University of Southern California Dissertations and Theses

Automatic tracking of protein vesicles

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Automatic tracking of protein vesicles

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Content AUTOMATIC TRACKING OF PROTEIN VESICLES
by
Min Xu
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Ful¯llment of the
Requirements for the Degree
MASTER OF ARTS
(APPLIED MATHEMATICS)
May 2009
Copyright 2009 Min Xu
Dedication
To Luge and Shiyou
ii
Acknowledgements
I realize this thesis with the help of several wonderful and kind people. First of all I
would like to thank my advisor, Professor Sergey Lototsky for his great help, patience
and indispensable advises. Also I would like to thank Professor Donald Arnold and Mr.
Sarmad Al-Bassam. My interest on designing methods of tracking protein vesicles is pri-
marily motivated from their research. Finally, I would like to thank Professor Remigijus
Mikulevicius for reviewing my thesis and attending my presentation.
iii
Table of Contents
Dedication ii
Acknowledgements iii
List Of Figures v
Abstract vi
Chapter 1: Introduction 1
Chapter 2: Literature review 4
2.1 Object tracking methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Object representation . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Object detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.3 Object tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Tracking methods applied to cellular dynamics . . . . . . . . . . . . . . . 9
Chapter 3: Tracking of single vesicle 11
3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 A dynamical programming approach . . . . . . . . . . . . . . . . . . . . . 12
3.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3.1 Tracking accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Chapter 4: Tracking of multiple vesicles 19
4.1 Track elimination and enumeration . . . . . . . . . . . . . . . . . . . . . . 19
4.1.1 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Improving track associations using Kalman ¯lter . . . . . . . . . . . . . . 22
4.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2.2 Introduction to Kalman ¯lter . . . . . . . . . . . . . . . . . . . . . 22
4.2.3 Improving tracking by incorporating Kalman ¯lter . . . . . . . . . 24
4.2.4 Combining dynamic programming and point detector . . . . . . . 27
4.2.5 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Chapter 5: Conclusion and discussion 30
References 31
iv
List Of Figures
1.1 Theintracellularcompartmentsoftheeucaryoticcellinvolvedinthebiosyn-
thetic secretory pathways. . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Three consecutive °uorescence microscopy images showing protein vesicles. 3
3.1 Examples of tracking two moving objects in a noisy video. . . . . . . . . . 14
3.2 Performance analysis of di®erent settings. . . . . . . . . . . . . . . . . . . 16
3.3 Simulated noise-free video that consists of one zigzag track and one linear
track. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.1 Left: Simulated video used in Section 3.3 (same as Figure 3.1 without
estimated track). Right: the video with the best track eliminated. . . . . 21
4.2 Wrong tracking of two objects due to cross of their real tracks. . . . . . . 23
4.3 Overcoming the wrong association problem by using Kalman ¯lter. . . . . 29
v
Abstract
With the advance of °uorescence imaging technologies, recently cell biologists are able to
record the movement of protein vesicles within a living cell. Automatic tracking of the
movementsofthesevesiclesbecomekeyforqualitativeanalysisofdynamicsofthesesvesi-
cles. In this thesis, weformulatesuchtrackingproblem as video objecttrackingproblem,
and design a dynamic programming method for tracking single object. Our experiments
on simulation data show that the method can identify a track with high accuracy which
is robust to the choose of tracking parameters and presence of high level noise. We then
extend this method to the tracking multiple objects using the track elimination strategy.
In multiple object tracking, the above approach often fails to correctly identify a track
when two tracks cross. We solve this problem by incorporating the Kalman ¯lter into
thedynamicprogrammingframework. Ourexperimentsonsimulateddatashowthatthe
tracking accuracy is signi¯cantly improved.
vi
Chapter 1
Introduction
Every cell must communicate with the world around it. Eucaryotic cells (i.e. cells of
animals, plants, and fungi) have internal membrane system that allows them to regulate
the delivery of newly synthesized proteins to the cell exterior. The biosynthetic-secretory
pathwayallowsthecelltomodifythemoleculesitproducesinaseriesofsteps,storethem
until needed, and then deliver them to the exterior. Such delivery is through protein
vesicles, which are small bubbles of liquid within a cell. Figure 1.1 conceptualize typical
biosynthetic-secretory pathways in a cell. In the ¯gure, each compartment encloses a
space, called a lumen, that is topologically equivalent to the outside of the cell, and all
compartmentsshowncommunicatewithoneanotherandtheoutsideofthecellbymeans
oftransportvesicles. Inthebiosynthetic-secretorypathway(redarrows)proteinmolecules
are transported from the endoplasmic reticulum (ER) to the plasma membrane or (via
late endosomes) to lysosomes. Some molecules are retrieved from the late endosome and
returned to the Golgi apparatus, and some are retrieved from the Golgi apparatus and
returned to the ER. The ¯gure and caption are adapted from [ABL
+
02].
1
Figure 1.1: The intracellular compartments of the eucaryotic cell involved in the biosyn-
thetic secretory pathways.
2
Fluorescence microscopy is a main tool to study the biosynthetic-secretory processes.
A cell is normally optical transparent. To visualize the molecules of a protein of interest,
they can be labeled using °uorescence dye. When excited using light of a particular
wavelength, the dye can emit light of another wavelength that can be detected. Thus,
the location of the protein molecules in a cell can be identi¯ed. In recent years, the
resolution of location identi¯cation is highly increased through the advance of confocal
microscopytechniques. Inaddition,thediscoveryof°uorescentproteinsenablesbiologists
to observe the dynamics of proteins in individual living cells. So far, little has been done
on automatic analysis of the dynamics of protein molecules. Here we focus on tracking
movement of protein vesicles from microscopy image sequence (i.e. a video).
Supposemicroscopyimagesaregreyscaleimages. Afterremovingstaticstructureina
