Processing, Analyzing and Learning of Images, Shapes, and Forms

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Contributors
Preface
Diffusion operators for multimodal data analysis
	Introduction
	Preliminaries: Diffusion Maps
	Alternating Diffusion
		Problem formulation: Metric spaces and probabilistic setting
		Illustrative example
		Algorithm
		Common manifold interpretation
			Observable operators
			Effective operators
	Self-adjoint operators for recovering hidden components
		Problem formulation
		Operator definition and analysis
		Discrete setting
	Applications
		Shape analysis
		Foetal heart rate recovery
		Sleep dynamics assessment
			Visualization
			Automatic annotation
	References
Intrinsic and extrinsic operators for shape analysis
	Introduction
	Preliminaries
		Extrinsic and intrinsic geometry
			Intrinsic geometry
			Extrinsic geometry
		Operators and spectra
	Theoretical aspects and numerical analysis
		Basics of linear operators
			Equivalent strong and weak forms
			Variational formulation
			Eigenvalue problem
		PDEs and Green\'s functions
			Laplace equation
			Heat equation
			Wave and Schrödinger equations
			Generalization
		Operator derivation and discretization
			Finite and boundary element methods
			Discrete exterior calculus (DEC)
			Mixed methods and compositions
			Graph affinity Laplacian
			Variants and generalizations
			Operator properties and desiderata
		Operators and geometry
		Inverse problems
			Shape-from-operator
			Shape-from-spectrum
	Spectral shape analysis and applications
		Spectral analysis: From Euclidean space to manifold
			A signal processing viewpoint
			Multiresolution and hierarchical representation
			Spectral shape analysis and processing
		Spectral data analysis
			Spectral graph theory with applications
			Manifold learning and data science
		Spectral analysis: Point embedding, signature, and geometric descriptors
			Spectral embedding and segmentation
			Spectral distances
			Point signatures and descriptors
		Shape analysis and geometry processing
			Segmentation
			Skeletonization (and segmentation)
			Parameterization and remeshing
			Data compression
			Mesh smoothing and fairing
			Feature extraction
			Shape matching and correspondence
			Shape differences
			Shape retrieval
			Geometric (deep) learning
			Structure and vibration analysis
			Acoustics
			Shape optimization
			Reduced simulation
		Other aspects of spectral shape analysis
			Localized bases
			Bases for shape collections
		Numerical aspects
			Convergence
			Eigensolvers
			Accelerations and preconditioners
			Multiresolution, subsampling, and approximation methods
			Robustness
			Padé approximation of heat kernel and distance
	Relevant geometric operators
		Identity operator, area form, and mass matrix
			Discretization: Full mass matrix
			Discretization: Lumped mass matrix
		Laplace-Beltrami (intrinsic Laplacian)
			Harmonic functions
			Discretization
		Combinatorial and graph Laplacians
		Restricted Laplacian
		Scale invariant Laplacian
		Affine and equi-affine invariant Laplacian
		Anisotropic Laplacian
			Discretization
		Hessian and normal-restricted Hessian: A family of linearized energies
		Modified Dirichlet energy
		Hamiltonian operator and Schrödinger operator
		Curvature Laplacian
		Concavity-aware Laplacian
		Extrinsic and relative Dirac operators
			Extrinsic Dirac operator Df
			Relative Dirac operator DN
		Intrinsic Dirac operator DI
		Volumetric (extrinsic) Laplacian
		Hessian energy
		Single layer potential operator and kernel method
			Boundary element method (BEM)
		Dirichlet-to-Neumann operator (Poincaré-Steklov operator)
		Other extrinsic methods
	Summary and experiments
		Experiments
		Eigenfunctions
		Heat kernel signatures
		Segmentation
		Distance or dissimilarity
	Conclusion and future work
		Summary
		Future work
			Better understanding of existing operators
			Synthesizing new operators
			Task-dependent and learnable operators
	Acknowledgements
	References
Operator-based representations of discrete tangent vector fields
	Abbreviations
	Abbreviations
	Abbreviations
	Introduction
		Organization
	Smooth functional vector fields
		Notation
		Directional derivative of functions
		Functional vector fields
		Flow maps
		Functional flow maps
		Lie bracket
	Discrete functional vector fields
		Notation
			Functions and vector fields
			Rotations
			Vectors to matrices
			Areas and inner products
			Interpolation
			Differential operators
			Divergence free vector fields
			Discrete integration by parts
		Directional derivative of functions
		Functional vector fields
			Duality
			Linearity
			Reconstruction
		Flow maps
		Functional flow maps
		Lie bracket
	Divergence-based functional vector fields
		Smooth DFVF
		Discrete DFVF
		Mixed Lie bracket operator
			Sparsity structure
			The Lie bracket of divergence-free vector fields
			Extraction of the commutator vector field
			Limitation: Loss of group structure
		The Lie bracket as a linear transformation on vector fields
			Application: Commutation-guided rescaling
			Application: As-commuting-as-possible vector fields
		Integrating the Lie bracket operator
			Application: Interpolation of vector fields
		Outlook
	Conclusion and future work
	References
