Functional and Shape Data Analysis

ISBN-13: 9781493981557 / Angielski / Miękka / 2018 / 447 str.

Anuj Srivastava; Eric P. Klassen

Opis

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.

Spis treści

Contents

1      Motivation for Function and Shape Analysis

1.1    Motivation

1.1.1    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 & Challenges

1.5    Organization of This Textbook  

2      Previous Techniques in Shape Analysis

2.1    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 Selection

2.3    Domain-Based Shape Representations

2.3.1    Level-Set Methods

2.3.2    Deformation-Based Shape Analysis

2.4    Exercises 

2.5    Bibliographic Notes  

3      Background: Relevant Tools from Geometry

3.1    Equivalence Relations 

3.2    Riemannian Structure and Geodesics  

3.3    Geodesics in Spaces of Curves on Manifolds

3.4    Parallel Transport of Vectors  

3.5    Lie Group Actions on Manifolds

3.5.1    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 Exercises

3.12  Bibliographic Notes  

4      Functional Data and Elastic Registration

4.1    Goals and Challenges 

4.2    Estimating Function Variables from Discrete Data 

4.3    Geometries of Some Function Spaces

4.3.1    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 Components

4.7.1    Amplitude of a Function 

4.7.2    Relative Phase Between Functions 

4.7.3    A Convenient Approximation

4.8    SRSF-Based Registration 

4.8.1    Registration Problem

4.8.2    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 Distances

4.10.1  Amplitude Space and A Metric Structure 

4.10.2  Phase Space and A Metric Structure 

4.11  Different Warping Actions and PDFs

4.11.1  Listing of Different Actions  

4.11.2  Probability Density Functions 

4.12  Exercises  

4.12.1  Theoretical Exercises 

4.12.2  Computational Exercises

4.13  Bibliographic Notes 

5      Shapes of Planar Curves

5.1    Goals & Challenges 

5.2    Parametric Representations of Curves 

5.3    General Framework

5.3.1    Mathematical Representations of Curves 

5.3.2    Shape-Preserving Transformations

5.4    Pre-Shape Spaces 

5.4.1    Riemannian Structure 

5.4.2    Geodesics in Pre-Shape Spaces

5.5    Shape Spaces

5.5.1    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 Matching

5.7.2    Geodesic Illustrations

5.8    Gradient-Based Optimization Over Re-Parameterization Group

5.9    Summary 

5.10  Exercises  

5.10.1  Theoretical Exercises

5.10.2  Computational Exercises

5.11  Bibliographic Notes

6      Shapes of Planar Closed Curves

6.1    Goals and Challenges

6.2    Representations of Closed Curves

6.2.1    Pre-Shape Spaces

6.2.2    Riemannian Structures 

6.2.3    Removing Parameterization  

6.3    Projection on a Manifold 

6.4    Geodesic Computation

6.5    Geodesic Computation: Shooting Method 

6.5.1    Example 1: Geodesics on S2

6.5.2    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 S2

6.6.4    Example 2: Geodesics in Elastic Pre-Shape Space 

6.7    Geodesics in Shape Spaces

6.7.1    Geodesics in Non-Elastic Shape Space  

6.7.2    Geodesics in Elastic Shape Space

6.8    Examples of Elastic Geodesics 

6.8.1    Elastic Matching: Gradient Versus Dynamic Programming Algorithm

6.8.2    Fast Approximate Elastic Matching of Closed Curves

6.9    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 Silhouettes

6.11  Symmetry Analysis of Planar Shapes

6.12  Exercises  

6.12.1  Theoretical Exercises

6.12.2  Computational Exercises

6.13  Bibliographic Notes 

7      Statistical Modeling on Nonlinear Manifolds

7.1    Goals & Challenges  

7.2    Basic Setup  

7.3    Probability Densities on Manifolds

7.4    Summary Statistics on Manifolds  

7.4.1    Intrinsic Statistics

7.4.2    Extrinsic Statistics  

7.5    Examples on Some Useful Manifolds

7.5.1    Statistical Analysis on S1

7.5.2    Statistical Analysis on S2

