Heat Kernel Smoothing - Moo K. Chung

Heat Kernel Smoothing on Anatomical Manifolds

(c) Moo K. Chung mkchung@wisc.edu

Department of Biostastics and Medical Infomatics
Waisman Laboratory for Brain Imaging and Behavior
University of Wisconsin-Madison

Description
May 26, 2010

Gaussian kernel smoothing has been widely used in 3D brain images to increase the signal-to-noise ratio. The Gaussian kernel weights observations according to their Euclidean distance. When the observations lie on a convoluted brain surface; however, it is more natural to assign the weight based on the geodesic distance along the surface [1] [2]. On the curved manifold, a straight line between two points is not the shortest distance so one may incorrectly assign less weights to closer observations. Therefore, smoothing data residing on manifolds requires constructing a kernel that is isotropic along the geodesic curves. With this motivation in mind, we constructed the kernel of a heat equation on manifolds that should be isotropic in the local conformal coordinates, and develop a framework for heat kernel smoothing on cortical manifolds. The codes have been tested under MATLAB 7.5.

If you are using the Matlab codes/sample data for your publication, please reference [1] or [2] in Referenes.

Gaussian kernel smoothing
August 16, 2011

Heat kernel smoothing genearlizes Gaussian kernel smoothing. Here we present a 2D version of Gaussian kernel smoothing as an example. The original image toy-key.tif is contaminated with Gaussian white noise N(0,2^2).

signal= imread('toy-key.tif');
signal=(double(signal)-219)/36; % makes the image into real numbers between 0 and 1.
figure; imagesc(signal); colormap('bone'); colorbar

noise= normrnd(0, 2, 596, 368);
f = signal + noise;
figure; imagesc(f); colormap('bone'); caxis([0 1]); colorbar

The noisy image went though Gaussian kernel smoothing with 10 and 20 FWHM to recover the signal.

F = gaussblur(f,100);
figure; imagesc(F); colormap('bone'); caxis([0 1]); colorbar

Figure 1. The orignal binary image was contaminated with Gaussian white noise N(0,2^2). Gaussian kernel smoothing with 10 and 20mm FWHM were performed to recover the original image.

Heat kernel smoothing
May 26, 2010

The reason that the code is short and simple is that it has been implemented as iterative kernel smoothing with very small bandwidth. For small bandwidth, a heat kernel converges to a Gaussian kernel. If you are using the following MATLAB codes in your research, please reference [1] or [2].

For MNI format, use the following lines to load a mesh into MATLAB. For this, you need to download the following codes:

hk_smooth.m heat kernel smoothing.

mni_getmesh.m load montreal neurological institute (MNI) mesh data into MATLAB.

thickness.data sample cortical thickness corresponding to the outer surface below.

outersurface.obj sample our cortical surface mesh.

[tri,coord,nbr,normal]=mni_getmesh('outersurface.obj');
surf.vertices = coord;
surf.faces=tri;

coord (3x40962) gives the x,y,z coordinates of nodes.

nbr (40962x6) gives neighboring node indices. Each node has up to 6 neighboring nodes.

normal (3x40962) gives the normal vectors of the cortex.

tri (81920x3) gives three node indices that forms each of 81920 triangles.

For FreeSurfer format, use the following code to load a mesh into MATLAB. FreeSurfer mesh data into MATLAB. The FreeSurfer package produces cortical meshes with different mesh topology for different subjects. The FreeSufer mesh format starts the face indexing from zero while the MATLAB can not handle zero indexing so we need to inrease the indexing by one.

mesh='lh.pial'
[surf.vertices, surf.faces] = read_surf(mesh);
surf.faces=surf.faces+1;
figure_wire(surf,'white','white');

To perform heat kernel smoothing, we generate 134506 uniform random numbers between 2 and 6mm. This simulated cortical thickness is smoothed out using heat kernel smoothing with bandwidth 1 and 20 iterations.

input=random('unif',2,6,[134506,1]);
figure_trimesh(surf,input,'rwb')
output=hk_smooth(input,surf,1,20);
figure_trimesh(surf,output,'rwb')

Figure 2. Heat kernel smoothing of real (top) and simulated (bottom) data with different number of iteratations. Simulation detail is given in [1].

References
January 6, 2008

Chung, M.K., Robbins,S., Dalton, K.M., Davidson, Alexander, A.L., R.J., Evans, A.C. 2005. Cortical thickness analysis in autism via heat kernel smoothing. NeuroImage 25:1256-1265
Chung, M.K., Robbins, S., Evans, A.C. 2005. Unified statistical approach to cortical thickness analysis. Information Processing in Medical Imaging (IPMI). Lecture Notes in Computer Science (LNCS) 3565:627-638. Springer-Verlag.
Chung, M.K. 2004. Heat kernel smoothing and its application to cortical manifolds. University of Wisconsin-Madison, Department of Statistics, Technical Report 1090.
Chung, M.K., Dalton, K.M., Davidson, R.J. 2008. Tensor-based cortical surface morphometry via weighed spherical harmonic representation. IEEE Transactions on Medical Imaging. 27:1143-1151.
Chung, M.K., Dalton, K.M., Shen, L., L., Evans, A.C., Davidson, R.J. 2007. Weighted Fourier series representation and its application to quantifying the amount of gray matter. IEEE Transactions on Medical Imaging. 26:566-581.

Created 02/05/2005. Update history 01/13/2006, 11/28/2006, 09/05/2007, 09/22/2007
05/26/2010 it can now smooth FreeSurfer mesh format.