Beltrami-Flow to Generate a Scale-Space of Cortical Feature Maps on the Sphere.

In this paper we define a scale-space for cortical mean curvature maps on the sphere, that offers a hierarchical representation of the cortical structures, useful in multi-scale algorithms. A spherical feature map is obtained through inflation of the cortical surface of one hemisphere. Using the Beltrami framework, we embed this spherical mesh in a higher dimensional space and the feature assigned to a mesh vertex becomes an additional component of its coordinates. This enhanced mesh then evolves under Beltrami flow. Imposing an appropriate aspect ratio for the feature components, we thus minimize an interpolation of the L_2 norm and the total variation (L_1 norm) of the map. The collection of all maps produced by this PDE forms a scale-space. Our results suggest that this scale-space provides a generalization of the brain map suitable for use e.g. within a multi-scale registration framework.