algorithms for tracking on the manifold of...
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ALGORITHMS FOR TRACKING ON THE MANIFOLD OF SYMMETRIC POSITIVEDEFINITE MATRICES
By
GUANG CHENG
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF THE PHILOSOPHY
UNIVERSITY OF FLORIDA
2012
c⃝ 2012 Guang Cheng
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To my wife Yandi, my daughter Lerong, and my parents
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ACKNOWLEDGMENTS
I would like to gratefully thank Dr. Baba Vemuri my major advisor, for his insightful
guidance, unflinching patience and encouragement throughout my PhD study. This
dissertation would not have been written without his guidance and support. I would
also like to thank my committee members, Dr. Jeffrey Ho, Dr. Anand Rangarajan, Dr.
Arunava Banerjee and Dr. Brett Presnell, not only for agreeing to be my committee,
but also for being always supportive during the entire academic program and the broad
exposure I gained through their course offerings.
I thank the generous research support provided by the NIH grants NS066340
and EB007082 to my advisor, Dr. Vemuri, that made it possible for me to have an
uninterrupted RAship during the course of my PhD I also have received travel grants
from the CISE department at the University of Florida.
I thank Dr. Dena Howland, Dr. John Forder, Dr. Min-Sig Hwang and Dr. Sarah
E. Mondello for providing the data and sharing the knowledge. I thank Dr. Bing Jian,
Dr. Angelos Barmpoutis and Dr. Santhosh Kodipaka for the productive discussions
and their help on the project. I thank all my lab-mates Yuchen Xie, Ting Chen, Meizhu
Liu, Wenxing Ye, Dohyung Seo, Sile Hu, Yuanxiang Wang, Yan Deng, Hesamodin
Salehian, Qi Deng and Theodore Ha, for all the helps they gave me. Special thanks are
extended to Hesamodin Salehian for his great work in our collaboration, especially the
experiments in the chapter of recursive Karcher expectation estimator.
I would like to thank my wife Yandi, for her patience support and love. I also thank
my parents, Kexin and Qixin, for the faith in me and allowing me to be what I want to be.
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TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1 Recursive Karcher Expectation Estimator . . . . . . . . . . . . . . 131.2.2 Intrinsic Recursive Filter . . . . . . . . . . . . . . . . . . . . . . . . 141.2.3 Intrinsic Unscented Kalman Filter . . . . . . . . . . . . . . . . . . . 15
1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 RIEMANNIAN GEOMETRY ON Pn . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 GL-invariant metric vs. Euclidean metric on Pn . . . . . . . . . . . . . . . 192.2 Log-Euclidean vs GL-invariance . . . . . . . . . . . . . . . . . . . . . . . 192.3 Algorithms on the Field of SPD Matrices . . . . . . . . . . . . . . . . . . . 20
3 RECURSIVE KARCHER EXPECTATION ESTIMATION . . . . . . . . . . . . . 22
3.1 Background and Previous Work . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1 The Recursive Karcher Expectation Estimator . . . . . . . . . . . . 233.2.2 Recursive form of the symmetrized KL-divergence mean . . . . . . 263.2.3 Recursive mean for the Log-Euclidean Metric . . . . . . . . . . . . 27
3.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.1 Performance of the Recursive Estimators . . . . . . . . . . . . . . 283.3.2 Application to DTI Segmentation . . . . . . . . . . . . . . . . . . . 29
4 INTRINSIC RECURSIVE FILTER ON Pn . . . . . . . . . . . . . . . . . . . . . . 34
4.1 Background and Previous Work . . . . . . . . . . . . . . . . . . . . . . . . 344.2 IRF: A New Dynamic Tracking Model on Pn . . . . . . . . . . . . . . . . . 36
4.2.1 Generalization of the Normal Distribution to Pn . . . . . . . . . . . 364.2.1.1 The mean and the variance of the generalized normal
distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.2 The Probabilistic Dynamic Model on Pn . . . . . . . . . . . . . . . 46
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4.3 IRF-based Tracking Algorithm on Pn . . . . . . . . . . . . . . . . . . . . . 464.3.1 The Bayesian Tracking Framework . . . . . . . . . . . . . . . . . . 464.3.2 The Tracking Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.4.1 The Synthetic Data Experiment . . . . . . . . . . . . . . . . . . . . 504.4.2 The Real Data Experiment . . . . . . . . . . . . . . . . . . . . . . . 52
5 INTRINSIC UNSCENTED KALMAN FILTER . . . . . . . . . . . . . . . . . . . 56
5.1 Background and Previous Work . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Intrinsic Unscented Kalman Filter for Diffusion Tensors . . . . . . . . . . . 57
5.2.1 The State Transition and Observation Models . . . . . . . . . . . . 575.2.2 The Intrinsic Unscented Kalman Filter . . . . . . . . . . . . . . . . 59
5.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6 ATLAS CONSTRUCTION FOR HARDI DATASET REPRESENTED BY GAUSSIANMIXTURE FIELDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.1 Background and Previous Work . . . . . . . . . . . . . . . . . . . . . . . . 646.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.2.1 Image Atlas Construction Framework . . . . . . . . . . . . . . . . . 656.2.2 L2 Distance and Re-orientation for GMs . . . . . . . . . . . . . . . 676.2.3 Mean GMF Computation . . . . . . . . . . . . . . . . . . . . . . . . 68
6.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.3.1 Synthetic Data Experiments . . . . . . . . . . . . . . . . . . . . . . 696.3.2 Real Data Experiments . . . . . . . . . . . . . . . . . . . . . . . . 71
7 DISCUSSION AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . 73
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
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LIST OF TABLES
Table page
3-1 Time (in seconds) for mean computation in the DTI segmentation on syntheticdataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3-2 Timing in seconds for segmentation of grey matter in a rat spinal cord . . . . . 32
4-1 Tracking result for the real data experiment. . . . . . . . . . . . . . . . . . . . . 53
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LIST OF FIGURES
Figure page
3-1 Accuracy and speed comparisons of the recursive versus non-recursive meancomputation algorithms for data on P3 . . . . . . . . . . . . . . . . . . . . . . . 29
3-2 Results for the DTI segmentation experiments on the synthetic dataset . . . . . 31
3-3 Segmentation results of grey matter in a rat spinal cord for 6 different methods. 32
3-4 Segmentation results of the molecular layer in a rat hippocampus for 3 differentmethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4-1 Mean estimation error from 20 trials for the synthetic data experiment. . . . . . 52
4-2 Head tracking result for video sequences with moving camera. . . . . . . . . . 55
5-1 Fiber tracking results on real datasets from rat spinal cords. c⃝[2012] IEEE . . 62
5-2 Biomarkers captured by computing density map for each fiber bundle. c⃝[2012]IEEE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6-1 Image registration results on synthetic dataset. c⃝[2011] IEEE . . . . . . . . . 70
6-2 Registration results for real dataset from rat spinal cord. c⃝[2011] IEEE . . . . 72
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LIST OF ABBREVIATIONS
DTI diffusion tensor imaging
DWMRI diffusion weighted magnetic resonance imaging
GMF Gaussian mixture field
HARDI high angular resolution diffusion imaging
RKEE recursive Karcher expectation estimator
IRF intrinsic recursive filter
IUKF intrinsic unscented Kalman filter
MRI magnetic resonance imaging
SPD symmetric positive definite
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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of the Philosophy
ALGORITHMS FOR TRACKING ON THE MANIFOLD OF SYMMETRIC POSITIVEDEFINITE MATRICES
By
Guang Cheng
May 2012
Chair: Baba C. VemuriMajor: Computer Engineering
The problem of tracking on the manifold of n × n symmetric positive definite (SPD)
matrices is an important problem and has many applications in several areas such as
computer vision and medical imaging. The aim of this dissertation is to develop novel
tracking algorithms on Pn for several different applications.
One of the basic tracking problems on Pn is to recursively estimate the Karcher
expectation – an generalization of the expectation to the Riemannian manifold,
which can be viewed as tracking a static system. In this dissertation, we proposed
a novel recursive Karcher expectation estimator (RKEE), and we further proved its
unbiasedness and L2-convergence to the Karcher expectation under symmetric
distribution on Pn. Synthetic experiments showed RKEE the similar accuracy as the
Karcher mean but more efficient for sequential data. We then developed a fast DTI
(diffusion tensor imaging) segmentation algorithm based RKEE. The experiments on
the real data of rat spinal cord and rat brain with comparison to Karcher mean and other
type of centres based algorithms demonstrated the accuracy and efficiency of RKEE.
To further tackle the dynamic system tracking on Pn, we studied and discovered
several properties of the generalized Gaussian distribution on Pn, based on which
a novel probabilistic dynamic model is proposed in conjunction with an intrinsic
recursive filter for tracking a time sequence of SPD matrix measurements in a Bayesian
framework. This newly developed filtering method can then be used for the covariance
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descriptor updating problem in covariance tracking, leading to new efficient video
tracking algorithms. To show the the accuracy and efficiency of our covariance tracker
in comparison to the state-of-the-art, we present synthetic experiments on Pn, and real
data experiments for tracking in video sequences.
To handle the non-Pn inputs and the non-linear observation model, a novel intrinsic
unscented Kalman filter tracking points on Pn is presented. With the combination of
the stream line tracking strategy, an efficient fiber tracking method is proposed to track
white matter fibers from diffusion weighted (DW) MR images of mammalian brains
specifically, human and rats. Different from the first method, the input of filter could be
the diffusion weighted MR signal, which makes it possible to track fibers directly without
the pre-process step commonly required by existing methods. Real data experiments on
data sets of human brain and rat spinal cords are presented and depicted the accuracy
and efficiency of the method. For group-wise analysis of the white matter fiber bundles
from our tracking algorithm, a novel group-wise registration and atlas construction
algorithm for the DW MR datasets represented by Gaussian mixture fields is proposed
and applied to the spinal cord dataset. The group-wise analysis result of the spinal cord
fiber bundle in this dissertation showed the significant difference between injured and
healthy rats.
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CHAPTER 1INTRODUCTION
1.1 Motivation
Tracking in general is a task to recursively estimate the current system state based
on a sequential dataset. It is very important task both computer vision and medical
imaging. In computer vision, tracking is crucial for video surveillance, augmented reality,
human-computer interaction, etc. It is also a necessary preprocessing step for high
level computer vision tasks such as visual scene analysis. In medical imaging, tracking
is very useful in not only analysing time sequences such as cardiac cycle in medical
imaging but also in neural fiber tractography from diffusion weight (DW) MRI dataset.
Tracking is traditionally a time series analysis problem, it is closely related to prediction.
Prediction has a wide clinical applications, such as disease prediction. Both tracking
and prediction take time sequences as data input, and are usually based on dynamic
models. Many tracking methods, such as the well known Kalman filter, are based on the
predict-update framework, where prediction is a crucial part in the tracking algorithm.
The main difference between tracking and prediction is that, in tracking we estimate the
current state of a certain process based on the current and previous observations, while
in prediction the estimation is for future state where no direct observation is available.
Most classical tracking techniques are in Euclidean space. However, in certain
applications, the problems might not naturally be in the Euclidean space. Instead,
they usually lie on a Riemannian manifold, but not in a vector space. Also, the input
data dimension in modern problems is usually huge. Many linear and non-linear
dimensional reduction techniques are hence used to find the meaningful lower
dimensional representations of the data, and these representations might not be in
the Euclidean space. Therefore, tracking and prediction algorithms on the Riemannian
manifold could be applied in many practical problems.
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This dissertation focuses the tracking problem in the space of n × nsymmetric
positive definite (SPD) matrices represented as Pn. Many feature descriptors such as
covariance matrices, Cauchy deformation tensors, diffusion tensors, metric tensors,
etc, can be represented in Pn. Thus algorithms on Pn could be widely applied to
practical problems in different areas such as computer vision, medical imaging, etc.
Pn is known to be a Riemannian space with non-positive curvature. Many researches
have reported on different problems on Pn such as the computation of intrinsic/extrinsic
mean, linear/non-linear dimensional reduction, statistics, etc with applications in many
different areas. This dissertation is primarily motivated by practical tracking problems
such as video tracking and other problems where tracking algorithms can be applied
including segmentation of DTI (diffusion tensor imaging) dataset and fiber tractography.
1.2 Main Contributions
1.2.1 Recursive Karcher Expectation Estimator
Finding the mean of a population of SPD matrices/tensors is an often encountered
problem in medical image analysis and computer vision, specifically in diffusion MRI
processing, tensor-based morphometry, texture analysis using the structure tensor
etc. The mean tensor can be used to represent a population of structure tensors in
texture analysis, diffusion tensors in diffusion tensor image (DTI) segmentation or for
interpolation of diffusion tensors or in clustering applications. A mean is usually used
as a good estimator of the expectation. If the data samples are given sequentially, the
mean finding problem can also be viewed as a tracking problem for a static process.
It is well known that computation of the mean can be posed as a minimization
problem in which one minimizes the sum of squared distances between the unknown
mean and the members of the set whose mean is being sought. Mathematically
speaking, we want to find, µ∗ = minµ∑n
i d2(xi ,µ), where, d is the chosen distance,
xi are the data samples whose mean is being sought and µ∗ is the mean. Depending on
the definition of distance d , one gets different kinds of means. For example, if we choose
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the Euclidean distance for d , we get the arithmetic mean, where as if we chose the
L1-norm instead of the L2 norm in the above formula, we get the median. If we choose
the geodesic distance in the domain of xi , we get the Karcher mean [38].
Currently, there are no existing closed form solution for Karcher mean computation
on Pn for more than two sample points [54]. A gradient based optimization algorithm [54]
is used in practice which is known to be inefficient. In this dissertation, we propose
a novel recursive Karcher expectation estimator (RKEE), which is an algorithm to
recursively estimate the Karcher expectation. The proof of the unbiasedness and
L2-convergence of RKEE under any symmetric distributions on Pn is also presented.
