© 2020

Handbook of Variational Methods for Nonlinear Geometric Data

  • Philipp Grohs
  • Martin Holler
  • Andreas Weinmann

Table of contents

  1. Front Matter
    Pages i-xxvi
  2. Processing Geometric Data

    1. Front Matter
      Pages 1-1
    2. Hanne Hardering, Oliver Sander
      Pages 3-49
    3. Thomas Vogt, Evgeny Strekalovskiy, Daniel Cremers, Jan Lellmann
      Pages 95-119
    4. Johannes Wallner
      Pages 121-152
    5. Henrik Schumacher, Max Wardetzky
      Pages 153-172
  3. Geometry as a Tool

    1. Front Matter
      Pages 173-173
    2. François Gay-Balmaz, Vakhtang Putkaradze
      Pages 175-205
    3. Ulrich Böttcher, Benedikt Wirth
      Pages 207-233
    4. Christoph Schnörr
      Pages 235-260
    5. André Uschmajew, Bart Vandereycken
      Pages 261-313 Open Access
  4. Statistical Methods and Non-linear Geometry

    1. Front Matter
      Pages 315-315
    2. Iddo Drori
      Pages 361-376
  5. Shapes Spaces and the Analysis of Geometric Data

    1. Front Matter
      Pages 377-377
    2. Xiaoyang Guo, Anuj Srivastava
      Pages 379-394
    3. Jingyong Su, Mengmeng Guo, Zhipeng Yang, Zhaohua Ding
      Pages 395-413
    4. Anirudh Som, Karthikeyan Natesan Ramamurthy, Pavan Turaga
      Pages 415-441

About this book


This book explains how variational methods 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


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

Editors and affiliations

  • Philipp Grohs
    • 1
  • Martin Holler
    • 2
  • Andreas Weinmann
    • 3
  1. 1.Faculty of MathematicsUniversity of ViennaWienAustria
  2. 2.Institute of Mathematics and Scientific ComputingUniversity of GrazGrazAustria
  3. 3.Department of Mathematics and Natural SciencesHochschule DarmstadtDarmstadtGermany

About the editors

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.

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