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Optimization of Surface Registrations Using Beltrami Holomorphic Flow

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An Erratum to this article was published on 16 September 2011

Abstract

In shape analysis, finding an optimal 1-1 correspondence between 3D surfaces within a large class of admissible bijective mappings is of great importance. Such a process is called surface registration. The difficulty lies in the fact that the space of all surface diffeomorphisms is a complicated functional space, making it challenging to exhaustively search for the best mapping. To tackle this problem, we propose a simple representation of bijective surface maps using Beltrami coefficients (BCs)—complex-valued functions defined on surfaces with supremum norm less than 1. Fixing any 3 points on a pair of surfaces, there is a 1-1 correspondence between the set of surface diffeomorphisms between them and the set of BCs. Hence, every bijective surface map may be represented by a unique BC. Conversely, given a BC, we can reconstruct the unique surface map associated with it using the Beltrami Holomorphic flow (BHF) method. Using BCs to represent surface maps is advantageous because it is a much simpler functional space, which captures many essential features of a surface map. By adjusting BCs, we equivalently adjust surface diffeomorphisms to obtain the optimal map with desired properties. More specifically, BHF gives us the variation of the associated map under the variation of BC. Using this, a variational problem over the space of surface diffeomorphisms can be easily reformulated into a variational problem over the space of BCs. This makes the minimization procedure much easier. More importantly, the diffeomorphic property is always preserved. We test our method on synthetic examples and real medical applications. Experimental results demonstrate the effectiveness of our proposed algorithm for surface registration.

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Authors and Affiliations

  1. Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

    Lok Ming Lui & Shing-Tung Yau

  2. Department of Mathematics, University of California, Los Angeles, USA

    Tsz Wai Wong

  3. Department of Computer Science, State University of New York at Stony Brook, New York, USA

    Wei Zeng & Xianfeng Gu

  4. Laboratory of Neuroimaging, UCLA Medical School, Los Angeles, USA

    Paul M. Thompson

  5. Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    Tony F. Chan

Authors
  1. Lok Ming Lui
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  2. Tsz Wai Wong
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  3. Wei Zeng
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  4. Xianfeng Gu
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  7. Shing-Tung Yau
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Correspondence to Lok Ming Lui.

Additional information

An erratum to this article can be found at http://dx.doi.org/10.1007/s10915-011-9541-z.

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Lui, L.M., Wong, T.W., Zeng, W. et al. Optimization of Surface Registrations Using Beltrami Holomorphic Flow. J Sci Comput 50, 557–585 (2012). https://doi.org/10.1007/s10915-011-9506-2

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  • DOI: https://doi.org/10.1007/s10915-011-9506-2

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