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Axioms 2015, 4(2), 156-176; doi:10.3390/axioms4020156

Diffeomorphism Spline

School of Computing and Information Sciences, Florida International University, 11200 SW 8th Street, Miami, FL 33199, USA
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Academic Editor: Angel Garrido
Received: 25 May 2014 / Revised: 17 February 2015 / Accepted: 27 March 2015 / Published: 10 April 2015
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Abstract

Conventional splines offer powerful means for modeling surfaces and volumes in three-dimensional Euclidean space. A one-dimensional quaternion spline has been applied for animation purpose, where the splines are defined to model a one-dimensional submanifold in the three-dimensional Lie group. Given two surfaces, all of the diffeomorphisms between them form an infinite dimensional manifold, the so-called diffeomorphism space. In this work, we propose a novel scheme to model finite dimensional submanifolds in the diffeomorphism space by generalizing conventional splines. According to quasiconformal geometry theorem, each diffeomorphism determines a Beltrami differential on the source surface. Inversely, the diffeomorphism is determined by its Beltrami differential with normalization conditions. Therefore, the diffeomorphism space has one-to-one correspondence to the space of a special differential form. The convex combination of Beltrami differentials is still a Beltrami differential. Therefore, the conventional spline scheme can be generalized to the Beltrami differential space and, consequently, to the diffeomorphism space. Our experiments demonstrate the efficiency and efficacy of diffeomorphism splines. The diffeomorphism spline has many potential applications, such as surface registration, tracking and animation. View Full-Text
Keywords: diffeomorphism; Beltrami differential; Beltrami coefficient; quasiconformal mapping; auxiliary metric; surface mapping; shape space; spline; animation diffeomorphism; Beltrami differential; Beltrami coefficient; quasiconformal mapping; auxiliary metric; surface mapping; shape space; spline; animation
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Zeng, W.; Razib, M.; Shahid, A.B. Diffeomorphism Spline. Axioms 2015, 4, 156-176.

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