Detecting Abnormal Shape Deformations using Yamabe Flow and Beltrami Coefficents

Project Description:
We address the problem of detecting the region of abnormal changes on surfaces. It is of great importance in shape analysis, such as detecting abnormalities on biological shapes. We propose an effective algorithm to detect the abnormal deformations, by generating quasi-conformal maps between the original and the deformed surfaces and study its Beltrami coefficient. We first conformally flatten the surfaces onto the rectangles using Yamabe flow method and compute the appropriate quasi-conformal map that matches features. By formulating abnormal changes as non-conformal deformations, we detect abnormalities by computing the Beltrami coeffient associated uniquely with the quasi-conformal map. The Beltrami coefficient is a complex-valued function defined on the surface which describes the conformality of the deformation at each points. By considering the norm of the Beltrami coefficient, we can effectively segment the regions of abnormal changes. Furthermore, by considering the argument of the Beltrami coefficient, we can captures the rotational changes of the abnormalities. We have tested the algorithm on synthetic surfaces and real brain surfaces. Experimental results show that our algorithm can effectively detect the abnormalities and capture their rotational changes.

Publication:

• Lok Ming Lui, Tsz Wai Wong, Wei Zeng, Xianfeng Gu, Paul Thompson, Tony Chan and Shing-Tung Yau, Detecting Shape Deformations using Yamabe Flow and Beltrami Coefficents,  Journal of Inverse Problem and Imaging (IPI): Special Issue in Medical Imaging, Volume 3, Number 4, December (2009)
  

Result: