  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 surface is assumed to be in correspondence with a subset of the
target surface. We call such a problem an {\it inconsistent surface
registration (ISR)} problem. This problem is challenging as the corresponding
regions on each surface and a meaningful bijection between them have to be
simultaneously determined. In this paper, we propose a variational model to
solve the ISR 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 surface. 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.