Synthetic data experiments showed the similar accuracy of RKEE and Karcher mean as
an estimator of Karcher expectation, but RKEE is more efficient especially for sequential
dataset. Further we applied RKEE to DTI segmentation problem and compared with
Karcher mean and other centres on real dataset of rat spinal cords and rat brains.
1.2.2 Intrinsic Recursive Filter
In recent years, the covariance region descriptor which is the covariance matrix
of the feature vectors at each pixel in the region, are shown to be robust and efficient
in video tracking and detections [61]. Several works [43, 75, 76, 83, 83, 84, 84] were
reported to address the problem of updating covariance descriptor in video tracking.
Here, a novel probabilistic dynamic model on Pn based on geometry and probability
theory is presented. The noisy state and observations are described by matrix-variate
random variables whose distribution is a generalized normal distribution to based on the
GL-invariant measure. Then an novel intrinsic recursive filter(IRF) on Pn is developed
based on the dynamic model, and applied to covariance tracking, which forms an real
time video tracking algorithm. Synthetic and real data experiments are presented to
support the effectiveness and the efficiency of the proposed algorithm.
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1.2.3 Intrinsic Unscented Kalman Filter
Diffusion-Weighted MR Imaging (DW-MRI) is a unique non-invasive technique that
can locally infer the imaged tissue structure in vivo by MR signal that is sensitive to the
water molecule diffusion. The DW-MRI dataset is then a 3D image that contains tissue
(such as brain white matter) directional information at each of its voxel. This directional
information at each voxel for single fiber case can be modelled by a 2nd order positive
definite tensor (SPD matrix) which is the classical diffusion tensor image (DTI). High
order models such as multi-tensor, high order tensor, etc have be reported in order to
handle more complex cases, e.g. fiber crossing. However, the 3 × 3 SPD matrix which
can be viewed as a point in P3 is still a very useful descriptor in representing the local
fiber information. To further visualize and analyse the tissue structure, the fiber tracking
technique is needed to reconstruct the imaged tissue which is very important in both
research and clinical applications in neuro science. The fiber tractography is formulated
as a tracking problem in Pn.
Here we present a novel intrinsic unscented Kalman filter (IUKF) on Pn, which to
the best of our knowledge is the first extension of the unscented Kalman filter to Pn. We
apply this filter to both estimate and track the tensors in multi-tensor model using the
intrinsic formulation to achieve as demonstrated through experiments. We perform real
demonstrate the accuracy and efficiency of our method. Also, a group-wise registration
and atlas construction method developed to register DW-MR datasets represented by
Gaussian Mixture Fields is proposed for group fiber analysis.
1.3 Outline
The remaining chapters are organized as follows: The basic properties of Pn and
common Riemannian manifolds can be found in Chapter 2. The RKEE and application
to DTI segmentation is introduced in Chapter 3, followed by the IRF on the space of
SPD matrices and its applications to covariance tracking is discussed in Chapter 4;
The Fiber tracking with intrinsic unscented Kalman filter is presented in Chapter 5; This
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is followed by the atlas construction method developed for the group fiber analysis in
Chapter 6. And finally the conclusion can be found in Chapter 7. A large part of this
thesis has been published in several papers [18, 20].
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CHAPTER 2RIEMANNIAN GEOMETRY ON PN
In this chapter we introduce the basic concepts of Riemannian geometry on Pn, and
refer the reader to [34, 53, 70] for details. Pn is the space of n × n symmetric positive
definite (SPD) matrices, which is a Riemannian manifold. It can be identified with the
quotient space O(n) \ GL(n) [70], where GL(n) denotes the General Linear group – the
group of (n × n) non-singular matrices, and O(n) is the orthogonal group – the group of
(n × n) orthogonal matrices. This makes Pn to be a homogeneous space with GL(n) as
the group that acts on it and the group action defined for any X ∈ Pn by X[g] = gXgt .
One can now define GL-invariant quantities such as the GL-invariant inner product
based on the group action defined above. We will now begin with inner product in the
tangent space of Pn. For tangent vectors U and V ∈ TXPn (the tangent space at point X,
which is the space of symmetric matrices of dimension (n + 1)n/2 and a vector space)
the GL invariant inner product is defined as ∀g ∈ GL(n), < U,V >X=< gUgt ,gVgt >gXgt .
On Pn this GL invariant inner product takes the form,
< U,V >X= tr(X−1/2UX−1VX−1/2). (2–1)
With metric/inner product defined on the manifold, the length of any curve in Pn, γ :
[0, 1] → Pn is defined as length(γ)2 =∫ 1
0< _γ, _γ >γ(t) dt. The distance between
∀X,Y ∈ Pn is defined as the length of the shortest curve between X and Y (Geodesic
distance). With the GL-invariant metric, the distance between X,Y ∈ Pn is
dist(X,Y)2 = tr(log2(X−1Y)) (2–2)
where log is the matrix log operator. Since this distance is induced from the GL-invariant
metric in Equation 2–1, this distance is naturally GL-invariant i.e.
dist2(X,Y) = dist2(gXgt ,gYgt) (2–3)
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With GL-invariant metric defined on Pn, the intrinsic or Karcher mean of a set of
elements Xi ∈ Pn can be computed by performing the following minimization:
µ∗ = argminµ∑i
dist2(Xi ,µ) (2–4)
using a gradient based technique, where the update equation in each iteration is
µnew = Expµold(α
∑i Logµold
(Xi)
N) (2–5)
where α is the step size, and Expµold() and Logµold
() are the Log and Expential maps at
point µold ∈ Pn.
The Log and Exponential maps [34] are very useful tools on the Riemannian
manifold. The Exponential map denoted as ExpX(·), where X ∈ Pn, maps a vector rooted
at the origin of the tangent space TXPn to a geodesic emanating from X. The Log map
(LogX(·)) is the inverse of the Exponential map. The Exponential and Log map on Pn are
given by:
ExpX(V) = X1/2 exp(X−1/2VX−1/2)X1/2
LogX(Y) = X1/2 log(X−1/2YX−1/2)X1/2
(2–6)
where X,Y ∈ Pn, V ∈ TXPn, and log and exp denote the matrix exp and log operators.
The Karcher mean can be viewed as an extension of the arithmetic mean from the
Euclidean space to the Riemannian manifold. Similarly, the expectation and the variance
can also be extended. Given a random variable M ∈ Pn with a probability density P(M)
E(M) = argminµ
∫Pn
dist(µ,X)2P(X)[dX] (2–7)
Var(M) =
∫Pn
dist(E(M),X)2P(X)[dX] (2–8)
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2.1 GL-invariant metric vs. Euclidean metric on Pn
There are two primary theoretical reasons for the choice of a GL-invariant metric
over the conventional Euclidean metric when doing operations on Pn and since our
dynamic model is on Pn, this is highly relevant.
Firstly, Pn is an open subset of the corresponding Euclidean space R(n+1)n/2, which
implies that Pn would be incomplete with a Euclidean metric since its possible to find
a Cauchy sequence which might not converge for this case. This implies that for some
of the optimization problems set in Pn, the optimum can not be achieved inside Pn.
This in turn means that the covariance updates could lead to matrices that are not
covariance matrices, an unacceptable situation in practice. This problem will not arise
when using the GL-invariant metric, since the space of Pn is geodesically complete with
a GL-invariant metric [70].
Secondly, in general, the feature vectors might contain variables from different
sources, e.g. object position, object color etc. In this case, a normalization of the
(in general) unknown scales of different variables would be necessary when using
the Euclidean distance, which is non trivial and may lead to use of ad hoc methods.
However, with a GL invariant metric, this scaling issue does not arise since, the
presence of different scales for the elements of a feature vector from which the
covariance matrix is constructed, is equivalent to multiplication of the covariance
matrix with a positive definite diagonal matrix. This operation is a GL group operation
and since GL-invariance implies invariance to GL group operations, the scaling issue is
a non issue when using a GL-invariant metric.
2.2 Log-Euclidean vs GL-invariance
Log-Euclidean defined in [3] is a framework that induces the metric from the
Euclidean space to the Riemannian manifold (called Log-Euclidean metric) through
the Log map at an arbitrarily chosen point on the manifold. From this definition we can
say that, in general for Pn (n > 1), the Log-Euclidean metric is not intrinsic. Moreover,
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it is not GL-invariant and is dependent on the aforementioned arbitrarily chosen point.
Hence, it is not a natural metric.
A typical Log-Euclidean operation is a three step procedure. In the first step, all
the data points on the manifold are projected to the tangent space at an arbitrarily
chosen point, usually the identity, through the Log map. Then, standard vector space
operations can be applied in the tangent space. In the last step, the result of vector
space operations which lie in the tangent space are projected back to the manifold via
the Exponential map. If one were to use the Log-Euclidean operations to compute the
intrinsic/Karcher mean of a given population of data on Pn, the result will not be the true
Karcher mean.
Log-Euclidean operations have been used in covariance tracking in [75], wherein,
the base point is arbitrarily chosen in the first frame and iteratively updated for
subsequent frames using a predefined and constant state-transition matrix. Hence,
the base point will never converge to any meaningful statistic of the dataset. Because
the Log-Euclidean framework is used to approximate the GL-invariant metric using the
Log-Euclidean metric, the approximation error will affect the tracking result, as shown in
the Section 4.4.1.
2.3 Algorithms on the Field of SPD Matrices
A field of SPD matrices is a map from 2D or 3D Euclidean space to Pn. In the
discrete case, it can be viewed as an image(volume) where the image value at each
pixel(voxel) is a SPD matrix. This field of SPD matrices is also referred as a tensor field.
The GL-invariance property is usually required for algorithms on tensor fields. This
is because in many applications, e.g. metric tensor field, deformation tensor field, etc,
the tensor values at each pixel(voxel) is directly related to the local coordinate system of
the image lattice in such a way that when ever the image is deformed, the tensor values
should changed linearly according to the Jacobian of the transformation. Assuming the
transformation is T on the image lattice, and the tensor value is I (x) = D at point x
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before the transformation. After the transformation, the tensor value would be
I (T (x)) = JT (x)DJT (x)t (2–9)
where JT (x) is the Jacobian of T at point x. (Note that in DTI (diffusion tensor
image) registration Equation 2–9 is called re-transformation [78] which is one of the
re-orientation strategies in DTI registration [19]). So operations such as interpolation
and dissimilarity measurements on these tensor are required to have the GL-invariant
property such that they can be preserved before and after the deformation. One
example is the DTI segmentation. The segmentation result could not guaranteed to be
the same before and after affine transformation if the distance used is not GL-invariant
[78].
21
CHAPTER 3RECURSIVE KARCHER EXPECTATION ESTIMATION
3.1 Background and Previous Work
Tomes of research has been published on finding the mean tensor using different
kinds of distances/divergences and has been applied to DTI as well as structure
tensor field segmentation, interpolation and clustering. In [79], authors generalized the
geometric active contour model-based piece-wise constant segmentation [17, 72] to
segmentation of DTIs using the Euclidean distance to measure the distance between
two SPD tensors. Authors in [27], present a geometric active contour [16, 49] based
approach for tensor field segmentation that used information from the diffusion tensors
to construct the so called structure tensor which is a sum of structure tensors formed
from each component of the diffusion tensor. A Riemannian metric on the manifold of
SPD matrices was used in [32, 40, 77] and [11, 54, 59] for DTI segmentation and for
computing the mean interpolant of diffusion tensors respectively. In [78, 80, 91] and
[54] the symmetrized KL-divergence was used for DTI segmentation and interpolation
respectively. The Log-Euclidean distance was introduced to simplify the computations
on the manifold of SPD matrices and this was achieved by using the principal Log-map
from the manifold to its tangent space and then using the Euclidean metric on the
Log-mapped matrices in the tangent space at the identity [4]. More recently, in [77],
a statistically robust measure called the total Bregman divergence (tBD) family was
introduced and used for interpolation as well as DTI segmentation.
None of the above methods for computing the mean of SPD matrices which are
used within the segmentation algorithms or in their own right for interpolation purposes
are in recursive form. A recursive formulation would be more desirable as it would
yield a computationally efficient algorithm for computing the means of regions in the
segmentation application. Also, in many applications such as DTI segmentation,
clustering and atlas construction, data are incrementally supplied to the algorithm for
22
classification or assimilation for updation of the mean and an algorithm that recursively
updates the mean rather than one that recomputes the mean in a batch mode would
be much more efficient and desirable. In this dissertation, we pursue this very task
of recursive mean computation. The key contributions are: (i) first, we present novel
theoretical results proving the L2-convergence of the recursive intrinsic Karcher
expectation computation of a set of SPD matrices to the true Karcher expectation. (ii)
Additionally, we present recursive formulations for computing the mean using commonly
used distance/divergence measures mentioned above and present experiments that
depict significant gains in compute time over their non-recursive counterparts. (iii) We
present synthetic and real data experiments depicting gains in compute time for DTI
segmentation using these recursive algorithms.
The rest of the section is organized as follows: in Section 3.2 we present novel
theoretical results leading to the intrinsic Karcher expectation computation algorithm. In
addition, we present the recursive formulations for the commonly used symmetrized
KL-divergence based mean computation as well as the Log-Euclidean distance
based mean. Section 3.3 contains synthetic and real data experiments depicting the
improvements in computation time of the DTI segmentation task.
3.2 Methods
3.2.1 The Recursive Karcher Expectation Estimator
We now develop an estimator for the intrinsic (Karcher) expectation that can be
used to represent a set of data points in Pn (the space of diffusion tensors) and can be
computed recursively. This recursive computation property is a very important property
especially for online problems where the data points are provided sequentially. This
is very pertinent to applications such as DTI and structure tensor field segmentation,
diffusion/structure tensor clustering etc.
