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E-raamat: Functional and Shape Data Analysis

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  • Formaat: PDF+DRM
  • Sari: Springer Series in Statistics
  • Ilmumisaeg: 03-Oct-2016
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9781493940202
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Functional and Shape Data AnalysisSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Springer Series in Statistics
  • Ilmumisaeg: 03-Oct-2016
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9781493940202
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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.

1 Motivation for Function and Shape Analysis
1(20)
1.1 Motivation
2(3)
1.1.1 Need for Function and Shape Data Analysis Tools
2(1)
1.1.2 Why Continuous Shapes?
3(2)
1.2 Important Application Areas
5(6)
1.3 Specific Technical Goals
11(6)
1.4 Issues and Challenges
17(2)
1.5 Organization of this Textbook
19(2)
2 Previous Techniques in Shape Analysis
21(18)
2.1 Principal Component Analysis
23(1)
2.2 Point-Based Shape-Analysis Methods
24(9)
2.2.1 ICP: Point Cloud Analysis
24(5)
2.2.2 Active Shape Models
29(1)
2.2.3 Kendall's Landmark-Based Shape Analysis
30(2)
2.2.4 Issue of Landmark Selection
32(1)
2.3 Domain-Based Shape Representations
33(3)
2.3.1 Level-Set Methods
33(2)
2.3.2 Deformation-Based Shape Analysis
35(1)
2.4 Exercises
36(1)
2.5 Bibliographic Notes
37(2)
3 Background: Relevant Tools from Geometry
39(34)
3.1 Equivalence Relations
40(1)
3.2 Riemannian Structure and Geodesies
41(7)
3.3 Geodesies in Spaces of Curves on Manifolds
48(2)
3.4 Parallel Transport of Vectors
50(2)
3.5 Lie Group Actions on Manifolds
52(4)
3.5.1 Actions of Single Groups
53(2)
3.5.2 Actions of Product Groups
55(1)
3.6 Quotient Spaces of Riemannian Manifolds
56(4)
3.7 Quotient Spaces as Orthogonal Sections
60(5)
3.8 General Quotient Spaces
65(2)
3.9 Distances in Quotient Spaces: A Summary
67(1)
3.10 Center of an Orbit
67(2)
3.11 Exercises
69(3)
3.11.1 Theoretical Exercises
69(3)
3.11.2 Computational Exercises
72(1)
3.12 Bibliographic Notes
72(1)
4 Functional Data and Elastic Registration
73(52)
4.1 Goals and Challenges
74(1)
4.2 Estimating Function Variables from Discrete Data
75(3)
4.3 Geometries of Some Function Spaces
78(7)
4.3.1 Geometry of Hilbert Spaces
78(4)
4.3.2 Unit Hilbert Sphere
82(1)
4.3.3 Group of Warping Functions
83(2)
4.4 Function Registration Problem
85(3)
4.5 Use of L2-Norm and Its Limitations
88(3)
4.6 Square-Root Slope Function Representation
91(2)
4.7 Definition of Phase and Amplitude Components
93(5)
4.7.1 Amplitude of a Function
94(2)
4.7.2 Relative Phase Between Functions
96(1)
4.7.3 A Convenient Approximation
97(1)
4.8 SRSF-Based Registration
98(6)
4.8.1 Registration Problem
98(1)
4.8.2 SRSF Alignment Using Dynamic Programming
99(1)
4.8.3 Examples of Functional Alignments
100(4)
4.9 Connection to the Fisher-Rao Metric
104(3)
4.10 Phase and Amplitude Distances
107(5)
4.10.1 Amplitude Space and a Metric Structure
107(2)
4.10.2 Phase Space and a Metric Structure
109(3)
4.11 Different Warping Actions and PDFs
112(6)
4.11.1 Listing of Different Actions
112(1)
4.11.2 Probability Density Functions
113(5)
4.12 Exercises
118(4)
4.12.1 Theoretical Exercises
118(3)
4.12.2 Computational Exercises
121(1)
4.13 Bibliographic Notes
122(3)
5 Shapes of Planar Curves
125(42)
5.1 Goals and Challenges
125(1)
5.2 Parametric Representations of Curves
126(2)
5.3 General Framework
128(5)
5.3.1 Mathematical Representations of Curves
128(4)
5.3.2 Shape-Preserving Transformations
132(1)
5.4 Pre-shape Spaces
133(8)
5.4.1 Riemannian Structure
135(3)
5.4.2 Geodesies in Pre-shape Spaces
138(3)
5.5 Shape Spaces
141(3)
5.5.1 Removing Parameterization
141(3)
5.6 Motivation for SRVF Representation
144(5)