7.5.3    Space of Probability Density Functions

7.5.4    Space of Warping Functions

7.6    Statistical Analysis on a Quotient Space M=G

7.6.1    Quotient Space as Orthogonal Section

7.6.2    General Case: Without Using Sections 

7.7    Exercises

7.7.1    Theoretical Exercises

7.7.2    Computational Exercises 

7.8    Bibliographic Notes 

8      Statistical Modeling of Functional Data

8.1    Goals and Challenges 

8.2    Template-Based Alignment & L2 Metric 

8.3    Elastic Phase-Amplitude Separation

8.3.1    Karcher Mean of Amplitudes 

8.3.2    Template: Center of the Mean Orbit

8.3.3    Phase-Amplitude Separation Algorithm  

8.4    Alternate Interpretation as Estimation of Model Parameters 

8.5    Phase-Amplitude Separation After Transformation

8.6    Penalized Function Alignment

8.7    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-FPCA

8.8.3    Joint Modeling of Principle Coefficients 

8.9    Joint Approach: Elastic FPCA

8.9.1    Model-Based Elastic FPCA in Function Space F

8.9.2    Elastic FPCA Using SRSF Representation  

8.10  Exercises  

8.10.1  Theoretical Exercises

8.10.2  Computational Exercises 

8.11  Bibliographic Notes 

9      Statistical Modeling of Planar Shapes

9.1    Goals & Challenges  

9.2    Clustering in Shape Spaces

9.2.1    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 S2

9.4.1    Truncated Wrapped-Normal (TWN) Model  

9.4.2    Learning TWN Model from Training Shapes in S2

9.5    Models for Planar Closed Curves 

9.6    Beyond TWN Shape Models

9.7    Modeling Nuisance Variables 

9.7.1    Modeling Re-Parameterization Function

9.7.2    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 Data

9.9.2    Maximum Likelihood Estimation of Model Parameters  

9.10  Problems

9.10.1  Theoretical Problems 

9.10.2  Computational Problems 

9.11  Bibliographic Notes 

 

10    Shapes of Curves in Higher Dimensions

10.1  Goals & Challenges  

10.2  Mathematical Representations of Curves 

10.3  Elastic and Non-Elastic Metrics

10.4  Shapes Spaces of Curves in Rn

10.4.1  Under Direction Function Representation 

10.4.2  Under SRVF Representation

10.4.3  Hierarchical Clustering of Elastic Curves  

10.4.4  Sample Statistics and Modeling of Elastic Curves in Rn

10.5  Registration of Curves 

10.5.1  Pairwise Registration of Curves in Rn

10.5.2  Registration of Multiple Curves 

10.6  Shapes of Closed Curves in Rn

10.6.1  Non-Elastic Closed Curves 

10.6.2  Elastic Closed Curves  

10.7  Shape Analysis of Augmented Curves

10.7.1  Joint Representation of Augmented Curves 

10.7.2  Invariances and Equivalence Classes

10.8  Problems

10.8.1  Theoretical Problems 

10.8.2  Computational Problems 

10.9  Bibliographic Notes 

11    Related Topics in Shape Analysis of Curves

11.1  Goals and Challenges 

11.2  Joint Analysis of Shape and Other Features 

11.2.1  Geodesics and Geodesic Distances on Feature Spaces

11.2.2  Feature-Based Clustering

11.3  Affine-Invariant Shape Analysis of Planar Curves 

11.3.1  Global Section Under the Affine Action  

11.3.2  Geodesics Using Path-Straightening Algorithm

11.4  Registration of Trajectories on Nonlinear Manifolds

11.4.1  Transported SRVF for Trajectories

11.4.2  Analysis of Trajectories on S2

11.5  Problems 

11.5.1  Theoretical Problems

11.5.2  Computational Problems

11.6  Bibliographic Notes  

A     Background Material

A.1   Basic Differential Geometry

A.1.1   Tangent spaces on a manifold

A.1.2   Submanifolds 

A.2   Basic Algebra 

A.3   Basic Geometry of Function Spaces

A.3.1   Hilbert Manifolds & Submanifolds 

B     The Dynamic Programming Algorithm

B.1   Theoretical Setup

B.2   Computer Implementation  

References

Index

O autorze

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.

Blurb

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 explanations

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.