Conformal Flattening Registration
The goal of this project is for better visualizing and computation of neural activity from fMRI brain imagery. Also, with this technique, shapes can be mapped to spheres for shape analysis, registration or other purposes. Our technique is based on conformal mappings which map genus-zero surface: in fMRI case cortical or other surfaces, onto a sphere in an angle preserving manner.
The explicit transform is obtained by solving a partial differential equation. Such transform will map the original surface to a plane(flattening) and then one can use classic stereographic transformation to map the plane to a sphere.
The process of the algorithm is briefly given below:
- The conformal mapping f is defined on the originla surface Σ as . In that u and v are the conformal coordinates defined on the surface and the δp is a Dirac function whose value is non-zero only at point p. By solving this partial differential equation the mapping f can be obtained.
- To solve that equation on the discrete mesh representation of the surface, finite element method(FEM) is used. The problem is turned to solving a linear system D.x = b. Since b is complex vector, the real and imaginary parts of the mapping f can be calculated separately by two linear system.
- Having the mapping f, the original surface can be mapped to a plane.
- Further, the plane can be mapped to a sphere by the stereographic projection.
Also, in the work of Multiscale Shape Segmentation, conformal flattening is used as the first step for remeshing the surface.
The road to ITK and Slicer3:
The conformal flattening algorithm is now in ITK/Code/Review/itkConformalFlatteningMeshFilter.h/txx. The following figures show how the surfaces are mapped conformally to a sphere.
During the project week 2008, it is further put into Slicer3 as a command line module. Below we show two screen shots of using it.
The major computation happens in the solution of the linear system using conjugate gradient method. This can be accelerated using the pre-conditioning technique.
- Georgia Tech Algorithms: Yi Gao, John Melonakos, Allen Tannenbaum