# Inconsistent Surface Registration via Optimization of Mapping Distortions

arXiv: https://arxiv.org/abs/1908.09098 · HTML: https://mlsanity.com/paper/1908.09098/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Aug 2019.

## Built on by (most cited first, 4 of 4)

- [A Deep Learning Framework for Diffeomorphic Mapping Problems via Quasi-conformal Geometry applied to Imaging](https://mlsanity.com/paper/2110.10580/) (2021)
- [Harmonic Beltrami Signature: A Novel 2D Shape Representation for Object Classification](https://mlsanity.com/paper/2103.16411/) (2021)
- [Free-Boundary Quasiconformal Maps via a Least-squares Operator in Diffeomorphism Optimization](https://mlsanity.com/paper/2511.11679/) (2025)
- [Deep Learning-Based Quasi-Conformal Surface Registration for Partial 3D Faces Applied to Facial Recognition](https://mlsanity.com/paper/2405.09880/) (2024)

## Abstract (arXiv)

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.