Cortical Surface Flattening: a Discrete Conformal Approach Using Circle Packings

The locations and patterns of functional brain activity in humans are difficult to compare across subjects because of individual differences in cortical folding and the fact that functional foci are often buried within cortical sulci. Cortical flat mapping is a tool which can address these problems by taking advantage of the two-dimensional sheet topology of the cortical surface. Flat mappings of the cerebral and cerebellar cortex may facilitate the recognition of structural and functional relationships that were not previously apparent. Mathematical and computational issues associated with new techniques for discrete conformal mapping are the subject of this paper. While it is mathematically impossible to flatten curved surfaces in 3-space without introducing metric and areal distortion, several algorithms can minimize such distortion; consequently, metric flattening has been central in brain mapping efforts to date. On the other hand, while it has been known for 150 years that it is mathematically possible to flatten surfaces without any angular distortion, until recently there has been no algorithm for approximating these so-called “conformal” flat maps. In this paper we describe an approach based on circle packings, the first computer realization of the 150-year-old Riemann Mapping Theorem, and provide both conceptual and practical arguments for conformal flattening. Conformal maps are particularly versatile, offering a variety of visual presentations and manipulations backed by a uniquely rich mathematical theory; using a circle packing algorithm we obtain discrete conformal mappings which exploit this versatility. Our discrete conformal maps of cortical surfaces can be displayed in three geometries, the Euclidean and hyperbolic planes and the sphere; they can be manipulated with Möbius transformations to zoom and focus on particular regions of interest; they respect canonical coordinate systems which can be used for intersubject registration, and, although they do not preserve linear or areal information, are locally Euclidean. We discuss the notion of a conformal structure on a surface, describe key features of the three geometries of constant curvature where our maps reside and their classical conformal automorphisms, and provide qualitative and quantitative details about our approximations. We demonstrate the practical advantages of this approach by producing quasiconformal flat maps with data from the Visible Man and from an MRI volume of the human cerebellum.