23
Let Xk ∈ Pn, k = 1, 2, ... be iid samples in Pn from probability measure P(M). The
recursive Karcher expectation estimator can be defined as:
M1 = X1 (3–1)
Mk+1(wk+1) =M12
k (M− 1
2
k Xk+1M− 1
2
k )wk+1M12
k (3–2)
Here we set wk+1 =1
k+1. We now prove the following properties of the recursive Karcher
expectation estimator presented here in the form of theorems with their proofs.
Theorem 1.: Let i.i.d. samples Xk be generated from a density P(X;µ) that is
symmetric w.r.t. to its expectation µ, then Mk is a unbiased estimator. By symmetry we
mean that ∀X ∈ Pn P(X;µ) = P(µX−1µ;µ), note that X,µ,µX−1µ are on the same
geodesic and dist(µ,X) = dist(µ,µX−1µ)
Proof. Without loss generality we assume that µ = I, where I is the identity matrix. Now
we can prove the theorem by induction. For k = 1,
E(M1) = E(X1) = I
where E denotes the Karcher expectation. Pm1(M1; I) is obviously symmetric.
Assuming E(Mk) = I, and the the posterior Pmk(Mk) is symmetric. Then,
Pmk+1(Mk+1) =∫
Pn
Px(Xk+1)Pmk(X
12
k+1(X− 1
2
k+1Mk+1X− 1
2
k+1)wk+1−1X
12
k+1)[dXk+1] =
Pmk+1(M−1
k+1)
since Px , Pmkare symmetric and
(X12
k+1(X− 1
2
k+1Mk+1X− 1
2
k+1)wk+1−1X
12
k+1)−1 = X
− 12
k+1(X12
k+1M−1k+1X
12
k+1)wk+1−1X
− 12
k+1
Thus, Pmk+1is symmetric with respect to I, and E(Mk+1) = I = µ since∫Pn
Log(M)Pmk+1(M)[dM] =
∫Pn
Log(N−1)Pmk+1(N−1)[dN−1] = 0
24
Theorem 2: ∀A,B ∈ Pn, and w ∈ [0, 1]
|Log(A12 (A− 1
2BA− 12 )wA
12 )||2 ≤ tr(((1− w)Log(A) + wLog(B))2) (3–3)
Note that the left side of the inequality is the square distance between the identity matrix
and a geodesic interpolation between A and B, and the right side of the inequality is the
square distance between the identity and the log linear interpolation. This inequality is
true based on the fact that Pn is a space with non-positive sectional curvature.
Proof. Let γ(w) = A12 (A− 1
2BA− 12 )wA
12 . Then γ(w) is a geodesic between A,B Since Pn
is a Hadamard space, based on Lurie’s notes on Hadamard space [46], we know that
lhs = dist(I, γ(w))2 ≤ (1− w)dist(I,A)2 + wdist(I,B)2 − w(1− w)dist(A,B)2 (3–4)
Also
rhs − ((1− w)dist(I,A)2 + wdist(I,B)2 − w(1− w)dist(A,B)2)
= w(1− w)(dist(A,B)2 − tr(LogA− LogB)2) ≥ 0
(3–5)
where the last inequality is based on the Cosine inequality in [6]. Thus, we have proved
that lhs ≤ rhs.
Theorem 3. Let wk = 1k, we then have, Var(Mk) ≤ 1
ku2, where u2 = Var(Xi), i =
1, 2....
Proof. : We can prove this theorem also by induction. We still assume that E(Xk) = I.
When k = 1, Var(M1) = Var(X1) = u2.
25
Assume that the claim is true for k = i , that is Var(Mi) ≤ 1iu2. When k = i + 1, using
the lemma above, we know that,
Var(Mi+1(w)) ≤∫Pn
∫Pn
||(1− w)Log(Mi) + wLog(Xi+1)||2P(Mi)P(Xi+1)[dMi ][dXi+1]
= (1− w)2Var(Mi) + w2Var(Xi+1)
≤ (1− w)21
iu2 + w2u2
(3–6)
(1− w)2 1iu2 + w 2u2 = 1
i+1u2 when w = 1
i+1.
From the theorems above, we can find that the recursive Karcher expectation
estimator is an unbiased estimator for the Karcher expectation when the samples are
drawn from a symmetric distribution on Pn. And it converges in the L2 sense to the
expectation. Of course, this recursive Karcher expectation estimator can be viewed as
an approximation of the Karcher sample mean. However, in our experiments, we find
that it actually has similar accuracy as the Karcher sample mean. Also, because it is a
recursive estimator, it would be far more computationally efficient to use our estimator
than the standard non-recursive Karcher mean algorithm when the diffusion tensors are
input sequentially to the estimation algorithm as in all the aforementioned applications.
3.2.2 Recursive form of the symmetrized KL-divergence mean
We now present a recursive formulation for computing the symmetrized KL-divergence
based mean. Let’s recall, the symmetrized KL divergence also called the J-divergence,
is defined by J(p, q) = 12(KL(p||q)+KL(q||p)) Using the square root of J, one can define
a divergence between two given positive definite tensors. The symmetrized KL (KLs)
divergence based mean of a set of SPD tensors is the minimizer of the sum of squared
KL divergences. This minimization problem has a closed form solution as shown in
Wang et al. [78] and repeated here for convenience,
MKL =√B−1[
√√BA
√B]
√B−1 (3–7)
26
where A = 1N
∑i
Ti is the arithmetic mean, B = 1N
∑i
T−1i is the harmonic mean,
T = {Ti} is the given tensor field and N is the total number of tensors. The closed form
Equation 3–7 can be computed in an recursive manner as follows. Let the arithmetic and
harmonic means at iteration n be denoted by (An) and (Bn), respectively. When a new
∗(n + 1)st tensor, Tn+1 augments the data set, the quantities An and Bn are recursively
updated via the following simple equations,
An+1 =n
n + 1An +
1
n + 1Tn+1and (3–8)
Bn+1 =n
n + 1Bn +
1
n + 1T−1n+1. (3–9)
Using the above recursive form of the arithmetic and harmonic means of a set of tensors
and the closed form expression (Equation 3–7), we can recursively compute the KLs
mean of a set of SPD tensors.
3.2.3 Recursive mean for the Log-Euclidean Metric
We now formulate the recursive form of the Log-Euclidean (LE) based mean. It is
well known that Pn can be diffeomorphically mapped to the Euclidean space using the
matrix Log function, which makes it possible to directly induce the Euclidean metric on
Pn called the Log-Euclidean metric [4]. Then, the Log-Euclidean distance can be defined
as,
DLE(T1,T2) = ||Log(T1)− Log(T2)|| (3–10)
where ||.|| is the Euclidean norm. The LE-mean on a set of SPD matrices, is obtained by
minimizing the sum of the squared LE distances which leads to a closed form solution
MLE = Exp(
n∑i=1
Log(Ti)) (3–11)
This closed form expression can be rewritten in a recursive form for more efficient
computation. Let Mn be the Log-Euclidean mean in the nth iteration. When the (n + 1)st
tensor, say Tn+1 is added, the current mean can be recursively updated using the
27
following equation,
Mn+1 = Exp(n
n + 1Log(Mn) +
1
n + 1Log(Tn+1)). (3–12)
3.3 Experiments
3.3.1 Performance of the Recursive Estimators
To justify the performance of the recursive estimators, we first generate i.i.d.
samples from the Log-normal distribution [64] on P3 with the expectation at the identity
matrix. Then, we input the 100 random samples sequentially to all estimators including
the recursive Karcher expectation estimator (RKEE ), Karcher mean (KM), recursive
KLs mean (RKLS), non-recursive KLs (KLS) mean, recursive Log-Euclidean mean
(RLEM) and the non-recursive Log-Eucildean mean (LEM) respectively. To compare
the accuracy of RKEE and KM, we evaluate the error of the estimator using the
squared distance in Equation 2–2 between the ground truth (the identity matrix) and
the computed estimate. The accuracy test of the remaining algorithms is not included
because for KLs and Log-Euclidean “metrics” , the recursive and non-recursive algo-
rithm will generate the exact same results. Also, the computation time for each step
(each sample) is recorded. For comparison, we have the same settings for all the mean
computation algorithms. We run the experiment 20 times and plot the average error and
the average computation time at each step in Figure 4-1. In Figure 4-1 (a), we see that
the accuracy of computed mean is nearly the same for both the non-recursive Karcher
mean and the recursive Karcher expectation estimators after they are given 10 samples.
The computation time (in CPU seconds on an I-7, 2.8GHZ processor) for the Karcher
mean however increases linearly with the number of steps, while that for the recursive
Karcher expectation estimator is nearly a constant and far less than the non-recursive
case. This means that the recursive Karcher expectation estimator is computationally far
superior especially for large size problems where data is input incrementally, for example
in algorithms for segmentation, clustering, classification and atlas construction. Similar
28
Figure 3-1. Accuracy and speed comparisons of the recursive versus non-recursivemean computation algorithms for data on P3. Figure (a) is the mean error ofthe Karcher mean (red dashed line) and the recursive Karcher expectationestimator (blue solid line) for each step. Figure (b) (c) (d) are thecomparisons of computation time (in seconds) between the recursive (reddashed line) and non-recursive (blue solid line) mean computationalgorithms for different “metrics”. Result for the Riemannian metric (Karchermean) is in Figure (b), KLs in Figure (c), Log-Euclidean in Figure (d).
conclusions can also be drawn in Figure 4-1 (c) and (d), where for sequentially input
data the recursive mean algorithm for KLs divergence and Log-Euclidean mean are
much more efficient than their own batch versions.
3.3.2 Application to DTI Segmentation
In this section, we present results of applying our recursive algorithms to the DTI
segmentation problem. In [78], the classical levelset based segmentation algorithm [17]
was extended to the field of diffusion tensors. In this algorithm, based on a piecewise
constant model, the segmentation procedure became a EM like algorithm, where at
each iteration, the mean tensor is computed over each region and the region boundary
is then evolved based on the mean tensor. In this section, we use this algorithm to
segment DTIs, and plug in different tensor field mean computation techniques for
comparison.
Firstly, experiments on DTI segmentation of synthetic datasets is presented here.
We manually generated an image region of size (64 × 64) which contains two different
kind of tensors (differing in orientation, one vertical and another horizontal). Then DW
MRsignal is generated based on [66] with 5 different level of Riccian noised added to
29
the DWMR signal σ = 0.1, 0.15, 0.2, 0.25, 0.3, where sigma2 is variance of the Gaussian
noise added to the real and image part of the DW MR signal. DTIs are constructed by
using the technique in [9]. Exact same dataset and same setting are used for all six
methods. The initialization curve overlayed on a noisy dataset is depicted in Figure
3-2 (a). To evaluate the segmentation result, the dice coefficient between the ground
truth segmentation and the estimated segmentation are computed. These results
are shown in Figure 3-2 with figure (b) depicting the dice coefficients and figure (c)
showing the comparison of the running times. From the figure (b) we can see that
the segmentation accuracies are very similar for the recursive and non-recursive
methods for the same “distance metric”. For different “distance metric”s, result of the
Riemannian (GL invariant) metric is the most accurate, since the GL invariant metric
is the natural metric on Pn. In Figure 3-2, we can find that segmentation using the
KM takes significantly longer time than other methods, this is because there is no
closed form computation formula for Karcher mean on Pn, and hence the Karcher
mean computation is very time consuming which can also be seen in Table 3-1. The
recursive Karcher expectation estimator is about 2 times faster and has the similar
accuracy. For KLs and Log-Euclidean “metrics”, the time saved by the recursive method
is not so significant as for the GL-invariant metric. This is because, although the mean
computation time for the recursive method is at least one tenth of the non-recursive
method (0.01 versus 0.1 in Table 3-1), the time used for curve evolution is about 1 4
seconds which makes the savings in total segmentation time not significant. From
these results we can find that the recursive Karcher expectation estimator is the most
attractive from an accuracy and efficiency viewpoint. For the real data experiment, the
DTI are estimated [78] from a DW-MR scan of a rat spinal cord. The DWMR data were
acquired using a PGSE with TR = 1.5s , TE = 28.3ms , bandwidth = 35 Khz, 21 diffusion
weighted images with a b-value of 1250s/mm2 were collected. The image size is
128× 128× 10. We used the same initialization for each segmentation. We applied all of
30
Figure 3-2. Results for the DTI segmentation experiments on the synthetic dataset.Figure (a) is the initialization curve overlayed on the synthetic dataset at oneof the noise levels used in the experiments. Figure (b) is the segmentationaccuracy evaluated by the dice coefficient of the segmentations from allmethods at all noise levels. Figure (c) is the total segmentation time (inseconds) for all segmentation methods at all noise levels.
Noise Level RKEE KM RKLS KLS RLEM LEM
0.1 0.01 0.75 0.01 0.12 0.01 0.130.15 0.007 1.02 0.006 0.23 0.01 0.150.2 0.02 1.55 0.01 0.42 0.005 0.22
0.25 0.01 2.22 0.01 0.63 0.006 0.280.3 0.01 4.52 0.008 0.55 0.01 0.45
Table 3-1. Time (in seconds) for mean computation in the DTI segmentation on syntheticdataset
the six methods (recursive and non-recursive for each of the three “distance” measures)
to perform this experiment. In order to compare the time efficiency, we report the whole
segmentation running time, including the total time required to compute the means.
