It really depends what scientific question you are trying to answer. If you are interested in one particular direction just subtract values of the two voxels in the corresponding volume. You can also calculate measures such as mean diffusivity or diffusivity in the main direction (first eigenvector) or the one perpendicular to it.

Hi Chris, Thank you for your answer.

In my case, I don't want to use tensor model. Actually, I want to use all the raw data, that is the 64 diffusivity values along each direction.

I think each voxel can be represented as a 3D shape with 64 vertices. Then the similarity can be measured by comparing two shapes.

The scientific (maybe not :)) question is I want to measure regional similarity of diffusivity shape. Have any suggestions？

I don't think you can "compare shapes" without using the tensor model or, at least a model.

Now, if you want to be creative there is a measure that we use in fMRI called regional homogeneity. It is a similarity measure. I am not sure exactly how to use it for diffusion imaging. Maybe you could arrange you 64 values for each voxel into a vector to mimic a time series and then proceed as usual.

Anyway, here is an introduction of the measure:

http://fcon_1000.projects.nitrc.org/indi/cpac/reho.html

First of all, do you really have the diffusivity in each tensor direction, or just the diffusion weighted signal in that direction? In the latter case you still need to solve the Stejskal-Tanner equation first D=ln(UW/DW)/b, where UW=unweighted (b=0) image, DW=diffusion-weighted image, b=b-value.

Then, for comparison it depends on you scientific question, as pointed out earlier. For scalar values you can compare the mean diffusivity (MD) or fractional anisotropy (FA), those have no directional dependency.

If you like to compare the "shape" of the orientation-distribution-function (ODF), i.e. the shape of the spatial diffusivity it gets tricky. E.g. in the tensor model you could compare direction (principal eigenvector) or the size of it (principal eigenvalue), or the FA, which is shape dependent (FA=0 sphere, FA=1 cylinder). You have 6 degrees of freedom (DOF) to work with (rotation and amplitude).

If you abandon the tensor model you have many possibilities, as with 64 directions you have a lot of DOF to work with and can compare many things. E.g. one could calculate the diffusion spectrum components (keyword DSI), and their size. Or you could create a fitting function for the radial values of each spatial angle, these could be correlation or some pattern matching (unrolling them for each angular position results in a vector again that can be compared). This is similar to the idea statet by Remi, which is suggesting to do a correlation analysis on that vector.

Or maybe you are interested in some shape features. Again there, the DSI or DTI components are of help (is it more of an ellipsoid, or flat shape).

So which one of these you use and their combination to come up with a statistical classifier is up to your question.

If it's the general ODF shape you are after, search for "shape comparison algorithms".

Let me know if you have a more specific scientific question, I may be able to help.

The tensor model is simply the interaction field of two vectors, the gradient direction and the direction of the physically constrained diffusing spins. One could as easily look at correlations between two different voxels diffusion values as well. There are an number of models that look at values from diffusion MR data. Diffusion spectrum imaging, or Q-space imaging or even Diffusion kurtosis imaging use different models to attach underlying meaning to the observed diffusion data. All these models acquire the diffusion data a little differently and process the data much differently.

You could easily look at the diffusion values in a statistical nature and deal with auto-correlation matrices. Anamolous diffusion investigations tend to start with this approach since Fickian type models do not account for the observed diffusion.

You can be applauded for opting out of the tensor model, despite its (unearned) popularity. It is well known by now that the tensor model is appropriate for only a small minority of white matter voxels, and for comparison across subjects it is much wiser not to make this assumption. Instead, you could use a similarity metric based on normalized feature vectors for each subject and voxel, consisting of the 64 values of signal attenuation produced by each combination of diffusion gradients. Such a metric could be a simple dot-product, which would be maximal for the greatest similarity.

If you feel free to use Spherical Harmonics, then it would provide a simple metric for similarity between the two angular functions. Since the DWI signal is an angular function, spherical harmonic decomposition of the voxel signals (each with 64 diffusion gradients) can be used for calculating angular correlation coefficient, i.e., similarity between the two angular structures. With 64 directions, you can expand up to 8th even orders utilizing 45 spherical harmonic coefficients. Hope this helps.

In addition to comparing normalized directional dependence of the diffusion, you should first compare the average diffisivity of all diffusion encoding directions. You do not need a tensor model for this.