View source for NA-MIC/Projects/Structural/Conformal Flattening in ITK

== Description ==

A general description of this project is given on page: [[NA-MIC/Projects/fMRI_Analysis/Conformal_Flattening_for_fMRI_Visualization|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 <span class="texhtml">''f''</span> is defined on the originla surface <span class="texhtml">Σ</span> as [[Image:89ff38a84b2fcde5fa7c5ea0f8c97a6b.png|\triangle f = (\frac{\partial}{\partial u} - i\frac{\partial}{\partial v})\delta_p]]. In that <span class="texhtml">''u''</span> and <span class="texhtml">''v''</span> are the conformal coordinates defined on the surface and the <span class="texhtml">δ<sub>''p''</sub></span> is a Dirac function whose value is non-zero only at point <span class="texhtml">''p''</span>. By solving this partial differential equation the mapping <span class="texhtml">''f''</span> 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 <span class="texhtml">''D''''x'' = ''b''</span>. Since b is complex vector, the real and imaginary parts of the mapping <span class="texhtml">''f''</span> can be calculated separately by two linear system.
# Having the mapping <span class="texhtml">''f''</span>, 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)

== Links ==

* [[NA-MIC/Projects/fMRI_Analysis/Conformal_Flattening_for_fMRI_Visualization|Conformal Flattening for fMRI Visualization]]