the event predicate with a probability / . We hope that this
error model encapsulates both calibration and measurement errors.
Further, we assume that the probing neighborhood is so “small” in
comparison to the area covered by the entire phenomenon that the
edge can be approximated by a straight line in this region.
As discussed in section 2.2, errors can be either false detections
or missed detections . False detections can arise in two ways.
One cause of false detections is when there is no edge in the probing
radius but the algorithm detects an edge due to sensor errors. The
second occurs when there is an edge in the probing radius but not
within the tolerance radius. We call the former kind of errors pure
false detections ( 0 ) and the latter unwanted detections ( ).
0 1 (6)
The reason we make this distiction is that, a high 0 can result in
a large number of sensors being deemed as edge sensors even when
they are in the “middle” of a phenomenon because of sensor errors
and local variations in sensor density. On the other hand a high simply increases the thickness of the edge. For certain applications,
the edge thickness may not be as harmful as identifying a sensor as
an edge sensor when it is far from an edge. For fixed values of % and . , the errors , 0 and depend on the choice of
%+’
.
Now, the number of sensors present in an area 2 can be modeled by
a Poisson random variable,
3 ! 4 6587
29. : (7)
and the number of sensors making an error ; among
! sensors can
be modeled by a binomial random variable,
3 <; >=
! ? @ ACB
D/ "
/ (8)
Based on these assumptions, one can numerically calculate the
probability density function for defined in (4), for given values
of % and . . The procedure for calculating the density function is
described in Appendix A.
Choosing
% ’ : An Example Figure 4, computed from our anal-
ysis, shows the variation of the percentage of true, unwanted and
false detections as
% ’ varies from 0 to 1, for two different values of
% (1.0 and 1.5). The sensor density is such that, the expected num-
ber of sensors in area of tolerance is 15. The sensor error probability
/ ) )FE
.
Suppose an application requires a true detection ratio of atleast 80%
and also wants false detections to be below 1%. Also suppose the
application can do with slightly thicker edges and allows about 20%
extra edge detections. For % G , corresponding to an error of
80%, the
’IH
E . Corresponding to this value of
’ , the false
4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
10
20
30
40
50
60
70
80
90
100
percentage
threshold
true
unwanted
false
R/r = 1.5
R/r = 1
R/r = 1.5
R/r = 1
R/r = 1.5
Figure 4: Variation of percentage true, unwanted and false detec-
tions as threshold
’ varies from 0 to 1 for % values of 1 and 1.5, an
expected value of 15 sensors in the tolerance region and / ) )FE
.
error 0 H ) and ) . The situation is considerably
improved when % E . If we choose a threshold of 0.4, one
can achieve 80% true detections, but now 0 less than 1% and
H . Hence, by increasing the probing radius we have im-
proved the performance of our scheme. However, by increasing the
probing area we incurred an increase in communication overhead
by 125% ( % ).
Clearly since performance depends on the choice of
% ’ and % , it be-
comes essential for sensors to be able to figure out a suitable setting
for these parameters for “satisfactory” operation. It is conceivable
that pre-calculated performance curves similar to those in Figure 4
are stored in the sensor nodes a priori. Nodes estimate the local
sensor density and based on the performance criterion desired, use
the performance curves to come up with a operating point.
3.2 The image processing approach
Numerous techniques for edge detection have been developed and
analyzed in the image processing literature [1]. It is therefore tempt-
ing to attempt to apply such techniques to localized edge detection
in sensor networks. In this paper we do not exhaustively examine
all possible image processing techniques, but simply pick a frame-
work that can incorporate a class of high pass filtering techniques (a
standard way of performing edge detection in images eg. Prewitt,
Sobel filters) for localized edge detection.
A high-pass filter allows only the high frquencies (abrupt changes
such as edges) present in the image to remain and removes all the
uniformities. Designing a filter with a desired frequency response
is a mature art and several different techniques exist. In general, if
a filter with a frequency response = & ) . & is drawn from a uniform
distribution over the entire -axis within the sensor field. = . is the slope of this line, generated by drawing . uniformly in ) 5 .
