Conformal Slit Mapping and Its Applications to Brain Surface Parameterization

Authors:
Yalin Wang;Xianfeng Gu;Tony F. Chan;Paul M. Thompson;Shing-Tung Yau
Affiliations:
Lab. of Neuro Imaging, UCLA School of Medicine, , Los Angeles, USA CA 90095 and Mathematics Department, UCLA, , Los Angeles, USA CA 90095;Comp. Sci. Department, SUNY at Stony Brook, Stony Brook, USA NY 11794;Mathematics Department, UCLA, , Los Angeles, USA CA 90095;Lab. of Neuro Imaging, UCLA School of Medicine, , Los Angeles, USA CA 90095;Department of Mathematics, Harvard University, Cambridge, USA MA 02138
Venue:
MICCAI '08 Proceedings of the 11th international conference on Medical Image Computing and Computer-Assisted Intervention - Part I
Year:
2008

Citing 4
Cited 6

A Numerical Solution to the Generalized Mapmaker's Problem: Flattening Nonconvex Polyhedral Surfaces

IEEE Transactions on Pattern Analysis and Machine Intelligence
Conformal Geometry and Brain Flattening

MICCAI '99 Proceedings of the Second International Conference on Medical Image Computing and Computer-Assisted Intervention
Computerized mappings of the cerebral cortex: A multiresolution flattening method and a surface-based coordinate system

Journal of Cognitive Neuroscience
Brain surface conformal parameterization with algebraic functions

MICCAI'06 Proceedings of the 9th international conference on Medical Image Computing and Computer-Assisted Intervention - Volume Part II

Teichmüller Shape Space Theory and Its Application to Brain Morphometry

MICCAI '09 Proceedings of the 12th International Conference on Medical Image Computing and Computer-Assisted Intervention: Part II
Multivariate Tensor-Based Brain Anatomical Surface Morphometry via Holomorphic One-Forms

MICCAI '09 Proceedings of the 12th International Conference on Medical Image Computing and Computer-Assisted Intervention: Part I
Human brain mapping with conformal geometry and multivariate tensor-based morphometry

MBIA'11 Proceedings of the First international conference on Multimodal brain image analysis
Hyperbolic ricci flow and its application in studying lateral ventricle morphometry

MBIA'12 Proceedings of the Second international conference on Multimodal Brain Image Analysis
Globally optimal cortical surface matching with exact landmark correspondence

IPMI'13 Proceedings of the 23rd international conference on Information Processing in Medical Imaging
Teichmüller Shape Descriptor and Its Application to Alzheimer's Disease Study

International Journal of Computer Vision

Quantified Score

Hi-index	0.00

Visualization

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.