imagesequence,proteinvesiclebecomesaspotlikeobjectthatisrelativebrighter(i.e. has
a higher value) than dark background. Figure 1.2 shows such an image. In the ¯gure,
majority of the static cell structure are suppressed. video obtained from [KGH
+
07].
There are two main challenges in such kind of tracking : 1) high level of noise in images
and 2) tracking of multiple objects.
Figure 1.2: Three consecutive °uorescence microscopy images showing protein vesicles.
3
Chapter 2
Literature review
2.1 Object tracking methods
Object tracking method is an important ¯rst step of automatic analyzing cellular dy-
namics. Object tracking is originally developed in the ¯eld of computer vision. Basically,
tracking can be de¯ned as the problem of estimating the trajectory of an object in the
imageplaneastheobjectmovesaroundascene. Inpractice,therearemanydi±cultiesto
successfulbuildingofatrackeralgorithm. Thedi±cultiesrelatedtoproteinvesicletrack-
ingare1)noiseinimages, 2)complexobjectmotion, 3)partialandfullobjectocclusions.
The above problems sometimes can be simpli¯ed by incorporating prior knowledge of the
objects. However, so far little is known about the characteristics of molecular dynamics
in cells.
Numerous methods for object tracking have been proposed in the ¯eld of computer
vision. They mainly di®er in the following aspects [YJS06]: 1) object representation 2)
image features used 3) modeling of motion, appearance, and shape of the object. These
4
methods address the above aspects according to the context/environment in which the
tracking is performed and the tracking information needed for subsequent analysis.
2.1.1 Object representation
The shape of an object can be represented in di®erent ways: 1) points, i.e., the centroid
[VRB01] or a ¯nite set of points [SMVG04]. 2) Primitive geometric shapes (eg ellipse,
rectangleetc) [CRM03]. 3)objectsilhouetteandcontour [YLS04]. 4)Articulatedshape
models, i.e., shapes held together with joints. 5) Skeleton [BB82]. Since in our case,
the protein vesicles only occupy small regions in an image, we normally represent these
vesicles as points.
Incombiningshaperepresentations,theareanumberofwaystorepresenttheappear-
ance features of objects. Following are a number of appearance features: 1) probability
densities,canbeeitherparametric [ZY96]ornonparametric [EDHD02];2)activeappear-
ance models [ETC98], which associate landmarks with feature vectors of color, texture,
etc; 3) multiview appearance models, which are generated from subspaces of given views
usingtechniqueslikePrincipalComponentAnalysis(PCA)andIndependentComponent
Analysis (ICA) [MP97].
2.1.2 Object detection
Atrackingmethodneedstobeabletodeterminetheexistenceofanobjectineveryframe
or in the frame that the object ¯rst appears in the video. Some detection methods use
the temporal information computed form a sequence of frames to reduce false detections.
Following are several types of common methods used for object detection.
5
Point Detectors: point detectors are used to identify points that belong to an object,
based on local context. Commonly used point detectors are: Moravec's interest operator,
Harris interest point detector, KLT detector, and SIFT detector [MS02].
Background Subtraction: object detection can also be achieved by constructing a
backgroundmodelandthen¯ndingdeviationsfromthemodel. Asigni¯cantchangeinan
image region from the background model would indicate a moving object. This process
is called background substraction. Due to the increasing abilities to e±ciently model
complex background, most of recent tracking methods for ¯xed cameras use background
subtraction methods to detect regions of interest (for example [HHD00], [CLFK01]).
Segmentation: imagesegmentationalgorithmsisusedtopartitiontheimageintoper-
ceptually similar regions. A segmentation algorithm normally de¯nes criteria for a good
partition and provides a method for e±cient partitioning. Several types of segmentation
methodshavebeendesignedfortrackingpurpose: 1)meanshiftclustering [CM99],which
performs clustering in the joint spatial and color space; 2) graph cutting, which converts
the image into a graph, and uses min-cut algorithm to ¯nd disjoint regions [SM00]; 3)
active contours, which evolves a closed contour to the objects' boundary, evaluated by
certain energy function (for example [KWT88]).
Supervised Learning: in this approach, object detection is formulated as a classi¯ca-
tion problem. The learning algorithm is used to generate models from the features of the
objects in the training video where the objects are known. Then these models are used
to predict the existence of objects in new video data. When a set of image features are
6
properly chosen, a number of di®erent classi¯cation methods can be used, such as neu-
ral networks [RBK98], adaptive boosting [VJS05], decision trees [GK95], and support
vector machines [POP98].
2.1.3 Object tracking
Object tracking is used to generate the trajectory of an object over time by locating its
position in every frame of the video. There are two tasks in object tracking: detecting
the object and establishing association of objects between frames. These two tasks can
be performed separately or jointly. For di®erent object representation, di®erent types of
tracking methods are developed. They fall into three categories: 1) point tracking, which
represents objects as points and only estimates the object's position in each frame; 2)
kernel tracking, which uses object shape and appearance and considers not only trans-
lation but also rotation of objects; 3) silhouette tracking, which uses template matching
to identify objects in each frame. Because in our project, the protein vesicles are mainly
represented as points, in this section, we will only describe point tracking methods.
In point tracking, tracking can be formulated as the association of detected objects
represented by points across frames. In general, there are two types of methods to asso-
ciate points: deterministic and statistical methods.
Deterministicmethodsusuallyde¯neacostofassociatingeachobjectinframet¡1to
asingleobjectinframetusingasetofmotionconstraints. Minimizationoftheassociation
costisformulatedasacombinatorialoptimizationproblem. Optimalassignmentmethods
are developed to obtain the best one-to-one association among all possible associations
(for example, Hungarian algorithm [Kuh55]). The association cost is usually de¯ned by
7
using a combination of the following constraints: 1) object displacement between frames;
2) maximum velocity; 3) small velocity change; 4) common motion, which requires the
