Fig 4. 2D histogram representation of three 3D MRI contrast images.

(A) Each voxel is considered as a 3 part composition in 3D real space. The barycentric coordinates of each composition which reside in 3D simplex space are represented in 2D real space after using a isometric log-ratio (ilr) transformation. (B) The ilr coordinates are used to create 2D histograms representing all voxels in the images. The blue lines are the embedded 3D real space primary axes. It should be noted that in this case the ilr coordinates are not easily interpretable by themselves but they are useful to visualize the barycentric coordinates which are interpretable via the embedded real space primary axes. Darker regions in the histogram indicate that many voxels are characterized by this particular scale invariant combination of the image contrasts. In this representation, brain tissue (WM and GM, red dashed lines) becomes separable from non-brain tissue (black dashed lines and arrows).