Interpolation

The sampling technique above relies on the local pdfs existing in continuous space. Unfortunately, we only have access to MR acquisitions, and hence these local pdfs, on a discrete acquisition grid. We need a technique to generate samples from the local pdfs at a point not on the grid.

An obvious solution to this problem would be to interpolate the original data (using a standard interpolation scheme, such as sinc or trilinear interpolation), and generate the local pdf on fiber direction given this new interpolated data. This would be extremely computationally costly, but also, on further consideration, may not conceptually be the best thing to do. In the middle of large fiber bundles, where neighboring voxels have similar fiber directions (each with low uncertainty), the choice of interpolation scheme will have very little effect. However, in places where neighboring voxels may have significantly different directions, such as at the edge of fiber bundles or where different bundles meet, such an interpolation scheme will generate a fiber direction in between the directions of the voxels on the grid. More over, the result of sinc or linear interpolation of data which is related to parameters in a highly nonlinear (e.g. exponent of trigonometric functions) manner is likely to produce interpolated data which does not fit well to the model, and thus the resulting most probable fiber direction will be highly dependent on the noise in the measurements at the grid locations. An alternative to interpolating the data in this fashion, is to choose an interpolation scheme which will pick a sample from one of the neighboring voxels on the grid. In a probabilistic system, we also have the opportunity to use a probabilistic interpolation scheme. That is, we can choose a scheme which chooses the data from a single neighboring point on the acquisition grid, but the probabilities of choosing each neighbor will be a function, $g$, of their positions relative to the interpolation site. There are many possible functions for $g$, but we have chosen one which is analogous to trilinear interpolation. That is, in the $x$-dimension, the probability of choosing data from $\tt {floor}(x)$ is $g(\tt {floor}(x)\vert x)=\frac{\tt {ceil}(x)-x}{\tt {ceil}(x)-\tt {floor}(x)}$, and from $\tt {ceil}(x)$ is $g(\tt {ceil}(x)\vert x)=1-g(\tt {floor}(x)\vert x)$, and the same in the $y$ and $z$-dimensions. If a streamline, $\vec{z}$, were to pass through the same point twice, different nearest neighbors may be chosen, reflecting our lack of knowledge of the true pdf at that point.