velocity of objects in a small neighborhood to be similar; 5) rigidity, which assumes that
objects are rigid.
Hereareanumberofsuchmethods. SethiandJain [SJ87]proposedagreedymethod
to solve the association problem. It is based on the proximity and rigidity constraints
applied on two consecutive frames. Salari and Sethi [SS90] designed a method that
establishesassociationforthedetectedpointsandthenextendthetrackingofthemissing
objectsbyaddinghypotheticalpoints. Fromtheirwork,Veenmanetal. [VRB01]further
added the common motion constraint. Sha¯que and Shah [SS05] uses the temporal
coherency of speed and position in multiple frames as constrain. In their approach, the
association problem is converted to ¯nding the best unique path for each point on a
graph.
Compared to deterministic methods, statistical association methods are used to re-
duce the e®ect of noise in the video and the perturbation of movements in objects by
incorporating the randomness into model. They use the state space approach to model
the object properties such as position, velocity, and acceleration.
For single object tracking, a typical statistical association method is Kalman ¯lter
[BC86], which assumes the transition of system states is linear and the noise is Gaussian.
WewilluseKalman¯ltertoenhanceourtrackingmethod. OnelimitationofKalman¯lter
is the assumption of Gaussian distribution of the state variables (see Method section for
details). Particle ¯ltering [Mac98] has been used to reduce the above limitation through
model estimation by importance sampling.
8
Multiobject association and state estimation are often carried out statistically. When
tracking multiple objects using Kalman or particle ¯lters, the association problem needs
to be solved before these ¯lters can be applied. There are two popular methods for data
association: Joint Probability Data Association Filtering (JPDAF) [CA91] and Multiple
Hypothesis Tracking (MHT) [Rei79]. JPDAF extends Kalman ¯lter by replacing its
innovation of single track with a sum of innovations of multiple tracks weighted by the
posterior probability that a measurement is associated with that track. On the other
hand, MHT iteratively improves associations. In each iteration, the algorithm starts
from a set of current track hypotheses, in form of collections of disjoint tracks. For each
hypothesis,thealgorithmpredictseachobject'spositioninthenextframe. Bycomparing
thepredictionswithactualmeasurements,associationsareestablishedforeachhypothesis
and a set of new hypotheses are formed for next iteration.
2.2 Tracking methods applied to cellular dynamics
The ¯eld of tracking of molecular dynamics in cells is relatively new. So far only a small
number of methods are developed for this purpose. They are summarized as follows.
Sbalzarini et al. [SK05] proposed a tracking method that consists of two steps:
feature detection and trajectory linking. In this approach, proteins are represented as
points. The feature detection step mainly consists of detection of re¯nement of points
according to local maxima values. Given the detected candidate locations, trajectory
linking then associates the points between each of two adjacent frames.
9
Godinezetal. [GLW
+
07]designedamethodfortrackingvirusparticles. Themethod
consists of virus particle detection and association. For the detection step, they use
Laplacian-of-Gaussian ¯ltering for detecting spots. Laplacian-of-Gaussian is a image
¯ltering technique that applies Gaussian blur and Laplacian operator to a image. They
then use Gaussian ¯tting to enhance spots. For the association step, they employed a
smoothmotionandnearestneighborconstraintstolinkdetectedparticlesbetweenframes.
Sage et al. [SNH
+
05] proposed a dynamic programming approach for tracking the
°uorescent markers attached in a single chromosome in a cell. Basically, given a discrete
scalar ¯eld, dynamic programming is a computational technique that can be used to
¯nd a curve such that integration along this curve would achieve optimal value. The
advantageofsuchapproachisthatobjectdetectionstepisnotrequired. Wewilldescribe
this approach in detail in the next section.
The above approaches are deterministic. A few stochastic tracking approaches also
appeared recently. For example, Yoon et al. [YBFK08] proposed to use particle ¯lter
to track a single molecule. Simply speaking, when the states of objects are modeled
as a Markov Chain, particle ¯lter obtains optimal Bayesian estimation of states given
noisy observations over time. Also using particle ¯lter, Smal et al. [SDG
+
07] designed
a method for tracking microtubles in a cell. In their approach, microtubes are modeled
using Gaussian functions for the detection. After detecting the molecules, particle ¯lter
is used for estimation of tracks.
10
Chapter 3
Tracking of single vesicle
3.1 Problem formulation
In this section, we study the tracking of a single vesicle in a video. We solve single vesicle
tracking problem through an optimization approach given a sequence of n-dimensional
images (normally n = 2). Denote X ½ R
n
as the set of all possible locations, which is
identical for all images. Let f(x;t) be the intensity of location x 2 X of the image at
time t, we want to ¯nd a track x
t
; t = 1;:::;T such that the following score function is
maximized:
s
T
=
T
X
t=1
f(x
t
;t)¡w(
T
X
t=2
kx
t
¡x
t¡1
k)) (3.1)
Intuitively, this score function tends to be high when the intensity along the track is
high and displacement is low. These two factors are balanced using a with a weight w.
11
3.2 A dynamical programming approach
Suchoptimizationproblemcanbesolvedusingdynamicprogrammingtechnique[CLRS01].
The above problem can be decomposed into subproblems, and optimal solutions of sub-
problems can be used to ¯nd the optimal solutions of the overall problem. Formally, let
s
t
(x) be the maximum score of all tracks of length t that ends up at positionx. That is:
s
t
(x),max
x
1
;:::;x
t¡1
ff(x
1
;1)+
t¡1
X
r=2
[f(x
r
;r)¡wkx
r
¡x
r¡1
k]+[f(x;t)¡wkx¡x
t¡1
k]g
(3.2)
Then s
t
(x) can be calculated using s
t¡1
(¢) as follows:
s
t
(x)=max
y2X
[s
t¡1
(y)+f(x;t)¡wkx¡yk] (3.3)
Thus given x, the calculation of s
t
(x) will automatically identify x
t¡1
that would
achieve maximum s
t
(x). In addition, argmax
x
s
T
(x) gives the location where the best
track terminates at time T. This provides foundation of tracking. The procedure of
optimal scores and trace-back can be summarized in the following algorithms. Similar
approach has been used by Sage et al. [SNH
+
05] to track a single particle in noisy
images.
