Anisotropic Gaussian kernel K(x) with bandwidth matrix H is given as a mean zero multivariate normal probability density function with HH' as the covariance matrix. Then the anisotropic kernel smoothing of signal f(x) is defined as the convolution F(x) = K*f(x). One simple way to match the covariance matrix to the diffusion tensor D in such a way that the signal would be smoothed along the white fiber track is by letting HH'=2tD, where t is the scaling parameter. Discretizing the convolution, we have a discrete Gaussian kernel smoothing F(x) = wj(x-xj)f(xj), where the summation is taken over the neighboring voxels around x and wj are positive weights that sum to 1. The scaling parameter t controls the amount of smoothing and isotropy. As t increases, the smoothing and isotropy increase.
As an application of the above anisotropic Gaussian kernel smoothing, we show how to represent and compute the connection probability along the white fiber tracks. The probability of going from p to q under diffusion is P(p,q)=K(q-p) when p and q are very close (Stevens, 1995). The probability of going from p to any q is the sum of the probabilities of going from p to q through all possible intermediate points x. Then it can be shown that the approximate connection probability between p and q is given by P(p,q)= K*P(p,q), where the convolution is taken over the first argument of P. Hence the connection probability is given as an iterative integral equation with the initial condition P(p,p)=1. A similar discretization gives P(p,q)= wj(p-xj)P(xj,q), where the summation is taken around the neighboring voxels of p.
To generate the connection probability map from a given p, we iteratively apply anisotropic kernel smoothing Fn+1(x)=K*Fn(x) with the initial condition F0(x)=1 if x=p and 0 otherwise (Chung et al., 2003). The iteration is repeated until it hits q. Our formulation is analogous to the diffusion equation approach: dF/dt = div(D grad F) with initial condition F(x,0)= 1 if x=p and 0 otherwise (Batchelor et al., 2001). However, our solution is not exactly the solution of the above diffusion equation although the resulting connection probability maps are quite similar. Compared to solving the diffusion equation, our approach is extremely fast.
Batchelor, P.G., Derek L.G.H., Calamante, F., Atkinson, D. (2001) Study of Connectivity in the Brain Using the Full Diffusion Tensor from MRI. IPMI 2001: 121-133.
Chung, M.K., Alexander, A.L., Lu, Y. (2003). Probabilistic representations of connectivity information in diffusion tensor imaging, TR Department of Statistics, University of Wisconsin-Madison. http://www.stat.wisc.edu/~mchung/papers/DTI_tech.pdf
Stevens, C.F. (1995). The six core theories of modern physics, The MIT Press.