Sensors with = & ) belong to the interior region ( )
and rest belong to the exterior region ( ) ). The edge (ground
truth) is defined by the line = & ) . The linear bound-
ary forms a baseline for evaluating our scheme; an acceptable edge
detection scheme should perform well for this data set.
The second, elliptical boundary data sets , consist of ellipses
<2 %$ ’ ’ . ) randomly chosen within the sensor field.
2 and
$ are lengths of the major and minor axes of the ellipse, uni-
formly drawn over the length of the sensor field. ’ ’ is the
center of the ellipse drawn uniformly over the entire sensor field.
. , which is the angle between the major axis of the ellipse and the
-axis is drawn uniformly in ) 5 . Let be the sensor coor-
dinates in a coordinate system “natural” to the ellipse (the major and
minor axes of the ellipse form the and axes). can be ob-
tained by first translating the origin to ’ ’ and then rotating the
axes by . in the anti-clockwise direction. If " 5 " # ) ,
the sensor is deemed to belong to the interior region ( ) and to
the exterior ( ) ) otherwise. The edge (ground truth) is defined
by the ellipse <2 %$ ’ ’ . ) . Ellipses of different eccen-
tricities represent continuously curved edges, and can be serve to
distinguish localized edge detection schemes.
Factors To examine the impact of density, we chose three values
of . : . ) sensors/sq.mt (low density - about 5 senors
within radio range), . ) sensors/sq.mt (moderate den-
sity - about 15 senors within radio range), and . ) sensors/sq.mt (high density - about 30 senors within radio range).
To capture the impact of sensor errors, we used a simple bit flipping
technique. In this model, a sensor toggles its event predicate value
from its true value with a probabiliy / . We used three different
choices for / . / (low), / E (moderate) and / ) (high).
Thus, for the linear boundary data set, a single simulation run repre-
sents one line chosen randomly, for one value of density and sensor
error. For a given density and sensor error probability, we average
the performance metrics for a localized edge detection scheme over
20 different runs corresponding to different randomly chosen lines.
The same is true for the ellipse.
Parameters We chose five different values of % , namely , E ,
,
E and
. We ran simulations for all the data sets, for each of the
three schemes, for the five values of % .
The statistical and the image processing schemes require choosing
% ’ , ) and
( ’-,
) respectively. This choice (as discussed
in section 3.1.2) can impact performance. To be fair to all schemes,
for a given simulation run, we chose the best threshold value (us-
ing the analysis in Appendix A and B) defined thus: “Choose the
threshold which satisfies 0 and minimizes ”. Thus,
for schemes that require thresholds, our simulations represent the
fewest possible missed detections.
We evaluate our schemes with respect to the metrics described in
Section 2.2.
4.2 Simulation results
In this section we discuss the results of our simulations. The space
of parameters and factors we have explored is large. Rather than
exhaustively present all of our results, we selectively describe the
simulation results in an effort to give the reader an understanding of
the main differences between the schemes.
We start by considering (Figure 6) which shows the variation of
(mean thickness error) and (detection probability) for
moderate error ( / E ) and moderate density ( . ) )
with % . It depicts the basic nature of the energy accuracy tradeoff .
As seen in Figure 6, the edge detection probability increases with
increase in % . However, increasing % increases the communication
overhead as & " and hence the energy consumption. For linear
data sets all the three schemes give similar detection ratios at % # ,
while the classifier gives a thinner edge. For elliptical data sets, at
% # , the classifer performs “slightly” inferior to the other two
schemes, however it gives a much “thinner” egde. The performance
of statistical and image processing based schemes is similar.
7
1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
R/r
Detection Probability
1 1.5 2 2.5 3
−0.5
0
0.5
1
1.5
R/r
Mean Thickness Error
1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
R/r
Detection Probability
1 1.5 2 2.5 3
0
1
2
3
4
R/r
Mean Thickness error
Stat
Img
Class
Data sets with linear edges
Data sets with ellipical edges
Figure 6: Energy accuracy tradeoff : As more and more neighbor-
hood is examined ( % the detection probability increases for all the
three detection schemes.)