Table 3-2 shows the result of this comparison, from which we can find that it is much
more efficient to use the recursive mean estimator in the segmentation than using the
batch mean estimator. Especially, in the case of the Karcher mean, which has no closed
form formula and takes nearly half of the total reported segmentation time, whereas,
using the recursive Karcher expectation estimator makes the computation much faster,
and also significantly reduces the total segmentation time. The segmentation results
are depicted in Figure 3-3 for each method. Each (3, 3) diffusion tensor in the DTI
data are illustrated as an ellipsoid whose axis directions and lengths correspond to
31
Segmentation Method RKEE KM RKLS KLS RLEM LEM
Mean computation time 0.02 3.56 0.01 0.56 0.01 0.4Total segmentation time 5.09 8.13 3.41 4.41 5.45 5.82
Table 3-2. Timing in seconds for segmentation of grey matter in a rat spinal cord
Figure 3-3. Segmentation results of grey matter in a rat spinal cord for 6 differentmethods. Figure (a) is RKEE based segmentation. Figure (b) issegmentation using the Karcher mean KM. Figure (c) and (d) are results forthe recursive and non-recursive KLs mean estimators respectively. Figure (e)and (f) are results for the recursive and non-recursive Log-Euclidean meanrespectively.
the eigen-vectors and eigen-values respectively. From the figure we can see that the
segmentation results are visually similar to each other, while our recursive Karcher
expectation based method takes much less time which would be very useful in practice.
32
Figure 3-4. Segmentation results of the molecular layer in a rat hippocampus for 3different methods. Figure (a) RKEE based segmentation. (b) Recursive KLsbased segmentation (c) Recursive Log-Euclidean based segmentation.
A second real data set from an isolated rat-hippocampus was used to test the
segmentation algorithms. Figure 3-4 depicts the segmentation of the molecular layer in
the rat hippocampus. For the sake of space, we present only the segmentation results
from the recursive algorithms presented and not the non-recursive counterparts as their
results are visually similar and the key difference is in the time savings.
33
CHAPTER 4INTRINSIC RECURSIVE FILTER ON PN
4.1 Background and Previous Work
Since Pn is a Riemannian manifold, but not a vector space, many operations and
algorithms in Euclidean space can not be applied directly to Pn and this has lead to a
flurry of research activity in the recent past. Several operations in Euclidean space have
been extended to Riemannian manifold. For example, the extension of arithmetic mean
to Riemannian manifold is Karcher Mean [38]; The extension of Principal Component
Analysis (PCA) is the Principle Geodesic Analysis [28, 29]; Mean shift [23] has also
been extended to Riemannian manifolds [69]. However, for filtering operations in
dynamic scenes such as the popular Kalman filter [67], an intrinsic extension does not
exist in literature to date.
Recursive filtering is a technique to reduce the noise in the measurements by
using theory of recursion applied to filtering. It is often used in time sequence data
analysis especially in the tracking problem where the model of the target needs to be
updated based on the measurement and previous tracking results. Many recursive
filtering techniques have been developed in the Euclidean space, such as Kalman filter,
Extended Kalman filter etc, where the inputs and outputs of the filter are all vectors
[37, 67]. However, several tracking problems are naturally set in Pn, a Riemannian
symmetric space [34]. Recent work reported in [61] on covariance tracking uses
a covariance matrix (constructed from pixel-wise features inside the object region)
that belongs to Pn in order to describe the appearance of the target being tracked.
This covariance descriptor has proved to be robust in both video detection [71, 74]
and tracking [21, 39, 43, 44, 60, 61, 75, 83]. The covariance descriptor is a compact
feature representation of the object with relatively low dimension compared to other
appearance models such as s the histogram model in [24]. In [73] an efficient algorithm
for generating covariance descriptors from feature vectors is reported based on the
34
integral image technique, which makes it possible to use covariance descriptors in real
time video tracking and surveillance.
One major challenge in covariance tracking is how to recursively estimate the
covariance template (a covariance descriptor that serves as the target appearance
template) based on the input video frames. In [61] and also in [44, 60, 61] the Karcher
mean of sample covariance descriptors from a fixed number of video frames is used
as the covariance template. This method is based on the natural Riemannian distance
– the GL-invariant distance in Pn. Currently, this Karcher mean can not be computed
in closed form, and the computation is achieved using a gradient based optimization
technique which is inefficient especially when the input contains a large number of
samples. To solve this problem, a Log-Euclidean metric is used in [39, 43], an arithmetic
mean like method is used in [83], and an extension of the optimal filter to Pn was
developed in [75]. However, none of these are intrinsic due to the use of approaches
that are extrinsic to Pn.
Recently, some methods were reported addressing the recursive filtering problem
on Riemannian manifolds other than Pn. For example, the geometric particle filter in
handling 2D affine motions (2-by-2 non-singular matrix) was reported in [39, 42], and
an extension to the Riemannian manifold was developed in [65]. However, since the
covariance descriptor is usually a high dimensional descriptor, e.g. the degrees of
freedom of a 5 × 5 covariance matrix are 15, the number of samples required for the
particle filter would be quite large in this case. Additionally, computing the intrinsic
(Karcher) mean on Pn is computationally expensive for large sample sizes. Thus, using
an intrinsic particle filter to update covariance descriptor would be computationally
expensive for the tracking problem. There are also existing tracking methods on
Grassmann manifolds [22, 68] however, it is non-trivial to extend these to Pn, since
the Grassmann manifolds and Pn have very different geometric properties, e.g.
Grassmann manifolds are compact and have a non-negative sectional curvature
35
when using an invariant Riemannian metric [81], while Pn is non-compact and has
non-positive sectional curvature when using an invariant (to the general linear group
(GL)) Riemannian metric [34].
In this dissertation, we focus on the problem of developing an intrinsic recursive
filter – abbreviated IRF for the rest of this chapter – on Pn. A novel probabilistic dynamic
model on Pn based on Riemannian geometry and probability theory is presented. Here,
the noisy state and observations are described by matrix-variate random variables
whose distribution is a generalized normal distribution to Pn based on the GL-invariant
measure. In [41, 58] authors provide a linear approximation of this distribution for cases
when the variance of the distribution is very small. In contrast, in this dissertation, we
explored several properties of this distribution for the arbitrary variance case. We then
develop the IRF based on this dynamic model and the Bayesian framework described
in [22]. By applying this recursive filter – to achieve covariance tracking – in conjunction
with a particle position tracker [5], we obtain a new efficient real time video tracking
algorithm described in Section 4.3.2. We present experiments with comparisons to
existing state-of-the-art methods and quantitative analysis that support the effectiveness
and efficiency of the proposed algorithm.
The remainder of this chapter is organized as follows: In Section 4.2 we introduce
the probabilistic dynamic model on Pn. Then the IRF and the tracking algorithms are
presented in Section 4.3, followed by the experiments in Section 4.4.
4.2 IRF: A New Dynamic Tracking Model on Pn
4.2.1 Generalization of the Normal Distribution to Pn
To define a probability distribution on a manifold, first we need to define a measure
on the manifold. Here we use the GL-invariant measure [dX] on Pn. GL invariance
here implies ∀g ∈ GL(n) and ∀X ∈ Pn, [dgXgt ] = [dX]. From [70], we know [dX] =
|X|−(n+1)/2∏
1≤i≤j≤n dxij , where xij is the element in the i -th row and j-th column of the
36
SPD matrix X. Also, this measure is consistent with the GL invariant metric defined on
Pn defined earlier and also presented in [58].
Similar to the Karcher mean, the Karcher Expectation for the random variable X in
any Riemannian Manifold M can be defined as the result of the following minimization
problem,
E(X) = argminY∈M
∫M
dist2(X,Y)dµ(X) (4–1)
where µ(X) is the probability measure defined in M. Similarly, the variance can be
defined based on this expectation by,
Var(X) =
∫M
dist2(X,E(X))dµ(X) (4–2)
Note that, in Euclidean space Rm which is also a Riemannian manifold, the Karcher
Expectation is equivalent to the traditional definition of expectation, and the variance
in Equation 4–2 is the trace of the covariance matrix. In Pn, by taking gradient of the
energy function in Equation 4–1 and setting it to zero, we find that the expectation of the
random variable will satisfy the following equation.∫Pn
log(E(X)−1/2XE(X)−1/2)p(X)[dX] = 0 (4–3)
The generalization of the normal distribution to Pn used here can be defined as
follows:
dP(X;M,ω2) = p(X;M,ω2)[dX] =1
Zexp(−dist(X,M)2
2ω2)[dX] (4–4)
where P(·) and p(·) are the probability distribution and density respectively of the
random variable X ∈ Pn, with two parameters M ∈ Pn and ω2 ∈ R+, and Z is the
scalar normalization factor. dist(·) is defined in Equation 2–2. As shown in [58], this
distribution has minimum information given the Karcher mean and variance. That is, in
the absence of any information this distribution would be the best possible assumption
from an information theoretic view point. Also, this distribution is different from the
Log-normal distribution which was used in [64, 75]. Actually, the two distributions have
37
very similar densities, but the density used here is based on GL invariant measure while
Log-normal density is based on the Lebesgue measure in the Euclidean space.
A very important property of the above generalized normal distribution is summarized
in the following theorem.
Theorem 4.1. The normalization factor Z in Equation 4–4 is a finite constant with
respect to parameter M ∈ Pn
The consequence of Theorem 4.1 is that, if the prior and the likelihood are both
based on the generalized normal distribution defined using the GL-invariant measure,
computing the mode of the posterior density can be achieved by minimizing the sum of
squared GL-invariant distances from the unknown expectation to the given samples.
To prove the Theorem 4.1, we need to first prove the following lemma:
Lemma 1.
W =
∫Pn
exp(−Tr(logX logXt)
2ω2)[dX] < ∞
Proof. This lemma indicates that the normalization factor Z is constant, and hence
p(X;M,ω2) is a probability density function on Pn. To prove this lemma, we first
represent X in polar coordinates {λi},R based on the eigen decomposition, X = R�Rt ,
where � = diag(λ1, ...λn),RRt = In×n. From [70] we know that
[dX ] = cn
n∏j=1
λ−(n−1)/2j
∏1≤i<j≤n
|λi − λj |n∏j=1
dλiλi
dR (4–5)
where dR is the invariant measure on the orthogonal group O(n) with∫O(n)
dR = 1, cn
is a constant depending on n, and dλi is the Lebesgue measure in R. With the following
38
change of variables, yi = log(λi), we get
W = cn
∫Rn
exp(−n∑i=1
(1
2ω2y 2i +
n − 1
2yi))
∏1≤i<j≤n
|exp(yi)− exp(yj)|dy
= cn
∫Rn
|∑γ∈Sn
sgn(γ)exp(−1
2
n∑i=1
(y 2i /ω2 + (n − 1− 2γ(i))yi))|dy
≤ cn(2πω2)
n2
∑γ∈Sn
exp(ω2
∑n
i=1 γ(i)2 − ω2n(n − 1)2/4
2) < ∞
(4–6)
where γ is an element of Sn which is the set of all permutations of 0, 1, ..., n − 1, and
sgn(γ) denotes the signature of γ which is 1 or −1 depending on the permutation.
We are now ready to present the proof of Theorem 4.1.
Proof. Assume Z is a function of M, ∀M ∈ Pn denoted by Z(M).
Z(M) =
∫Pn
exp(−dist2(X,M)
2ω2)[dX] (4–7)
Since the GL group action is transitive on Pn, ∀N ∈ Pn, ∃g ∈ GL(n) such that N = gMgt .
Thus,
Z(N) =
∫Pn
exp(−dist2(X, gMgt)
2ω2)[dX] =
∫Pn
exp(−dist2(g−1Xg−t ,M)
2ω2)[dX]
Let Y = g−1Xg−t , so X = gYgt . Substituting this into the above equation we get
Z(N) =
∫Pn
exp(−dist2(Y,M)
2ω2)[dgYgt] =
∫Pn
exp(−dist2(Y,M)
2ω2)[dY] = Z(M)
Thus, ∀M,N ∈ Pn, Z(M) = Z(N). From Lemma 1 we know that Z(I) < ∞, by
substitution as in the above, we obtain the result, Z is finite and constant with respect to
M.
One direct consequence of Theorem 4.1 is the following corollary.
Corollary 1. Given a set of i.i.d samples {Xi} drawn from the distribution dP(X;M,ω2),
the MLE of the parameter M is the Karcher mean of all samples.
39
Proof.
−log(p(X1,X2, ....Xm;M,ω2)) = −∑i
log(p(Xi;M,ω2))
= nlogZ +
∑i dist
2(Xi,M)
2ω2
Since Z is constant with respect to M as proved the Theorem 4.1, we have
argmaxMp(X1,X2, ....Xm;M,ω2) = argminM∑i
dist2(Xi ,M)
Thus, MLE of the parameter M of the distribution dP(X;M) equals to the Karcher mean
of samples.
From Theorem 4.1 we know that the normalization factor Z in Equation 4–4 is a
function of ω. The integral in Equation 4–6 is non-trivial, and currently no exact solution
is available for arbitrary n. For n = 2 we can have,
Z2(ω) = 2c2
∫ ∞
−∞
∫ y1
−∞exp(−
2∑i=1
(1
2ω2y 2i +
1
2yi))(exp(y1)− exp(y2))dy2dy1
=√2πc2ωexp(
1
4ω2)
∫ ∞
−∞(exp(−(y1 − 0.5ω2)2
2ω2)(1 + erf (
y1 + 0.5ω2
√2ω2
))
−exp(−(y1 + 0.5ω2)2
2ω2)(1 + erf (
y1 − 0.5ω2
√2ω2
))))dy1
= 4πc2ω2exp(
1
4ω2)erf (
ω
2)
(4–8)
where erf (x) = 2√π
∫ x
0exp(−t2)dt is the error function.
4.2.1.1 The mean and the variance of the generalized normal distribution
Similar to the normal distribution in Euclidean space, the mean and the variance
of the generalized normal distribution on Pn in Equation 4–4 are controlled by the
parameters M and ω2 respectively. The relation between M and dP(X;M,ω2) is given
by the following theorem.
Theorem 4.2. M is the Karcher Expectation of the generalized normal distribution
dP(X;M,ω2).
40
Proof. To prove this, we need to show that dP(X;M,ω2) satisfies Equation 4–3. Let
=
∫Pn
log(M−1/2XM−1/2)dP(X;M,ω2)
then in the integral, using a change of variable, X to Y = MX−1M (X = MY−1M).
