Functional and Shape Data Analysis

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 Exercises6.13 Bibliographic Notes 7 Statistical Modeling on Nonlinear Manifolds7.1 Goals & Challenges 7.2 Basic Setup 7.3 Probability Densities on Manifolds7.4 Summary Statistics on Manifolds 7.4.1 Intrinsic Statistics7.4.2 Extrinsic Statistics 7.5 Examples on Some Useful Manifolds7.5.1 Statistical Analysis on S17.5.2 Statistical Analysis on S27.5.3 Space of Probability Density Functions7.5.4 Space of Warping Functions7.6 Statistical Analysis on a Quotient Space M=G7.6.1 Quotient Space as Orthogonal Section7.6.2 General Case: Without Using Sections 7.7 Exercises7.7.1 Theoretical Exercises7.7.2 Computational Exercises 7.8 Bibliographic Notes 8 Statistical Modeling of Functional Data8.1 Goals and Challenges 8.2 Template-Based Alignment & L2 Metric 8.3 Elastic Phase-Amplitude Separation8.3.1 Karcher Mean of Amplitudes 8.3.2 Template: Center of the Mean Orbit8.3.3 Phase-Amplitude Separation Algorithm 8.4 Alternate Interpretation as Estimation of Model Parameters 8.5 Phase-Amplitude Separation After Transformation8.6 Penalized Function Alignment8.7 Function Components, Alignment and Modeling 8.8 Sequential Approach 8.8.1 FPCA of Amplitude Functions: A-FPCA 8.8.2 FPCA of Phase Functions: P-FPCA8.8.3 Joint Modeling of Principle Coefficients 8.9 Joint Approach: Elastic FPCA8.9.1 Model-Based Elastic FPCA in Function Space F8.9.2 Elastic FPCA Using SRSF Representation 8.10 Exercises 8.10.1 Theoretical Exercises8.10.2 Computational Exercises 8.11 Bibliographic Notes 9 Statistical Modeling of Planar Shapes9.1 Goals & Challenges 9.2 Clustering in Shape Spaces9.2.1 Hierarchical Clustering 9.2.2 A Minimum-Dispersion Clustering 9.3 A Finite Representation of Planar Shapes 9.3.1 Shape Representation: A Brief Review 9.3.2 Finite Shape Representation: Planar Curves 9.3.3 Finite Representation: Planar Closed Curves 9.4 Models for Planar Curves as Elements of S29.4.1 Truncated Wrapped-Normal (TWN) Model 9.4.2 Learning TWN Model from Training Shapes in S29.5 Models for Planar Closed Curves 9.6 Beyond TWN Shape Models9.7 Modeling Nuisance Variables 9.7.1 Modeling Re-Parameterization Function9.7.2 Modeling Shape Orientations 9.8 Classification of Shapes With Contour Data 9.8.1 Nearest-Neighbor Classification 9.8.2 Probabilistic Classification 9.9 Detection/Classification of Shapes in Cluttered Point Clouds 9.9.1 Point Process Models for Cluttered Data9.9.2 Maximum Likelihood Estimation of Model Parameters 9.10 Problems9.10.1 Theoretical Problems 9.10.2 Computational Problems 9.11 Bibliographic Notes 10 Shapes of Curves in Higher Dimensions10.1 Goals & Challenges 10.2 Mathematical Representations of Curves 10.3 Elastic and Non-Elastic Metrics10.4 Shapes Spaces of Curves in Rn10.4.1 Under Direction Function Representation 10.4.2 Under SRVF Representation10.4.3 Hierarchical Clustering of Elastic Curves 10.4.4 Sample Statistics and Modeling of Elastic Curves in Rn10.5 Registration of Curves 10.5.1 Pairwise Registration of Curves in Rn10.5.2 Registration of Multiple Curves 10.6 Shapes of Closed Curves in Rn10.6.1 Non-Elastic Closed Curves 10.6.2 Elastic Closed Curves 10.7 Shape Analysis of Augmented Curves10.7.1 Joint Representation of Augmented Curves 10.7.2 Invariances and Equivalence Classes10.8 Problems10.8.1 Theoretical Problems 10.8.2 Computational Problems 10.9 Bibliographic Notes 11 Related Topics in Shape Analysis of Curves11.1 Goals and Challenges 11.2 Joint Analysis of Shape and Other Features 11.2.1 Geodesics and Geodesic Distances on Feature Spaces11.2.2 Feature-Based Clustering11.3 Affine-Invariant Shape Analysis of Planar Curves 11.3.1 Global Section Under the Affine Action 11.3.2 Geodesics Using Path-Straightening Algorithm11.4 Registration of Trajectories on Nonlinear Manifolds11.4.1 Transported SRVF for Trajectories11.4.2 Analysis of Trajectories on S211.5 Problems 11.5.1 Theoretical Problems11.5.2 Computational Problems11.6 Bibliographic Notes A Background MaterialA.1 Basic Differential GeometryA.1.1 Tangent spaces on a manifoldA.1.2 Submanifolds A.2 Basic Algebra A.3 Basic Geometry of Function SpacesA.3.1 Hilbert Manifolds & Submanifolds B The Dynamic Programming AlgorithmB.1 Theoretical SetupB.2 Computer Implementation References Index