Special Section on High-Speed 3-D Optical Metrology and Applications

Shape signature based on Ricci flow and optimal mass transportation

[+] Author Affiliations
Wei Luo

Zhejiang University, Mathematics Department, 38 Yugu Road, Hangzhou 310027, China

Zengyu Su

Stony Brook University, Computer Science Department, Stony Brook, New York 11794-4400

Min Zhang

Stony Brook University, Computer Science Department, Stony Brook, New York 11794-4400

Wei Zeng

Florida International University, School of Computing & Information Sciences, Miami, Florida 33199

Junfei Dai

Zhejiang University, Mathematics Department, 38 Yugu Road, Hangzhou 310027, China

Xianfeng Gu

Stony Brook University, Computer Science Department, Stony Brook, New York 11794-4400

Opt. Eng. 53(11), 112209 (Apr 25, 2014). doi:10.1117/1.OE.53.11.112209
History: Received January 27, 2014; Revised March 19, 2014; Accepted March 21, 2014
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Abstract.  A shape signature based on surface Ricci flow and optimal mass transportation is introduced for the purpose of surface comparison. First, the surface is conformally mapped onto plane by Ricci flow, which induces a measure on the planar domain. Second, the unique optimal mass transport map is computed that transports the new measure to the canonical measure on the plane. The map is obtained by a convex optimization process. This optimal transport map encodes all the information of the Riemannian metric on the surface. The shape signature consists of the optimal transport map, together with the mean curvature, which can fully recover the original surface. The discrete theories of surface Ricci flow and optimal mass transportation are explained thoroughly. The algorithms are given in detail. The signature is tested on human facial surfaces with different expressions accquired by structured light 3-D scanner based on phase-shifting method. The experimental results demonstrate the efficiency and efficacy of the method.

© 2014 Society of Photo-Optical Instrumentation Engineers

Citation

Wei Luo ; Zengyu Su ; Min Zhang ; Wei Zeng ; Junfei Dai, et al.
"Shape signature based on Ricci flow and optimal mass transportation", Opt. Eng. 53(11), 112209 (Apr 25, 2014). ; http://dx.doi.org/10.1117/1.OE.53.11.112209


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