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
R/r
False detection probability
Linear
Elliptical
Figure 7: False detections for the classifier based scheme decrease
with increase in % The statistical and image processing based schemes, allow one to
restrict the false detection probability by selecting an “appropri-
ate” choice of threshold (
% ’ and
( ’ ). In all our simulations, the
choices restricted false detection to below 1%. In the classifier based
scheme, there is no such direct way to restrict false error probability.
(Figure 7 ) shows the variation of false detections made by the clas-
sifier scheme with increase in % for the moderate density, moderate
sensor errors data sets. The false detection probability decreases
with increase in % .
The classifier scheme behaves qualitatively different from the sta-
tistical and image processing schemes. For the latters, as the % ratio increases, the edge thickness increases while for the classifier
based schemes the edge thickness decreases. Edge thickness error
comes from two reasons, namely i)pure false detections ( 0 )and
ii)unwanted detections ( ). Since, we restricted 0 * for
ths statistical and image processing based schemes, edge thickness
error mostly arises out of ). In the classifier based scheme 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
0
0.5
1
1.5
R/r
Unwanted detections
Line
Ellipse
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
0
0.5
1
1.5
R/r
Unwanted detections
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
0
0.1
0.2
R/r
Unwanted detections
Statistical Scheme
Image Processing Scheme
Classifier Based Scheme
Figure 8: Variation of with increase in % for the three schemes
for moderately dense sensor fields with moderate errors.
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
0
0.5
1
R/r
Detection Probability
160
360
720
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
0
0.5
1
R/r
Detection Probability
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
0.6
0.8
1
R/r
Detection Probability
Statistical Scheme
Image Processing Scheme
Classifier Based Scheme
Figure 9: Variation of (detection probability), with increase
in density for the three schemes for linear edge data sets.
is very “small” and does not change significantly with % . This is
depicted in figure 8. The increase in causes an increase in thick-
ness error for the statistical and image processing based schemes,
while a decrease in false detections leads to a decrease in thickness
error for the classifier based scheme. For this regime, then, the clas-
sifier based scheme represents a low-energy technique for achieving
thin edges with high likelihood of true detections.
What happens when we change density but keep sensor error con-
stant? Predictably, as the detection probability increases with in-
crease in sensor density for both kinds of data sets. This is de-
picted in figure 9. There was no
( ’ which gave an 0 * at . ) , hence this point is missing. We also found
that while the thickness error increases with increase in density for
the statistical and image processing based schemes, it decreases for
the classifier based scheme. The reason is that, unwanted errors,
which dictate thickness error for the statistical and image process-
ing schemes increase with increase in sensor density. However, the
false detections which dictate the edge thickness error for the clas-
sifier based scheme decrease with increase in density.
How sensitive are the schemes with respect to sensor errors? Keep-
ing density fixed, we notice an expected qualitative trend. The de-
8
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
0
0.5
1
R/r
Detection probability
1%
5%
10%
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
0
0.5
1
R/r
Detection probability
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
0.7
0.8
0.9
1
R/r
Detection probabilty
Statistical Scheme
Image Processing Scheme
Classifier Based Scheme
Figure 10: Variation of (detection probability), with increase
in sensor error / for the three schemes for linear edge data sets.
tection probabilty decreases with increase in sensor errors for all
the three schemes. This is depicted in figure 10. We also found that
the thickness error increases with increase in sensor error for all the
three schemes and both kinds of data sets.
Finally, how critical is the choice of apporpriate thresholds? The
classifier-based scheme does not require selection of a threshold.
The other schemes do, and in the results we have presented, we have
chosen the best threshold possible for each particular scenario. The
threshold based schemes might have been acceptable if there existed
one or a small range of thresholds that were acceptable over the
density and error ranges we consider. However, we found that the
thresholds in the statistical scheme vary very widely with changes
in % and / , especially at “low” densities. For instance the optimal
value of
% ’ is 0.79 for % , / ) )9E
and . ) .
The optimal value of
% ’ is 0.17 for % , / E and . ) . For the image processing based scheme, it turns out
that the variation in the choice of
( ’ is very small (within 10%) with
respect to % and / at low densities. However the scheme exhibits
variations similar to the statistical scheme at higher sensor densities
for changes in % .
