Functional Characterization of Intrinsic and Extrinsic Geometry 

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Consistent Functional Maps

Our method is based on the functional map representation introduced in [88]. In this section, we give a brief overview of the representation and the method used in [88] to construct a functional map for a given pair of shapes. While our method is general, throughout the chapter we assume that all shapes are represented as triangle meshes, and all functions are expressed as vectors in the basis of the eigenfunctions of the Laplace-Beltrami operator. This basis needs to be computed beforehand on each shape. The objective is to output a uniquely defined functional map.

Functional Map Representation

The functional map representation is based on the observation that given two surfaces M0 and Mi, a point-to-point map ϕi : Mi → M0 induces a map between function spaces Ci : L2(M0) → L2(Mi), where L2(M) is the set of square integrable functions defined on the surface M. The functional map Ci is defined by composition with ϕi as Cif = f ◦ ϕi. The operator Ci is a linear transformation and given a basis it can be represented as a matrix in the discrete setting. This matrix can be easily computed if the map ϕ is known. The basic method described in [88] approximates the functional map Ci using a set of linear constraints. The first type of constraints is given by a set of pairs of functions, which we refer to below as “probe functions”, that are expected to be preserved by the deformation. The second is a regularization term coming from the deformation model. This leads to the least squares problem:

Selection of the Best Functional Correspondences

The idea developed here is to assign a weight to each pair of probe functions. These weights can then be tuned according to their consistency in the matching. Since a priori there is no reason to choose one probe function over another, we propose to learn the optimal weights given a training set of shapes. As input we need a set of n triangulated meshes with known correspondences representing the same object undergoing a set of deformations. Our main assumption is that the optimal weights on the probe functions should be stable across the shapes in the collection. Thus, if we are given a new deformation of the same shape, the learned weight should also select the consistent probe functions. The output of our algorithm will be a set of weights for the probe functions, which, as we will show below, can then be used to find correspondences between new, unseen shape instances.

Basis function extraction

Since the probe functions can give redundant information in some shape parts and incomplete information in others, the resulting functional map will map some subspaces of L2(M0) with more confidence than others. Using a collection of shapes we would like to extract the most stable subspaces. For this purpose we propose to use the learned optimal weights D and the resulting estimated functional maps Xi(D) and to identify stably mapped functional subspaces by comparing Xi(D) to the reference maps Ci. The input here is the same as in the previous section. We need n shapes with known ground truth functional maps Ci and a set of consistent probe functions Gi. The output will be Y an orthonormal basis of L2(M0) ordered with decreasing confidence. As we demonstrate in Section 5.5, in most cases this order remains stable even for maps that are estimated to previously unseen shapes.

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Table of contents :

1 Introduction 
2 Introduction en français 
3 Context in Differential Geometry and its Discrete Equivalents 
3.1 Differential Geometry
3.1.1 Manifold
3.1.2 Tangent Vectors and Local Coordinates
3.1.3 Mappings
3.1.4 Riemannian Manifold
3.1.5 Integral on Manifold
3.1.6 Gradient, Divergence and Laplacian
3.2 Discretization
3.2.1 Discrete Manifold
3.2.2 Finite Element Method and Cotangent Weight Formula
3.2.3 Discrete Local Coordinates
4 Operator Representation 
4.1 Functional Map
4.1.1 Mathematical Properties
4.1.2 Discrete Functional Maps
4.2 Shape Differences
4.2.1 Definition
4.2.2 Fundamental Properties
4.2.3 Algebraic Properties
4.2.4 Discrete Shape Differences
4.3 Organization of the Thesis
I Shape to Deformation 
5 Supervised Descriptor Learning for Non-Rigid Shape Matching 
5.1 Introduction
5.1.1 Related Work
5.2 Consistent Functional Maps
5.2.1 Functional Map Representation
5.2.2 Main Challenge
5.2.3 Algorithm Outline
5.3 Selection of the Best Functional Correspondences
5.3.1 Weighting the probe functions
5.3.2 Finding the best weights
5.4 Basis function extraction
5.4.1 Identifying stable subspaces
5.4.2 Functional map to a test shape using a reduced basis
5.5 Experimental Results
5.5.1 Functional correspondences
5.5.2 Isometric Shape Matching
5.5.3 Non-Isometric Shape Matching
5.6 Conclusion
6 Continuous Matching via Vector Field Flow 
6.1 Introduction
6.2 Related Work
6.3 Functional Maps Conversion
6.3.1 Main Challenges
6.3.2 Algorithm Overview
6.4 Family of Diffeomorphisms
6.5 Optimal vector field
6.5.1 Optimization Problem
6.5.2 Properties
6.5.3 Practical Choice of the Norm
6.6 Vector Field Flow on Manifold
6.7 Results
6.7.1 Symmetry Blending
6.7.2 Error using a computed functional map
6.7.3 Parameters Dependence
6.7.4 Performance
6.8 Conclusion, Limitations and Future Work
II Deformation to Shape 
7 Functional Characterization of Intrinsic and Extrinsic Geometry 
7.1 Introduction
7.2 Related Work
7.3 Overview
7.4 Structure of Discrete Inner Products
7.4.1 Discrete Inner Products
7.5 Encoding Extrinsic Structure
7.5.1 Extrinsic Alternatives
7.5.2 Offset Surfaces
7.5.3 Recovery of Embedding
7.5.4 Discussion
7.6 From Inner Products to Shape Differences
7.6.1 Discrete Shape Differences
7.6.2 Source-Truncated Correspondence
7.6.3 Source- and Target-Truncated Correspondence
7.7 Recovery of Intrinsic and Extrinsic Structure
7.7.1 Triangle Area Computation
7.7.2 Edge Length Computation
7.7.3 Global Extrinsic Reconstruction
7.8 Experiments
7.8.1 Shape Space
7.8.2 Effects of Truncation
7.8.3 Intrinsic Recovery
7.8.4 Reconstruction
7.8.5 Timings
7.9 Discussion & Conclusion
8 Functional Characterization of Deformation Fields 
8.1 Introduction
8.2 Related Work
8.3 Overview
8.4 Extrinsic Vector Fields as Operators
8.4.1 Isometric Shape Difference Operator
8.4.2 Infinitesimal Deformations as Operators
8.4.3 Main Properties of Infinitesimal Shape Differences
8.5 Discrete Setting
8.5.1 Discrete unified shape difference
8.5.2 Infinitesimal shape difference
8.5.3 Properties
8.6 Representation in basis
8.6.1 Basis for the Function Space
8.6.2 Vector field basis
8.7 Experiments
8.7.1 Deformation transfer
8.7.2 Functional map inference
8.8 Conclusion and Future Work
9 Conclusion 


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