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
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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 Introduction 3.1.1 Functional lifting in Euclidean spaces
3.1.2 Manifold-valued functional lifting
3.1.3 Further related work 3.2 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: SO¹3°
3.3.3 Normals fields from digital elevation data
3.3.4 Denoising of high resolution InSAR data 3.4 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
About the Author: Prof. Dr. Philipp Grohs was born on July 7, 1981 in Austria and has been a professor at the University of Vienna since 2016. In 2019, he also became a group leader at RICAM, the Johann Radon Institute for Computational and Applied Mathematics in the Austrian Academy of Sciences in Linz. After studying, completing his doctorate and working as a postdoc at TU Wien, Grohs transferred to King Abdullah University of Science and Technology in Thuwal, Saudi Arabia, and then to ETH Zürich, Switzerland, where he was an assistant professor from 2011 to 2016. Grohs was awarded the ETH Zurich Latsis Prize in 2014. In 2020 he was selected for an Alexander-von-Humboldt-Professorship award, the highest endowed research prize in Germany. He is a member of the board of the Austrian Mathematical Society, a member of IEEE Information Theory Society and on the editorial boards of various specialist journals.
Martin Holler was born on May 21, 1986 in Austria. He received his MSc (2010) and his PhD (2013) with a "promotio sub auspiciis praesidentis rei publicae" in Mathematics from the University of Graz. After research stays at the University of Cambridge, UK, and the Ecole Polytechnique, Paris, he currently holds a University Assistant position at the Institute of Mathematics and Scientific Computing of the University of Graz. His research interests include inverse problems and mathematical image processing, in particular the development and analysis of mathematical models in this context as well as applications in biomedical imaging, image compression and beyond.
Andreas Weinmann was born on July 18, 1979 in Augsburg, Germany. He studied mathematics with minor in computer science at TU Munich, and received his Diploma degree in mathematics and computer science from TU Munich in 2006 (with highest distinction). He was assistant at the Institute of Geometry, TU Graz. He obtained his Ph.D. degree from TU Graz in 2010 (with highest distinction). Then he worked as a researcher at Helmholtz Center Munich and TU Munich. Since 2015 he holds a position as Professor of Mathematics and Image Processing at Hochschule Darmstadt. He received his habilitation in 2018 from University Osnabruck. Andreas's research interests include applied analysis, in particular variational methods, nonlinear geometric data spaces, inverse problems as well as computer vision, signal and image processing and imaging applications, in particular Magnetic Particle Imaging.
