Heat Kernel Smoothing on
(c) Moo K. Chung email@example.com
Department of Biostastics and Medical Infomatics
Waisman Laboratory for Brain Imaging and Behavior
University of Wisconsin-Madison
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
codes have been tested under MATLAB 7.5.
If you are using the Matlab
codes/sample data for your publication, please
reference  or  in References.
Gaussian kernel smoothing
August 16, 2011
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).
% makes the image into real numbers between 0 and
imagesc(signal); colormap('bone'); colorbar
normrnd(0, 2, 596, 368);
= signal + noise;
imagesc(f); colormap('bone'); caxis([0 1]);
The noisy image went
though Gaussian kernel smoothing with 10 and 20 FWHM to recover the
signal (Figure 1).
F = gaussblur(f,100);
imagesc(F); colormap('bone'); caxis([0 1]);
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
kernel smoothing on brain surfaces
May 26, 2010
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  or .
- For MNI
format, use the following lines to load a mesh into
MATLAB. For this, you need to download the following codes:
- surf.vertices = coord;
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.
- 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.
[surf.vertices, surf.faces] = read_surf(mesh);
To perform heat kernel smoothing using function hk_smooth,
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
Figure 2. Heat kernel
smoothing of real (top) and simulated (bottom) data with different
number of iterations. Simulation detail is given in .
kernel smoothing of mandible surface
October 26, 2015
As an application of heat kernel smoothing, we show how to
smooth a closed surface topologically equivalent to a
hk_smooth2 is a
modified version of hk_smooth
vector data in the first argument.
Now we take three coordinates in the form [x y z].
figure; figure_patch( surf,[0.74
May 26, 2010
- Chung, M.K., Robbins,S., Dalton, K.M., Davidson, Alexander,
A.L., R.J., Evans, A.C. 2005. Cortical
autism via heat kernel smoothing. NeuroImage 25:1256-1265
- Chung, M.K., Robbins, S., Evans, A.C. 2005. Unified
cortical thickness analysis. Information Processing in
Medical Imaging (IPMI). Lecture Notes in Computer Science (LNCS)
- Chung, M.K. 2004. Heat
its application to cortical manifolds. University of
Wisconsin-Madison, Department of Statistics, Technical Report
Update history 01/13/2006, 11/28/2006, 09/05/2007, 09/22/2007
05/26/2010 it can now smooth FreeSurfer mesh format.