1Xianfeng Gu, 2Yalin Wang, 2Tony F. Chan, 3Paul M. Thompson, 4Shing-Tung Yau
1Division of Engineering and Applied Science, Harvard University
2Department of Mathematics, UCLA
3Laboratory of Neuro Imaging and Brain Research Institute, UCLA Medical Center
4Department of Mathematics, Harvard University
Any genus zero surface can be mapped conformally onto the sphere and any local portion thereof onto a disc. This mapping, a conformal equivalence, is bijective and angle preserving, and the first fundamental form is only scaled. For this reason, conformal mappings are often described as being similarities in the small. Since the cortical surface of the brain is genus zero, conformal mapping is an ideal way to preserve its local geometric information. Indeed, several brain mapping groups have used conformal mappings to flatten surface data or transform them to a canonical space. However, these approaches are either not strictly angle preserving, or there are areas near the poles with large geometric distortion. In this paper, we propose a new genus zero surface conformal mapping algorithm, which minimizes these problems, and demonstrate its application to the mapping of brain surfaces.
Brief Introduction to the Algorithm
A mapping between any two genus zero surfaces is conformal if and only if the mapping is harmonic. Our algorithm minimizes the harmonic energy of a degree one mapping by moving image points in the tangent spaces until the tangential Laplacian is zero. All conformal mappings form a so-called Mobius transformation group, which is six dimensional and can be formulated explicitly. By utilizing Mobius transformations, different regions of the brain can be zoomed and examined. We add further constraints so that the algorithm converges to a unique conformal map.
Algorithm Benefits and Conclusion
Our algorithm computes conformal mappings accurately and stably. It depends only on the surface geometry and is invariant to differences in resolution and the specific triangulation used. More importantly, our algorithm can straughtforwardly incorporate other constraints such as gyral/sulcal landmark information. It can be generalized to compute the conformal mapping between any two genus zero surfaces, so it can register two brain surfaces directly, without using any intermediate canonical space. Our experimental results, applied to cortical surfaces extracted from MRI data, show that our algorithm is accurate and stable, and offers advantages for analyzing brain surface data.
Paul Thompson, Ph.D.
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