Cortical Surface Flattening: A Quasi-Conformal Approach Using Circle Packings
M. K. Hurdal, K. Stephenson, P. L. Bowers, D. W. L. Summers, D. A. Rottenberg
Comparing the location and size of functional brain activity across subjects is difficult due to individual differences in folding patterns and 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 cortex assist in simplifying complex information and may reveal spatial relationships in functional and anatomical data that were not previously apparent. Metric and areal flattening algorithms have been central to brain flattening efforts to date. However, it is mathematically impossible to flatten a curved surface in 3-space without introducing metric and areal distortion. Nevertheless, the Riemann Mapping Theorem of complex function theory implies that it is theoretically possible to preserve conformal (angular) information under flattening. In this paper we present a novel approach for creating flat maps of the brain that involves a computer realization of the 150-year-old Riemann Mapping Theorem. This approach uses a circle packing algorithm to compute an essentially unique (i.e. up to Möbius transformations), discrete approximation of a conformal mapping from the cortical surface to the plane or the sphere. Conformal maps are very versatile and offer a variety of visual presentations and manipulations. Maps can be displayed in three geometries: the Euclidean and hyperbolic planes, and the sphere. A wide variety of Möbius transformations can be used to zoom and focus the maps in a particular region of interest. Conformal maps are mathematically unique and canonical coordinate systems can also be specified on these maps. Although conformal maps do not attempt to preserve linear or areal information, locally they appear Euclidean. Conformal information allows shape to be preserved. In this paper we describe our approach and present some of advantages of conformal flattening using circle packings. We discuss the notion of a conformal structure on a surface, and describe the three geometries of constant curvature surfaces where our maps reside, as well as classical conformal automorphisms (Möbius transformations) of these surfaces. We describe how circle packing can be used to obtain quasi-conformal mappings of surfaces and demonstrate the advantages of this approach by producing quasi-conformal flat maps with data from the Visible Man and from an MRI volume of the human cerebellum.