# American Institute of Mathematical Sciences

May  2010, 4(2): 311-333. doi: 10.3934/ipi.2010.4.311

## Detection of shape deformities using Yamabe flow and Beltrami coefficients

 1 Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02143, United States, United States 2 Department of Mathematics, UCLA, 520 Portola Plaza, Math Sciences Building 6363, Los Angeles, CA 90095, United States, United States 3 Department of Computer Sciences, Computer Science Building, State University of New York at Stony Brook, Stony Brook, New York 11794-4400, United States 4 Department of Computer Sciences, Room 2425, Computer Science Building, State University of New York at Stony Brook, Stony Brook, New York 11794-4400, United States 5 Lab of Neuro Imaging UCLA School of Medicine, 635 Charles E. Young Drive South Suite 225E, Los Angeles, CA 90095-7332, United States

Received  March 2009 Revised  November 2009 Published  May 2010

We address the problem of detecting deformities on elastic surfaces. This is of great importance for shape analysis, with applications such as detecting abnormalities in biological shapes (e.g., brain structures). We propose an effective algorithm to detect abnormal deformations by generating quasi-conformal maps between the original and deformed surfaces. We firstly flatten the 3D surfaces conformally onto 2D rectangles using the discrete Yamabe flow and use them to compute a quasi-conformal map that matches internal features lying within the surfaces. The deformities on the elastic surface are formulated as non-conformal deformations, whereas normal deformations that preserve local geometry are formulated as conformal deformations. We then detect abnormalities by computing the Beltrami coefficient associated uniquely with the quasi-conformal map. The Beltrami coefficient is a complex-valued function defined on the surface. It describes the deviation of the deformation from conformality at each point. By considering the norm of the Beltrami coefficient, we can effectively segment the regions of abnormal changes, which are invariant under normal (non-rigid) deformations that preserve local geometry. Furthermore, by considering the argument of the Beltrami coefficient, we can capture abnormalities induced by local rotational changes. We tested the algorithm by detecting abnormalities on synthetic surfaces, 3D human face data and MRI-derived brain surfaces. Experimental results show that our algorithm can effectively detect abnormalities and capture local rotational alterations. Our method is also more effective than other existing methods, such as the isometric indicator, for locating abnormalities.
Citation: Lok Ming Lui, Tsz Wai Wong, Wei Zeng, Xianfeng Gu, Paul M. Thompson, Tony F. Chan, Shing Tung Yau. Detection of shape deformities using Yamabe flow and Beltrami coefficients. Inverse Problems & Imaging, 2010, 4 (2) : 311-333. doi: 10.3934/ipi.2010.4.311
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