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A Riemannian Framework for Shape Analysis of Subcortical Brain Structures

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A Riemannian Framework for Shape Analysis of Subcortical Brain Structures A Riemannian Framework for Shape Analysis of Subcortical Brain Structures A dissertation presented to the faculty of the Russ College of Engineering and Technology of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Shuisheng Xie August 2013 © 2013 Shuisheng Xie. All Rights Reserved. 2 This dissertation titled A Riemannian Framework for Shape Analysis of Subcortical Brain Structures by SHUISHENG XIE has been approved for the School of Electrical Engineering and Computer Science and the Russ College of Engineering and Technology by Jundong Liu Associate Professor of Electrical Engineering and Computer Science Dennis Irwin Dean, Russ College of Engineering and Technology 3 Abstract XIE, SHUISHENG, Ph.D., August 2013, Electrical Engineering and Computer Science A Riemannian Framework for Shape Analysis of Subcortical Brain Structures (106 pp.) Director of Dissertation: Jundong Liu Measuring the volumetric and morphological changes of brain structures in MR images can provide a reliable basis for diagnosing and studying Alzheimer’s Disease (AD) and Mild Cognitive Impairment (MCI). Currently, the prevailing morphometric analysis frameworks are voxel-based morphometry (VBM), deformation-based morphometry (DBM) and tensor-based morphometry (TBM), which all involve a low-order spatial normalization procedure to project the computed voxel tissue map or motion field into a standard space. Shape and deformation information in local areas tend to be smoothed or even wiped out during the averaging process. In order to address the drawbacks of VBM/DBM/TBM, a novel local shape-based morphometry (SBM) framework on individual structures is developed in this dissertation, to measure the difference among a collection of anatomical images for individual subjects. SBM consists of three components: (1) a shape representation; (2) shape matching and comparison under Riemannian shape spaces; and (3) shape classifications and group analysis based on a well defined shape distance metric. Two different sets of solutions have been developed for the three components. The first approach relies on an information rich shape representation based on skeleton (IRS), and the dissimilarity between two shapes is defined as the geodesic distance connecting their projections on a shape manifold. The second model is based on a Riemannian meridian shape (RMS) representation. Stemmed from spectral graph theory, RMS possesses the merit of capturing the salient structural properties along the direction of maximal shape variation. Group clustering, based on pair-wise shape distances, is carried out through advanced manifold learning techniques. Experiments with synthetic shapes 4 and subcortical brain structures demonstrate the effectiveness of our framework in calculating the distances among different shapes and serving as a potential anatomical biomarker for neurodegenerative diseases, including AD. 5 To my parents. 6 Acknowledgments This dissertation would never have come to fruition without the supports from many professors, group mates, classmates and friends, and I would like to take this opportunity to express my sincere appreciation of their efforts. My deepest gratitude is to my advisor, Dr. Jundong Liu. His constant guidance, patience and support opened the door and guided me through a very rigorous yet enjoyable Ph.D. research training experience. His enormous enthusiasm and passion for work will always be an inspiration to me for many years to come. I would like to thank Dr. Chelberg for the helps he provided over the years. His solid background and deep understanding of the technology constantly enlighten my way to tackle the research problems I was facing. I really appreciate Dr. Marling’s encouragements to develop myself into an independent, mature and well-rounded researcher. Thanks to Dr. Drews. His suggestions to improve my research and dissertation are very valuable and insightful. I am grateful to the external committee members, Drs. Melkonian, McCall, Xu and Lin. It is really kind of them to spend their precious time serving on my committee, evaluating my research, and providing comments to improve my dissertation. Thanks to my group mates, Huihui Xu, Bibo Shi, David Days, Kevin H. Hobbs, Zhe Yu, Xin Qian, Pin Zhang and Yani Chen for the enjoyable discussions and collaborations over many interesting research topics. I am also indebted to many professors and friends at OU who have helped me over the years. I feel thankful to Dr. Shen. She has been a mentor for me in many aspects of my life. Thanks to Dr. Aizicovici and other professors in the Math department. Their kindness and considerateness towards the students are really admirable. 