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Elastic Statistical Shape Analysis with Landmark Constraints

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Elastic Statistical Shape Analysis with Landmark Constraints Elastic Statistical Shape Analysis with Landmark Constraints Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Justin Strait, M.S. Graduate Program in Statistics The Ohio State University 2018 Dissertation Committee: Dr. Sebastian Kurtek, Advisor Dr. Oksana Chkrebtii Dr. Steven N. MacEachern c Copyright by Justin Strait 2018 Abstract Due to mathematical and computational advances, the study of shape data is of great interest in numerous fields, including biology, medicine, computer vision, and biometrics. Shape can be defined as a property of an object which remains after removing variability associated with shape-preserving transformations, including translation, scale, rotation, and, in some cases, re-parameterization. This type of data presents mathematical challenges, as objects may have identical shape despite appearing differently in Euclidean space. The complex structure of shape data requires tools from fields such as differential geometry, algebra, and functional analysis. One challenge in analyzing shape data is the choice of shape representation used for subsequent comparison and statistical modeling. Two of the primary choices in existing lit- erature are landmark-based and function-based. Landmarks are a finite collection of labeled points pre-specified by the researcher. Once shape-preserving transformations are accounted for, landmark sets can be analyzed using multivariate statistical techniques. More recently, there has been a push to develop infinite-dimensional shape representations, which treat an object's outline using a continuous function. This thesis explores the interplay between elastic shape representations (a special type of function-based representation) and landmark sets. A primary contribution is the development of a joint landmark-elastic shape represen- tation, which allows researchers to represent shape by a function, while also incorporating landmark constraints provided by subject-matter experts. We demonstrate improvement in ii shape comparison, and present some available statistical tools using this representation. In many cases, we show improved performance in tasks such as clustering and classification. Under the aforementioned landmark-constrained elastic shape representation, this thesis also introduces a weighted metric. This allows one to emphasize features which are deemed important, which in turn affects optimal pairwise registration of shapes. If weights cannot be easily specified, this metric can allow for inference of these weights through optimiza- tion over a particular statistical task. Lastly, this thesis presents a model for inference of landmark locations, if they are unknown to the researcher. Annotating curves with land- marks is subject to uncertainty, and can be time-consuming for larger datasets. We present a hierarchical model for automatic detection of landmark locations, as well as the number of landmarks. Computational Bayesian techniques provide efficient posterior inference and uncertainty quantification. All methods discussed are applied to simulated curves, as well as data provided from various disciplines, including biology, medical imaging, and forensic analysis. iii Acknowledgments First of all, I would like to thank my parents, Kerry and Yongmi Strait, for being an integral part of my life. Without their love and support, I would not be where I am today. I also want to acknowledge my Ohio relatives, who I have gotten to interact with much more during my time here { they have made this place feel like home. There are several people in the Department of Statistics at Ohio State that I am grateful for. First, I must thank my advisor, Dr. Sebastian Kurtek, who has been instrumental in my success throughout my graduate school career. He has been a tremendous mentor, full of enthusiasm for both the field of statistics and the role of an academic. His selflessness has allowed me numerous opportunities that I would have never expected, and his ability to motivate and encourage has inspired self-belief in becoming a better researcher. I should also thank him for always having his office door open; I certainly appreciated being able to stop by for short or long conversations (even if he did not), and hope to be as interactive with students that I mentor in the future. I am also thankful for the other members of my candidacy and dissertation committee, Dr. Oksana Chkrebtii, Dr. Steven MacEachern, and Dr. Yoonkyung Lee, for their con- structive feedback throughout the progress of this work. I would like to thank Dr. Michelle Everson for being an outstanding teaching role model. Her positive outlook and desire to constantly improve