NA-MIC/Projects/Structural/Conformal Flattening in ITK
Description
A general description of this project is given on page: Conformal Flattening for fMRI Visualization.
This page outlines the steps we will take to code the Conformal Flattening mapping in ITK.
For the detail of the algorithm please refer to the paper: S. Angenent, S. Haker, A. Tannenbaum, and R. Kikinis, “On the Laplace-Beltrami operator and brain surface flattening,” IEEE Trans. on Medical Imaging, Vol. 18, pp. 700-711, 1999.
An angle preserving flattening mapping is proposed in this paper. 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 brifly given below:
- The conformal mapping f is defined on the originla surface Σ as \triangle f = (\frac{\partial}{\partial u} - i\frac{\partial}{\partial v})\delta_p. 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.
The code for doing this is currently local codes, not having been embeded into ITK class hierarchy. This is what we are working on now.
Current Status
- We have written and tested the itkConformalFlatteningFilter
- Our paper on this was accepted for oral presentation at the 2006 MICCAI Open-source Workshop
Next Steps
- We will help integrate this into the ITK CVS repository
Members
- Yi Gao (Gatech)
- John Melonakos (Gatech)
- Jim Miller (GE)
- Luis Ibanez (Kitware)
