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Part of: Synthesis digital library of engineering and computer science.

Includes bibliographical references (pages 155-165).

1. Problem introduction and motivation -- 1.1 Problem area: 3D shape analysis -- 1.2 General goals and challenges -- 1.3 Past approaches and their limitations -- 1.4 Our approach: elastic shape analysis -- 1.5 Organization of this book -- 1.6 Notation --

2. Elastic shape analysis: metrics and representations -- 2.1 Shapes -- 2.2 Elastic shape analysis -- 2.2.1 Encoding of registration -- 2.2.2 Riemannian metric and optimal registration -- 2.2.3 Geometric invariance -- 2.3 Background: elastic framework for curves -- 2.3.1 Elastic metric for curves -- 2.3.2 Geometric invariance -- 2.3.3 Summary of elastic framework for curves -- 2.4 Elastic framework for surfaces -- 2.4.1 Square-root map -- 2.4.2 Generalizing the elastic metric for curves -- 2.4.3 Elastic metric for surfaces -- 2.4.4 Reduced elastic metric: square-root normal field -- 2.4.5 Geometric invariance -- 2.4.6 SRNF inversion problem -- 2.5 Summary and next steps -- 2.6 Bibliographic notes --

3. Computing geometrical quantities -- 3.1 Computing in shape space -- 3.1.1 Optimal registration and alignment -- 3.1.2 Optimal deformation -- 3.1.3 Putting it all together -- 3.1.4 Simplifying the computations using SRNFs -- 3.2 Registration and alignment using SRNFs -- 3.2.1 Optimization over the rotation group -- 3.2.2 Optimization over the reparameterization group -- 3.3 Geodesic computation techniques on general manifolds -- 3.3.1 Geodesic computation via path-straightening -- 3.3.2 Geodesic computation via shooting -- 3.4 Elastic geodesic paths between surfaces using pullback metrics -- 3.4.1 Path-straightening under pullback metrics -- 3.4.2 Shooting geodesics under SRNF pullback metric -- 3.5 Elastic geodesic paths between surfaces using SRNF inversion -- 3.5.1 Geodesics using SRNF inversion -- 3.5.2 Parallel transport in SRNF space -- 3.6 Elastic geodesic path examples -- 3.6.1 Discretization -- 3.6.2 Path-straightening -- 3.6.3 Shooting method -- 3.6.4 SRNF inversion -- 3.7 Summary and next steps -- 3.8 Bibliographic notes --

4. Statistical analysis of shapes -- 4.1 Statistical summaries of 3D shapes -- 4.1.1 Pullback metric approach -- 4.1.2 SRNF inversion approach -- 4.2 Statistical models on shape spaces -- 4.2.1 Tangent space and pullback metric approach -- 4.2.2 SRNF inversion approach -- 4.3 Clustering and classification -- 4.4 Bibliographic notes --

5. Case studies using human body and anatomical shapes -- 5.1 Clustering and classification -- 5.1.1 Attention deficit hyperactivity disorder (ADHD) classification -- 5.1.2 Clustering of identity and pose of human body shapes -- 5.2 Geodesic deformation -- 5.2.1 Geodesics -- 5.2.2 Deformation transfer -- 5.2.3 Reflection symmetry analysis and symmetrization -- 5.3 Statistical summaries of shapes -- 5.3.1 Means and modes of variation -- 5.3.2 Random sampling from shape models -- 5.4 Bibliographic notes --

6. Landmark-driven elastic shape analysis -- 6.1 Problem statement -- 6.2 Landmark-guided registration -- 6.2.1 Initial registration using landmarks -- 6.2.2 Registration using landmark-constrained diffeomorphisms -- 6.2.3 Landmark-constrained basis for registration -- 6.3 Elastic geodesics under landmark constraints -- 6.3.1 Illustration of geodesic paths -- 6.3.2 Evaluation of performance and computational cost -- 6.4 Landmark-constrained 3d shape atlas -- 6.5 Bibliographic notes --

A. Differential geometry -- Differentiable manifolds and tangent spaces -- Riemannian manifolds, geodesics, and the exponential map -- Geodesics -- Exponential map -- Lie group actions and quotient spaces -- B. Differential geometry of surfaces -- C. Spherical parametrization of triangulated meshes -- Conformal spherical mapping -- Coarse-to-fine minimal stretch embedding -- D. Landmark detection -- Landmark detection using heat kernel signatures -- Landmark correspondences -- Bibliography -- Authors' biographies.

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Statistical analysis of shapes of 3D objects is an important problem with a wide range of applications. This analysis is difficult for many reasons, including the fact that objects differ in both geometry and topology. In this manuscript, we narrow the problem by focusing on objects with fixed topology, say objects that are diffeomorphic to unit spheres, and develop tools for analyzing their geometries. The main challenges in this problem are to register points across objects and to perform analysis while being invariant to certain shape-preserving transformations. We develop a comprehensive framework for analyzing shapes of spherical objects, i.e., objects that are embeddings of a unit sphere in R3 , including tools for: quantifying shape differences, optimally deforming shapes into each other, summarizing shape samples, extracting principal modes of shape variability, and modeling shape variability associated with populations. An important strength of this framework is that it is elastic: it performs alignment, registration, and comparison in a single unified framework, while being invariant to shape-preserving transformations. The approach is essentially Riemannian in the following sense. We specify natural mathematical representations of surfaces of interest, and impose Riemannian metrics that are invariant to the actions of the shape-preserving transformations. In particular, they are invariant to reparameterizations of surfaces. While these metrics are too complicated to allow broad usage in practical applications, we introduce a novel representation, termed square-root normal fields (SRNFs), that transform a particular invariant elastic metric into the standard L2 metric. As a result, one can use standard techniques from functional data analysis for registering, comparing, and summarizing shapes. Specifically, this results in: pairwise registration of surfaces; computation of geodesic paths encoding optimal deformations; computation of Karcher means and covariances under the shape metric; tangent Principal Component Analysis (PCA) and extraction of dominant modes of variability; and finally, modeling of shape variability using wrapped normal densities. These ideas are demonstrated using two case studies: the analysis of surfaces denoting human bodies in terms of shape and pose variability; and the clustering and classification of the shapes of subcortical brain structures for use in medical diagnosis. This book develops these ideas without assuming advanced knowledge in differential geometry and statistics. We summarize some basic tools from differential geometry in the appendices, and introduce additional concepts and terminology as needed in the individual chapters.

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