Abstract
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 of shapes, whose mathematical structure is inherently relevant to the classification task. One intriguing metric comes from using conformal mappings of 2D shapes into each other, via the theory of Teichmuller spaces. In this 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 Mobius 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 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 of shapes and we show how this can be used to define shape transformations, like for instance `adding a protruding limb' to any shape.
Highlights
Many different representations for the collection of all 2D shapes, and many different measures of similarity between them have been studied recently [15, 16, 22, 1, 23, 20, 17, 12, 13, 11, 2, 3, 21, 7, 10, 10, 14]
We propose the study of a new approach to the collection of all shapes by applying the mathematical theory of complex analysis
In this paper we concentrate on representing the 2D shapes in the space we suggest, demonstrating how to move back and forth between the two presentations of 2D shapes: a simple, closed curve and a representative diffeomorphism from its corresponding coset in the quotient space
Summary
Many different representations for the collection of all 2D shapes, and many different measures of similarity between them have been studied recently [15, 16, 22, 1, 23, 20, 17, 12, 13, 11, 2, 3, 21, 7, 10, 10, 14]. In this paper we concentrate on representing the 2D shapes in the space we suggest, demonstrating how to move back and forth between the two presentations of 2D shapes: a simple, closed curve and a representative diffeomorphism from its corresponding coset in the quotient space. Note that this means that every shape, up to scaling and translation, can be naturally described by a “fingerprint”, which is a diffeomorphism of the circle – a 1D, real-valued periodic function of [0, 2π] to itself
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