Discrete conformal methods for cortical brain flattening

Locations and patterns of functional brain activity in humans are difficult to compare across subjects because of differences in cortical folding and functional foci are often buried within cortical sulci. Unfolding a cortical surface via flat mapping has become a key method for facilitating the recognition of new structural and functional relationships. Mathematical and other issues involved in flat mapping are the subject of this paper. It is mathematically impossible to flatten curved surfaces without metric and area distortion. Nevertheless, "metric" flattening has flourished based on a variety of computational methods that minimize distortion. However, it is mathematically possible to flatten without any angular distortion--a fact known for 150 years. Computational methods for this "conformal" flattening have only recently emerged. Conformal maps are particularly versatile and are backed by a uniquely rich mathematical theory. This paper presents a tutorial level introduction to the mathematics of conformal mapping and provides both conceptual and practical arguments for its use. Discrete conformal mapping computed via circle packing is a method that has provided the first practical realization of the Riemann Mapping Theorem (RMT). Maps can be displayed in three geometries, manipulated with Möbius transformations to zoom and focus on particular regions of interest, they respect canonical coordinates useful for intersubject registration and are locally Euclidean. The versatility and practical advantages of the circle packing approach are shown by producing conformal flat maps using MRI data of a human cerebral cortex, cerebellum and a specific region of interest (ROI).