5.6.1 What Is an Elastic Metric?
144(4)
5.6.2 Significance of the Square-Root Representation
148(1)
5.7 Geodesic Paths in Shape Spaces
149(7)
5.7.1 Optimal Re-Parameterization for Curve Matching
152(1)
5.7.2 Geodesic Illustrations
152(4)
5.8 Gradient-Based Optimization Over Re-Parameterization Group
156(4)
5.9 Summary
160(1)
5.10 Exercises
160(4)
5.10.1 Theoretical Exercises
160(3)
5.10.2 Computational Exercises
163(1)
5.11 Bibliographic Notes
164(3)
6 Shapes of Planar Closed Curves
167(66)
6.1 Goals and Challenges
167(1)
6.2 Representations of Closed Curves
168(10)
6.2.1 Pre-shape Spaces
170(1)
6.2.2 Riemannian Structures
171(3)
6.2.3 Removing Parameterization
174(4)
6.3 Projection on a Manifold
178(1)
6.4 Geodesic Computation
179(1)
6.5 Geodesic Computation: Shooting Method
180(9)
6.5.1 Example 1: Geodesies on S2
182(3)
6.5.2 Example 2: Geodesies in Non-elastic Pre-shape Space
185(4)
6.6 Geodesic Computation: Path-Straightening Method
189(15)
6.6.1 Theoretical Background
189(5)
6.6.2 Numerical Implementation
194(3)
6.6.3 Example 1: Geodesies on S2
197(1)
6.6.4 Example 2: Geodesies in Elastic Pre-shape Space
197(7)
6.7 Geodesies in Shape Spaces
204(11)
6.7.1 Geodesies in Non-elastic Shape Space
204(4)
6.7.2 Geodesies in Elastic Shape Space
208(7)
6.8 Examples of Elastic Geodesies
215(3)
6.8.1 Elastic Matching: Gradient Versus Dynamic Programming Algorithm
217(1)
6.8.2 Fast Approximate Elastic Matching of Closed Curves
218(1)
6.9 Elastic Versus Non-elastic Deformations
218(1)
6.10 Parallel Transport of Shape Deformations
219(6)
6.10.1 Prediction of Silhouettes from Novel Views
223(1)
6.10.2 Classification of 3D Objects Using Predicted Silhouettes
224(1)
6.11 Symmetry Analysis of Planar Shapes
225(3)
6.12 Exercises
228(3)
6.12.1 Theoretical Exercises
228(1)
6.12.2 Computational Exercises
229(2)
6.13 Bibliographic Notes
231(2)
7 Statistical Modeling on Nonlinear Manifolds
233(36)
7.1 Goals and Challenges
233(1)
7.2 Basic Setup
234(1)
7.3 Probability Densities on Manifolds
235(1)
7.4 Summary Statistics on Manifolds
236(6)
7.4.1 Intrinsic Statistics
236(5)
7.4.2 Extrinsic Statistics
241(1)
7.5 Examples on Some Useful Manifolds
242(19)
7.5.1 Statistical Analysis on S1
242(6)
7.5.2 Statistical Analysis on S2
248(7)
7.5.3 Space of Probability Density Functions
255(4)
7.5.4 Space of Warping Functions
259(2)
7.6 Statistical Analysis on a Quotient Space M/G
261(4)
7.6.1 Quotient Space as Orthogonal Section
262(1)
7.6.2 General Case: Without Using Sections
263(2)
7.7 Exercises
265(2)
7.7.1 Theoretical Exercises
265(1)
7.7.2 Computational Exercises
266(1)
7.8 Bibliographic Notes
267(2)
8 Statistical Modeling of Functional Data
269(36)
8.1 Goals and Challenges
270(1)
8.2 Template-Based Alignment and L2 Metric
271(2)
8.3 Elastic Phase-Amplitude Separation
273(8)
8.3.1 Karcher Mean of Amplitudes
273(1)
8.3.2 Template: Center of the Mean Orbit
274(2)
8.3.3 Phase-Amplitude Separation Algorithm
276(5)
8.4 Alternate Interpretation as Estimation of Model Parameters
281(1)
8.5 Phase-Amplitude Separation After Transformation
282(2)
8.6 Penalized Function Alignment
284(2)
8.7 Function Components, Alignment, and Modeling
286(2)
8.8 Sequential Approach
288(6)
8.8.1 FPCA of Amplitude Functions: A-FPCA
289(1)
8.8.2 FPCA of Phase Functions: P-FPCA
290(2)
8.8.3 Joint Modeling of Principle Coefficients
292(2)
8.9 Joint Approach: Elastic FPCA
294(7)
8.9.1 Model-Based Elastic FPCA in Function Space
294(3)
8.9.2 Elastic FPCA Using SRSF Representation
297(4)
8.10 Exercises
301(2)