Since Pn is a symmetric space and the metric/measure is GL-invariant we know that
[dX] = [dY], and dist(X,M) = dist(Y,M). Thus we have,
=
∫Pn
log(M−1/2XM−1/2)1
Zexp(−dist2(X,M)
2ω2)[dX]
=
∫Pn
log(M1/2Y−1M1/2)1
Zexp(−dist2(Y,M)
2ω2)[dY] = − = 0.
Since Pn has non-positive curvature, the solution of Equation 4–1 is unique [38]. Thus M
is the Karcher Expectation of dP(X;M,ω2).
The variance of dP(X;M,ω2) is controlled by the parameter ω2. Unlike the
multi-variate normal distribution in the Euclidean space, where the Karcher variance(Equation 4–2)
is equal to nω2, the relation between the variance and ω2 of the generalized normal
distribution is much more complex. Without loss of generality we assume X ∈
Pn is a matrix-valued random variable from dP(X; I,ω2). The variance Var(X) =
1Z
∫Pn
||log(X)||2exp(− ||log(X)||22ω2 )[dX]. As in Equation 4–6, by using the Polar coordinates
and taking log of the eigen values, we can get
Var(X) = ω2Varq(y) (4–9)
where y is a random variable in Rn having the distribution with density function,
q(y) =1
z(ω)exp(−1
2
∑i
y 2i )∏
1≤i<j≤n
2|sinh(ω(yi − yj)
2)|, (4–10)
where z(ω) is the normalization factor. Currently there are no analytic solutions for
Varq(y) for arbitrary n. When n = 2 we can compute Var(y) using the similar technique
41
as in Equation4–8
Varq(y) =ω√
πexp(14ω2)erf (ω
2)+ 2(1 +
ω2
4) (4–11)
From Equation 4–11 we can find that, in P2 when ω is closed to zero, Var(X) ≈ 3ω2,
and when ω is large Var(X) ≈ ω4
2. This is because Pn can be locally approximated by an
Euclidean space. When ω is closed to zero, the major portion of the distribution would
be in a small region in Pn, where the Euclidean approximation is relatively accurate.
Hence, Var(X) is proportional to ω2 which is similar to the normal distribution in the
Euclidean space. When ω2 is not closed to zero, the the Euclidean approximation is no
longer accurate, and the Var(X) becomes a complicated function of ω2. This property
has been used to get the approximation of the generalized normal distribution with small
ω2 in [41, 58].
The following two theorems show that the above stated approximations will still be
satisfied for n > 2.
Theorem 4.3.
limω→0
Var(X)
ω2=
n(n + 1)
2(4–12)
Proof. Let
v(y,ω) =∑γ∈Sn
sgn(γ)exp(−1
2
n∑i=1
(y 2i + ω(n − 1− 2γ(i))yi)) (4–13)
where γ, Sn and sgn(γ) are related to the permutation of 0, 1, ..., n − 1 which is defined
to be the same as in Equation 4–6. Also we can find that q(y) = |v(y,ω)|z(ω)
and z(ω) =∫Rn |v(y,ω)|dy. The Taylor expansion of v(y,ω) up to n(n−1)
2-th order with respect to ω
around zero is
v(y,ω) =∑γ∈Sn
sgn(γ)
n(n−1)2∑
k=0
(−ω)k
k!exp(−
∑ni=1 y
2i
2)(
n∑i=1
(n − 1
2− γ(i))yi)
k +O(ωn2−n+2
2 )
= C(−ω)n(n−1)
2 exp(−∑n
i=1 y2i
2)
∏1≤i<j≤n
(yi − yj) +O(ωn2−n+2
2 )
(4–14)
42
where C is a constant. The Equation 4–14 above used the fact that given n non-negative
integers κi , and∑n
i=1 κi ≤n(n−1)
2,
∑γ
sgn(γ)n∏i=1
γ(i)κi = 0 if {κi} /∈ Sn (4–15)
So, in the Taylor expansion all the terms with degree less than n(n−1)2
are zeros. In then(n−1)
2-th order terms, only terms with powers in Sn will be non-zero.
Let the density q(y) = 1zexp(−
∑ni=1 y
2i
2)∏
1≤i<j≤n |yi − yj |, which is exactly the joint
distribution of the eigen values of a Gaussian Orthogonal Ensemble [52] which is a
symmetric random matrix with each of its elements being independent random variables
drawn from a zero mean Gaussian. In this case, the variance of the diagonal elements
in the random matrix is 1 and that of the off diagonal elements is 12. Recall that we are
now in polar coordinates. By transforming q to the Cartesian coordinates of the space of
symmetric matrices we get
Varq(y) =1
z
∫Rn
ytyq(y)dy
= (2π)−n(n−1)
4
∫A∈Sym(n)
tr(A2)exp(−tr(A2)
2)dA =
n(n + 1)
2
(4–16)
where Sym(n) is the space of n × n symmetric matrices, and dA is the Lebesgue
measure in Sym(n).
From above we know that,
limω→0
Var(X)
ω2= lim
ω→0Varq(y) = Varq(y) =
n(n + 1)
2(4–17)
Note that this theorem could also be got using the approximation of the generalized
normal distribution with small ω2 in [41, 58]. Further more, from the proof above we
can get that since the Log-Normal is a projection of a normal distribution from the
tangent space (can be identified with Sym(n)) to Pn, and here the random variable y
43
is the normalized log of the eigen values of X, we can see that when ω is close to
zero the generalized normal distribution can be approximated by a Log-Normal
distribution.
Theorem 4.4.
limω→∞
Var(x)
ω4=
(n3 − n)
12(4–18)
Proof. We first define the upper bound and lower bound on q(y).
qu(y) =1
zu(ω)
∑γ
exp(−1
2
n∑i=1
(y 2i + ω(n − 1− 2γ(i))yi))
qι(y) =1
zι(ω)exp(−1
2
∑i
y 2i )∏
1≤i<j≤n
2(cosh(ω(yi − yj)
2)− 1)
(4–19)
with zu and zι being the normalization factors respectively. Note that both qu and qι
are both Gaussian mixtures. In qu all mixing weights are positive, while in qι there are
negative weights. After expansion we have,
qι(y) =1
zι(ω)
∑β∈Bn
wβexp(−1
2
n∑i=1
(yi + ω(n − 1− 2β(i))/2)2)
wβ = αβexp(ω2
∑n
i=1 β(i)2 − ω2(n(n − 1)2/4)
2)
(4–20)
where Bn is the set of all possible power combinations of polynomial
∑β∈Bn
αβ
n∏i=1
xβ(i)i =
∏1≤i<j≤n
(xi + xj − 2√xixj) (4–21)
and αβ are the coefficients. We can prove that
maxβ∈Bn
n∑i=1
β(i)2 =n∑i=1
γ(i)2 =(2n − 1)(n2 − n)
6(4–22)
The maximum can be achieved only at β ∈ Sn, and αβ = 1, ∀β ∈ Sn.
44
From the definition we can compute the normalization constants and the variances
of qι and qu in a closed form.
zu = (2π)n/2∑γ
exp(ω2
∑n
i=1 γ(i)2 − ω2n(n − 1)2/4
2)
= (2π)n/2n!exp(ω2n(n2 − 1)
24)
zι = (2π)n/2∑β∈Bn
αβexp(ω2
∑n
i=1 β(i)2 − ω2n(n − 1)2/4
2)
= zu + (2π)n/2∑
β∈Bn\Sn
αβexp(ω2
∑n
i=1 β(i)2 − ω2n(n − 1)2/4
2)
Varqu(y) = n +
n3 − n
12ω2
Varqι(y) = n − n(n − 1)2
4ω2
+(2π)n/2
zιω2
∑β∈Bn
exp(ω2
∑n
i=1 β(i)2 − ω2n(n − 1)2/4
2)(
n∑i=1
β(i)2)
(4–23)
Since 0 ≤ cosh(x) − 1 ≤ |sinh(x)| and |∑
i xi | ≤∑
i |xi |, we have ∀y ∈ Rn,
zιqι(y) ≤ zq(y) < zuqu(y) and also zι ≤ z < zu. We can then get the following bounds for
Varq(y).zι
zuVarqι(y) ≤ Varq(y) ≤
zu
zιVarqu
(y) (4–24)
From Equation 4–23 we can show that,
limω→∞
zι
zu= lim
ω→∞(1 +
∑β∈Bn\Sn
αβexp(ω2
2(
n∑i=1
β(i)2 − (2n − 1)(n2 − n)
6)) = 1, (4–25)
because ∀β ∈ Bn \ Sn,∑
i β(i)2 < (2n−1)(n2−n)
6. Similarly,
limω→∞
Varqι(y)
ω2= −n(n − 1)2
4
+ limω→∞
∑β∈Bn
αβexp(ω2
2(
n∑i=1
β(i)2 − (2n − 1)(n2 − n)
6))(
n∑i=1
β(i)2)
=n3 − n
12= lim
ω→∞
Varqu(y)
ω2= lim
ω→∞
Varq(y)
ω2= lim
ω→∞
Var(X)
ω4
(4–26)
45
4.2.2 The Probabilistic Dynamic Model on Pn
To perform tracking on Pn, obviously the observation Yk and the state Xk at time k
∈ Pn respectively. The state transition model and the observation model can then be
defined as
p(Xk |Xk−1) =1
Zs
exp(−dist2(Xk ,gXk−1gt)
2ω2)
p(Yk |Xk) =1
Zo
exp(−dist2(Yk ,hXkht)
2ϕ2)
where g,h ∈ GL(n). ω2,ϕ2 > 0 are the parameters that control the variance of the state
transition and the observation noise. The above two densities are both based on the
GL invariant measure on Pn, unlike in [64, 75] where they are based on the Lebesgue
measure. What does this imply? The key implication of this is that the normalization
factor in the densities is a constant for the GL invariant measure and not so for the
Lebesgue measure case. If the normalization factor was not a constant, one does not
have a valid density.
4.3 IRF-based Tracking Algorithm on Pn
4.3.1 The Bayesian Tracking Framework
For simplicity we use the Bayesian tracking framework described in [22]. The
tracking problem can be viewed as, given a time sequence of observations Ys =
{Y1,Y2, ......,Ys} from time 1 to time s, how can one compute the state Xs at time s? To
solve this problem, first we make two assumptions: (1) The state transition is Markovian,
i.e., the state Xs depends only on Xs−1, or say
p(Xs |Xs−1,Ys−1) = p(Xs |Xs−1) (4–27)
(2) The observation Ys is dependent only on the state Xs at the current time point s, in
other words,
p(Ys |Xs ,Ys−1) = p(Ys |Xs) (4–28)
46
And hence, p(Xs |Xs−1) is called the state transition model and p(Ys |Xs) is called the
observation model.
The goal of tracking can thus be viewed as computing the posterior p(Xs |Ys). First
we have
p(Xs |Ys) = p(Xs ,Ys)/p(Ys) ∝ p(Xs ,Ys) (4–29)
And also,
p(Xs ,Ys) = p(Ys |Xs)p(Xs |Xs−1)p(Xs−1,Ys−1)
=
s∏k=1
p(Yk |Xk)p(Xk |Xk−1).
So, if Xk−2,Xk−3, ...,X0 have already been computed, we can then compute Xk , Xk−1 by
solving the following optimization problem:
Xk , Xk−1 = argmaxXk ,Xk−1
k∏j=k−1
p(Yj |Xj)p(Xj |Xj−1)
= argminXk ,Xk−1Ek(Xk ,Xk−1)
where
Ek(Xk ,Xk−1) = ϕ−2dist2(h−1Ykh−t,Xk) + ω−2dist2(gXk−1g
t,Xk)+
ϕ−2dist2(h−1Yk−1h−t ,Xk−1) + ω−2dist2(Xk−1,gXk−2g
t)
This problem can be solved by using the gradient descent method on Pn, where at
each step we compute the gradient which lies in the tangent space, and get the new
state by moving along the geodesic in the corresponding direction. At the i -th iteration
step, ∂Ek
∂X(i)k
∈ TPX(i)k
, and ∂Ek
∂X(i)k−1
∈ TPX(i)k−1
. Therefore, X(i+1)k = Exp
X(i)k
(δ ∂Ek
∂X(i)k
) and,
X(i+1)k−1 = Exp
X(i)k−1
(δ ∂Ek
∂X(i)k−1
), where δ is the step size and Exp(.) is the Exponential map as
47
defined in Chapter 2. The gradient is given by,
∂Ek
∂X(i)k
= ϕ−2LogXk(h−1Ykh
−t) + ω−2LogXk(gXk−1g
t)
∂Ek
∂X(i)k−1
= ϕ−2LogXk−1(h−1Yk−1h
−t)
+ω−2LogXk−1(g−1Xkg
−t) + ω−2LogXk−1(gXk−2g
t)
It is easy to show that, the state update here is an estimation of the mode of
the posterior p(Xs |Ys), which is different from the usual Kalman filter and particle
filter methods, where the state update is the mean of the posterior p(Xs |Ys). In the
proposed update process, the covariance of the posterior is not necessary for updating
the state. We do not provide an update of the covariance here, partly because the
covariance update is hard to compute for this distribution on Pn. Actually, there’s no
existing closed form solution for the covariance matrices even for the distribution
p(Xk |Xk−1) =1Zsexp(−dist2(Xk ,gXk−1g
t)
2ω2 ).
4.3.2 The Tracking Algorithm
The recursive filter for covariance matrices (descriptors) on Pn presented above can
be used in combination with many existing tracking techniques. Many algorithms based
on covariance descriptors like those in [61, 75] can use our IRF as the model updating
method for covariance descriptors. Here we combine the IRF with a particle position
tracker and get a real-time video tracking algorithm.
Feature Extraction: Assume we have an rectangular region R with width W and
height H which represents the target object in a certain image I in the video sequence.