Active contour methods on arbitrary graphs based on partial differential equations
	Introduction
	Background and related work
	Active contours on graphs via geometric approximations of gradient and curvature
		Geometric gradient approximation on graphs
			Formulation
			Convergence for random geometric graphs
			Asymptotic analysis of approximation error
			Practical application
		Geometric curvature approximation on graphs
			Formulation
			Practical application
		Gaussian smoothing on graphs
	Active contours on graphs using a finite element framework
		Problem formulation and numerical approximation
		Locally constrained contour evolution
	Experimental results
	Conclusion
	References
Fast operator-splitting algorithms for variational imaging models: Some recent developments
	Introduction
	Regularizers and associated variational models for image restoration
		Generalities
		Total variation regularization
		The Euler elastica regularization
		L1-Mean curvature and L1-Gaussian curvature energies
		Willmore bending energy
		Summary
	Basic results, notations and an introduction to operator-splitting methods
		Basic results and notations
		The Lie and Marchuk-Yanenko operator-splitting schemes for the time discretization of initial value problems
			Generalities
		Time discretization of the initial value problem (17) by the Lie scheme
		Asymptotic properties of the Lie and Marchuk-Yanenko schemes
	Operator splitting method for Euler elastica energy functional
		Formulation of the problem and operator-splitting solution methods
		On the solution of problem (44)
		On the solution of problem (45)
		On the solution of problem (46)
	An operator-splitting method for the Willmore energy-based variational model
		Formulation of the problem and operator-splitting solution methods
		On the solution of problem (77)
		On the solution of problem (78)
		Estimating γ
	An operator-splitting method for the L1-mean curvature variational model
	Operator-splitting methods for the ROF model
		Generalities: synopsis
		A first operator-splitting method
		A second operator-splitting method
	Conclusion
	Acknowledgements
	References
From active contours to minimal geodesic paths: New solutions to active contours problems by Eikonal equations
	Introduction
		Outline
	Active contour models
		Edge-based active contour model
			The computation of the image gradient features
			The original active contour model
			Balloon force
			Geodesic active contour model
			Geometric active contour models with alignment terms
		The piecewise smooth Mumford-Shah model and the piecewise constant reduction model
			The piecewise smooth Mumford-Shah model
			Piecewise constant reduction of the full Mumford-Shah model
			Minimization of the piecewise constant active contour model
	Minimal paths for edge-based active contours problems
		Cohen-Kimmel minimal path model
		Finsler and Randers minimal paths
			Randers minimal paths
			Riemannian minimal paths
	Minimal paths for alignment active contours
		Randers alignment minimal paths
		Riemannian alignment minimal paths
	Orientation-lifted Randers minimal paths for Euler-Mumford elastica problem
		Euler-Mumford elastica problem and its Finsler metric interpretation
		Finsler elastica geodesic path for approximating the elastica curve
		Data-driven Finsler elastica metric
	Randers minimal paths for region-based active contours
		Hybrid active contour model
		A Randers metric interpretation to the hybrid energy
		Practical implementations
		Application to image segmentation
	Conclusion
	Acknowledgements
	References
Computable invariants for curves and surfaces
	Introduction
	Scale invariant metric
		Scale invariant arc-length for planar curves
		Scale invariant metric for implicitly defined planar curves
		Scale invariant metric for surfaces
		Approximating the scale invariant Laplace-Beltrami operator for surfaces
	Equi-affine invariant metric
		Equi-affine invariant arc-length
		Equi-affine metric for surfaces
	Affine metric
		Affine invariant arc-length
		Affine metric for surfaces
			Voronoi diagram
		Approximating the Gaussian curvature
	Applications
		Self functional maps: A song of shapes and operators
			A question of representation
			Hearing shapes using invariant operators
			Shape from eigenfunctions
			A tale of two metrics
			Introduction to functional maps
			Self functional maps
			Self functional maps as surface signatures
		Object recognition
	Summary
	Acknowledgements
	References
Solving PDEs on manifolds represented as point clouds and applications
	Introduction
	Solving PDEs on manifolds represented as point clouds
		Moving least square methods
		Local mesh method
	Solving Fokker-Planck equation for dynamic system
		Double well potential
		Rugged Mueller potential
	Solving PDEs on incomplete distance data
	Geometric understanding of point clouds data
		Construction of skeletons from point clouds
		Construction of conformal mappings from point clouds
		Nonrigid manifolds registration using LB eigenmap
	Conclusion
	Acknowledgements
	References
Tighter continuous relaxations for MAP inference in discrete MRFs: A survey
	LP relaxation
		Tightening the polytope
	Cluster pursuit algorithms
	Cycles in the graph
		Searching for frustrated cycles
		Efficient MAP inference in a cycle
		Planar subproblems
