Conformal Slit Mapping and Its Applications to Brain Surface Parameterization

Yalin Wang, Xianfeng Gu, Tony F. Chan, Paul M. Thompson, and Shing-Tung Yau

Abstract

We propose a method that computes a conformal mapping from a multiply connected mesh to the so-called slit domain, which consists of a canonical rectangle or disk in which 3D curved landmarks on the original surfaces are mapped to concentric or parallel lines in the slit domain. In this paper, we studied its application to brain surface parameterization. After cutting along some landmark curve features on surface models of the cerebral cortex, we obtain multiple connected domains. By computing exact harmonic one-forms, closed harmonic one-forms, and holomorphic one-forms, we are able to build a circular slit mapping that conformally maps the surface to an annulus with some concentric arcs and a rectangle with some slits. The whole algorithm is based on solving linear systems so it is very stable. In the slit domain parameterization results, the feature curves are either mapped to straight lines or concentric arcs. This representation is convenient for anatomical visualization, and may assist statistical comparisons of anatomy, surface-based registration and signal processing. Preliminary experimental results parameterizing various brain anatomical surfaces are presented.

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