On the Multi-Dimensional Extension of the Quincunx Subsampling Matrix
D. Van De Ville, T. Blu, M. Unser
IEEE Signal Processing Letters, in press.
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The dilation matrix associated with the 3D FCC sublattice is often considered to be the natural 3D extension of the 2D quincunx dilation matrix. However, we demonstrate that both dilation matrices are of different nature: while the 2D quincunx matrix is a similarity transform, the 3D 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 3D 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 3D data before processing using such a subsampling matrix.