The feature vector f (x , y) ,where (x , y) ∈ R, can be extracted to include the information
of appearance, position and etc to describe information at the point (x , y). In [61],
the feature vector was chosen to be f = [x , y , I (x , y), |Ix(x , y)|, |Iy(x , y)|] where Ix and
Iy are the components of the gradient ∇I . For color images, I (x , y) = [R,G ,B] is a
vector. With the feature vectors at each point in the region of the object, the covariance
48
matrix can be computed as CR = 1WH
∑k∈R(fk − µR)(fk − µR)
t. This covariance matrix
can be computed in constant time with respect to the size of the region R by using the
technique called the integral image as was done in [73]. We can also add the mean µR
in to the covariance descriptor and still obtain a symmetric positive definite matrix in the
following manner,
CR =
CR + λ2µµt λµ
λµt 1
(4–30)
where λ is a parameter used to balance the affect of the mean and variance in the
descriptor (in the experiments λ = 0.001).
As in [73] we use several covariance descriptors for each object in the scene. Very
briefly, each region enclosing an object is divided into 5 regions and in each of these, a
covariance descriptor is computed and tracked individually. A matching score (likelihood)
is computed using 4 of them with relatively small distance to the corresponding template
in the template matching stage described below. This approach is used in order to
increase the robustness of our algorithm.
Tracking and Template Matching: We use a sampling importance re-sampling
(SIR) particle filter [5] as a position and velocity tracker. The state vector of the particle
filter is now given by, (x , y , vx , vy , log(s))t , where x , y , vx .vy denote the position and
velocity of the object in the 2D image, and log(s) is the log of the scale. The state
transition matrix is defined based on Newton’s first law
F =
1 0 0.01 0 0
0 1 0 0.01 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
(4–31)
The variance of the state transition is a diagonal matrix and in our work reported
here, they were set to, (42, 42, 202, 202, 0.0152). These state transition parameters are
49
dependent on the videos being tracked. They could be learned from the manually
labeled training sets. The likelihood for the particle filter is based on the generalized
normal distribution given in Equation 4–4.
At step k , we first compute the prediction of the object covariance template
using Yk = hgXk−1gtht , and the prediction of the position & scale of the object
represented in the set of particles based on the state transition matrix (Equation 4–31).
Covariance descriptors then are computed for each of the predicted particle state
at the corresponding object regions. The likelihood for each covariance descriptor
is computed based on the generalized normal distribution centered at the predicted
corresponding covariance template. And the likelihood for each particle’s state is
computed as multiplication of the likelihoods of covariance descriptors that are closer to
their corresponding template as mentioned above. After multiplying the likelihood with
the weight of each particle, the mean of the sample set is computed. This is followed
by computation of covariance descriptors at the location of the mean of the particle set.
This covariance descriptor then forms the input to our IRF. In our experiments, we use
300 particles for the particle set. Our tracking algorithm runs in around 15Hz for videos
with a frame size of 352× 288, on a desktop with a 2.8GHz CPU.
4.4 Experiments
In this section, we present the results of applying our intrinsic recursive filter to
both synthetic and real data sets. The real data sets were taken from standard video
sequences used in the computer vision community for testing tracking algorithms. First
we present the synthetic data experiments and then the real data.
4.4.1 The Synthetic Data Experiment
To validate the proposed filtering technique, we first performed synthetic data
experiments on P3, the space of 3 × 3 SPD matrices. A time sequence of i.i.d samples
of SPD matrices were randomly drawn from the Log-normal distribution [64] centered
at the identity matrix. This was done by first drawing samples {vi}in R6(isomorphic to
50
Sym(3)) from the Normal distribution N(0,σ2I6). Then, these samples are projected
to P3 (denoted by {Xi}) using the exponential map at the point I3(identity matrix).
Thus, {Xi} can be viewed as a time sequence of random measurements of the identity
matrix. Our recursive filter can then be used as an estimator of this random process.
The estimation error at time point k can be computed as the Riemannian distance by
Equation 2–2 between the estimate Xk and the ground truth (the identity matrix).
We tested our recursive filter (IRF) and evaluated by comparing its performance
with the optimal recursive filter for linear systems on Pn (denoted by ORF) reported
in [75]. The parameters of ORF are set to be exactly the same as in [75] except for the
initial base point Xb, where all the samples are projected to the tangent space TXbPn
and then processed with ORF. We set Xb to be the observation in the first step. In this
problem, setting Xb to be the ground truth would give the best result for ORF, because
in this case ORF would reduce to the Kalman filter on the tangent space. Since in
the practical case, the ground truth is unknown, here we set Xb as the observation at
the first step which is the best information we know about the data sequence before
tracking. We also tried to randomly set Xb, and this did not lead to any observable
big differences. For the proposed method, the GL actions g,h were both set to be the
identity, and ϕ2/ω2 = 200. We performed experiments with three different noise levels,
σ2 = 0.1, 1 and 2. At each noise level we execute the whole process 20 times and
computed mean error for the corresponding time step.
The results are summarized in Figure 4-1. From the figure, we can see that the
ORF performs better when σ2 = 0.1, and our method (IRF) performs better when the
data is more noisy (σ2 = 1, 2). The reason is that ORF uses several Log-Euclidean
operations, which is in fact an approximation. For low noise level data, the data points
are in a relatively small region around the ground truth (Identity), in which case, the
Log-Euclidean approximation is quite accurate. But for higher noise levels in the data,
the region becomes larger and the approximation becomes inaccurate which leads to
51
Figure 4-1. Mean estimation error from 20 trials for the synthetic data experiment. Thex-axis denotes the time step. The y-axis denotes the estimation errormeasured using the Riemannian distance between the estimates and theground truth. In all three sub-figures the red curves denote the estimationerror for our IRF, the blue curves for ORF with Xb set as the observation inthe first step.
large estimation error. In contrast, our filtering method is fully based on the Riemannian
geometry without any Log-Euclidean approximation, so it performs consistently, and
correctly converges for all the three noise levels. In conclusion, although our recursive
filter might converge a little bit slower than the ORF, it is more robust to larger amounts
noise which is common in real tracking situations.
4.4.2 The Real Data Experiment
For the real data experiment, we applied our IRF to more than 3000 frames in
different video sequences. Two other covariance descriptor updating methods were
also applied to these sequences for comparison namely, (1) the ORF reported in [75];
(2) The updating method using Karcher mean of tracking results in previous frames
(KM) reported in [61]. The image feature vectors for the target region were computed
as reported in [61]. The buffer size T in the KM method were set to be 20 which means
the Karcher mean of covariance descriptors in 20 previous frames were used for the
prediction of the covariance descriptor in current frame. The parameters for the ORF
were set to values given in the paper [75]. The parameter controlling the state transition
and observation noise in our IRF are set to ω2 = 0.0001 and ϕ2 = 0.01. Since our IRF is
52
Table 4-1. Tracking result for the real data experiment.Seq. Obj. Start End Err(IRF) Err(ORF) Err(KM)
C3ps1 1 200 700 4.3213 10.8467 10.7666C3ps1 7 200 700 4.1998 7.3524 4.7929C3ps1 8 200 700 2.6258 5.5844 6.2928C3ps2 7 500 800 2.7605 10.6478 12.4289C3ps2 8 500 800 3.4451 7.2113 11.6432
Cosow2 1 500 900 3.7612 4.613 6.0196Cosow2 3 500 900 4.9871 5.8552 7.9788
combined with a particle filter as a position tracker, for the purpose of comparisons, the
KM and ORF are also combined with exactly the same particle filter.
Firstly, we used three video sequences from the dataset CAVIAR : 1. ThreePastShop1cor
(C3ps1); 2. ThreePastShop2cor (C3ps2); 3. OneShopOneWait2cor (Cosow2). All three
sequences are from a fixed camera and a frame size of 384 × 288. 7 “objects” were
tracked separately. Given ground truth was used to quantitatively evaluate the tracking
results. To measure the error for the tracking results, we used the distance between the
center of the estimated region and the ground truth. With all the three methods having
the same initialization, the tracking results are shown in the Table 4-1, where all the
errors shown are the average errors over all the tracked frames. From the table we can
see that ORF is more accurate than KM based methods in most of the results, and our
IRF outperforms both these methods. The KM drifts away from the target because it is
based on a sliding window approach. If the number of consecutive noisy frames is close
to the window size, the tracker will tend to track the noisy features. For ORF, since it is
a non-intrinsic approach, the approximation of the GL-invariant metric would introduce
errors that accumulate over time across the frames causing it to drift away. Since IRF
is an intrinsic recursive filter, which uses the GL-invariant metric, there is less error
introduced in the covariance tracker updates. This in turn leads to higher accuracy in the
experiments above.
In the second experiment, we performed head tracking in video sequences with a
moving camera. Two video sequences were used: (i) Seq mb sequence(tracking face)
53
and (ii) Seq sb. Each of the sequences have 500 frames with frame size 96 × 128. Both
sequences are challenging because of complex background, fast appearance changes
and occlusions. The results are summarized in Figure 4-2.
In Seq mb, KM fails at frame 450 where the occlusion occurs, while ORF and IRF
do not lose track. Both KM and ORF produce relatively large errors in capturing the
position of the girl’s face after the girl turns around the first time between frames 100 to
150 due to the complete change in appearance of the target (girl’s face). ORF produces
relatively larger error in estimating the scale (compared to the initialization) of the face
between frames 400 to 500, which can be found in the snap shots included in Figure
4-2. The result of our method (IRF) has relatively larger error at around frame 100 and
180, because at this time, the camera is tracking the hair of the girl where no feature can
be used to locate the position of the face. However, for other frames, IRF tracks the face
quite accurately.
In Seq sb both KM and ORF fail at frame 200,and IRF however successfully tracked
the whole sequence with relatively high accuracy even with fast appearance changes
and occlusions, as shown in the quantitative analysis in Figure 4-2. These experiments
thus demonstrate the accuracy of our method in both moving camera and fixed camera
cases.
54
Figure 4-2. Head tracking result for video sequences with moving camera. Top andbottom rows depict snap shots and quantitative evaluations of the resultsfrom the Seq mb and Seq sb respectively. The tracking error is measured bythe distance between the estimated object center and the ground truth.Tracking results from the three methods are shown by using different coloredboxes superposed on the images and different colored lines in the plots.Results from our method (IRF) are in red, ORF in green and KM in blue.
55
CHAPTER 5INTRINSIC UNSCENTED KALMAN FILTER
5.1 Background and Previous Work
Diffusion Weighted MR Imaging (DW MRI) is the technique that can measure the
local constrained water diffusion properties in different spatial directions in MR signals
and thus infer the underlying tissue structure. It is a unique non-invasive technique
that can reveal the neural fiber structures in-vivo. The local water diffusion property
can be described either via a diffusivity function or a diffusion propagator function. The
diffusivity function can be estimated from the DW-MR signals and represented by a 2nd
order tensor at each image voxel yielding the so called Diffusion Tensor Imaging (DTI)
pioneered in [12]. It is now well known that DTI fails to accurately represent locations
the data volume containing complex tissue structures such as fiber crossings. To solve
this problem, several higher order models were proposed such as [1, 26, 51].
To further reveal the fibrous structures such as brain white matter, fiber tracking
methods were proposed to analyze the connectivities between different regions in the
brain. Existing fiber tracking methods fall mainly in two categories, deterministic and
probabilistic. One popular deterministic tracking method is the stream line method [13,
50], where the tracking problem is tackled using a line integration. The deterministic
tracking method can also based on the (Riemannian or Finsler) geometry imposed
by the diffusivity function [62] where the tracking problem is posed as a shortest path
problem. In probabilistic fiber tracking methods [14, 63, 89], a probabilistic dynamic
model is first built and then a filtering technique such as particle filter is applied. Most of
the existing fiber tracking methods are based on two stages namely, first estimating the
tensors from DWI and then tracking using these estimated tensors.
Recently, in [48] a filtered multi-tensor tractography method was proposed in which
the fiber tracking and the multi-tensor reconstruction was performed simultaneously.
There are mainly two advantages of this approach: (1) The reconstruction is performed
56
only at locations where it is necessary, which would significantly reduce the computational
complexity compared to the approaches that first reconstruct whole tensor field and then
apply tractography, (2) fiber tracking is used as a regularization in the reconstruction
i.e., the smoothness of the fiber path is used to regularize the reconstruction. However,
in [48] the filtering is applied only to the tensor features (major eigen vectors etc.) all
of which have strict mathematical constraints that ought to be satisfied but not all of
the constraints were enforced. For example, the constraint on eigen vectors to lie on
the unit sphere was not enforced. In general it would be more favorable to track the full
tensor and enforce the necessary constraints. It is known that diffusion tensors are in
the the space of symmetric positive definite(SPD) matrices denoted as Pn, which is not
a Euclidean space but a Riemannian manifold. Vector operations are not available on
Pn. So algorithms that are based on vector operations can not be applied directly to
these spaces, and non-trivial extensions are needed. In this dissertation, we propose
a novel intrinsic unscented Kalman filter on Pn, which to the best of our knowledge
is the first extension of the unscented Kalman filter to Pn. We apply this filter to both
estimate and track the tensors in the multi-tensor model using the intrinsic formulation
to achieve better accuracy as demonstrated through experiments. We perform real data
experiments to demonstrate the accuracy and efficiency of our method.
The rest of the chapter is organized as follows: the intrinsic unscented Kalman filter
is described in Section 5.2, where a novel dynamic model defined for the multi-tensor
model. We then present the intrinsic unscented Kalman filter algorithm and finally the
experiments are presented in Section 5.3.