3.3 Experimental results
Figure 3.1 shows the experiments of applying the dynamic programming approach. The
left to right sub-¯gures correspond to experiment 1, 2 and 3 respectively. Two red curves
12
Algorithm 1 DPScoring
Input: (1) X, the set of all locations
(2) T, total number of time points
(3) f(x;t), the intensity function of image sequences of T time points
Output: (1) s
t
(x), the function of maximum score from all tracks that terminates
at x at time t
(2) b
t
(x), the trace back function that indicates the location of track
(corresponding to s
t
(x)) at time t¡1
1: for all x in X do
2: s
1
(x) Ã f(x;1); f In discrete space case, s
t
(x) is an n dimensional array. Its
elements are to be determined in this algorithm. Same is for b
t
(x).g
3: end for
4: for t=2 to T do
5: for all x in X do
6: s
t
(x)Ãmax
y2X
[s
t¡1
(y)+f(x;t)¡wkx¡yk];
7: b
t
(x)Ãargmax
y2X
[s
t¡1
(y)+f(x;t)¡wkx¡yk];
8: end for
9: end for
Algorithm 2 DPTraceback
Input: (1) s
t
(x), the function of maximum score from all tracks that terminates
at x at time t
(2) b
t
(x), the trace back function that indicates the location of track
(corresponding to s
t
(x)) at time t¡1
Output: x
1
;:::;x
T
, the locations in the track that achieves maximum score s
T
1: x
T
Ãargmax
x2X
s
T
(x)
2: for t=T ¡1 to 1 do
3: x
t
Ãb
t+1
(x
t+1
)
4: end for
indicate the true positions of the objects. Green curve indicates the inferred positions of
a track using the dynamic programming approach. The objects move up and down, and
the video proceeds from left to right.
In the ¯rst experiment, we assume n = 1. That is, the object is moving in an one
dimensional discrete space. Then each image can be represented as a column vector
(of length 200). They are put together from left to right according to order in time.
The images contain two objects moving over time of length 200. The objects start from
13
¯xed positions, and their movements are modeled as discretized Brownian motion, i.e. a
normal distribution N(0;2). The true intensity of the object is 0.5, and the noise follows
a normal distribution of N(0:2;0:2).
Because the dynamic programming approach does not assume any movement model,
it can be used to track more complicated movements. In the second experiment, addition
tothe¯rstexperiment,wefurtheraddaconstantshift¡0:2tothemotion,i.e. themotion
follows N(¡0:1;2). In the third experiment, we assume there is a constant acceleration
¡0:003, i.e. the motion follows N(¡0:003t;2). In all the three experiments, the tracking
algorithm correctly inferred one of the tracks.
To obtain another track, it is possible to trace back from the position of the second
best score. However, in practice, in general we don't know the number of objects. Also,
objects may emerge or disappear during image recording.
Figure 3.1: Examples of tracking two moving objects in a noisy video.
3.3.1 Tracking accuracy
Performance measure: we use the Root Mean Squared Error (RMSE) to measure the
tracking accuracy [SNH
+
05]. This measure is de¯ned as RMSE(^ x) =
pP
t
k^ x
t
¡x
t
k
2
,
14
which is the di®erence between the estimated track ^ x and true track x. Here the expec-
tation is simply approximated by averaging di®erent realization across time.
Averageperformanceon di®erentparametersand con¯gurations: Starting
from the setting of the experiment 1 in Figure 3.1, we simulated the data tracking by 1)
using di®erent norm (speci¯ed by di®erent powers) in the dynamical programming track
estimation algorithm, 2) using di®erent weight w, 3) adding di®erent amount of noise in
the images, 4) di®erent variance in the Brownian motion.
For each con¯guration, the simulation and tracking are repeated 100 times and the
average RMSE is calculated. Since there are two objects in the model but only one track
is estimated, for each simulation, the smaller of the two RMSEs are chosen for averaging.
The results are summarized in Figure 3.2. Note that the scales of di®erent plots are
di®erent.
It can be seen from the ¯gure, generally, 1) the tracking is insensitive to the choice
of norms, 2) the smaller the magnitude of weight, the more accurate the tracking; 3) the
tracking performance is similar when the level of noise (indicated by mean and standard
deviation of the normal distribution of noise) is less than 0.3, but decreases quickly when
noise level increases from 0.3; 4) the faster the movement of objects (indicated by the
standard deviation of Brownian motion), the higher the tracking errors.
Selection of the optimal track: In the experiment 1 of Figure 3.1, only one of
the two tracks is selected. To measure the contribution of noise to the choice of one track
against another, we ¯xed the object tracks as in the experiment, and repeated tracking
of the videos with di®erent instances of noise for 1000 times. We found the lower track
is selected with a frequency of 39.8%, the upper one is with frequency 60.2%.
15
1 2 3 4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Different norm
Power of norm
Average RMSE
−0.2 0 0.2 0.4 0.6 0.8 1 1.2
0
10
20
30
40
50
60
Different weight
Weight
Average RMSE
−0.2 0 0.2 0.4 0.6 0.8 1 1.2
0
5
10
15
20
25
30
Different amont of noise
Standard deviation and mean of noise distibution
Average RMSE
−1 0 1 2 3 4 5 6
0
2
4
6
8
10
12
14
16
18
20
Different Standard deviation of Brownian motion
Standard deviation of Brownian motion
Average RMSE
Figure 3.2: Performance analysis of di®erent settings.
16
The reason on why there is a slightly higher chance for the upper track be selected is
probably due to the shapes of the two tracks: the upper track has a smaller overall drift
than the lower track. In such case, the noise nearby the track, if realized in a high value,
can be included into the estimated track to reduce the cost of displacement, therefore
generating a higher dynamic programming score.
Todemonstratethise®ect,werepeatedtheaboveexperimentonanewsetofvideos(of
200 pixels in size and 50 time points) containing two objects. The moving displacements
of both objects are 1 between two consecutive frames. But one object keep change
moving direction, therefore following a zigzag track. While the other object does not
change direction, therefore following a linear track. Figure 3.3 shows a noise free video
that consists these two tracks. Without the presence of noise, the two tracks would
result in the same dynamic programming scores. However, the noise provided positive
contributions to those tracks with small drift. Our experiment shows the frequency of
the two tracks being selected are 68.8% and 31.2% respectively.
17
