Software - Jacobian Image Analysis
Jacobian images can be computed from the high-dimensional warps of one image onto another. At each voxel, the Jacobian value represents the local volume expansion or contraction computed by the warp. No local change in volume would result in a Jacobian of 1.0. Expansions yield values greater than 1 and contractions have Jacobians smaller than 1, but still positive. A 0 or negative Jacobian value would indicate a singularity (folding or tear) in the warp and is avoided by prohibiting any such local volume alterations during the warp computation. Log-transformed Jacobian images may be displayed so that contractions are shown to the same scale as expansions.
Jacobian images are typically noisy. Accordingly we have implemented an algorithm due to Colin Studholme (pdf) which filters and smoothes the Jacobians by underlying tissue type. The output is an image showing more coherent volume change patterns with distinct boundaries at tissue boundaries. Thus, for example, relatively uniform expansion of the ventricles can be displayed together with contraction in the neighboring brain tissue.
The following figure illustrates these concepts. It shows the results of warping a template onto an ADC subject and the resulting areas of contraction and expansion.
Figure Unsmoothed (left) and smoothed, filtered (right) Jacobian images of template warp onto ADC subject. Light areas indicate local expansion of template to match subject. Dark areas indicate local contractions. Ventricle expansion and contractions of hippocampi and cortical areas are more easily seen in the smoothed image.