5.2 Intrinsic Unscented Kalman Filter for Diffusion Tensors
5.2.1 The State Transition and Observation Models
The state transition model on Pn here is based on the GL operation and the
LogNormal distribution. For the bi-tensor (sum of two Gaussians) model, the state
57
transition model at step k is given by,
D(1)k+1 = Exp
FD(1)kFt(v
(1)k )
D(2)k+1 = Exp
FD(2)kFt(v
(2)k )
(5–1)
where, D(1)k , D(2)
k are the two tensor states at step k , F is the state transition GL based
operation, v(1)k and v(2)k are the Gaussian distributed state transition noise for D(1)k and
D(2)k in the tangent space T
D(1)k
P3 and TD(2)k
P3 respectively. Here we assume that the
two state transition noise models are independent from each other and the previous
states. The covariance matrices of the two state transition noise models are Q(1)k and
Q(2)k respectively. The covariance matrix Q(i)
k i = 1, 2 is a 6 × 6 matrix defined for the
tangent vectors in TD(1)k
P3. Note that Q(i)k is not invariant to GL coordinate transform on
Pn. Assume a random variable X = Expµ(v) in Pn, where v is a random vector from a
zero mean Gaussian with Q being the covariance matrix. Then, after a GL coordinate
transform g ∈ GL(n), the new random variable Y = gXgt = Expgµgt(u). The covariance
matrix of u is
Q(g) = (g⊗ g)−1Q(g⊗ g)−t (5–2)
where ⊗ denotes the Kronecker product. Here we first define the covariance matrix at
the identity QI3×3= qI6×6, where q is a positive scalar. And the covariance matrix at point
X can be computed using Equation 5–2 by setting g = X12 . With this definition the state
transition noise is independent with respect to the system state.
The observation model is based on the bi-tensor diffusion model.
S(n)k = S0(e
−bngtnD
(1)kgtn + e−bngt
nD(2)kgtn) (5–3)
where gn denotes the direction of n-th magnetic gradient, and bn is the corresponding
b-value, and S(n)k is the MR signal for n-th gradient at iteration step k . The covariance
matrix of the observation model for all the magnetic gradients is a diagonal matrix
58
denoted by R. This assumes that the measurements from distinct gradient directions are
independent.
5.2.2 The Intrinsic Unscented Kalman Filter
Just as in the standard Kalman filter, at each iteration step of the unscented Kalman
filter [48] there are two stages, the prediction and update stages respectively. In the
prediction stage, the state of the filter at the current iteration is predicted based on
the result from the previous step and the state transition model. In the update step,
the information from the observation at the current iteration is used in the form of the
likelihood to correct the prediction. Since the states are now diffusion tensors which
are in the space of Pn, where no vector operations are available, we need a non-trivial
extension of the Unscented Kalman filter, especially for the prediction stage to be valid
on Pn.
To begin with, we define the augmented state for the bi-(diffusion) tensor state at
iteration step k to be
Xk = [u(1),tk ,u
(2),tk , v
(1),tk , v
(2),tk ]t (5–4)
where v(i)k i = 1, 2 is the state transition noise vector for diffusion tensor state D(i)k and
u(i)k = Log
EK(D(i)k)(D
(i)k ) which is the representation of the state random variable in the
tangent plane at its Karcher expectation(EK(·)). Xk is zero mean and with covariance
matrix denoted by Pak . The covariance matrix for the state [u
(1),tk ,u
(2),tk ]t is denoted by
Pk,DD . Note that Pak is a block-wise diagonal matrix composed from Pk,DD , Q(1)
k and Q(2)k .
In the prediction stage, 2L + 1 weighted samples from the distribution of Xtk are first
computed by a deterministic sampling scheme given below. Here, L = 24 and denotes
the dimension of Xtk .
Xk,0 = 0, w0 = κ/(L+ κ) (5–5)
Xk,j = (√
(L+ κ)Pak)j , wj = 1/2(L+ κ) (5–6)
Xk,j+L = −(√
(L+ κ)Pak)j , wj+n = 1/2(L+ κ) (5–7)
59
where wj is the weight for the corresponding sample, κ ∈ R is a parameter to
control the scatter of the samples, and (√
(L+ κ)Pak)j is the j-th column vector of matrix√
(L+ κ)Pak . Since samples Xk,j = [u
(1),tk,j ,u
(2),tk,j , v
(1),tk,j , v
(2),tk,j ]t are generated from the joint
distribution of posterior and state transition at frame k , we can get the samples from the
distribution of prediction in frame k + 1 based on Xk,j through a two-step procedure. First
we can get the samples from the posterior
D(i)k,j = Exp
D(i)k
(u(i)k,j) (5–8)
where D(i)k is the state estimate from the last iteration (the estimator of EK(D
(i)k )). And
then the samples from the predicted distribution can be generated based on D(i)k,j and
v(i)k,j ,
D(i)k+1,j = Exp
D(i)k,j
(v(i)k,j) (5–9)
where D(i)k+1,j denotes the j-th sample from the distribution of the prediction. The
predicted mean is computed as the weighted Karcher mean,
D(i)k+1 =
∑jwjD(i)
k+1,j (5–10)
The predicted covariance of the states is computed in the product space TD(1)k+1
P3 ×
TD(2)k+1
P3,
Pk+1,DD =∑j
wjUjU tj (5–11)
where U tj = [LogD(1)
k+1
(D(1)k+1,j),LogD(2)
k+1
(D(2)k+1,j)] is a concatenation of the two vectors
obtained from the Log-map of each predicted sample.
Applying the observation model defined in Equation 5–3 to the predicted state
samples we get the predicted vector of MR signals for different magnetic gradients
denoted by Sk+1,j . Because this is in a vector space, we can use standard vector
operations to compute the predicted mean Sk+1 as the average of Sk+1,j . Using the
observation noise covariance R, the predicted observation covariance can be computed
60
as
Pk+1,SS = R+∑j
wj(Sk+1,j − Sk+1)(Sk+1,j − Sk+1)t (5–12)
Also the cross-correlation matrix between the observation and the states is given by,
Pk+1,DS =∑j
wj(Uj(Sk+1,j − Sk+1)t) (5–13)
In the update step, the Kalman gain is computed as Kk+1 = Pk+1,DSP−1k+1,SS.
Knowing the Kalman gain we can update of the states and covariance which are given
by:
D(i)k+1 = ExpD(i)
k+1
z(i)k+1
Pk+1,DD = Pk+1,DD −Kk+1Pk,SSKtk+1
(5–14)
where [z(1),tk+1 , z
(2),tk+1 ]
t = Kk+1(Sk+1 − Sk+1), and Sk+1 is the observation (MR signal vector)
at step k + 1.
5.3 Experiments
To validate our tractography, we applied IUKF to HARDI scans of rat cervical spinal
cord at C3 C5. In this experiment, 8 different rats were included 6 of them healthy
and 2 injured with the injury in the thoracic spinal cord. The HARDI scan for each rat
was acquired with 1 s0 image (taken with b closed to zero), and 21 different diffusion
gradients with b = 1000s/mm2, � = 13.4ms and δ = 1.8ms . The voxel size of the
scan is 35µm × 35µm × 300µm, and the image resolution is 128x128 in the x − y plane
and in the z-direction the resolution is 24 to 34. All HARDI datasets where aligned into
the same coordinate system by a similarity transform before tracking. To initialize the
algorithm, for each scan we first placed a seed point at each voxel of the grey matter,
and then a 2nd order tensor estimation is employed as an initialization for the algorithm.
In the experiment, various parameters were set to: the state transition noise variance
in Equation 5–1 Q1 = Q2 = 0.1I, the observation noise variance R = 0.03I and the
size of each tacking step δt = 0.01mm. The algorithm stops if the angle between two
61
Figure 5-1. Fiber tracking results on real datasets. Figure (a) is the region of interestoverlayed with the S0 image. Figure (b) & (c) are the fiber tracking result of ahealthy (injured) rat overlayed on S0 where the fibers are colored by its localdirection with xyz being encoded by RGB. c⃝[2011] IEEE
consecutive tangent vectors becomes larger than 60 degree or the fiber tract arrives at
the boundary of the spinal cord.
The fiber bundle of interest is the motoneuron which starts from the gray matter
and ends at the boundary of the spinal cord. To visualize the motoneuron fiber
bundle, we took corresponding ROIs such that only the fiber passing through the
ROIs are displayed. The results are shown in Figure 5-2, where we can find fiber
bundles starting from the gray matter and end at the boundary of the spinal cord. The
differences between the injured and control rats are not easily seen directly. To visualize
the difference between the healthy and injured rats, we first computed the axonal
fiber density map for each rat by counting the number of fibers passing through the
3-by-3 neighborhood of each voxel. We then non-linearly deform the density map to a
spinal cord atlas derived from HARDI data [20] and do voxel-wise t-test analysis. The
result are shown in the Figure 5-2, where we can find significant differences between
the healthy and the injured rats in the motoneuron region, which demonstrates the
effectiveness of our tracking method.
62
Figure 5-2. Biomarkers captured by computing density map for each fiber bundle. Figure(a) & (b) show a sample slice of fiber density maps obtained for each controland injured rats, respectively. Figure (c) is the region in which the p-value isless than 0.005, overlaid on the S0 image. c⃝[2012] IEEE
63
CHAPTER 6ATLAS CONSTRUCTION FOR HARDI DATASET REPRESENTED BY GAUSSIAN
MIXTURE FIELDS
6.1 Background and Previous Work
Groupwise image registration and image atlas construction is a very important
and challenging task in medical image analysis. It has many applications in image
segmentation and statistical analysis of a group of subject images. Several researcher
groups have tackled variations of this problem and reported their results in literature
[36, 47, 57, 82]. Most of these are on groupwise registration and atlas construction from
scalar images or segmented shapes.
Diffusion-Weighted MR Imaging (DW-MRI) is a powerful noninvasive technique
that can capture the information of water diffusion in tissues and thus infer its structure
in vivo. Serveral methods have been reported in litrature to model and estimate the
diffusivity functions from the MRI signals. One popular method is the so called Diffusion
Tensor Imaging(DTI) [12] which approximates the diffusivity function at a voxel by a
positive definite matrix. A DTI based atlas will obviously provide more information
than conventional scalar image based atlas [55] since DTI contains both scalar and
directional information. Atlas construction requires the DTI data to be groupwise
registered and in this regard, until recently, most of the DTI registration techniques
reported in literature were pairwise registration methods [10, 15, 86, 88]. Some of the
existing DTI based atlases are built by coregistration techniques as in [45] but a DTI
based groupwise registration and atlas construction methods was reported in [90].
It is however well known that the DTI model cannot resolve complex tissue structure
such as fiber crossings. To handle this problem, several higher order models [7, 26,
35, 56] based on High Angular Resolution Diffusion Imaging(HARDI) dataset were
reported in literature. Serveral recent works were reported for the HARDI pairwise
registrations [8, 10, 19, 30], and shown to outperform DTI based registration especially
in aligning fiber crossing regions [10]. But very few works have been reported in
64
the groupwise registration for HARDI dataset, except a 4-th order tensor field based
groupwise registration reported in [8] which extended the unbiased atlas construction
technique in [36] to handle 4-th order tensor fields by using novel distances.
In this dissertation, we present a novel atlas construction method for HARDI
datasets represented by a Gaussain Mixture Field (GMF) generated by the algorithm
described in [35]. GMF is a field of zero mean 3D Gaussian mixture models one each
at each lattice point of the field. We use the L2 distance between GMFs to measure
the dissimilarity between two Gaussian mixture fields as was defined in [19]. And we
significantly extended the framework in [36] to construct the atlas from a set of GMFs.
A novel mean GMF computation method is also presented along with the groupwise
registration process. The key contributions are: 1. A GMF based groupwise registration
is proposed which is the first of its kind; 2. An objective function involving the L2 distance
between Gaussian mixtures is used that leads to a closed form expression for the
distance and the gradient computation. 3. A minimal distance projection is defined and
used to obtain a sharp (non-blurry) atlas which is useful in atlas based segmentation.
Experiments along with comparisons are presented to demonstrate the performance of
our algorithm.
The rest of the chapter is organized as follows: The method for atlas construction
on GMF is presented in Section 6.2, along with the atlas construction framework in
Section 6.2.1, followed by the distance metric in Section 6.2.2, and the implementation
and mean GMF computation in Section 6.2.3. In Section 6.3, we present synthetic and
real data experiments along with comparisons to other existing methods.
6.2 Methods
6.2.1 Image Atlas Construction Framework
An atlas of a set of images/shapes etc. is commonly defined as an average over the
set, which is taken to be a representative of the set. The problem with simply taking an
average as the atlas is that the average tends to be rather blurred and is not effective
65
for use in tasks such as, atlas-based segmentation or atlas-based registration etc.
Mathematically speaking, this can be caused due to the fact that the average may not
necessarily belong to the same abstract space (e.g., space of brain images) defined
by the original data set. For instance, the technique described in [36] searches for the
average in the image space without a constraint on the space of images, which can lead
to a blurred atlas image. To solve this problem, in [47], the atlas is constrained to be
deformed diffeomorphically from a super template which needs to be pre-selected; And
in [31, 57, 85] the structure of the subject image space is learned from the dataset,
using which the atlas is computed. These methods need registrations between
all the image pairs in the dataset (O(N2) registrations), which makes the approach
computationally expensive for large datasets.
Here we define the space of images of interest to us (spinal cord images) to be
spanned by a set of GMFs {In}Nn=1 and denoted by S =∪
nO(In), where O(In) = {J :
J = In ◦ T,T ∈ Di� } is the orbit spanned by the image In and all the diffeomorphic
deformations Tn : → In , where denotes the domain of the image. Thus, finding the
atlas I can be viewed as solving the following problem
m∗,T∗1, ... = argminm,T1,...
∑n
E(In ◦ Tn, Im ◦ Tm) + ϕ(Tn)
And the final atlas could be defined as I = �−1m∗[Im∗ ◦ Tm∗], where �m∗ is the Jacobian
of the deformation Tm∗, and �−1m∗[] denotes the re-orientation operation discussed in
Section 6.2.2. Solving this problem directly would make the computational complexity
similar to O(N2) pairwise registrations. What we would like to do, is to achieve an
approximate solution using a two step procedure.