is contagious, and I would not have had the teaching experiences I was afforded without her mentorship and support. I am extremely grateful for the rest of the iv Department of Statistics at Ohio State, including the friendly staff and faculty. Finally, I would like to thank Dr. Lajos Horvath at the University of Utah for agreeing to write a letter of recommendation for graduate school if and only if I received an A in his course. His Mathematical Statistics I class was my first statistics course ever, and inspired me to continue my studies in the field. I sometimes wonder where my life would have taken me had I gotten a B in that class. Finally, I want to acknowledge my friends here in the past five years of graduate school. Even though people have come and gone, I have always felt grateful to have a stable, amazing group of people to interact with. First, to Corey, Matt, Tim, Emily, and Xiaofei, thank you for being great partners in crime from day 1 of 2013. We made it through the first year classes together, celebrated and commiserated, and have so many stories and memories that I look forward to reminiscing on in the future. To Abhijoy, Deborah, Jason, Michael, Nate, Shreyan, and Sophie (my fellow co-president!), you all came into my life in so many different ways, but I am lucky to have such a close knit group of friends. While I may not always explicitly say so, I could not have made it without your friendship. v Vita 2013 . .B.S., Mathematics, B.S., Atmospheric Science, University of Utah 2016 . .M.S., Statistics, The Ohio State University 2013{2017 . Graduate Teaching Associate, Department of Statistics, The Ohio State University 2017 . .Graduate Research Associate, Department of Statistics, The Ohio State University 2017{2018 . Presidential Fellow, The Ohio State University Publications Research Publications Strait, J. and Kurtek, S. \Landmark-constrained statistical shape analysis of elastic curves and surfaces." Book chapter in New Advances in Statistics and Data Science, ICSA Book Series in Statistics. Springer, 2018. Strait, J., Kurtek, S., Bartha, E., and MacEachern, S. Landmark-constrained elastic shape analysis of planar curves, Journal of the American Statistical Association, 112(518), 521-533, 2017. Fields of Study Major Field: Statistics vi Table of Contents Page Abstract ........................................... ii Acknowledgments ...................................... iv Vita ............................................. vi List of Tables ........................................ xii List of Figures ....................................... xiv 1. Introduction ...................................... 1 1.1 Overview .................................... 1 1.2 Literature Review ............................... 3 1.2.1 Landmark-Based Representations ................... 4 1.2.2 Function-Based Representations .................... 5 1.3 Organization and Contribution of Thesis ................... 7 1.4 Computing ................................... 8 1.5 Description of Data .............................. 8 1.5.1 MPEG-7 Computer Vision ....................... 8 1.5.2 Mice Vertebrae ............................. 9 1.5.3 Hawaiian Drosophila Fly Wings .................... 10 1.5.4 Signatures ................................ 10 1.5.5 Handwritten Digits ........................... 11 1.5.6 Brain Substructures .......................... 11 2. Overview of Shape Spaces .............................. 13 2.1 Kendall's Landmark Representation ..................... 14 2.2 Square Root Velocity Function Representation . 16 vii 2.2.1 Parameterized Curves ......................... 16 2.2.2 Definition of Square Root Velocity Function (SRVF) . 19 2.2.3 Benefits of SRVF Representation ................... 20 2.2.4 Connection to Elastic Metric ..................... 20 2.3 Removing Translation and Scale Variability . 23 2.3.1 Pre-Size-and-Shape Space ....................... 24 2.3.2 Pre-Shape Space ............................ 25 2.4 Removing Rotation and Parameterization Variability ............ 26 2.4.1 Limitation of Standardizing Parameterization . 26 2.4.2 Metrics on Quotient Spaces ...................... 29 2.4.3 Size-and-Shape Space ......................... 31 2.4.4 Shape Space .............................. 32 2.5 Registration ................................... 33 2.5.1 Optimizing over SO(2) ......................... 35 2.5.2 Optimizing over Γ ........................... 35 2.5.3 Comparing Geodesics on Pre-Shape and Shape Space . 36 2.6 Closed Curves .................................. 37 2.7 Summary .................................... 38 3. Landmark-Constrained Shape Spaces ........................ 40 3.1 Motivation ................................... 40 3.2 Introducing Landmark Constraints ...................... 42 3.2.1 Soft Landmark Constraints .....................
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