Diffusion Smoothing

© 2017 Moo K. Chung
University of Wisconsin-Madison


Description

 Diffusion smoothing based on the finite element method (FEM) is introduced in Chung et al (2001).  The following self-contained package (diffusion_smooth.zip) perform diffusion smoothing. Simply run DEMO.diffusion.m line by line. You should be able to generate Figure 1. The diffusion_smooth.m routine requires FEM.m that computes A and C matrices associated with the finite element discritization of the given surface mesh (Chung & Taylor, 2004).

Instead of solving heat diffusion using FEM, it is possible to solve it as a series expansion involving  a heat kernel (Seo et al. 2010, Seo & Chung, 2011). Heat kernel smoothing approach is more stable and robust. This is the approach I strongly recommend to use to smooth surface data and surface itself.


Figure 1. (A) Triangle mesh of a torus. The isosruface algorithm is used to construct the mesh. (B) Simulated data with random noise (C) Diffusion smoothing with  sigma=0.1 and 100 iterations (D) Diffusion smoothing with  sigma=1 and 100 iterations. The step size 1/100 is too big to converge. (E) Diffusion smoothing with  sigma=1 and 2000 iterations.The step size 1/2000 is small enough for convergence.









 

References

[1] Chung, M.K. 2001. Statistical Morphometry in Neuroanatomy, PhD Thesis, McGill University.

[2] Chung, M.K., Taylor, J. 2004. Diffusion Smoothing on Brain Surface via Finite Element Method,  IEEE International Symposium on Biomedical Imaging (ISBI). 562.

[3] Seo, S., Chung, M.K., Vorperian, H. K. 2010. Heat kernel smoothing using Laplace-Beltrami eigenfunctions. 13th International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI). Lecture Notes in Computer Science (LNCS). 6363:505-512.

[4] Seo, S., Chung, M.K. 2011. Laplace-Beltrami eigenfunction expansion of cortical manifolds. IEEE International Symposium on Biomedical Imaging (ISBI).