In the first step, we try to find a intermidiate atlas in the space of all images, which
can be viewed as solving the optimization problem (similar as [36] but generalized to
GMFs)
I ,T1, ... = argminI ,T1,...
∑n
E(In ◦ Tn, I ) + ϕ(Tn) (6–1)
66
where the data term energy function E(, ) is defined as a sum of squared voxelwise
distance (details in Section6.2.2)
E(In ◦ Tn, I ) =
∫I
dist2�n(In ◦ Tn(x), I (x))dx (6–2)
ϕ is the penalty term used to enforce the smoothness constraint on the deformation.
The deformation can be modeled as a diffeomorphism, and parametrized by a velocity
field ∂Tn
∂t= vn(Tn(x, t), t). Thus the deformation can be computed as Tn(x) = x+dn(x) =
x +∫ 1
0vn(x(t), t)dt , where dn represents the displacement field. The smoothness
constraint we use here is given by,
ϕ(Tn) = λ ∗ log(det(�n)) ∗ (1− det(�n)) +
∫I
∫ 1
0
||Lvn(x, t)||2dt. (6–3)
Where, L is a linear operator, and the first term in Equation 6–3 imposes additional
smoothness as in [87].
In the second step, we project the intermidiate atlas to the space S by solving
another distance minimization
I = �−1m∗ [Im∗ ◦ T∗]
m∗,T∗ = argminn,TE(In ◦ T, I )(6–4)
and the projection result I is our final atlas.
6.2.2 L2 Distance and Re-orientation for GMs
We use the L2 distance as a dissimilarity measure between two Gaussian
mixture densities (GMs), which can be computed in a closed form [19]. Let f (r) =
�Mi=1ηiG(r; 0,�i) and g(r) = �N
j=1ρjG(r; 0,�j) be two Gaussian mixture density functions,
where r ∈ R3 is the displacement vector and ηi , ρj denote the mixture weights of
the corresponding Gaussian components G(r; 0,�i) and G(r; 0,�j) with covariance
matrices �i and �j respectively. The L2 distance between f and g can be written as a
quadratic function of the mixture weights dist2(f , g) = ηtAη + ρtBρ − 2ηtCρ, where
67
η = (η1, ..., ηM)t and ρ = (ρ1, ..., ρN)
t , and AM×M ,BN×N and CM×N are the matrices
generated by the Gaussian components, see [19] for details.
A re-orientation operation is needed for image transformation, when the image
value at each pixel/voxel is related to the image coordinates and is not rotationally
invariant [2]. In this case, the image value should change according to the image
coordinates. Otherwise, an artifact might be introduced due to this transformation.
As in [19], the Preservaion of Principal Direction(PPD) re-orientation is extended to
GMs, and it is the only re-orientation strategy that can capture the change of angle
between fiber crossings during a non-rigid transformation. In this dissertation, we
adopt this re-orientation strategy, and the energy function with reorientation would be
dist2�(f , g) = ηtA�−1η + ρtBρ − 2ηtC�−1ρ, where computation of A�−1 and C�−1 can be
found in [19].
6.2.3 Mean GMF Computation
We employ an (iterative) greedy algorithm to solve the problem in Equation 6–1.
Given the initialization of atlas I and {Tn}, in each iteration step, we first fix the
deformations and update the atlas by optimizing w.r.t. I . I new = argminI∑
n
∫Idist2�n
(In ◦
Tn(x), I (x))dx Since we use an L2 distance, the global minimum can be found as
I new =∑
�−1n [In◦T]N
. However, from this formula, I new(x) would have many more
components than In(x) would have had, which would make the algorithm computational
expensive. To solve this problem, we fixed the group of mixture components of I new .
Since each mixture component is zero mean, and cylindrically symmetric with fixed
eigenvalues. All we need to do is to decide on the eigen vectors. In this dissertation,
we discretize the hemi-sphere using 46 different directions, and used them as the eigen
vectors (the same approach has been used in the reconstruction method [35]). Then,
the only thing left is to compute the mixing weights, which is equivalent to solving the
linear system at each voxel Aρ =∑
n C�−1nηn. This would be easy to solve since A is low
dimensional and full rank.
68
After updating the atlas, the force field can be first computed as the first order
variation of data term of the objective function plus the first term on the right (enforcing
additional smoothness) in Equation 6–3; And then the velocity field is updated as the
Gaussain kernel based smoothed version of the force field [8], and the deformation field
T(x) is updated using the following update equation Tnew(x) = Told(x+ ϵv).
The derivative of the objective function is computed by applying the derivative chain
rule as in [19]. And the derivative of the regularization term in Equation 6–3 can be
computed directly using the derivative for the determinent.
We employ a coarse-to-fine strategy in our registration algorithm for an efficient
implementation. With initialization of the deformation set to identity, the algorithm yields
satisfactory results in 200 steps. After we get the atlas I , we can project it to S to get I
by using Equation 6–4.
6.3 Experiments
6.3.1 Synthetic Data Experiments
To validate the registration framework used in our atlas construction method, we
first apply our method to the pairwise registration problem. For the two images case,
Equation 6–1 would reduce to
T1,T2 = argminT1,T2E(I1 ◦ T1, I2 ◦ T2) + ϕ(T1) + ϕ(T2). (6–5)
By using the displacement field inverse method [25], we can get T = T1 ◦ T−12 : I2 →
I1. Thus we have a pairwise registration algorithm. We applied this algorithm to the
synthetic dataset and then compared it to a GA based registration algorithm (using an
SSD cost) and a DTI based registration algorithm in [88] with the same dataset.
To generate the synthetic dataset, a 3D synthetic image (64 × 64 × 4) with two
crossing fiber bundles was generated and then, 20 randomly deformed images were
synthesized from this by using a bspline-based non-rigid deformation. The method
described in [66] was used to generate the simulated MR signals from the fiber bundles.
69
Figure 6-1. Experimental results on synthetic dataset. Figure (a) and (b) are the meanand standard deviation of error for the 20 registrations from all the threemethods at different noise levels in the two different ROIs. c⃝[2011] IEEE
Rician noise was added to simulate data at 4 different noise levels with SNR = 50, 20, 10
and 5. The method in [35] was used to generate the GMF from the MR signals with 46
Gaussian components at each voxel.
After the data generation, we registered each of the randomly deformed images
(source image) to the original image (target image) separately. To evaluate the
registration, the resulting deformation obtained from the registration was applied
to the noise free source image, and then the dissimilarity between the deformed
source and target images were computed as the error in registration. The dissimilarity
measure we used here was Hellinger distance between the displacement probability
profiles (represented in spherical harmonic coefficients) at corresponding lattice
pointsr =∫S2(
√f (x) −
√g(x))2dx . Also, we computed the registration error in two
different regions: (1) the whole image, (2) the region that contains only the crossing
fibers.
The data sets and the results are displayed in Figure 6-1. Figure (a) and (b) show
that our method yields a slightly lower mean and standard deviation of the registration
errors for the whole image, and much lower error in the fiber crossing region for all four
noise levels. This demonstrates the accuracy of our HARDI registration method.
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6.3.2 Real Data Experiments
For the real data experiments, we apply our method to the HARDI scans of a rat
spinal cord at C3 − 5 for 7 different rats. In each scan 21 diffusion gradients were
used (with� and the δ was set to be 13.4ms and 1.8ms) with b = 1000s/mm2. Also,
one S0 image is taken with b close to zero. The image resolution in the x − y plane
was 128 × 128, with the number of slices varying from 25 to 34. The voxel size is
35µm× 35µm× 300µm. [35] was used to generate the GMF from the MR signals with 46
Gaussian components at each voxel.
We first apply a similarity registration to quotient out the translation, rotation and
scaling factors, and apply our groupwise registration algorithm to the dataset to get
an atlas I . In the following, we projected I to the space spaned by the given data
samples using Equation 6–4 to get the sharp (non-blurry) atlas I . The results of our
atlas construction method are depicted in Figure 6-2 (via S0 images even though the
algorithm was applied to the the GMF representation of the HARDI data), where the
(a) is the voxelwise mean of S0 images before registration, and (b) after registration.
We can see that (a) is fuzzy because the structure is not well aligned, and (b) is not.
This indicates the effectiveness of our groupwise registration method. (c) is the Im∗ in
Equation 6–4, and (d) is the final atlas I . We can see that (b) is much more blurry than
(d), and the shape of the boundary between white and grey matter is nearly the same for
(b) and (d). This indicates that I could be a good representative for the whole dataset,
and thus justifies the effectiveness of our method.
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Figure 6-2. Experimental results on real datasets depicted using S0 images. Figure(a)-(d) are all S0 images, with (a) the voxelwise mean before groupwiseregistration, and (b) after registration. (c) is the S0 image for Im∗ and (d) is theS0 image for the final atlas I . Figure (f) is the diffusion profile of I , and iscolored by the max direction of the diffusion profile, with xyz direcitionsmapped to RGB. c⃝[2011] IEEE
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CHAPTER 7DISCUSSION AND CONCLUSIONS
Tracking on the manifold of Pn is an important research problem and has many
applications in computer vison and medical learning. In this disseration, three novel
intrinsic tracking algorithms in Pn are presented. The first one is a novel recursive
Karcher expectation estimator (RKEE). The unbiasedness and L2-convergence under
the symmetric distribution is also presented. Real and synthetic data experiments
showed the similar accurarcy between the RKEE and the classical batch Karcher
expectation estimator – Karcher mean, but the RKEE is much more efficient than
Karcher mean especially for recursive estimation problem. The second one is a novel
intrinsic recursive filter (IRF) for covariance tracking, which proved to be more robust
to noise than existing methods reported in literature. IRF is based on the intrinsic
geometry of the space of covariance matrices and a GL-invariant metric that are used in
developing the dynamic model and the recursive filter. The third one is a novel intrinsic
unscented Kalman filter which then used to recursively estimate the Gaussian mixture
model on DW MR data sets. In combination of the stream line tracking strategy, this
forms a novel fiber tracking algorithm. Experiments on the human brain and rat spinal
cord demonstrated the accuracy of the proposed algorithm.
A generalization of the normal distribution to Pn is presented and used to model the
system in IRF and the observation noise in the IRF. Several properties of this distribution
in Pn were also presented in this dissertation, which to the best of our knowledge have
never been addressed in the literature. Note that our generalization of the normal
distribution to Pn is rotationally invariant, and the variance of the distribution is controlled
by a scalar (ω2 in Equation 4–4) rather than a variance control tensor which is a more
general form. One main reason for using this specific form is that the scalar variance
control parameter is GL-invariant, while the variance control tensor is not as shown
through the following simple calculation. Suppose V ∈ TXPn is a tangent vector (which is
73
a symmetric matrix) at point X ∈ Pn, and � is a variance control tensor. The value of the
density function on expX(V) would depend upon the quadratic form vec(V)t�−1vec(V)
where vec(·) is the vectorization operation on the matrix argument and � is a second
order tensor. In practice X would be the Karcher expectation of the aforementioned
distribution and V would be the tangent vector corresponding to the geodesic from X
to a sample point from the distribution. If we change the coordinates by using a GL
operation g, the Karcher expectation becomes gXgt , the vector becomes gVgt , and the
quadratic form becomes vec(gVgt)t�−1vec(gVgt). If we want to keep the value of the
density unchanged, we need to change � according to g which means that � is not GL-
invariant. However, in contrast, it is easy to show that ω2 in Equation 4–4 is GL-invariant.
Further, the IRF is quite different from the Kalman filter which is known to be an
optimal linear filter based on an additive Gaussian noise assumption. One reason for
the Kalman filter to be optimal is that it actually tracks the distribution of the object state
(posterior) based on a Bayesian tracking framework. If a filter doesn’t track the whole
distribution, usually it would explicitly or implicitly approximate the posterior based on the
state variables it has tracked. However, the approximation error might accumulate in the
system. From a geometric point of view, Kalman filter is highly dependent on geometric
properties of the Euclidean space. This is because Kalman filter is based on the fact
that the convolution of two Gaussians is a Gaussian. And this property of the Gaussian
stems from the fact that the Gaussian is the limit distribution in the central limit theorem.
One key problem in extending the Kalman filter intrinsically to Pn is finding two densities
pA(X; θA), pB(X|Y) with following properties
pA(X; θA) =
∫Pn
pB(X|Y)pA(Y; θ)[dY] (7–1)
where θA is the parameter of density pA. pA here is usually the posterior and pB is
the state transition noise distribution. The equation above means that after the state
transition the form of the posterior remains the same. Without this property, even if the
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whole distribution is tracked, the filter is implicitly approximating the true distribution after
the state transition by using the same form as the posterior from the last step which
still would lead to errors being accumulated in the system. However, it is non-trivial to
find such distributions on Pn. In [33, 70], a central limit theorem was presented in Pn
for rotationally invariant probability measures based on the Helgason-Fourier transform
[34]. However, currently the probability measure in the limit does not have a closed
form in the space domain. Thus, intrinsically extending the Kalman filter to Pn is still an
open problem. IRF instead only tracks the mode of the distribution. It is not an optimal
filter, but is intrinsic and mathematically consistent with respect to the noise model
used. We also presented a real time covariance tracking algorithm based on this filter
which is combined with an existing particle position tracker from literature [5]. Finally,
experiments on synthetic and real data favorably demonstrated the accuracy of our
method over rival methods.
75
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BIOGRAPHICAL SKETCH
Guang Cheng was born in Xingjiang, China in 1981. He graduated from Tsinghua
University with a Bachelor of Engineering degree in automation in 2003 and a Master
of Engineering degree in constrol science and engineering in 2006. He came to the
University of Florida to pursue a Doctor of Phylosophy degree in computer engineering
in August 2006.
Guang met Yandi Fan at UF, and they were married in December of 2007. Their first
daughter, Lerong Cheng, was born in August 2010.
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