This discussion leads to the following conclusions. Over a range of
sensor error rates and densities, all the three scheme can achieve
true detection rates of 90% or better by using a two-hop probing
radius. Among the three schemes the classifier provides the thinnest
edges and performs better with increasing probing radius. The clas-
sifier based scheme does not require choosing appropriate thresh-
olds. Thus, from a practical perspective, the classifier-based scheme
represents a low energy approach to accurate localized edge detec-
tion. Even though the classifier based scheme does not provide a
direct control over false detection errors (as thresholds in other two
schemes do), one can increase the probing radius (eg.3 hop prob-
ing) and achieve lower false detections at the cost of more commu-
nication cost. Recall that localized edge detection will usually be a
component of a larger system that, for example, computes bound-
aries of a phenomenon. We observe that false detections can be dis-
ambiguated at the level of these boundary finding algorithm. If the
algorithm that constructs the boundary from the observed edges also
uses information about each edge sensor’s partition line, it should
be able to detect inconsistent partition line orientations and loca-
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure 11: Classifier based edge detection on a low density low error
data set. Each unit on and axis represent 100mts.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure 12: Image processing based edge detection on a moderate
density moderate error data set. Each unit on and axis represent
100mts.
tions among neighboring edge sensors caused by false detections.
We believe that a better scheme which relies on edge continuity in-
formation will result in fewer false detections.
For this reason, we suggest that, of the three schemes we consider,
the classifier is the most promising for localized edge detection. Fig-
ures 11,12,13 show three examples of the three edge detection algo-
rithms at work.’ & ’ represent edge sensors and the interior.
5 Conclusion
In this paper we introduced the problem of localized edge detec-
tion in a sensor field. We discussed an “edge” and proposed met-
rics to assess edge detection algorithms. We proposed three qual-
itatively different approaches to edge detection namely statistical,
image processing based and classifier based approaches. We pro-
posed an example scheme for each of these approaches. Through
numerous simulations we compared the three schemes with respect
to the energy accuracy-tradeoff, sensitivity to choice of parameters
and performance.
9
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure 13: Statistical scheme based edge detectin on a high density
high error data set. Each unit on and axis represent 100mts.
Our results indicate that the classifier scheme performs much bet-
ter than the other schemes. Under higher sensor error conditions, it
is susceptible to more false detections than other schemes. These
false detections can be reduced at the expence of higher commu-
nication cost or can probably be disambiguated by a higher-level
boundary finding algorithms. The statistical and image processing
based scheme can exhibit similar performance but only if detection
thresholds are correctly set. The correct detection thresholds vary
widely with density and sensor error and we believe that dynami-
cally setting thresholds by empirically observing densities will be
hard to do. As such, then, of the schemes we consider, the classifier
based scheme seems to be the most promising for localized edge
detection.
References
[1] Bernd J¨ ahne, Digital Image Processing,, Springer, 4th edition,
1997.
[2] R. O. Duda, P. E. Hart, D. G. Stork, Pattern Classification,
John Wiley and Sons, 2nd edition, 2000.
APPENDIX A
In this appendix we calculate , and 0 to analyze the
scheme described in section 3.1.1.
We assume that the phenonmenon is ”large” enough and the edge
can be approximated by a line segment
# within the probing
neighborhood. Let
# be
units distant from the sensor. We assume
that line segments at all values of
are equally likely. Suppose an
edge passes at a distance
from the sensor ( as depicted in figure
14). The sensor collects information about the exterior from area
2 (ABC) and about the interior from 2 (ADC).
2 " @ " B " (A-1)
2 5" 2 (A-2)
Edge
Exterior
R
l
a
a
1
2
cos
−1
( l/R )
Interior
B
C
D
A
Figure 14:
The number of sensors
! and
! in these regions can be modeled
as possion random variables.
3 ! ? 65 7 <2 . : (A-3)
The number of sensor errors (in ABC) and (in ADC), can
be modeled by a binomial random variable with / as sesnor error
probabilty.
