A Metric Approach to Elastic Reformations
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A Metric Approach to Elastic Reformations Luca Granieri · Francesco Maddalena
Received: 30 July 2013 / Accepted: 5 December 2013 © Springer Science+Business Media Dordrecht 2013
Abstract We study a variational framework to compare shapes, modeled as Radon measures on RN , in order to quantify how they differ from isometric copies. To this purpose we discuss some notions of weak deformations termed reformations as well as integral functionals having some kind of isometries as minimizers. The approach pursued is based on the notion of pointwise Lipschitz constant leading to a matric space framework. In particular, to compare general shapes, we study this reformation problem by using the notion of transport plan and Wasserstein distances as in optimal mass transportation theory. Keywords Calculus of variations · Shape analysis · Mass transportation theory · Geometric measure theory · Elasticity Mathematics Subject Classification (2000) 37J50 · 49Q20 · 49Q15 1 Introduction One of the main goal in shape analysis relies in detecting and quantifying differences between shapes. The interest for such studies concerns a wide range of applications, especially
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L. Granieri ( ) Dipartimento di Matematica e Applicazioni, Università Federico II di Napoli, Via Cintia, Monte S. Angelo 80126 Napoli, Italy e-mail: [email protected] L. Granieri e-mail: [email protected] L. Granieri e-mail: [email protected] L. Granieri · F. Maddalena Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy F. Maddalena e-mail: [email protected]
L. Granieri, F. Maddalena
those within the computer vision community, in particular in pattern recognition, image segmentation, and computation anatomy (see [13, 21, 49]). In recent years many authors have focused their attention on the notions of shape space and shape metric to the aim of establishing a general framework in which the analysis of shapes crucially depends on their invariance with respect to suitable geometric transformations (see [13, 26, 30, 57]). In the interesting papers [13, 21, 30] the authors model shapes as mathematical currents (linear functionals on the space of differential forms) defining a suitable distance which captures relevant variations of geometry. In particular, in [13] the authors introduce functional currents to take also into account the presence of signals (scalar or tensor valued fields) attached to the geometric shapes. A natural suggestion in this direction comes from continuum mechanics since the variational theory of elasticity can be used to compare the initial and final shape of a deformable material body, i.e. to establish how the two shapes differ from an isometry of the Euclidean space. Therefore some authors begin to study links between elastic energies and distances in shape spaces (see [26, 64, 65]). In the paper [50] the author introduces a transport problem with gradient penalization in which the transport cost depends also on the Jacobian matrix of the transport map. Though the result
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