Axioms 2014, 3(2), 222-243; doi:10.3390/axioms3020222
Article

Conformal-Based Surface Morphing and Multi-Scale Representation

Received: 11 February 2014; in revised form: 9 April 2014 / Accepted: 23 April 2014 / Published: 20 May 2014
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.
Abstract: This paper presents two algorithms, based on conformal geometry, for the multi-scale representations of geometric shapes and surface morphing. A multi-scale surface representation aims to describe a 3D shape at different levels of geometric detail, which allows analyzing or editing surfaces at the global or local scales effectively. Surface morphing refers to the process of interpolating between two geometric shapes, which has been widely applied to estimate or analyze deformations in computer graphics, computer vision and medical imaging. In this work, we propose two geometric models for surface morphing and multi-scale representation for 3D surfaces. The basic idea is to represent a 3D surface by its mean curvature function, H, and conformal factor function λ, which uniquely determine the geometry of the surface according to Riemann surface theory. Once we have the (λ, H) parameterization of the surface, post-processing of the surface can be done directly on the conformal parameter domain. In particular, the problem of multi-scale representations of shapes can be reduced to the signal filtering on the λ and H parameters. On the other hand, the surface morphing problem can be transformed to an interpolation process of two sets of (λ, H) parameters. We test the proposed algorithms on 3D human face data and MRI-derived brain surfaces. Experimental results show that our proposed methods can effectively obtain multi-scale surface representations and give natural surface morphing results.
Keywords: surface morphing; multi-scale representation; conformal parameterization; conformal factor; mean curvature
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MDPI and ACS Style

Lam, K.C.; Wen, C.; Lui, L.M. Conformal-Based Surface Morphing and Multi-Scale Representation. Axioms 2014, 3, 222-243.

AMA Style

Lam KC, Wen C, Lui LM. Conformal-Based Surface Morphing and Multi-Scale Representation. Axioms. 2014; 3(2):222-243.

Chicago/Turabian Style

Lam, Ka C.; Wen, Chengfeng; Lui, Lok M. 2014. "Conformal-Based Surface Morphing and Multi-Scale Representation." Axioms 3, no. 2: 222-243.

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