Mathematics of Shapes and Applications

Mathematics of Shapes and Applications

Sergey Kushnarev, Anqi Qiu, Laurent Younes


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Understanding how a single shape can incur a complex range of transformations, while defining the same perceptually obvious figure, entails a rich and challenging collection of problems, at the interface between applied mathematics, statistics and computer science. The program on Mathematics of Shapes and Applications, was held at the Institute for Mathematical Sciences at the National University of Singapore in 2016. It provided discussions on theoretical developments and numerous applications in computer vision, object recognition and medical imaging.

The analysis of shapes is an example of a mathematical problem directly connected with applications while offering deep open challenges to theoretical mathematicians. It has grown, over the past decades, into an interdisciplinary area in which researchers studying infinite-dimensional Riemannian manifolds (global analysis) interact with applied mathematicians, statisticians, computer scientists and biomedical engineers on a variety of problems involving shapes.

The volume illustrates this wealth of subjects by providing new contributions on the metric structure of diffeomorphism groups and shape spaces, recent developments on deterministic and stochastic models of shape evolution, new computational methods manipulating shapes, and new statistical tools to analyze shape datasets. In addition to these contributions, applications of shape analysis to medical imaging and computational anatomy are discussed, leading, in particular, to improved understanding of the impact of cognitive diseases on the geometry of the brain.

  • Variational Second-Order Interpolation on the Group of Diffeomorphisms with a Right-Invariant Metric (François-Xavier Vialard)
  • Riemannian Geometry for Shape Analysis and Computational Anatomy (Martins Bruveris)
  • Diffeomorphic Registration of Discrete Geometric Distributions (Hsi-Wei Hsieh and Nicolas Charon)
  • Stochastic Metamorphosis with Template Uncertainties (Alexis Arnaudon, Darryl D Holm and Stefan Sommer)
  • Piecewise Rigid Motion in Diffeomorphism Groups with Strong Right-Invariant Metrics (Dai-Ni Hsieh and Laurent Younes)
  • Existence and Continuity of Minimizers for the Estimation of Growth Mapped Evolutions for Current Data Term and Counterexamples for Varifold Data Term (Irène Kaltenmark and Alain Trouvé)
  • 3D Normal Coordinate Systems for Cortical Areas (J Tilak Ratnanather, Sylvain Arguillère, Kwame S Kutten, Peter Hubka, Andrej Kral and Laurent Younes)
  • Heat Kernel Smoothing in Irregular Domains (Moo K Chung and Yanli Wang)

Readership: Graduate students, applied mathematicians, statisticians, computer scientists and biomedical engineers.Shape Analysis;Global Analysis;Optimal Control Theory;Statistics on Manifolds;Computational Anatomy0Key Features:
  • Features exciting new developments and applications in an interdisciplinary manner on the mathematics of shapes