NeuroImage 9, Number 6, 1999, Part 2 of 2 Parts


Generating Conformal Flat Maps of the Cortical Surface via Circle Packing

Monica K. Hurdal, De Witt L. Sumners, Ken Stephenson,
Philip L. Bowers, David A. Rottenberg

Department of Mathematics, Florida State University, Tallahassee, U.S.A.
Department of Mathematics, University of Tennessee, Knoxvillee, U.S.A.
PET Imaging Center, VA Medical Center, Minneapolis, U.S.A.
Department of Radiology, University of Minnesota, Minneapolis, U.S.A.

Abstract. A circle packing is a configuration of circles with a specified pattern of tangencies. We present a novel method which uses circle packing to create flat maps of the cortical surface. These maps exhibit conformal behavior in that angular distortion is controlled.

Methods. Let K represent a simply-connected triangulated surface which is a 2-manifold. Then there exists a circle packing for K on the Riemann sphere. This circle packing is unique up to Möbius transformations and inversions of the Riemann sphere. Furthermore, this packing represents a discrete conformal map (1). A packing for K is created as follows. Each vertex, v, of K is represented by a circle cv with the center of cv located at v. If vertices v and w are connected by an edge, then the circles cv and cw are tangent.

Vertex v belongs to k triangle faces, giving k angles at v. The sum of these angles is defined to be the angle sum of v, (v). A set of circles can be "flattened" so they fit together in the plane if (v) = 2Pi. This is accomplished by iteratively refining the circle packing and adjusting each angle at v. This process is repeated for all interior vertices until a nominated accuracy in the angle sums is achieved (2).

Results and Conclusions. This is a novel method that seeks to preserve conformal information and produce, as a first approximation, a flat quasi-conformal mapping of the 2D piecewise cortical surface in the conventional Euclidean plane. No cuts in the original cortical surface are required. This mapping can also be created in the hyperbolic plane which allows the map origin to be transformed interactively so that different anatomical landmarks can be used as the map focus. Color can be used to identify anatomical features. Results obtained from this approach which use triangulated macaque (3) and human cortical surfaces are presented.
Reconstruction of Macaque Cortex Hyperbolic Flat Map Hyperbolic Map with Different Map Origin
Reconstruction of Macaque Cortex (3) Hyperbolic Flat Map Hyperbolic Map with Different Map Origin

1. Dubejko, T., Stephenson, K., Experimental Mathematics, 1995, 4:307-348.
2. Stephenson, K., Computation Methods in Function Theory, 1997, "The Approximation of Conformal Structures via Circle Packing", Available at URL
3. Drury, H. A, Van Essen, D. C., Anderson, C. H., Lee, C. W., Coogan, T. A., Lewis, J. W., 1996, J. Cog. Neurosci., 8:1-28.

Acknowledgments. The authors would like to gratefully acknowledge Heather Drury and David Van Essen from Washington University Medical School, St. Louis, U.S.A., for their kind assistance in providing some of the data used in this presentation. This work has been supported in part by NIH grants MH57180 and NS33718.