Gaussian kernel smoothing has been widely used in either 2D flat or 3D volume images, but it does not work on the curved cortical surface. However, by reformulating Gaussian kernel smoothing as a solution to a diffusion equation on a 2D manifold, we can generalize it to the cortical surface. This generalization is called

Figure 1: A typical triangulation | Figure 2: The mean curvature from the outer cortex is mapped onto an ellipsoid during the diffusion smoothing. (a) Before the iteration (b) After 40 iterations (c) After 100 iterations |

**Methods**

Gaussian kernel smoothing of the signal *f(x)* in
n-dimension with
FWHM=4(ln2)^{1/2}*t*^{1/2} is defined as the
convolution of the n-dimensional Gaussian kernel *G(x;t)*
with the
signal *f(x)*, i.e. *F(x,t)=f*G(x;t)*. It can be shown
that
the convoluted signal *F* is the solution of a diffusion
equation *dF/dt=L[F]*
with the n-dimensional Euclidean Laplacian *L*. Since the
cortical
surface in non-Euclidean, the Euclidean Laplacian is not well
defined on
the cortical surface. The generalization of the Laplace operator *L*
to an arbitrary curved surface is called the *Laplace-Beltrami
operator* and it is defined in terms of the Riemmanian metric
tensors.

In order to estimate the Laplace-Beltrami operator on a
triangulated
cortical surface, we have used the *finite element method*[2].
Let *F(p _{i})*
be the signal on the

with the weights

**Results**

The ASP method [3] is used to extract the outer cortical surfaces
each
consisting of 81920 triangles from MR scans. At this surface
sampling
rate, the average intervertex distance is about 4mm. The mean
curvature *f(p _{i})*
of the cortical surface is computed based on the least-squares
estimation of a local quadratic surface [2]. Figure 2 shows the
diffusion smoothing of the mean curvature with 5mm FWHM. If the
smoothing were based on simple inter-nodal averaging, such sulcal
pattern is not possible to obtain.

**Conclusion**

Gaussian kernel smoothing can be generalized to cortical surfaces
enabling surface-based statistical analysis. The numerical
implementation will be freely available as Matlab code on the web
at
http://www.math.mcgill.ca/keith/BICstat.

**References**

[1] Andrade A et al., *Detection of fMRI Activation on the
Cortical
Surface*, NeuroImage, 2000 (in press).

[2] Chung MK et al.,
http://www.math.mcgill.ca/chung/diffusion/diffusion.pdf, 2000

[3] Chung MK et al., Statistical Analysis of Cortical Surface Area
Change, with an Application to Brain Growth, HBM2001 Conference

[4] MacDonald D et al., NeuroImage **12**:340-356, 2000