Octahedral Transforms for 3-D Image Processing

The octahedral group is one of the finite subgroups of the rotation group in 3-D Euclidean space and a symmetry group of the cubic grid. Compression and filtering of 3-D volumes are given as application examples of its representation theory. We give an overview over the finite subgroups of the 3-D rotation group and their classification. We summarize properties of the octahedral group and basic results from its representation theory. Wide-sense stationary processes are processes with group theoretical symmetries whose principal components are closely related to the representation theory of their symmetry group. Linear filter systems are defined as projection operators and symmetry-based filter systems are generalizations of the Fourier transforms. The algorithms are implemented in Maple/Matlab functions and worksheets. In the experimental part, we use two publicly available MRI volumes. It is shown that the assumption of wide-sense stationarity is realistic and the true principal components of the correlation matrix are very well approximated by the group theoretically predicted structure. We illustrate the nature of the different types of filter systems, their invariance and transformation properties. Finally, we show how thresholding in the transform domain can be used in 3-D signal processing.