This textbook for courses on function data analysis and shape data analysis describes how to define, compare, and mathematically represent shapes, with a focus on statistical modeling and inference. It is aimed at graduate students in analysis in statistics, engineering, applied mathematics, neuroscience, biology, bioinformatics, and other related areas. The interdisciplinary nature of the broad range of ideas covered-from introductory theory to algorithmic implementations and some statistical case studies-is meant to familiarize graduate students with an array of tools that are relevant in developing computational solutions for shape and related analyses. These tools, gleaned from geometry, algebra, statistics, and computational science, are traditionally scattered across different courses, departments, and disciplines; Functional and Shape Data Analysis offers a unified, comprehensive solution by integrating the registration problem into shape analysis, better preparing graduate students for handling future scientific challenges.

Recently, a data-driven and application-oriented focus on shape analysis has been trending. This text offers a self-contained treatment of this new generation of methods in shape analysis of curves. Its main focus is shape analysis of functions and curves-in one, two, and higher dimensions-both closed and open. It develops elegant Riemannian frameworks that provide both quantification of shape differences and registration of curves at the same time. Additionally, these methods are used for statistically summarizing given curve data, performing dimension reduction, and modeling observed variability. It is recommended that the reader have a background in calculus, linear algebra, numerical analysis, and computation.

• Presents a complete and detailed exposition on statistical analysis of shapes that includes appendices, background material, and exercises, making this text a self-contained reference
• Addresses and explores the next generation of shape analysis
• Focuses on providing a working knowledge of a broad range of relevant material, foregoing in-depth technical details and elaborate mathematical explanationsAnuj 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.

Autorentext

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.

Inhalt

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 Exercises4.13 Bibliographic Notes 5 Shapes of Planar Curves5.1 Goals & Challenges 5.2 Parametric Representations of Curves 5.3 General Framework5.3.1 Mathematical Representations of Curves 5.3.2 Shape-Preserving Transformations5.4 Pre-Shape Spaces 5.4.1 Riemannian Structure 5.4.2 Geodesics in Pre-Shape Spaces5.5 Shape Spaces5.5.1 Removing Parameterization 5.6 Motivation for SRVF Representation 5.6.1 What is an Elastic Metric?5.6.2 Significance of the Square-Root Representation 5.7 Geodesic Paths in Shape Spaces 5.7.1 Optimal Re-Parameterization for Curve Matching5.7.2 Geodesic Illustrations5.8 Gradient-Based Optimization Over Re-Parameterization Group5.9 Summary 5.10 Exercises 5.10.1 Theoretical Exercises5.10.2 Computational Exercises5.11 Bibliographic Notes 6 Shapes of Planar Closed Curves6.1 Goals and Challenges6.2 Representations of Closed Curves6.2.1 Pre-Shape Spaces6.2.2 Riemannian Structures 6.2.3 Removing Parameterization 6.3 Projection on a Manifold 6.4 Geodesic Computation6.5 Geodesic Computation: Shooting Method 6.5.1 Example 1: Geodesics on S26.5.2 Example 2: Geodesics in Non-Elastic Pre-Shape Space 6.6 Geodesic Computation: Path Straightening Method 6.6.1 Theoretical Background 6.6.2 Numerical Implementation 6.6.3 Example 1: Geodesics on S26.6.4 Example 2: Geodesics in Elastic Pre-Shape Space 6.7 Geodesics in Shape Spaces6.7.1 Geodesics in Non-Elastic Shape Space 6.7.2 Geodesics in Elastic Shape Space6.8 Examples of Elastic Geodesics 6.8.1 Elastic Matching: Gradient Versus Dynamic Programming Algorithm6.8.2 Fast Approximate Elastic Matching of Closed Curves6.9 Elastic versus Non-Elastic Deformations 6.10 Parallel Transport of Shape Deformations 6.10.1 Prediction of Silhouettes from Novel Views 6.10.2 Classification of 3D Objects Using Predicted Silhouettes6.11 Symmetry Analysis of Planar Shapes6.12 Exercises 6.12.1 Theoretical Exercises6.12.2 Computational Exercises…