Overview
- As a first feature, the book provides a comprehensive collection of works describing the state of the art in their respective discipline. The benefit is a standard reference work for researchers (such a reference is presently missing).
- As a second feature, there are contributions from different communities. A benefit resulting from this is that the book provides cross discipline information.
- A third feature is that it contains introductory contents in connection with works previously only available in research articles. As a resulting benefit it may serve as an introduction/overview for newcomers.
- Finally, the book addresses both theoretical aspects as well as applications
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Table of contents (24 chapters)
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Processing Geometric Data
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Statistical Methods and Non-linear Geometry
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Shapes Spaces and the Analysis of Geometric Data
Keywords
- Geometric nonlinear data
- manifold valued data
- variational methods
- total variation
- denoising
- optimization in manifolds
- statistics in manifolds
- diffusion tensor imaging
- medical imaging
- applied differential geometry
- geometric finite elements
- curvature regularization
- labeling
- optical flow
- geometry processing
- functional lifting techniques
- metamorphosis models
About this book
This book covers different, current research directions in the context of variational methods for non-linear geometric data. Each chapter is authored by leading experts in the respective discipline and provides an introduction, an overview and a description of the current state of the art.
Non-linear geometric data arises in various applications in science and engineering. Examples of nonlinear data spaces are diverse and include, for instance, nonlinear spaces of matrices, spaces of curves, shapes as well as manifolds of probability measures. Applications can be found in biology, medicine, product engineering, geography and computer vision for instance.
Variational methods on the other hand have evolved to being amongst the most powerful tools for applied mathematics. They involve techniques from various branches of mathematics such as statistics, modeling, optimization, numerical mathematics and analysis. The vast majority of research on variational methods, however, is focused on data in linear spaces. Variational methods for non-linear data is currently an emerging research topic.
As a result, and since such methods involve various branches of mathematics, there is a plethora of different, recent approaches dealing with different aspects of variational methods for nonlinear geometric data. Research results are rather scattered and appear in journals of different mathematical communities.
The main purpose of the book is to account for that by providing, for the first time, a comprehensive collection of different research directions and existing approaches in this context. It is organized in a way that leading researchers from the different fields provide an introductory overview of recent research directions in their respective discipline. As such, the book is a unique reference work for both newcomers in the field of variational methods for non-linear geometric data, as well as for established experts that aim at to exploit new research directions or collaborations.
Chapter 9 of this book is available open access under a CC BY 4.0 license at link.springer.com.
Editors and Affiliations
About the editors
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). Thenhe 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.
Bibliographic Information
Book Title: Handbook of Variational Methods for Nonlinear Geometric Data
Editors: Philipp Grohs, Martin Holler, Andreas Weinmann
DOI: https://doi.org/10.1007/978-3-030-31351-7
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2020
Hardcover ISBN: 978-3-030-31350-0Published: 04 April 2020
Softcover ISBN: 978-3-030-31353-1Published: 26 August 2021
eBook ISBN: 978-3-030-31351-7Published: 03 April 2020
Edition Number: 1
Number of Pages: XXVI, 701
Number of Illustrations: 34 b/w illustrations, 125 illustrations in colour
Topics: Computational Mathematics and Numerical Analysis, Math Applications in Computer Science, Image Processing and Computer Vision, Mathematical Applications in Computer Science