Software for Interacting with Quasi-Conformal Flat Maps


1 Monica K. Hurdal, 2 Ken Stephenson, 1 Phil Bowers, 1 De Witt Sumners,
3 Kirt Schaper, 3,4 Kelly Rehm, 3,4 David Rottenberg

1 Department of Mathematics, Florida State University, Tallahassee
2 Department of Mathematics, University of Tennessee, Knoxville
3 PET Imaging Center, VA Medical Center, Minneapolis
4 Department of Radiology, University of Minnesota, Minneapolis

Previously we have presented an approach for creating quasi-conformal flat maps of a cortical surface [1,2]. It is impossible to flatten a curved surface in 3D space without metric and areal distortion; however, the Riemann Mapping Theorem [3] implies that it is theoretically possible to preserve conformal (angular) information under flattening. The maps that we produce exhibit conformal behavior in that angular distortion is controlled. Conformal maps are mathematically unique and our maps can be displayed in the Euclidean and hyperbolic planes and on a sphere.

We present some of the flat maps we have produced of the human cerebellum and demonstrate software which we have developed that allows the user to manipulate the flat maps and cortical surface. This includes interacting between the original 3D cortical surface and the flat map, such as picking a point on a flat map and displaying this point on the 3D surface, and vice versa. The user can also interactively change the map focus to bring any desired area into sharp focus, relegating the map distortion to map boundary. Various map-coloring schemes have been implemented which enable the maps to be colored according to any parcellation. This software is written with public-domain tools and a tutorial web version will also be available. A major benefit of (quasi-)conformal flat maps is that they can be used to generate canonical surface-based coordinate systems for a cortical surface. This software will assist users in imposing a unique coordinate system on the flat map based on two landmarks. We believe this will improve our ability to localize functional activity on the cerebellar cortex and to quantify anatomical and functional differences between individual subjects and groups of subjects.


1. Hurdal, M.K., Bowers, P.L., Sumners, D.W.L., Stephenson, K., Rottenberg, D.A., Lecture Notes in Computer Science, 1999, 1679:279-286.

2. Hurdal, M.K., Sumners, D.W.L., Stephenson, K., Bowers, P.L., Rottenberg, D.A., Neuroimage, 1999, 9:S195.

3. Ahlfors, L.V., Complex Analysis, McGraw-Hill Book Company, New York, 1996.

This presentation will be a computer demonstration of this software.