Inconsistent Surface Registration via Optimization of Mapping Distortions
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Inconsistent Surface Registration via Optimization of Mapping Distortions Di Qiu1 · Lok Ming Lui1 Received: 22 August 2019 / Revised: 26 May 2020 / Accepted: 30 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 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 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. Keywords Inconsistent surface registration · Mapping distortions · Beltrami coefficient · Mapping optimization problem · Quasiconformal theories
1 Introduction Surface registration aims to find meaningful point-wise correspondence between two surfaces embedded in R3 . It has important applications in various fields, such as in computer graphics, computer vision and medical imaging. For instances, the surface registration problem in computer vision and graphics aims to find point-wise correspondence in order to perform shape analysis, relational learning, to transfer motions, textures between shapes; in medical imaging, it is necessary to find one-to-one point-wise correspondence between the target and
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Lok Ming Lui [email protected] Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong 0123456789().: V,-vol
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the template anatomical surfaces so that the data defined on the surfaces can be compared meaningfully. Very often, a desirable registration map should be much more complex than a global rigid or affine motion. Due to its importance, various registration models have been proposed. Existing approaches usually assume a global bijection between the two surfaces to be registered if the surfaces are closed without boundary. For the registration between open domains, prescribed boundary condition is imposed. In a practical situation, a natural bijection betwee
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