Dynamic Programming for Detecting, Tracking, and Matching Deformable Contours
IEEE Transactions on Pattern Analysis and Machine Intelligence
International Journal of Computer Vision
Algorithm 756: a MATLAB toolbox for Schwarz-Christoffel mapping
ACM Transactions on Mathematical Software (TOMS)
Measurement of Visual Motion
IEEE Transactions on Pattern Analysis and Machine Intelligence
Modal Matching for Correspondence and Recognition
IEEE Transactions on Pattern Analysis and Machine Intelligence
Shape Matching and Object Recognition Using Shape Contexts
IEEE Transactions on Pattern Analysis and Machine Intelligence
Large Deformation Diffeomorphic Metric Curve Mapping
International Journal of Computer Vision
Joint appearance and deformable shape for nonparametric segmentation
Proceedings of the 2nd conference on Human motion: understanding, modeling, capture and animation
Shape analysis of planar objects with arbitrary topologies using conformal geometry
ECCV'10 Proceedings of the 11th European conference on Computer vision: Part V
Metrics, connections, and correspondence: the setting for groupwise shape analysis
EMMCVPR'11 Proceedings of the 8th international conference on Energy minimization methods in computer vision and pattern recognition
On Moduli of Rings and Quadrilaterals: Algorithms and Experiments
SIAM Journal on Scientific Computing
SMI 2012: Full Canonical conformal mapping for high genus surfaces with boundaries
Computers and Graphics
Image and Vision Computing
Spaces and manifolds of shapes in computer vision: An overview
Image and Vision Computing
A Reparameterisation Based Approach to Geodesic Constrained Solvers for Curve Matching
International Journal of Computer Vision
SIAM Journal on Imaging Sciences
The mean boundary curve of anatomical objects
ACIVS'12 Proceedings of the 14th international conference on Advanced Concepts for Intelligent Vision Systems
Alignment and morphing for the boundary curves of anatomical organs
SSPR'12/SPR'12 Proceedings of the 2012 Joint IAPR international conference on Structural, Syntactic, and Statistical Pattern Recognition
Geodesic Warps by Conformal Mappings
International Journal of Computer Vision
Teichmüller Shape Descriptor and Its Application to Alzheimer's Disease Study
International Journal of Computer Vision
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The study of 2D shapes and their similarities is a central problem in the field of vision. It arises in particular from the task of classifying and recognizing objects from their observed silhouette. Defining natural distances between 2D shapes creates a metric space of shapes, whose mathematical structure is inherently relevant to the classification task. One intriguing metric space comes from using conformal mappings of 2D shapes into each other, via the theory of Teichmüller spaces. In this space every simple closed curve in the plane (a "shape") is represented by a `fingerprint' which is a diffeomorphism of the unit circle to itself (a differentiable and invertible, periodic function). More precisely, every shape defines to a unique equivalence class of such diffeomorphisms up to right multiplication by a Möbius map. The fingerprint does not change if the shape is varied by translations and scaling and any such equivalence class comes from some shape. This coset space, equipped with the infinitesimal Weil-Petersson (WP) Riemannian norm is a metric space. In this space, the shortest path between each two shapes is unique, and is given by a geodesic connecting them. Their distance from each other is given by integrating the WP-norm along that geodesic. In this paper we concentrate on solving the "welding" problem of "sewing" together conformally the interior and exterior of the unit circle, glued on the unit circle by a given diffeomorphism, to obtain the unique 2D shape associated with this diffeomorphism. This will allow us to go back and forth between 2D shapes and their representing diffeomorphisms in this "space of shapes". We then present an efficient method for computing the unique shortest path, the geodesic of shape morphing between each two end-point shapes. The group of diffeomorphisms of S1 acts as a group of isometries on the space of shapes and we show how this can be used to define shape transformations, like for instance `adding a protruding limb' to any shape.