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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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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

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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 1.5 dilation . 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 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 S in Equation 6. Using non- −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 = {} ([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 T genLattices may contain several equivalent copies of the same lattice. A 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 , 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 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. The lattice on the left is the well known quincunx sampling with a θ =45◦ rotation. The other two are new schemes with different rotation angles. The black markers show the sample positions that are retained after subsampling by K.

1: R = [−0 0.93;1.1 −0.54] 2: R = [0 0.84;−1.2 0] 3: R = [0 0.74;−1.3 0.22] 4: R = [0.66 0;1.1 −1.5] K = [−1 2;−2 1] θ=90 K = [1 −1;2 1] θ=54.74 K = [1 −1;3 0] θ=73.22 K = [−3 4;−3 3] θ=90 5 5 5 5 4 4 4 4 3 3 3 3 2 2 2 2 1 1 1 1 0 0 0 0 0 5 0 5 0 5 0 5

Figure 3: Three non-equivalent 2D lattices obtained for a design with dilation matrices having |det K| =3. The lattice on the left is the well known hexagonal lattice with a θ =30◦ rotation. The other three are new schemes with different rotation angles. of the set q(A)={kT Ak : k ∈ Zn} using the Gram ma- kissing # = 2, # f = 14, # v = 24, G(P) = 0.081904, # zones = 6 trices of the respective lattices. If the sorted lists q(A1) q(A2) R1 R2 and disagree in any element, and are not 0.8 equivalent ([5], p. 60). It is possible to restrict the set of 0.6 n indices k ∈ Z to the Voronoi relevant neighbours [1]. 0.4 Further, since these neighbours determine the hyperplanes 0.2 0 bounding the Voronoipolytope of the lattice, they can also −0.2 be used for a sufficient test for equivalence. −0.4 −0.6

−0.8 0.5

−0.5 0 3. Constructions for different dimensions 0 −0.5 and subsampling ratios 0.5 Figure 4: The best 3D lattice obtained for a design with For the 2D case we have created lattices permitting a re- dilation matrices having |det K| =2. The letters f and v 2 3 duction rate in Figure 2 and rate in Figure 3. In both in the title line indicate faces and vertices, respectively. cases, familiar examples arise in the quincunx and the hex lattice for the respective ratios. A search of 3D lattices enjoying the self-similar subsam- teed to produce all possible solutions. For 2D and 3D we 2 53 pling property with rate dilations resulted in non- also employed an exhaustive search over a range of integer equivalent cases. These lattices were compared in terms of matrices with values in [−3, 3] resulting in the same num- their dimensionless second order moments, corresponding ber of non-equivalent 2D cases as the construction via K T . to the expected squared vector quantization error ([2], p. However, for dimensionality n>3 the exhaustive search 451). When performing the continuous optimization men- had to be replaced by a random sampling of integer matri- tioned at the end of Section 2.1, all of these cases con- ces ultimately rendering the method infeasible for n>5. verged to the same optimum lattice shown in Figure 4. In that light the current construction via scaled eigenvec- The dimensionless second order moment for the Voronoi tors of the companion matrix is a significant improvement G =0.081904 Cell of this lattice is . For comparison, the as it allows to produce a large number of non-equivalent G =0.0833 Cartesian cube has cc and the truncated octa- lattices for any dimensionality. hedron of the BCC lattice has Gbcc =0.0785. Our subsampling schemes may have applications for mul- tidimensional wavelet transforms [7]. Another direction 4. Discussion and potential applications for possible investigation is the construction of sparse grids that are employed in the context of high-dimensional The current formation of candidate matrices K based on integration and approximation adapting to smoothness similarity transforms of one valid example is not guaran- conditions of the underlying function space [3]. Appendix: Characteristic polynomial of a Thus, if scaled rotation matrix in Rn n d(λ)= c λk The similarity relationship between K and Q in Equa- k k=0 tion 2 implies that they share the same characteristic poly- n α2 k λ n nomial d(λ)=det(K − λI)=det(Q − λI) leading to = − c d(λ )=0 k λ α an agreement in eigenvalues k and determinant k=0 (13) d(0) ([6], p. 184). Further, since K is an n d(λ) ∈ Z[λ] c n−2k k the polynomial has integer coefficients k. = − cn−kα λ In order to find integer matrices K with the eigenvalues k=0 of a scaled rotation matrix, it will be important to distin- n−2k 1− 2k ⇔ ck = −α cn−k = −δ n cn−k. guish the two different forms of the diagonal matrix ∆ in n = Equation 4 and 5 for the case even By the same reasoning as for the even case, ck =0for all n−1 jθ1 −jθ1 jθn/2 −jθn/2 k =1, 2,... ∆ = diag[e e ...e e ] 2 resulting in only one possible character- istic polynomial and the case n = odd d(λ)=λn − αn. ∆ = diag[1 ejθ1 e−jθ1 ...ejθ(n−1)/2 e−jθ(n−1)/2 ] (14) with analogue block-wise constructions for Jn. To refer to the above procedure we will invoke a function For dimensionality n = even the characteristic polyno- compoly(n, α, C) that returns a companion matrix (Equa- mial of K and Q fulfills tion 7) with a characteristic polynomial as in Equation 11 or 14. n/2 d(λ)= (αejθk − λ)(αe−jθk − λ) k=1 References: n/2 2 2 = (α − 2λα cos θk + λ ) [1] E. Agrell, T. Eriksson, A. Vardy, and K. Zeger. Clos- est point search in lattices. Information Theory, IEEE k=1 (9) n/2 Transactions on, 48(8):2201–2214, August 2002. α4 α3 λ2 = ( − 2 cos θ + α2) [2] J.H. Conway and N.J.A. Sloane. Sphere Packings, 2 k 2 λ λ α Lattices and Groups. – 3rd ed. Springer, 1999. k=1 α2 λ n [3] M. Griebel. Sparse grids and related approxima- = d λ α tion schemes for higher dimensional problems. In L. Pardo, A. Pinkus, E. Suli, and M.J. Todd, ed- Thus, if itors, Foundations of Computational Mathematics n (FoCM05), Santander, pages 106–161. Cambridge k d(λ)= ckλ University Press, 2006. k=0 [4] Y.M.Lu, M.N. Do, and R.S. Laugesen. A Computable n k n α2 λ Fourier Condition Generating Alias-Free Sampling = c k λ α Lattices. IEEE Transactions on Signal Processing, k=0 (10) 57(5):(15 pages), May 2009. n n−2k k = cn−kα λ [5] M. Newman. Integral Matrices. Academic Press, k=0 1972. See http://www.dleex.com/read/ n−2k 1− 2k ?3907 for a digital copy. ⇔ ck = α cn−k = δ n cn−k. [6] L.N. Trefethen and D. Bau III. Numerical Linear Al- 1− 2k If ck =0and ck,δ ∈ Z then δ n ∈ Q. This is impos- gebra. SIAM, 1997. sible for 0 < 2k