Conformal and Area-Preserving Maps for Surface Warping and Image
Registration


In this talk, we will outline some recent work using conformal
and area-preserving mappings for problems in surface warping and
image registration. First, we consider a novel 3D
visualization technique based on surface flattening for virtual
colonoscopy. Such visualization methods could be important in
virtual colonoscopy since they have the potential for non-invasively 
determining the presence of polyps and other pathologies.  
Further, we demonstrate a method which presents a 
surface scan of the entire colon to demonstrate 
a method which presents a surface scan of the entire colon as
a cine, and affords the viewer the opportunity 
to examine each point on the surface without distortion. 
This conformal flattening method will also be employed for flattening 
the brain on the sphere for visualization in functional MR.

The conformal mapping is an attempt to preserve the 
appearance of the surface through the preservation of angles.  
However, in some applications it is desirable to be able to 
preserve areas instead of angles, so that the sizes of 
surface structures are accurately represented in the plane. This 
leads us to the theory of area-preserving mappings, and Monge-Kantorovich 
optimal transport for surface warping and image registration.

This is joint work with Steve Haker, Sigurd Angenent, and Ron Kikinis.