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 sub-grid 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. 8341-8435, 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. 475-487, Davis, CA, June 2001

4. The fast marching method used in this project is implemented by Xiao Han.