(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 [1-3]. On a 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 construct 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
manifolds. The
codes have been tested under MATLAB 7.5.

August 16, 2011

Heat kernel smoothing generalizes Gaussian kernel smoothing. Here we present a 2D version of Gaussian kernel smoothing as an example. The original binary 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 (Figure 1).

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.

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. For details on heat
kernel smoothing, please read [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.

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 using function

input=random('unif',2,6,[134506,1]);

figure_trimesh(surf,input,'rwb')

output=

figure_trimesh(surf,output,'rwb')

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

October 26, 2015

As an application of heat kernel smoothing, we show how to smooth a closed surface topologically equivalent to a sphere.

sigma=1;

n_smooth=10;

ss.vertices=hk_smooth2(surf.vertices,surf, sigma, n_smooth);

figure; figure_patch( surf,[0.74 0.71 0.61],0.7);

References

May 26, 2010

- 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.

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.