Estimation of Probability Distributions on Multiple Anatomical Objects and Evaluation of Statistical

159 Pages·2009·9.63 MB·English

Estimation of Probability Distributions on Multiple
Anatomical Objects and Evaluation of Statistical Shape
Models
Ja-Yeon Jeong
A dissertation submitted to the faculty of the University of North Carolina at Chapel
Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy in
the Department of Computer Science.
Chapel Hill
2009
Approved by:
Stephen M. Pizer, Advisor
Surajit Ray, Reader
J. S. Marron, Reader
Martin A. Styner, Reader
Edward L. Chaney, Reader
(cid:13)c 2009
Ja-Yeon Jeong
ALL RIGHTS RESERVED
ii
Abstract
Ja-Yeon Jeong: Estimation of Probability Distributions on Multiple Anatomical
Objects and Evaluation of Statistical Shape Models
(Under the direction of Stephen M. Pizer)
Theestimationofshapeprobabilitydistributionsofanatomicstructuresisamajorresearch
area in medical image analysis. The statistical shape descriptions estimated from training
samples provide means and the geometric shape variations of such structures. These are key
components in many applications.
This dissertation presents two approaches to the estimation of a shape probability distri-
bution of a multi-object complex. Both approaches are applied to objects in the male pelvis,
and show improvement in the estimated shape distributions of the objects. The first approach
is to estimate the shape variation of each object in the complex in terms of two components:
the object’s variation independent of the effect of its neighboring objects; and the neighbors’
effect on the object. The neighbors’ effect on the target object is interpreted using the idea on
which linear mixed models are based.
The second approach is to estimate a conditional shape probability distribution of a target
object given its neighboring objects. The estimation of the conditional probability is based on
principal component regression.
This dissertation also presents a measure to evaluate the estimated shape probability dis-
tribution regarding its predictive power, that is, the ability of a statistical shape model to
describe unseen members of the population. This aspect of statistical shape models is of key
importance to any application that uses shape models. The measure can be applied to PCA-
based shape models and can be interpreted as a ratio of the variation of new data explained by
the retained principal directions estimated from training data. This measure was applied to
shape models of synthetic warped ellipsoids and right hippocampi. According to two surface
distance measures and a volume overlap measure it was empirically verified that the predictive
measure reflects what happens in the ambient space where the model lies.
iii
Acknowledgments
I would like to express my gratitude to people who have been important to this work and
to my life as a graduate student at UNC.
FirstandforemostIamthankfultoStevePizer,mydoctoraladvisor. Withouthisguidance
and encouragement I do not think I would be able to finish this dissertation. His patience in
reading draft after draft of my dissertation chapters amazed me. I sometimes felt he had more
confidence in me than I had in myself. I thank him for his willingness to make time for me
whenever I needed his help. I have learned a great deal from him, not only from his immense
knowledge and understanding of the field but also from his positive attitude, patience, and
passion for his work. He has been a great mentor to me.
I am grateful to my committee members for their comments and suggestions. Surajit Ray
has been actively involved in my dissertation research and has been of great help in developing
major ideas in this dissertation. Steve Marron has lent me freely his invaluable insight into
statistical issues. I believe he can visualize any statistical concepts with several color pens and
white papers. I want to thank Ed Chaney for his critical comments and advices as a medical
physicist and Martin Styner for his feedback and practical advices. I also want to thank Keith
Muller at UF for investing time and energy discussing ideas with me. Many of the ideas in
this dissertation originated from discussions with them.
I have been fortunate to have the people in MIDAG as my colleagues. They are good,
hard-working, and smart. Especially, I want to thank Josh Levy, Joshua Stough, Qiong Han,
Graham Gash, and Gregg Tracton who I pestered with questions, and my last office mate
Rohit Saboo, two female members Xiaoxiao Liu, and Ilknur Kabul whose office I frequently
visited whenever I wanted to take break from work.
I would like to thank faculty members and staffs in the department. My special thanks go
to Russ Taylor and Leandra Vicci for their encouragements during my gloomy first year, and
to Janet, Tammy, Linda, and Tim for their help in all (non-research related) problems that I
encountered as a graduate student.
iv
I wish to thank elders and my friends in First Baptist Korean Church of Raleigh for
providing a loving and supportive environment for me.
Lastly, and most importantly, I am grateful to my parents who have supported me with
constant love and understanding. To them I dedicate this thesis.