Figure 3.3: Simulated noise-free video that consists of one zigzag track and one linear
track.
18
Chapter 4
Tracking of multiple vesicles
4.1 Track elimination and enumeration
The dynamic programming approach can only generate track of one object. To track
multiple objects, in practice, one can generate the estimation of multiple tracks by repet-
itively eliminating the signal along the track inferred by dynamical programming. The
following MultiTrack algorithm gives an example of generating multiple tracks.
Algorithm 3 MultiTrack
Input: (1) n
trk
, the number of tracks to enumerate
(2) X, the set of all locations
(3) T, total number of time points
(4) f(x;t), the intensity function of image sequences of T time points
Output: (x
(1)
1
;:::;x
(1)
T
);:::;(x
(n
trk
)
1
;:::;x
(n
trk
)
T
), the best tracks
1: for i=1 to n
trk
do
2: (s;b)ÃDPScoring(X;T;f)
3: (x
(1)
1
;:::;x
(1)
T
)ÃDPTraceback(s;b)
4: f ÃTrackElimination(X;T;f;(x
(1)
1
;:::;x
(1)
T
))
5: end for
The MultiTrack calls TrackElimination algorithm to eliminate a track in the video.
It is described as follows.
19
Algorithm 4 TrackElimination
Input: (1) X, the set of all locations
(2) T, total number of time points
(3) f(x;t), the intensity function of image sequences of T time points
(4) (x
1
;:::;x
T
), one track
Output: f(x;t), the intensity function of image sequences with eliminated track
1: for t=1 to T do
2: x
0
Ã a random location in X
3: f(x
t
;t)Ãf(x
0
;t)
4: end for
For simpli¯cation, the above algorithm only stops when a ¯xed number of n
trk
is
obtained. A more rigorous stopping criterion may be obtained by comparing the scor-
ing function s
T
obtained from true video against s
T
from a permutated video through
randomly shu²ing its pixels.
In addition, the above TrackElimination algorithm assumes the object is only of one
pixel. In practice, one object on the track may occupy a small consecutive region instead
of just one pixel, we can assume that the track is the trajectory of the center of the
object. To remove this object, we can approximate the object using a Gaussian function
and subtract the values of this function from the image. To be more speci¯c, denote
the gaussian function as f
t
(x) = a
t
exp(¡(x¡x
t
)
T
§
¡1
t
(x¡x
t
)), where x
t
are obtained
from trace back information, but a
t
and §
t
are to be estimated so that f
t
would have
best least square ¯t to the image. McKenna et al. used similar idea for the modeling
of object intensities for tracking [MRG99], where they model the pixel intensities of an
object as random variables that follow bivariate Gaussian distribution. They then used
expectation maximization to estimate the mean and covariance matrix of the Gaussian
distribution, obtaining the best approximation of the object.
20
4.1.1 Experimental results
An example of applying TrackElimination to the simulation generated in Section 3.3 is
shown in Figure 4.1.
Figure 4.1: Left: Simulated video used in Section 3.3 (same as Figure 3.1 without
estimated track). Right: the video with the best track eliminated.
The accuracy of second track after eliminating ¯rst track: We use the fol-
lowing steps to test the tracking performance of the second object by following steps: 1)
simulatevideosusingthesamemodelasinSection 3.3; 2)usealgorithmsDPScoringand
DPTraceback to obtain the track of one object; 3) use TrackElimination to eliminate the
track from the video; 4) use DPScoring and DPTraceback to get the second estimated
track. We repeated the above steps 1000 times. We ¯nd the RMSE of the ¯rst track-
ing from step 3 is 2:29§0:89, and the average RMSE of the second tracking from step
4 is 2:30§0:55. These two RMSEs are very comparable. We conclude that the track
elimination strategy can successfully estimate tracks of multiple objects.
21
4.2 Improving track associations using Kalman ¯lter
In this section, we propose to improve the tracking accuracy by incorporating Kalman
¯lter into dynamic programming framework.
4.2.1 Motivation
In practice, when a video contains multiple moving objects, the trajectories of these
objects can often get very close or even cross each other. In this case, the previous
dynamicalprogrammingapproachmayinferatrackthatisactuallyamixtureofmultiple
real tracks. Figure 4.2 shows such an example, resulting from a video of 100 pixels and
the100timepoints. Inthisexample, therealtrack(indicatedbyredlines)oftwoobjects
are two cross straight lines plus small random displacements following discritized normal
distribution. The green curve corresponds to the estimated track. It can be seen in this
¯gure that the estimated track is actually a combination of ¯rst part of one track and
the second part of another track. To avoid such problem, we modify the score function
in Equation 3.1 to incorporate Kalman ¯lter which provides estimate of current object
state.
4.2.2 Introduction to Kalman ¯lter
Kalman ¯lters are based on linear dynamical systems in discrete time domain. Let the
state of the system at time t be represented as a real vectors
t
. The Kalman ¯lter model
assumes the true state at time t is evolved from the state at (t¡1) according to
s
t
=F
t
s
t¡1
+B
t
u
t
+w
t
(4.1)
22
Figure 4.2: Wrong tracking of two objects due to cross of their real tracks.
where
² F
t
is the state transition operator applied to the previous state s
t¡1
;
² B
t
is the control-input operator applied to the external control vector u
t
;
² w
t
is the process noise. It is assumed to be sampled from a zero mean multivariate
normal distribution with covariance Q
t
.
At time t an indirect measurement z
t
of the true state s
t
is observed according to
z
t
=H
t
s
t
+v
t
(4.2)
where H
t
is the operator that transforms the true state space into the observed space
and v
t
is the observation noise. The noise is assumed to be zero mean Gaussian white
noise with covariance R
t
.
v
t
»N(0;R
t
) (4.3)
23
In addition, the initial state, and the noise vectors at each step fs
0
;w
1
;:::;w
t
;v
1
:::v
t
g
are all assumed to be mutually independent.
The Kalman ¯lter method consists of two phases: predict and update. In the predict
phase, the current state estimate is generated using previous state estimates:
² Predicted state ^ s
tjt¡1
=F
t
^ s
t¡1jt¡1
+B
t¡1
u
t¡1
² Predicted estimate covariance P
tjt¡1
=F
t
P
t¡1jt¡1
F
T
t
+Q
t¡1
Intheupdatephase,thecurrentlyobservedmeasurementinformationisusedtore¯ne
the prediction:
² Innovation or measurement residual ~ y
t
=z
t
¡H
t
^ s
tjt¡1
² Innovation (or residual) covariance S
t
=H
t
P
tjt¡1
H
T
t
+R
t
² Optimal Kalman gain K
t
=P
tjt¡1
H
T
t
S
¡1
t
² Updated state estimate ^ s
tjt
=^ s
tjt¡1
+K
t
