Detecting Shape Deformations using Yamabe Flow and Beltrami Coefficients

Ronald Lok Ming Lui Harvard University Tsz Wai Wong University of California, Los Angeles Wei Zeng Stony Brook University Xianfeng Gu Stony Brook University Paul M. Thompson University of California, Los Angeles Tony F. Chan The Hong Kong University of Science and Technology Shing-Tung Yau Harvard University

Computational Geometry mathscidoc:1609.09037

Journal of Inverse Problem and Imaging, 4, (2), 311 - 333, 2010.5
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.
Beltrami coefficient, Quasi-conformal map, Conformal map, Surface registration, Shape Analysis
[ Download ] [ 2016-09-06 12:27:40 uploaded by lmlui ] [ 787 downloads ] [ 0 comments ] [ Cited by 4 ]
@inproceedings{ronald2010detecting,
  title={Detecting Shape Deformations using Yamabe Flow and Beltrami Coefficients},
  author={Ronald Lok Ming Lui, Tsz Wai Wong, Wei Zeng, Xianfeng Gu, Paul M. Thompson, Tony F. Chan, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160906122740643474649},
  booktitle={Journal of Inverse Problem and Imaging},
  volume={4},
  number={2},
  pages={311 - 333},
  year={2010},
}
Ronald Lok Ming Lui, Tsz Wai Wong, Wei Zeng, Xianfeng Gu, Paul M. Thompson, Tony F. Chan, and Shing-Tung Yau. Detecting Shape Deformations using Yamabe Flow and Beltrami Coefficients. 2010. Vol. 4. In Journal of Inverse Problem and Imaging. pp.311 - 333. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160906122740643474649.
Please log in for comment!
 
 
Contact us: office-iccm@tsinghua.edu.cn | Copyright Reserved