	Tighter subgraph decompositions
		Global higher-order cliques
	Semidefinite programming-based relaxation
		Rounding schemes for SDP relaxation
		Problems where SDP relaxation has helped
	Characterizing tight relaxations
	Conclusion
	References
Lagrangian methods for composite optimization
	Introduction
	The Lagrangian framework
		Lagrangian-based methods: Basic elements and mechanism
			Main difficulties with ALS
		Proximal mappings and minimization
		Application examples
	The convex setting
		Preliminaries on the convex model (CM)
		Proximal method of multipliers and fundamental Lagrangian-based schemes
		One scheme for all: A perturbed PMM and its global rate analysis
		Special cases of the perturbed PMM: Fundamental schemes
	The nonconvex setting
		The nonconvex nonlinear composite optimization-Preliminaries
		ALBUM-Adaptive Lagrangian-based multiplier method
		A methodology for global analysis of Lagrangian-based methods
		ALBUM in action: Global convergence of Lagrangian-based schemes
			A proximal multipliers method (PMM)
			A proximal alternating direction method of multipliers (PADMM)
	Acknowledgements
	References
Generating structured nonsmooth priors and associated primal-dual methods
	Introduction
		Context
		Main contributions and organization of this chapter
	Nonsmooth priors
		Total Variation
		Total generalized variation
		Dualization
		Dualization of the variational regularization problems
	Numerical algorithms
	Bilevel optimization
		Background
		Bilevel optimization-A monolithic approach
	Numerical examples
		Discrete operators for (PTV)
		Bilevel TV numerical experiments
		Discrete operators for (PTGV)
		Bilevel TGV numerical experiments
	References
Graph-based optimization approaches for machine learning, uncertainty quantification and networks
	Introduction
	Graph theory
	Recent methods for semisupervised and unsupervised data classification
		Semisupervised learning and the Ginzburg-Landau graph model
		The graph MBO scheme for data classification and image processing
		Heat kernel pagerank method
		Unsupervised learning and the Mumford-Shah model
		Imposing volume constraints
	Total variation methods for semisupervised and unsupervised data classification
	Uncertainty quantification within the graphical framework
	Networks
	Conclusion
	Acknowledgements
	References
Survey of fast algorithms for Euler\'s elastica-based image segmentation
	Introduction
	Piecewise constant representation and interface problems to illusory contour with curvature term
	Euler\'s elastica-based segmentation models and fast algorithms
	Discussion
	References
Recent advances in denoising of manifold-valued images
	Introduction
	Preliminaries on Riemannian manifolds
		General notation
		Convexity and Hadamard manifolds
	Intrinsic variational restoration models
	Minimization algorithms
		Subgradient descent
		Half-quadratic minimization
		Proximal point and Douglas-Rachford algorithm
			Proximal mapping
			Cyclic proximal point algorithm
			Douglas-Rachford algorithm for symmetric Hadamard spaces
	Numerical examples
	Conclusions
	References
Image and surface registration
	Introduction
	Mathematical background
		Continuous and discrete images
		A mathematical framework for image registration
	Distance measures
		Volumetric differences
		Feature-based differences
	Regularization
		Regularization by ansatz-spaces, parametric registration
		Quadratic regularizer
			Diffusion regularizer
			Elastic regularizer
			Curvature regularizer
		Nonquadratic regularizer
			Mean curvature regularizer
			Gaussian curvature regularizer
			Fractional derivative based regularizer
			Hyperelastic regularizer
		Registration penalties and constraints
		Penalties for locally invertible maps
		Diffeomorphic registration
		Registration by inverse consistent approach
	Surface registration
		Brief introduction to surface geometry
		Parameterization-based approaches
		Laplace-Beltrami eigenmap approaches
		Metric approaches
		Functional map approaches
		Relationship between SR and IR
	Numerical methods
	Deep learning-based registration
	Conclusions
	References
Metric registration of curves and surfaces using optimal control
	Introduction
	Building metrics via submersions
	Optimal control framework
	Chordal metrics on shapes
		Motivation
		General principle
		Oriented varifold distances
		Numerical aspects
	Intrinsic metrics
		Reparametrization-invariant metrics on parametrized shapes
		The metric on the space of unparametrized shapes
		The induced geodesic distance
		The geodesic equation
		An optimal control formulation of the geodesic problem on the space of unparametrized shapes
		Numerical aspects
	Outer deformation metric models
	A hybrid metric model
	Conclusion
	Acknowledgements
	References
Efficient and accurate structure preserving schemes for complex nonlinear systems
	Introduction
	The SAV approach
		Suitable energy splitting
		Adaptive time stepping
	Several extensions of the SAV approach
		Problems with global constraints
		L1 minimization via hyper regularization
		Free energies with highly nonlinear terms
		Coupling with other physical conservation laws
		Dissipative/conservative systems which are not driven by free energy
	Conclusion
	Acknowledgements
	References
Index
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