~ y
t
² Updated estimate covariance P
tjt
=(I¡K
t
H
t
)P
tjt¡1
4.2.3 Improving tracking by incorporating Kalman ¯lter
Generalidea: WeincorporateKalman¯lterintoourdynamicprogrammingframework
as follows: Let z
t
be the vector of observed state of an object, which is usually a combi-
nation of the object's location, velocity etc. Assumez
t
is observable at all time. Kalman
¯lter can provide an estimation ^ z
tjt¡1
of the object state at time t, given observations up
to time t¡1. Supposez
t
is of length m, Equation 4.4 gives a modi¯ed scoring function,
24
s
T
=
T
X
t=1
f(z
t
;t)¡
m
X
i=1
[w
i
(
T
X
t=2
kz
(i)
t
¡^ z
(i)
tjt¡1
k)] (4.4)
where z
(i)
t
is the ith element of z
t
, and f(z
t
;t) only uses the location elements of z
t
.
Designofthedynamicmodel: Weproposeasimpledesignofthedynamicsmodelfor
the construction of the Kalman ¯lter for video of one dimensional images. In this case,
the true location x
t
is a scalar x
t
. Assume there is no control on the objects, so we have
B
t
=0 andu
t
=0. Also, assumeF,H,R, andQ are time invariant. We de¯ne the state
vector as location and velocity of a vesicle.
s
t
=
2
6
6
4
x
t
_ x
t
3
7
7
5
(4.5)
We assume that between the t¡1 and t timestep the vesicle undergoes a constant
acceleration of a
t
that is normally distributed, with mean 0 and standard deviation ¾
a
.
Assuming object motion follows Newton's laws, we have
s
t
=Fs
t¡1
+Ga
t
(4.6)
where
F=
2
6
6
4
1 ¢t
0 1
3
7
7
5
(4.7)
and
25
G=
2
6
6
4
¢t
2
2
¢t
3
7
7
5
(4.8)
with ¢t=1. We ¯nd that
Q=cov(Ga)=E[(Ga)(Ga)
T
]=GE[a
2
]G
T
=G[¾
2
a
]G
T
=¾
2
a
GG
T
(4.9)
Ateachtimestep,anoisymeasurementofthetruepositionofthevesicleismade. Assume
the noise is also normally distributed, with mean 0 and standard deviation ¾
z
.
z
t
=H s
t
+v
t
(4.10)
where
H=
2
6
6
4
1 0
0 0
3
7
7
5
(4.11)
and
R=E[v
t
v
T
t
]=
·
¾
2
z
¸
(4.12)
In dynamic programming, we assume to know the initial starting state of the vesicle with
perfect precision, so we initialize
^ s
0j0
=
2
6
6
4
x
0
_ x
0
3
7
7
5
(4.13)
26
and to tell the ¯lter that we do not know the exact position and speed, we give it a zero
covariance matrix:
P
0j0
=
2
6
6
4
B 0
0 B
3
7
7
5
(4.14)
with some large number B. The ¯lter will then prefer the information from the ¯rst
measurements over the information already in the model. Given this above dynamic
model,wemodifythealgorithmDPScoringbyreplacingthedisplacementwiththeobject
position predicted from Kalman ¯lter, as in Equation 4.4, where
^ z
tjt¡1
=
2
6
6
4
^ z
tjt¡1
0
3
7
7
5
=H
t
^ s
tjt¡1
(4.15)
4.2.4 Combining dynamic programming and point detector
We compare the above integrating Kalman ¯lter with dynamic programming approach
withtheapproachthatdirectlyusesKalman¯ltertoassociatetheobjectstatesestimated
by object detectors. Since in our case, the protein vesicles are small, we regard them as
points. So we use point detector. A simple point detector that is robust to the noise
in the images is based on Gaussian ¯lter. Let G(x;§) be a Gaussian function with
covariance matrix §, the Gaussian ¯ltering is the convolution of an image (at time t) and
the Gaussian function. g(x;t) =
R
f(y;t)G(x¡y;§)dy. After applying the Gaussian
¯lter, the noise in the image are reduced. Then the pixels of g(x;t) with high intensities
may correspond to objects.
27
Given the detected possible locations of a object at time 0, we can then use greedy
approach through Kalman ¯lter to associate the locations to form a track, as shown in
the following algorithm.
Algorithm 5 DectorKalman
Input: (1) X, the set of all locations
(2) T, total number of time points
(3) g(z;t), the Gaussian ¯ltered intensity function of image sequences of
T time points
(4) F;G;Q;H;R , the parameters for the dynamic model
Output: y
1
;:::;y
T
, the locations in the estimated track
1: y
1
Ãargmax
a2X
g(a;t);
2: for t=2 to T do
3: calculate ^ z
tjt¡1
using Kalman ¯lter
4: y
t
Ãargmax
a2X
[g(a;t)¡wk^ z
tjt¡1
¡ak];
5: end for
4.2.5 Experimental results
Inourexperiment,weassumetheproteinvesiclesareundersmallamountofacceleration,
therefor we choose ¾
a
= 0:01. We assume ¾
z
= 1 and B = 1. We set ±t = 1. Given
these parameters, we tested the above new scoring method on the example data shown
in Figure 4.2. Figure 4.3 shows that the method correctly ¯nds one of the two tracks.
Given the same true tracks as in the above example, we simulated 1000 noisy videos
and compare the performance between tracking using 1) dynamic programming with
Kalman¯lter2)dynamicprogrammingwithoutusingKalman¯lterand3)pointdetector
with Kalman ¯lter (with ¾ =1 for Gaussian ¯ltering). The ¯rst method gives an RMSE
of 2:9§ 2:1, the second method gives RMSE of 8:3§ 5:8 and the third method gives
17:6§ 14:7, suggesting that, compared to the pure dynamic programming approach,
the integration of Kalman ¯lter with the dynamic programming greatly reduced the
28
Figure 4.3: Overcoming the wrong association problem by using Kalman ¯lter.
association mistake induced by the cross of two tracks. On the other hand, use Kalman
¯lter alone substantially rely on the accuracy of the point detector. However, in our case,
the objects are very small, and the noise is very strong, it is very hard to detect the
objects from single images even after ¯ltering. So the third approach resulted in very
inaccurate tracking.
29
Chapter 5
Conclusion and discussion
Automatic tracking of protein vesicle's movements is key to qualitative analysis of the
dynamics of these vesicles. The main challenge of such tracking is that the video data
is very noisy and the vesicles are very small. In this thesis, after providing an overview
of the ¯eld of object tracking and their application to the tracking of molecules in cells,
we studied the tracking of single and multiple vesicles using dynamic programming and
Kalman ¯lter based approaches. Our experiments on simulation data show that dynamic
programmingapproachcanachievehightrackingaccuracyforsinglevesicletrackingeven
there are high levels of noise in the video, and the integration of Kalman ¯lter further
signi¯cantly increased tracking accuracy by in the case of tracking of multiple vesicles.
Due to the complexity of the vesicle movements, many issues in such tracking remain
to be explored. For example, all methods used in this thesis assume the existence of
the vesicles in all video frames. In real videos, the vesicles could emerge or disappear in
some frames. The vesicles may also split or merge. Therefore, more complex association
methods like Multiple Hypothesis Testing or Particle Filter may be used to handle this
situation.
30
References
[ABL
+
