Contents1 Motivation for Function and Shape Analysis1.1 Motivation1.1.1 Need for Function and Shape Data Analysis Tools 1.1.2 Why Continuous Shapes? 1.2 Important Application Areas 1.3 Specific Technical Goals 1.4 Issues & Challenges1.5 Organization of This Textbook
2 Previous Techniques in Shape Analysis2.1 Principal Component Analysis (PCA)2.2 Point-Based Methods 2.2.1 ICP: Point Cloud Analysis 2.2.2 Active Shape Models 2.2.3 Kendall's Landmark-Based Shape Analysis 2.2.4 Issue of Landmark Selection2.3 Domain-Based Shape Representations2.3.1 Level-Set Methods2.3.2 Deformation-Based Shape Analysis2.4 Exercises 2.5 Bibliographic Notes
3 Background: Relevant Tools from Geometry3.1 Equivalence Relations 3.2 Riemannian Structure and Geodesics 3.3 Geodesics in Spaces of Curves on Manifolds3.4 Parallel Transport of Vectors 3.5 Lie Group Actions on Manifolds3.5.1 Actions of Single Groups 3.5.2 Actions of Product Groups 3.6 Quotient Spaces of Riemannian Manifolds 3.7 Quotient Spaces as Orthogonal Sections 3.8 General Quotient Spaces 3.9 Distances in Quotient Spaces: A Summary 3.10 Center of An Orbit 3.11 Exercises 3.11.1 Theoretical Exercises 3.11.2 Computational Exercises3.12 Bibliographic Notes
4 Functional Data and Elastic Registration4.1 Goals and Challenges 4.2 Estimating Function Variables from Discrete Data 4.3 Geometries of Some Function Spaces4.3.1 Geometry of Hilbert Spaces 4.3.2 Unit Hilbert Sphere 4.3.3 Group of Warping Functions 4.4 Function Registration Problem 4.5 Use of L2-Norm And Its Limitations 4.6 Square-Root Slope Function (SRSF) Representation 4.7 Definition of Phase & Amplitude Components4.7.1 Amplitude of a Function 4.7.2 Relative Phase Between Functions 4.7.3 A Convenient Approximation4.8 SRSF-Based Registration 4.8.1 Registration Problem4.8.2 SRSF Alignment Using Dynamic Programming 4.8.3 Examples of Functional Alignments 4.9 Connection to the Fisher-Rao Metric 4.10 Phase and Amplitude Distances4.10.1 Amplitude Space and A Metric Structure 4.10.2 Phase Space and A Metric Structure 4.11 Different Warping Actions and PDFs4.11.1 Listing of Different Actions 4.11.2 Probability Density Functions 4.12 Exercises 4.12.1 Theoretical Exercises 4.12.2 Computational Exe
About the Author: Anuj Srivastava is a Professor in the Department of Statistics and a Distinguished Research Professor at Florida State University. His areas of interest include statistical analysis on nonlinear manifolds, statistical computer vision, functional data analysis, and statistical shape theory. He has been the associate editor for the Journal of Statistical Planning and Inference, and several IEEE journals. He is a fellow of the International Association of Pattern Recognition (IAPR) and a senior member of the Institute for Electrical and Electronic Engineers (IEEE).
Eric Klassen is a Professor in the Department of Mathematics at Florida State University. His mathematical interests include topology, geometry, and shape analysis. In his spare time, he enjoys playing the piano, riding his bike, and contra dancing.