Landmark constrained genus zero surface conformal mapping and its application to brain mapping research

In order to compare and integrate brain data more effectively, 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. Cortical surface is a genus zero surface. Therefore, conformal mapping offers a convenient method to parameterize cortical surfaces without angular distortion, generating an orthogonal grid on the cortex that locally preserves the metric. Although conformal mapping preserves the local geometry well, the important anatomical features, such as the sulci landmarks, are usually not aligned consistently. 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 create 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 describe two methods to accomplish the task. The first approach is based on pursuing an optimal Möbius transformation to minimize the landmark mismatch error. The second approach is 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 landmark mismatch energy can be significantly reduced while effectively preserving conformality. The key advantage of these conformal parameterization approaches is that any local adjustments of the mapping to match landmarks do not affect the conformality of the mapping significantly. A detailed comparison between the two approaches will be discussed. The first approach can generate a map which is exactly conformal, although the landmark mismatch error is not reduced as effective as the second approach. The second approach can generate a map which significantly reduces the landmark mismatch error, but some conformality will be lost.