Detection of shape deformities using Yamabe flow and Beltrami coefficients
The Beltrami coefficient is a complex-valued function defined on the surface. It describes the deviation of the deformation from conformality at each point. The authors 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, for instance brain structures. The authors show an effective algorithm to detect abnormal deformations by generating quasi-conformal maps between the original and deformed surfaces. They 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. Then they detect abnormalities by computing the Beltrami coefficient associated uniquely with the quasi-conformal map. By considering the norm of the Beltrami coefficient, they 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, the authors can capture abnormalities induced by local rotational changes. They tested the algorithm by detecting abnormalities on synthetic surfaces, 3D human face data and MRI-derived brain surfaces. Experimental results show that their algorithm can effectively detect abnormalities and capture local rotational alterations.
scientific article; zbMATH DE number 5776174