Conformal Surface Mapping of Planar Texture
(IMA Logo for the 2000-2001 year on "Mathematics in Multimedia")

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The conformal mapping method is based on the work of Sigurd Angenent (University of Wisconsin), Steven Haker (University of Minnesota), Allen Tannenbaum (University of Minnesota), and Ron Kikinis (Brigham and Women's Hospital). It appears in their paper "On the Laplace-Beltrami Operator and Brain Surface Flattening" (IEEE Trans. on Medical Imaging, 18, (1999), pp. 700-711).

In this paper, using certain conformal mappings from uniformization theory, we give an explicit method for flattening highly undulated surfaces such as the brain cortical surface in a way which preserves angles. From a triangulated surface representation of the cortex, we indicate how the procedure may be implemented using finite elements. The technique also may be applied in a straightforward manner to automatic texture mappings as well. Flattened representations of various organs have become increasingly important in 3D medical visualization in applications ranging from functional magnetic resonance to virtual colonoscopy.

In our work, the key observation is that the flattening or texture mapping function may be obtained as the solution of a second order elliptic partial differential equation (PDE) on the surface to be flattened. For triangulated surfaces, there exist powerful, reliable finite element procedures which can be employed to numerically approximate the flattening function. We incorporate some key modifications to accomodate the special boundary conditions of our problem. In the end, the mapping is obtained as the solution of two sparse systems of linear equations.

2000-2001 Program: Mathematics in Multimedia