Spaces and manifolds of shapes in computer vision: An overview
Image and Vision Computing
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Representation and comparison of shapes is a central problem in computer vision. In this dissertation, we continue investigation of a relatively recent approach, based on conformal mapping, which was first introduced by Mumford and Sharon in 2006. We first study the way this representation encodes the geometry. One perceptually salient attribute of a shape is its medial axis. By careful study of the boundary derivatives of conformal maps, we demonstrate how our representation encodes a sort of continuous version of the medial axis. This line of work provides medial-axis-based variants of results like the Ahlfors distortion theorem, and also provides a new proof and refinement of the conjecture, attributed to Thurston and first proven by V. Markovic, that the nearest-point retraction map is 2-Lipschitz in the hyperbolic metrics. In addition, we obtain explicit estimates demonstrating how conformal maps encode the local Euclidean curvature of the boundary. The machinery involved in making our geometric estimates comes both from classical function theory and hyperbolic geometry. We next investigate the differential of the isomorphism between quasisymmetric circle maps and shapes; we obtain explicit formulas for the differential in terms of the Hilbert transform. We then apply our results to develop an adaptive compressor for plane curves based on the "fingerprint" construction explored by Sharon and Mumford. We demonstrate that our compressor is optimal on certain classes of functions.