Inconsistent Surface Registration via Optimization of Mapping Distortions

We address the problem of registering two surfaces, of which a natural bijection between them does not exist. More precisely, only a partial subset of the source domain is assumed to be in correspondence with a subset of the target domain. We call such a problem an {\it inconsistent surface registration} problem. This problem is challenging as the corresponding regions on each surfaces and a meaningful bijection between them have to be simultaneously determined. In this paper, we propose a variational model to solve the inconsistent surface registration problem by minimizing mapping distortions. Mapping distortions are described by the Beltrami coefficient as well as the differential of the mapping. Registration is then guided by feature landmarks and/or intensities, such as curvatures, defined on each surfaces. The key idea of the approach is to control angle and scale distortions via quasiconformal theory as well as minimizing landmark and/or intensity mismatch. A splitting method is proposed to iteratively search for the optimal corresponding regions as well as the optimal bijection between them. Bijectivity of the mapping is easily enforced by a thresholding of the Beltrami coefficient. We test the proposed method on both synthetic and real examples. Experimental results demonstrate the efficacy of our proposed model.