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Modeling of the spatial covariance structure of the brain using variograms with a non-Euclidean metric

2(sill), r (range) and ρ (nugget) are the parameters to be estimated to fit the curve. The quantity d is the distance between voxels with locations u and v. Sill σ 2 , range r and nugget ρ, as well as the trend µ may be estimated by the non-linear least squares method applied to the estimated semi-variogram. It should be noted that by definition the semi-variogram function takes the value 0 when d is 0. But it may not be the case when d is very small but not identically 0. This is the reason for the appearance of the nugget in the model. We should also mention that the choice of exponential covariance function is made only for the sake of simplicity. There are numerous other models which may give us a better fit. We have already mentioned that the conventional Euclidean metric is not an appropriate choice in the present scenario. We now describe how we measure the distance function with a non-Euclidean metric. Ideally, the distance function for a pair of points within a segment should be the minimum length of all paths that connect the points and lie entirely within the segment. However, it is not feasible from a computational standpoint to calculate such a metric for all possible pairs of points in the segment of interest. Instead we make use of the fact that we have a finite number of voxels which may be considered as a subset of a finite rectangular graph with nodes at voxel locations. Note that this is only a subset of the rectangular graph due to the missing nodes at locations where there the voxels are missing. We can now make it a connected graph by joining adjacent vertical and horizontal nodes. Each single edge is assigned a length of one voxel unit. We can then compute the shortest distance between any two points in this graph along the edges [2]. The algorithm can be further refined by including adjacent diagonal edges as well, but this comes at a higher computational price and is not performed here. Results For computational simplicity, instead of the whole brain, we only consider a 256×256 segmented gray matter slice in the sagittal plane from an image acquired with a standard SPGR protocol. In Figure 1, we plot the semi-variogram for the data. For very large values of distance, the semi-variogram appears to be unstable. There are two reasons behind that. Large distance in general corresponds to voxels which are far apart from one another and predictive ability is expected to be low in such a case. More importantly, since we are only considering a single slice, there may be points with large non-Euclidean distances among them which would be significantly reduced if the paths are considered in 3D. Variogram analysis using a 3-D data is expected to improve the results significantly. Nevertheless, we observe a nice exponential behavior if we restrict ourselves to distances less than 80 voxel units, which is more than sufficient for any sort of prediction involving the data. In Figure 2, we plot the semi-variogram restricted to distances less than 80 voxel units and the best nonlinear least square fit of the exponential model. The fit was excellent with an R 2