8.10.1 Theoretical Exercises
301(1)
8.10.2 Computational Exercises
302(1)
8.11 Bibliographic Notes
303(2)
9 Statistical Modeling of Planar Shapes
305(44)
9.1 Goals and Challenges
306(1)
9.2 Clustering in Shape Spaces
307(4)
9.2.1 Hierarchical Clustering
307(2)
9.2.2 A Minimum-Dispersion Clustering
309(2)
9.3 A Finite Representation of Planar Shapes
311(6)
9.3.1 Shape Representation: A Brief Review
311(2)
9.3.2 Finite Shape Representation: Planar Curves
313(3)
9.3.3 Finite Representation: Planar Closed Curves
316(1)
9.4 Models for Planar Curves as Elements of S2
317(5)
9.4.1 Truncated Wrapped-Normal (TWN) Model
317(1)
9.4.2 Learning TWN Model from Training Shapes in S2
318(4)
9.5 Models for Planar Closed Curves
322(5)
9.6 Beyond TWN Shape Models
327(2)
9.7 Modeling Nuisance Variables
329(5)
9.7.1 Modeling Re-Parameterization Function
330(3)
9.7.2 Modeling Shape Orientations
333(1)
9.8 Classification of Shapes With Contour Data
334(5)
9.8.1 Nearest-Neighbor Classification
335(1)
9.8.2 Probabilistic Classification
335(4)
9.9 Detection/Classification of Shapes in Cluttered Point Clouds
339(6)
9.9.1 Point Process Models for Cluttered Data
341(2)
9.9.2 Maximum Likelihood Estimation of Model Parameters
343(2)
9.10 Problems
345(2)
9.10.1 Theoretical Problems
345(1)
9.10.2 Computational Problems
346(1)
9.11 Bibliographic Notes
347(2)
10 Shapes of Curves in Higher Dimensions
349(36)
10.1 Goals and Challenges
349(1)
10.2 Mathematical Representations of Curves
350(1)
10.3 Elastic and Non-elastic Metrics
351(2)
10.4 Shape Spaces of Curves in Rn
353(12)
10.4.1 Direction Function Representation
353(3)
10.4.2 Under SRVF Representation
356(5)
10.4.3 Hierarchical Clustering of Elastic Curves
361(1)
10.4.4 Sample Statistics and Modeling of Elastic Curves in Rn
362(3)
10.5 Registration of Curves
365(4)
10.5.1 Pair wise Registration of Curves in Rn
366(2)
10.5.2 Registration of Multiple Curves
368(1)
10.6 Shapes of Closed Curves in Rn
369(7)
10.6.1 Non-elastic Closed Curves
369(3)
10.6.2 Elastic Closed Curves
372(4)
10.7 Shape Analysis of Augmented Curves
376(6)
10.7.1 Joint Representation of Augmented Curves
378(1)
10.7.2 Invariances and Equivalence Classes
379(3)
10.8 Problems
382(2)
10.8.1 Theoretical Problems
382(2)
10.8.2 Computational Problems
384(1)
10.9 Bibliographic Notes
384(1)
11 Related Topics in Shape Analysis of Curves
385(32)
11.1 Goals and Challenges
385(1)
11.2 Joint Analysis of Shape and Other Features
386(5)
11.2.1 Geodesies and Geodesic Distances on Feature Spaces
387(2)
11.2.2 Feature-Based Clustering
389(2)
11.3 Affine-Invariant Shape Analysis of Planar Curves
391(10)
11.3.1 Global Section Under the Affine Action
393(3)
11.3.2 Geodesies Using Path-Straightening Algorithm
396(5)
11.4 Registration of Trajectories on Nonlinear Manifolds
401(12)
11.4.1 Transported SRVF for Trajectories
405(6)
11.4.2 Analysis of Trajectories on S2
411(2)
11.5 Problems
413(2)
11.5.1 Theoretical Problems
413(2)
11.5.2 Computational Problems
415(1)
11.6 Bibliographic Notes
415(2)
Background Material
417(18)
A.1 Basic Differential Geometry
417(10)
A.1.1 Tangent Spaces on a Manifold
421(4)
A.1.2 Submanifolds
425(2)
A.2 Basic Algebra
427(4)
A.3 Basic Geometry of Function Spaces
431(4)
A.3.1 Hilbert Manifolds and Submanifolds
432(3)
The Dynamic Programming Algorithm
435(4)
B.1 Theoretical Setup
435(1)
B.2 Computer Implementation
436(3)
References 439(6)
Index 445
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.
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