
Extract curves
on triangulated meshes
1. Overview
The
fast marching method, proposed by J.A. Sethian, is a numerical
technique designed to track the evolution of interfaces. It has
proven to be powerful in finding geodesic path, or path with certain
properties. In this project, we use this method to find the sulcal
fundi on the human brain cortex, which is represented as a triangulated
mesh.
2. Method
Given an interface dividing a domain into two regions
at time t=0. Imagine that each point on the interface is
a source of spherical wave. The wave propogates in the domain with
certain speed, F(x,y). We want to find the time at which the wave
front crosses any point (x,y), i.e., solve for the arrival time
for each point, T(x,y). This is a "boundary value problem":
The fast marching method is a fast method to solve the
above problem numerically. Once the arrival time T(x,y) is computed,
we can take its gradient and follow the negative gradient direction
from any point in the domain and eventually get to a point on the
interface. In this way, we can get the geodesic path between any
two point by setting F(x,y)=constant, or tracking some other
desired features by using appropriate speed functions.
3. Results
This is a piece of central sulcus viewed from inside
the brain. The blue curve lies on the deepest part of the sulcus.
In this case, the speed function F(x,y) is set so that it is large
at points that are deeply buried or have large curvature, and small
at points with small curvature. The curves generated using this
method is smooth and have subgrid resolution.
Zoom in of part of the above image.
Geodesic curve shown on a triangulated mesh. Black
lines show the mesh of the surface.
References
1. J.A. Sethian, "Level Set Methods and Fast Marching
Methods Evolving Interfaces in Computational Geometry, Fluid Mechanics,
Computer Vision, and Materials Science", Cambridge University Press,
1999.
2. Kimmel, R., and Sethian, J.A., "Fast Marching Methods
on Triangulated Domains", Proc. Nat. Acad. Sci., 95, pp. 83418435,
1998.
3. Xiaodong Tao, Xiao Han, Maryam E. Rettmann,
Jerry L. Prince, and Christos Davatzikos, "Statistical Study on
Cortical Sulci of Human Brains", Proc.
of Information Processing in Medical Imaging (IPMI),
p.p. 475487, Davis, CA, June 2001
4. The fast marching method used in this project is
implemented by Xiao Han.