v
Table of Contents
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Statistical Shape Analysis of Multiple Objects . . . . . . . . . . . . . . 4
1.1.2 Quality Measures of Statistical Shape Models . . . . . . . . . . . . . . . 7
1.2 Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Overview of Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Statistical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1.2 Expectations of a Random Variable . . . . . . . . . . . . . . . 16
2.1.1.3 Special Distributions . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1.4 Statistics: Sample Mean and Variance . . . . . . . . . . . . . . 18
2.1.1.5 Multivariate Distributions. . . . . . . . . . . . . . . . . . . . . 19
2.1.1.6 The Multivariate Normal Distribution . . . . . . . . . . . . . . 20
2.1.2 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.3 The Multivariate Linear Model . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.3.1 Principal Component Regression . . . . . . . . . . . . . . . . . 27
2.2 The Statistical Theory of Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.1 Kendall’s Shape Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.2 Estimation of a Shape Probability Distribution . . . . . . . . . . . . . . 31
vi
2.2.3 Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.3.1 Alignment for multi-object complexes . . . . . . . . . . . . . . 34
2.2.4 Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Probabilistic Deformable Models . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.1 Deformable Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.2 ProbabilisticDeformableModels: SegmentationbyPosteriorOptimization 37
2.3.3 Object Representations via Landmarks. . . . . . . . . . . . . . . . . . . 38
2.3.3.1 Point Distribution Model . . . . . . . . . . . . . . . . . . . . . 39
2.3.4 Object Representations via Basis Functions . . . . . . . . . . . . . . . . 39
2.3.4.1 Elliptic Fourier Representation . . . . . . . . . . . . . . . . . . 40
2.3.4.2 Spherical Harmonics Shape Description . . . . . . . . . . . . . 40
2.4 Medial Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4.1 The Medial Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4.2 M-rep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4.2.1 Discrete M-reps . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4.3 Statistical Shape Analysis of M-reps . . . . . . . . . . . . . . . . . . . . 50
2.4.3.1 Shape Space of M-reps and PGA . . . . . . . . . . . . . . . . . 50
2.4.3.2 Tangent Space at a Point of a Manifold . . . . . . . . . . . . . 50
2.5 Construction of Training Models . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.6 Segmentation of M-reps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Evaluation of Statistical Shape Models 1 . . . . . . . . . . . . . . . . . . . 55
3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.1.1 Decomposition of the Covariance Matrix . . . . . . . . . . . . . . . . . . 56
3.1.2 PCA for Statistical Shape Analysis . . . . . . . . . . . . . . . . . . . . . 57
3.2 Goodness of Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.1 PCA Input to Multivariate Regression . . . . . . . . . . . . . . . . . . . 58
3.2.2 Measure of Association: Second Moment Accuracy . . . . . . . . . . . . 59
3.2.3 Procedure for Iterative Calculation of ρ2 . . . . . . . . . . . . . . . . . . 62
3.3 Derivation of B-reps from M-reps . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 Application of ρ2 to Models in Linear Space . . . . . . . . . . . . . . . . . . . . 63
vii
3.4.1 Simulated Ellipsoid M-reps . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4.2 Experiments on Simulated Ellipsoid B-reps . . . . . . . . . . . . . . . . 65
3.4.3 Right Hippocampus M-reps . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4.4 Experiments on Right Hippocampus B-reps . . . . . . . . . . . . . . . . 68
3.5 Distance Measures for ρ2 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5.1 Application of D2 to Right Hippocampus B-reps . . . . . . . . . . . . . 71
∗
3.6 Goodness of Prediction ρ2 for Curved Manifolds . . . . . . . . . . . . . . . . . 72
d
3.6.1 Two Possible Extensions of ρ2 . . . . . . . . . . . . . . . . . . . . . . . 72
3.6.2 ρ2 for Nonlinear Shape Models . . . . . . . . . . . . . . . . . . . . . . . 73
d
3.7 Application of ρ2 on Models in Nonlinear Space . . . . . . . . . . . . . . . . . . 74
d
3.7.1 Deformed Binary Ellipsoids . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.7.2 Experiment on M-rep Fits to Deformed Binary Ellipsoids . . . . . . . . 75
3.7.3 Experiment on Right Hippocampus M-rep . . . . . . . . . . . . . . . . . 76
3.7.4 Evaluation of a Coarse-to-fine Shape Prior . . . . . . . . . . . . . . . . . 77
3.8 Conclusion & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4 Multi-Object Statistical Shape Models2 . . . . . . . . . . . . . . . . . . . . 82
4.1 M-rep Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.1.1 Residues of the Object Variations. . . . . . . . . . . . . . . . . . . . . . 86
4.2 Inter-Object Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2.1 Augmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2.2 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3 Propagation of Sympathetic Changes . . . . . . . . . . . . . . . . . . . . . . . . 89
4.3.1 Residues of Objects in Order . . . . . . . . . . . . . . . . . . . . . . . . 92
4.3.2 Training the Probability Distribution per Object . . . . . . . . . . . . . 92
4.3.3 Geometrically Proper Objects . . . . . . . . . . . . . . . . . . . . . . . . 93
4.4 Decomposition of Shape Variations . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.4.1 Self Variations & Neighbor Effects . . . . . . . . . . . . . . . . . . . . . 98
viii
4.4.2 An Iterative Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4.3 Joint Probability of Multiple Objects. . . . . . . . . . . . . . . . . . . . 101
4.4.4 Shape Prior in MAP-Based Segmentation . . . . . . . . . . . . . . . . . 102
4.4.5 Segmentation of Male-Pelvis Model . . . . . . . . . . . . . . . . . . . . . 103
4.4.5.1 Probability Density Estimation . . . . . . . . . . . . . . . . . . 103
4.4.5.2 Segmentation Results . . . . . . . . . . . . . . . . . . . . . . . 105
4.5 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5 Conditional Shape Statistics3 . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1 Conditional Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2 Estimation of Conditional Shape Distribution using PCR . . . . . . . . . . . . 113
5.3 Applications of CSPD to Deformable M-rep Segmentation . . . . . . . . . . . . 115
5.3.1 Simulated Multi-objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.3.1.1 Training Results . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.3.2 Objects in the Pelvic Region of Male Patients . . . . . . . . . . . . . . . 119
5.3.2.1 Results of Within-patient Successive Segmentations . . . . . . 120
5.4 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.2 Evaluation of Statistical Shape Models . . . . . . . . . . . . . . . . . . . . . . . 131
6.2.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.3 Multi-Object Statistical Shape Models . . . . . . . . . . . . . . . . . . . . . . . 137
6.3.1 Discussion and Future Work. . . . . . . . . . . . . . . . . . . . . . . . . 137
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
ix
List of Tables
4.1 Total variations of self and neighbor effects per bladder, prostate, and rectum . 104
5.1 Number of fractions (sample size) per patient . . . . . . . . . . . . . . . . . . . 118
5.2 Mean volume overlaps of figure 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.3 Number of principal geodesics estimated from patient 3106 data . . . . . . . . 121
x