7 I need to thank the School of Electrical Engineering and Computer Science for the well designed program, as well as the financial support provided over the years. Without them, I would not have been able to complete my study and research. Last but not least, I would like to say thank you to my parents and other family members. Thanks for their understanding and supports. 8 Table of Contents Page Abstract.........................................3 Dedication........................................5 Acknowledgments....................................6 List of Tables...................................... 10 List of Figures...................................... 11 1 Introduction..................................... 13 2 Technical Background............................... 20 2.1 Riemannian Shape Space........................... 20 2.1.1 Basic Concepts............................ 20 2.1.2 Geodesics on Spheres........................ 22 2.1.3 Riemannian Statistics......................... 24 2.1.4 Some Recent Work of Shape Analysis on Manifolds........ 26 2.2 Skeletons for Shape Representation..................... 28 2.2.1 Curve Skeleton............................ 29 2.2.2 M-Rep and CM-Rep ......................... 33 2.3 Mesh Generation............................... 34 2.4 Spectral Graph Theory............................ 39 2.4.1 Graph Laplacian........................... 39 2.4.2 Laplacian Eigenmaps Embedding.................. 42 3 IRS Model for Shape-Based Morphometry..................... 44 3.1 Parameterization and Sphere Mapping.................... 45 3.2 IRS for 2D Shapes.............................. 46 3.3 IRS for 3D Shapes.............................. 47 3.4 Experimental Results............................. 47 3.5 Discussion................................... 53 4 Riemannian Shape Analysis based on Meridian Curves.............. 57 4.1 Related hippocampal structure analysis work................ 58 4.2 Mesh Generation............................... 61 4.3 Fiedler value and Fiedler vector....................... 63 4.4 Meridians Extraction based on Fiedler Vector................ 66 4.4.1 Meridians Extraction......................... 67 9 4.5 Riemannian Shape Space based on Meridians................ 71 4.6 Distances between RMSs........................... 74 4.7 Shape Clustering based on Inter-subject Distances.............. 75 4.7.1 Manifold Learning for RMS Dimensional Reduction........ 75 4.7.2 Shape Clustering........................... 77 4.8 Experimental Results............................. 78 4.8.1 Synthetic Examples.......................... 78 4.8.2 Subcortical structures from brain MR images............ 80 4.8.3 Clustering of AD Patients based on Hippocampi Shapes...... 84 4.8.4 Comparison with State-of-the-art Solutions............. 86 4.9 Discussion................................... 91 5 Summary and Future Work............................. 93 5.1 Summary................................... 93 5.2 Future Work.................................. 94 References........................................ 96 10 List of Tables Table Page 3.1 Pairwise shape distances using IRS model..................... 51 3.2 Pairwise shape distances using Joshi’s model [1]................. 52 4.1 Pairwise shape distances between shapes A-E .................. 79 4.2 Pairwise shape distances between the first 15 structures.............. 82 11 List of Figures Figure Page 1.1 Amyloid plaques and neurofibrillary tangles [2].................. 13 1.2 Brain atrophy in Alzheimer’s Disease (AD) [3].................. 14 1.3 Dynamic biomarkers of the Alzheimer’s pathological cascade [4]........ 15 1.4 Illustration of the VBM analysis procedure. [3].................. 17 1.5 Hippocampus as a subcortical component of human brains [5].......... 18 2.1 Tangent space, tangent vector and the curve.................... 21 2.2 Row-wise geodesic paths between the pair of curves shown to the left [1].... 28 2.3 Level curves and the curvilinear coordinate system in [6]............. 29 2.4 Skeleton of a 2D object............................... 30 2.5 3D objects and their corresponding surface skeletons [7]............. 31 2.6 Examples of curve-skeletons of different 3D objects [8]............. 31 2.7 Skeletonization results using the algorithm in [9]................. 33 2.8 The m-rep in [10]: Medial atom with a cross section of the boundary surface it implies (left). An m-rep model of a hippocampus and its boundary surface (right)........................................ 34 2.9 Quadtree decomposition [11]........................... 35 2.10 The advancing front triangulation method [11].................. 35 2.11 The empty circle property in Delaunay triangulation method [11]........ 36 2.12 Diagonal swapping in the Lawson’s Algorithm.................. 37 2.13 Bowyer-Watson Algorithm’s “void boundary” procedure............
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