We present a novel approach of obtaining anatomical connection probability in diffusion tensor imaging (DTI) via anisotropic Gaussian kernel smoothing. This method can be also used in smoothing functional and structural signals along the white fiber tracks. Our approach is computationally faster than solving a diffusion equation, which has been used in probabilistic representation of white matter connectivity by others.

**Anisotropic Smoothing**

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**)
= *w*_{j}(**x**-**x**_{j})*f*(**x**_{j}),
where the summation is taken over the neighboring voxels around **x**
and *w*_{j} 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.

**Connection Probability**

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**)=
*w*_{j}(**p**-**x**_{j})*P*(**x**_{j},**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 *F*_{n+1}(**x**)=*K***F*_{n}(**x**)
with the initial condition *F*_{0}(**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:
d*F*/d*t* = 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.

**References**

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.