02] B. Alberts, D. Bray, J. Lewis, M. Ra®, K. Roberts, and J.D. Watson. Molec-
ular Biology of the Cell. New York, 2002.
[BB82] D.H. Ballard and C.M. Brown. Computer Vision. Prentice Hall Professional
Technical Reference, 1982.
[BC86] TJ Broida and R. Chellappa. Estimation of object motion parameters from
noisyimages. Pattern Analysis and Machine Intelligence, IEEE Transactions
on, 8(1):90{99, 1986.
[CA91] Y.L. Chang and JK Aggarwal. 3D structure reconstruction from an ego mo-
tion sequence usingstatistical estimation and detection theory. In Visual Mo-
tion, 1991., Proceedings of the IEEE Workshop on, pages 268{273, 1991.
[CLFK01] R.T. Collins, A.J. Lipton, H. Fujiyoshi, and T. Kanade. Algorithms for coop-
erative multisensor surveillance. Proceedings of the IEEE, 89(10):1456{1477,
2001.
[CLRS01] T.H. Corman, C.E. Leiserson, R.L. Rivest, and C. Stein. Introduction to
Algorithms. MIT Press and McGraw-Hill, New York, USA, 2001.
[CM99] D.ComaniciuandP.Meer. Meanshiftanalysisandapplications. InThe Pro-
ceedings of the Seventh IEEE International Conference on Computer Vision,
1999, volume 2, 1999.
[CRM03] D. Comaniciu, V. Ramesh, and P. Meer. Kernel-Based Object Tracking.
PatternAnalysisandMachineIntelligence,IEEETransactionson,pages564{
577, 2003.
[EDHD02] A. Elgammal, R. Duraiswami, D. Harwood, and L.S. Davis. Background and
foregroundmodelingusingnonparametrickerneldensityestimationforvisual
surveillance. Proceedings of the IEEE, 90(7):1151{1163, 2002.
[ETC98] GJ Edwards, CJ Taylor, and TF Cootes. Interpreting face images using
activeappearancemodels. InAutomatic Face and Gesture Recognition, 1998.
Proceedings. Third IEEE International Conference on, pages 300{305, 1998.
[GK95] L. Grewe and A.C. Kak. Interactive learning of a multiple-attribute hash
tableclassi¯erforfastobjectrecognition. CVIU: Computer Vision and Image
Understanding, 61(3):387{416, 1995.
31
[GLW
+
07] WJ Godinez, M. Lampe, S. Worz, B. Muller, R. Eils, and K. Rohr. Tracking
of Virus Particles in Time-Lapse Fluorescence Microscopy Image Sequences.
In Biomedical Imaging: From Nano to Macro, 2007. ISBI 2007. 4th IEEE
International Symposium on, pages 256{259. Springer, 2007.
[HHD00] I. Haritaoglu, D. Harwood, and L.S. Davis. W4: Real-Time Surveillance
of People and Their Activities. Pattern Analysis and Machine Intelligence,
IEEE Transactrions on, pages 809{830, 2000.
[KGH
+
07] M. Kockx, D.L. Guo, T. Huby, P. Lesnik, J. Kay, T. Sabaretnam, E. Jary,
M. Hill, K. Gaus, J. Chapman, et al. Secretion of Apolipoprotein E From
MacrophagesOccursviaaProteinKinaseAandCalcium-DependentPathway
Along the Microtubule Network. Circulation Research, 101(6):607, 2007.
[Kuh55] HWKuhn. Thehungarianmethodforsolvingtheassignmentproblem. Naval
Research Logistics Quarterly, 2:83{97, 1955.
[KWT88] M. Kass, A. Witkin, and D. Terzopoulos. Snakes: Active contour models.
International Journal of Computer Vision, 1(4):321{331, 1988.
[Mac98] DJC MacKay. Introduction to Monte Carlo methods. Learning in Graphical
Models, pages 175{204, 1998.
[MP97] B. Moghaddam and A. Pentland. Probabilistic visual learning for object rep-
resentation. Pattern Analysis and Machine Intelligence, IEEE Transactions
on, 19(7):696{710, 1997.
[MRG99] S.J. McKenna, Y. Raja, and S. Gong. Tracking colour objects using adaptive
mixture models. Image and Vision Computing, 17(3-4):225{231, 1999.
[MS02] K. Mikolajczyk and C. Schmid. An A±ne Invariant Interest Point Detector.
Lecture Notes in Computer Science, pages 128{142, 2002.
[POP98] CP Papageorgiou, M. Oren, and T. Poggio. A general framework for object
detection. In Computer Vision, 1998. Sixth International Conference on,
pages 555{562, 1998.
[RBK98] HA Rowley, S. Baluja, and T. Kanade. Neural network-based face detection.
Pattern Analysis and Machine Intelligence, IEEE Transactions on, 20(1):23{
38, 1998.
[Rei79] D. Reid. An algorithm for tracking multiple targets. Automatic Control,
IEEE Transactions on, 24(6):843{854, 1979.
[SDG
+
07] I. Smal, K. Draegestein, N. Galjart, W. Niessen, and E. Meijering. Rao-
Blackwellized Marginal Particle Filtering for Multiple Object Tracking in
Molecular Bioimaging. Lecture Notes in Computer Science, 4584:110, 2007.
[SJ87] IK Sethi and R. Jain. Finding trajectories of feature points in a monocular
image sequence. Pattern Analysis and Machine Intelligence, IEEE Transac-
tions on, 9(1):56{73, 1987.
32
[SK05] IF Sbalzarini and P. Koumoutsakos. Feature point tracking and trajectory
analysis for video imaging in cell biology. Journal of Structural Biology,
151(2):182{195, 2005.
[SM00] J. Shi and J. Malik. Normalized Cuts and Image Segmentation. Pattern
Analysis and Machine Intelligence, IEEE Transactrions on, pages 888{905,
2000.
[SMVG04] D. Serby, EK Meier, and L. Van Gool. Probabilistic object tracking using
multiple features. In Pattern Recognition, 2004. ICPR 2004. Proceedings of
the 17th International Conference on, volume 2, 2004.
[SNH
+
05] D. Sage, FR Neumann, F. Hediger, SM Gasser, and M. Unser. Auto-
matic Tracking of Individual Fluorescence Particles: Application to the
Study of Chromosome Dynamics. Image Processing, IEEE Transactions on,
14(9):1372{1383, 2005.
[SS90] V. Salari and IK Sethi. Feature point correspondence in the presence of
occlusion. Pattern Analysis and Machine Intelligence, IEEE Transactions
on, 12(1):87{91, 1990.
[SS05] K. Sha¯que and M. Shah. A Noniterative Greedy Algorithm for Multiframe
Point Correspondence. Pattern Analysis and Machine Intelligence, IEEE
Transactions on, pages 51{65, 2005.
[VJS05] P. Viola, M.J. Jones, and D. Snow. Detecting Pedestrians Using Patterns
of Motion and Appearance. International Journal of Computer Vision,
63(2):153{161, 2005.
[VRB01] C.J.Veenman,M.J.T.Reinders,andE.Backer. ResolvingMotionCorrespon-
denceforDenselyMovingPoints. Pattern Analysis and Machine Intelligence,
IEEE Transactions on, pages 54{72, 2001.
[YBFK08] J.W. Yoon, A. Bruckbauer, W.J. Fitzgerald, and D. Klenerman. Bayesian
Inference for Improved Single Molecule Fluorescence Tracking. Biophysical
Journal, 94(12):4932, 2008.
[YJS06] A. Yilmaz, O. Javed, and M. Shah. Object tracking: A survey. ACM Com-
puting Surveys (CSUR), 38(4), 2006.
[YLS04] A. Yilmaz, X. Li, and M. Shah. Contour-Based Object Tracking with Oc-
clusion Handling in Video Acquired Using Mobile Cameras. Pattern Analysis
and Machine Intelligence, IEEE Transactions on, pages 1531{1536, 2004.
[ZY96] S.C. Zhuand A. Yuille. Region competition: unifying snakes, region growing,
and Bayes/MDLfor multiband image segmentation. Pattern Analysis and
Machine Intelligence, IEEE Transactions on, 18(9):884{900, 1996.
33

