Combination of Brain Conformal Mapping and Landmarks: A Variational Approach

Yalin Wang, L.M. Lui, T.F. Chan, and P.M. Thompson (USA)


Conformal mapping, variational method, brain mapping


To compare and integrate brain data, data from multiple subjects are typically mapped into a canonical space. One method to do this is to conformally map cortical surfaces to the sphere. It is well known that any genus zero Riemann surface can be mapped conformally to a sphere. Since the cortical surface of the brain is a genus zero surface, con formal mapping offers a convenient method to parameter ize cortical surfaces without angular distortion, generating an orthogonal grid on the cortex that locally preserves the metric. To compare cortical surfaces more effectively, it is advantageous to adjust the conformal parameterizations to match consistent anatomical features across subjects. This matching of cortical patterns improves the alignment of data across subjects, although it is more challenging to cre ate a consistent conformal (orthogonal) parameterization of anatomy across subjects when landmarks are constrained to lie at specific locations in the spherical parameter space. Here we propose a new method, based on a new energy functional, to optimize the conformal parameterization of cortical surfaces by using landmarks. Experimental results on a dataset of 40 brain hemispheres showed that the land mark mismatch energy can be significant reduced while ef fectively preserving conformality. The key advantage of this conformal parameterization approach is that any local adjustments of the mapping to match landmarks do not af fect the conformality of the mapping significantly. We also examined how the parameterization changes with different weighting factors. As expected, the landmark matching er ror can be reduced if it is more heavily penalized, but con formality is progressively reduced.

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