[hal-00453440, v1] On sampling lattices with similarity scaling relationships

We provide a method for constructing regular sampling lattices in arbitrary dimensions together with an integer dilation matrix. Subsampling using this dilation matrix leads to a similarity-transformed version of the lattice with a chosen density reduction. These lattices are interesting candidates for multidimensional wavelet constructions with a limited number of subbands. 1. Primer on sampling lattices and related work A sampling lattice is a set of points {Rk : k ∈ Zn} ⊂ R that is closed under addition and inversion. The nonsingular generating matrix R ∈ Rn×n contains basis vectors in its columns. Lattice points are uniquely indexed by k ∈ Z and the neighbourhood around each sampling point is identical. This makes them suitable sampling patterns for the reconstruction in shift-invariant spaces. Subsampling schemes for lattices are expressed in terms of a dilation matrix K ∈ Zn×n forming a new lattice with generating matrix RK. The reduction rate in sampling density corresponds to |detK| = α = δ ∈ Z. (1) Dyadic subsampling discards every second sample along each of the n dimensions resulting in a δ = 2 reduction rate. To allow for fine-grained scale progression we are particularly interested in low subsampling rates, such as δ = 2 or 3. As discussed by van de Ville et al. [8], the 2D quincunx subsampling is an interesting case permitting a twochannel relation. With the implicit assumption of only considering subsets of the Cartesian lattice it is shown that a similarity two-channel dilation may not extend for n > 2. Here, we show that by permitting more general basis vectors in R the desired fixed-rate dilation becomes possible for any n. Our construction produces a variety of lattices making it possible to include additional quality criteria into the search as they may be computed from the Voronoi cell of the lattice [9] including packing density and expected quadratic quantization error (second order moment). Agrell et al. [1] improve efficiency for the computation by extracting Voronoi relevant neighbours. Another possible sampling quality criterion appears in the R = [ 0 −0.3307 1 −0.375 ] , K = [ 2 −1 4 −1 ] , θ = 69.3◦ −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 Figure 1: 2D lattice with basis vectors and subsampling as given by R and K in the diagram title. The spiral shaped points correspond to a sequence of fractional subsamplings RK for s = 0..1 with the notable feature that for s = 1 one obtains a subset of the original lattice sites shown as thick dots. This repeats for any further integer power of K, each time reducing the sample density by |detK| = 2. work of Lu et al. [4] in form of an analytic alias-free sampling condition that is employed in a lattice search. 2. Lattice construction We are looking for a non-singular lattice generating matrix R that, when sub-sampled by a dilation matrix K with reduction rate δ = α, results in a similarity-transformed version of the same lattice, that is, it can be scaled and rotated by a matrix Q with QQ = αI. An illustration of a subsampling resulting in a rotation by θ = arccos 1 2 √ 2 in 2D is given in Figure 1. Formally, this kind of relationship can be expressed as