Abstract (if available)

Abstract With the advance of fluorescence imaging technologies, recently cell biologists are able to record the movement of protein vesicles within a living cell. Automatic tracking of the movements of these vesicles become key for qualitative analysis of dynamics of theses vesicles. In this thesis, we formulate such tracking problem as video object tracking problem, and design a dynamic programming method for tracking single object. Our experiments on simulation data show that the method can identify a track with high accuracy which is robust to the choose of tracking parameters and presence of high level noise. We then extend this method to the tracking multiple objects using the track elimination strategy. In multiple object tracking, the above approach often fails to correctly identify a track when two tracks cross. We solve this problem by incorporating the Kalman filter into the dynamic programming framework. Our experiments on simulated data show that the tracking accuracy is significantly improved.

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Asset Metadata

Creator Xu, Min (author)

Core Title Automatic tracking of protein vesicles

School College of Letters, Arts and Sciences

Degree Master of Arts

Degree Program Applied Mathematics

Publication Date 04/21/2009

Defense Date 03/06/2009

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

Tag automatic tracking,cell biology,dynamical programming,Kalman filters,OAI-PMH Harvest,protein transportation

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Lototsky, Sergey V. (committee chair), Arnold, Donald B. (committee member), Mikulevicius, Remigijus (committee member)

Creator Email mxu@usc.edu,xumin100@hotmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m2098

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