3 A ! > @ A B / "
/ (A-4)
The value of the statistic in (4) can now can be written in terms of
and
! as,
! ! ! ! (A-5)
Using equations (A-1)-(A-5), we can numerically calculate the
probability density function,
3 % .
Calculation of and : A miss-detection occurs when
and *&% ’ .
3 (A-6)
3 < % ’ 3 % (A-7)
’ 3 % % (A-8)
An unwanted detection occurs when " # # and #&% ’ .
3 " # # " (A-9)
3 % " # # % 3 % " (A-10)
" 3 % " # # % (A-11)
Calculation of 0 : Now suppose there is no edge with the probing
neighborhood (
# " ). can assume non-zero values only because
of sensor errors. The pdf of number of sensors in the neighborhood
region
! can be expressed as a poisson random variable.
3 ! ?
# " % 7 5" . : (A-12)
10
θ
l
R
A
B
C
D
E
F
G
H
I
Exterior
Interior
a
a
a
a
1
2
3
4
Figure 15: Figure for appendix B
The number of sensor errors in the probing neighborhood can be
expressed as a binomial random variable.
3 A ! ? @ ACB
/ / (A-13)
The value of the statistic in (4) can now can be written in terms of
and
! as,
! ! (A-14)
We numerically calculate the pdf
3 < % # " . A pure false
error occurs when, #
% ’ .
0 ! . 3 < % # " % (A-15)
! is the total number of sensors in the field.
APPENDIX B
In this appendix we calculate the errors , and 0 for Pre-
witt filter example in section 3.2.1. Let the line segment
# . approximate the edge (as argued in Appendix A) intersect the prob-
ing neighborhood. Here,
is its distance from the sensor and . is
the angle the -axis makes with the normal from the sensor (see fig-
ure 15). We divide the neighborhood into 4 areas, 2 (DCHF), 2 (ABCD), 2 (ADEG) and 2 (EDFI) as depicted in figure 15. The
four areas 2( for . , ) can be calculated by the equations,
2 % % . . " . * % ) . # % (B-1)
(B-2)
2 % % 4. . " . * % % % " . # % (B-3)
2 5 " 2 (B-4)
2 5 " 2 (B-5)
The number of sensors ! in the four regions and the number
of sensor errors can be modeled as in A-3 and A-4. The
value of
( in (18) can now can be written in terms of and
! as,
( , , , , ! ! ! ! ! ! ! ! , , , , (B-6)
The pdf
3 ( . can be numerically calculated using, (B-1)-
(B-6), (A-3) and (A-4).
It turns out that
3 ( . is same as
3 ( . . The pdf
3 ( . can be calculated numerically from
( ) ( ( .
Calculation of and : A miss-detection occurs when
but
( * ( ’ .
3 ( 5 ’ ’ 3 ( . . (B-7)
’ 3 ( * (B-8)
An unwanted detection occurs when " # # but
( # ( ’ .
3 ( " # # 5 " ’ % 3 ( . . (B-9)
" 3 ( " # # (B-10)
Here, we integrate on . only over ) since,
3 ( . is
identical in any section (
.
Calculation of 0 : Now suppose there is no edge with the probing
neighborhood (
# " ) and a non-zero value of
( occurs due to
sensor errors. Let
! be the number of sensors in the quadrant
(quadrants numbered in anti-clockwise direction). Let be the
number of sensor errors in the quadrant. Then,
3 ! > 7 % . : (B-11)
3 A ! > @ A B D/ / (B-12)
The value of
( can be calculated in terms of ! and .
( , , , , , !8
!8
! "
, , , , , (B-13)
( , , , , , ! ! ! , , , , , (B-14)
( ) ( ( (B-15)
Using (B-11)-(B-15), we can numerically calculate the pdf
3 ( # " . A pure false error occurs when,
( # ( ’ .
0 ! . 3 ( # " (B-16)
! is the total number of sensors in the field.
11
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Description
Krishna Kant Chintalapudi, Ramesh Govindan. "Localized edge detection in sensor fields." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 773 (2002).
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