On the multidimensional extension of the quincunx subsampling matrix

The dilation matrix associated with the three-dimensional (3-D) face-centered cubic (FCC) sublattice is often considered to be the natural 3-D extension of the two-dimensional (2-D) quincunx dilation matrix. However, we demonstrate that both dilation matrices are of different nature: while the 2-D quincunx matrix is a similarity transform, the 3-D FCC matrix is not. More generally, we show that is impossible to obtain a dilation matrix that is a similarity transform and performs downsampling of the Cartesian lattice by a factor of two in more than two dimensions. Furthermore, we observe that the popular 3-D FCC subsampling scheme alternates between three different lattices: Cartesian, FCC, and quincunx. The latter one provides a less isotropic sampling density, a property that should be taken into account to properly orient 3-D data before processing using such a subsampling matrix.