Synopsis
Part I Processing geometric data
1 Geometric Finite Elements
Hanne Hardering and Oliver Sander
1.1 Introduction
1.2 Constructions of geometric finite elements
1.2.1 Projection-based finite elements
1.2.2 Geodesic finite elements
1.2.3 Geometric finite elements based on de Casteljau's algorithm
1.2.4 Interpolation in normal coordinates
1.3 Discrete test functions and vector field interpolation
1.3.1 Algebraic representation of test functions
1.3.2 Test vector fields as discretizations of maps into the tangent bundle
1.4 A priori error theory
1.4.1 Sobolev spaces of maps into manifolds
1.4.2 Discretization of elliptic energy minimization problems
1.4.3 Approximation errors . .
1.5 Numerical examples
1.5.1 Harmonic maps into the sphere
1.5.2 Magnetic Skyrmions in the plane
1.5.3 Geometrically exact Cosserat plates
2 Non-smooth variational regularization for processing manifold-valued
data
M. Holler and A. Weinmann
2.1 Introduction
2.2 Total Variation Regularization of Manifold Valued Data
vii
viii Contents
2.2.1 Models
2.2.2 Algorithmic Realization
2.3 Higher Order Total Variation Approaches, Total GeneralizedVariation
2.3.1 Models
2.3.2 Algorithmic Realization
2.4 Mumford-Shah Regularization for Manifold Valued Data
2.4.1 Models
2.4.2 Algorithmic Realization
2.5 Dealing with Indirect Measurements: Variational Regularization
of Inverse Problems for Manifold Valued Data
2.5.1 Models
2.5.2 Algorithmic Realization
2.6 Wavelet Sparse Regularization of Manifold Valued Data
2.6.1 Model
2.6.2 Algorithmic Realization
3 Lifting methods for manifold-valued variational problems
Thomas Vogt, Evgeny Strekalovskiy, Daniel Cremers, Jan Lellmann
3.1 Introduction3.1.1 Functional lifting in Euclidean spaces
3.1.2 Manifold-valued functional lifting
3.1.3 Further related work3.2 Submanifolds of RN
3.2.1 Calculus of Variations on submanifolds
3.2.2 Finite elements on submanifolds
3.2.3 Relation to [47]
3.2.4 Full discretization and numerical implementation
3.3 Numerical Results
3.3.1 One-dimensional denoising on a Klein bottle
3.3.2 Three-dimensional manifolds: SO¹3º
3.3.3 Normals fields from digital elevation data
3.3.4 Denoising of high resolution InSAR data3.4 Conclusion and Outlook
4 Geometric subdivision and multiscale transforms
Johannes Wallner
4.1 Computing averages in nonlinear geometries
The Fréchet mean
The exponential mapping
Averages defined in terms of the exponential mapping
4.2 Subdivision
4.2.1 Defining stationary subdivision
Linear subdivision rules and their nonlinear analogues
4.2.2 Convergence of subdivision processes
4.2.3 Probabilistic interpretation of subdivision in metric spaces
4.2.4 The convergence problem in manifolds
4.3 Smoothness analysis of subdivision rules
4.3.1 Derivatives of limits
4.3.2 Proximity inequalities
4.3.3 Subdivision of Hermite data
4.3.4 Subdivision with irregular combinatorics
4.4 Multiscale transforms
4.4.1 Definition of intrinsic multiscale transforms
4.4.2 Properties of multiscale transforms
Conclusion
5 Variational Methods for Discrete Geometric Functionals
Henrik Schumacher and Max Wardetzky
5.1 Introduction
5.2 Shape Space of Lipschitz Immersions
5.3 Notions of Convergence for Variational Problems
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