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On Sampling Lattices with Similarity Scaling Relationships Steven Bergner, Dimitri Van De Ville, Thierry Blu, Torsten Möller

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On Sampling Lattices with Similarity Scaling Relationships Steven Bergner, Dimitri Van De Ville, Thierry Blu, Torsten Möller On sampling lattices with similarity scaling relationships Steven Bergner, Dimitri van de Ville, Thierry Blu, Torsten Möller To cite this version: Steven Bergner, Dimitri van de Ville, Thierry Blu, Torsten Möller. On sampling lattices with similarity scaling relationships. SAMPTA’09, May 2009, Marseille, France. pp.General session. hal-00453440 HAL Id: hal-00453440 https://hal.archives-ouvertes.fr/hal-00453440 Submitted on 4 Feb 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On sampling lattices with similarity scaling relationships Steven Bergner (1), Dimitri Van De Ville(2), Thierry Blu(3), and Torsten Moller¨ (1) (1) GrUVi-Lab, Simon Fraser University, Burnaby, Canada. (2) BIG, Ecole Polytechnique Fed´ erale´ de Lausanne, Switzerland. (3) The Chinese University of Hong Kong, Hong Kong, China. [email protected], [email protected], [email protected], [email protected] 0 −0.3307 2 −1 R = K = θ =69.3◦ Abstract: 1 −0.375 , 4 −1 , We provide a method for constructing regular sampling lattices in arbitrary dimensions together with an integer 1.5 dilation matrix. Subsampling using this dilation matrix leads to a similarity-transformed version of the lattice with 1 a chosen density reduction. These lattices are interest- ing candidates for multidimensional wavelet constructions 0.5 with a limited number of subbands. 0 1. Primer on sampling lattices and related work −0.5 A sampling lattice is a set of points {Rk : k ∈ Zn}⊂Rn that is closed under addition and inversion. The non- −1 singular generating matrix R ∈ Rn×n contains basis vec- −1.5 tors in its columns. Lattice points are uniquely indexed −1.5 −1 −0.5 0 0.5 1 1.5 by k ∈ Zn and the neighbourhood around each sampling point is identical. This makes them suitable sampling pat- Figure 1: 2D lattice with basis vectors and subsampling terns for the reconstruction in shift-invariant spaces. as given by R and K in the diagram title. The spiral Subsampling schemes for lattices are expressed in terms n×n shaped points correspond to a sequence of fractional sub- of a dilation matrix K ∈ Z forming a new lattice with samplings RKs for s =0..1 with the notable feature that generating matrix RK. The reduction rate in sampling for s =1one obtains a subset of the original lattice sites density corresponds to shown as thick dots. This repeats for any further integer |det K| = αn = δ ∈ Z+. (1) power of K, each time reducing the sample density by |det K| =2. Dyadic subsampling discards every second sample along each of the n dimensions resulting in a δ =2n reduction work of Lu et al. [4] in form of an analytic alias-free sam- rate. To allow for fine-grained scale progression we are pling condition that is employed in a lattice search. particularly interested in low subsampling rates, such as δ =2or 3. As discussed by van de Ville et al. [8], the 2D quin- 2. Lattice construction cunx subsampling is an interesting case permitting a two- channel relation. With the implicit assumption of only We are looking for a non-singular lattice generating matrix R K considering subsets of the Cartesian lattice it is shown that, when sub-sampled by a dilation matrix with re- δ = αn that a similarity two-channel dilation may not extend for duction rate , results in a similarity-transformed n>2. version of the same lattice, that is, it can be scaled and ro- tated by a matrix Q with QT Q = α2I. An illustration of a Here, we show that by permitting more general basis vec- 1 n θ = arccos √ tors in R the desired fixed-rate dilation becomes possi- subsampling resulting in a rotation by 2 2 in ble for any n. Our construction produces a variety of lat- 2D is given in Figure 1. Formally, this kind of relationship tices making it possible to include additional quality cri- can be expressed as teria into the search as they may be computed from the QR = RK (2) Voronoi cell of the lattice [9] including packing density and expected quadratic quantization error (second order leading to the observation that subsampling K and scaled moment). Agrell et al. [1] improve efficiency for the com- rotation Q are related by a similarity transform putation by extracting Voronoi relevant neighbours. An- other possible sampling quality criterion appears in the R−1QR = K. (3) 1 j uses a construction of T = LU from several random in- Using a matrix J2 = it is possible to diago- 1 −j teger lower and upper triangular matrices having ones on nalize a 2D rotation matrix by the following similarity their diagonal. transform It is not guaranteed that all possible K for a given charac- cos θ − sin θ ejθ 0 teristic polynomial can be generated through a similarity = J−1 J = J−1∆J . sin θ cos θ 2 0 e−jθ 2 2 2 transform with some T. However, formRQ(KT ) provides R (4) numerous non-equivalent T lattice generators. Among Using this observation to replace the scaled rotation matrix them it is possible to apply further criteria to select the Q in Equation 3 leads to “best” lattice. An alternative to transforming K is the eigenvector scal- K = R−1QR ing by diagonal matrix S in Equation 6. Using non-unit −1 −1 −1 K = αR Jn S∆S JnR (5) scaling allows to produce further lattices for any given K K = αP∆P−1 resulting in an n-dimensional continuous search space. with −1 −1 R = Jn SP 2.2 Construction Algorithm Q = αJ−1∆J . (6) n n The steps for constructing lattices with the desired Thus, given a matrix K that has an eigen-decomposition subsampling matrices are summarized in algorithm 1. corresponding to that of a uniformly scaled rotation ma- The function compoly(n, α, C) is defined in the trix, we can compute the lattice generating matrix R as in Equation 6. The elements of the diagonal matrix S in- Algorithm 1 genLattices(n, δ) serted in the construction of R scale the otherwise unit 1: Llist ←{} eigenvectors in the columns of P. Below, we will refer to 2: Ks ← genKompans(n, δ) this construction as function formRQ(K, S) using S = I 3: Ts ← genUnimodular(n) ∪{I} by default. 4: for all K ∈ Ks do 5: for all T ∈ Ts do −1 2.1 Constructing suitable dilation matrices K 6: KT = TKT 7: (RT , QT ) ← formRQ(KT ) The eigenvalues of K, ∆ and Q impose restrictions on 8: Llist← Llist∪{(KT , RT , QT )} their shared characteristic polynomial d(λ)=det(K − n k 9: end for λI)= k=0 ckλ as discussed in the appendix. For 10: end for the case n = even with the only non-zero integer coef- 2 11: return Llist ficients c0 = δ, cn/2 < 4δ, cn =1this leaves a finite number of different options for c n/2. The case n = odd permits a single possible polynomial with non-zero coef- n, δ ficients c0 = −δ, cn =1. For these monic polynomials Algorithm 2 genKompans( ) it is possible to directly construct a candidate K via the 1: Ks = {} companion matrix ([6], p. 192) 2: if n is even then 2 ⎡ ⎤ 3: for all C ∈ Z : C < 4δ do 1 0 −c0 4: Ks ← Ks ∪ compoly(n, δ n ,C) ⎢ ⎥ ⎢ 10 −c1 ⎥ 5: end for ⎢ . ⎥ K = ⎢ 10 . ⎥ . 6: else {n is odd} ⎢ . ⎥ (7) 1 ⎢ ⎥ 7: Ks ←{compoly(n, δ n )} ⎣ .. .. ⎦ . −cn−2 8: end if 1 −cn−1 9: return Ks This allows to construct a lattice fulfilling the self-similar subsampling condition for any dimensionality n, one for appendix. A possible implementation for the func- tion genUnimodular(n) is described in Section 2.1 and every possible characteristic polynomial. K With this starting point it is possible to construct additional formRQ( ) is defined below Equation 6. suitable dilation matrices via a similarity transform with a It should be noted that the list of lattices returned by unimodular matrix T genLattices may contain several equivalent copies of the same lattice. A Gram matrix implicitly represents angles −1 T −1 A = R R R1 KT = TKT = PT ∆PT . (8) between basis vectors as . Two lattices and R2, scaled to have the same determinant, are equivalent if T Using a unimodular rather than any non-singular T guar- their Gram matrices are related via A1 = T A2T with antees that T−1 is also unimodular following from the fact a unimodular matrix T ∈ Zn×n and |det T| =1. Deter- that T−1 can be constructed from the adjugate (the trans- mining this unimodular matrix is known to be a difficult posed co-factor matrix) of T. Thus, K T remains an inte- problem, as it for instance also occurs when relating the ger matrix by this transform. Possible generators for this adjacency matrices of two supposedly isomorphic graphs. unimodular group are discussed in ([5], pp. 23). Our im- Hence, our current method employs a simpler necessary plementation, referred to as function genUnimodular(n), test for equivalence by comparing the first few elements 1: R = [0.71 −0;−0.71 1.4] 2: R = [0 0.58;−1.7 0.65] 3: R = [0 0.84;−1.2 0] K = [2 −2;1 0] θ=45 K = [2 −1;4 −1] θ=69.3 K = [0 −1;2 0] θ=90 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 0 2 4 0 2 4 0 2 4 Figure 2: Three non-equivalent 2D lattices obtained for a design with dilation matrices having |det K| =2.
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