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Multidimensional Lattices in Signal Processing

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Multidimensional Lattices in Signal Processing Multidimensional Lattices in Signal Processing Riccardo Bernardini [email protected] https://www.linkedin.com/in/riccardobernardini/ March 5, 2019 Abstract This document collects few notes about the use of lattices as multidimensional sampling/periodicity structures, pointing out the connection with the Unified Theory. Contents 1 What is a lattice?2 1.1 Change of basis and sub-lattices..............................4 1.1.1 Change of basis and elementary transformations.................5 1.2 Rational Lattices......................................5 2 Lattices and unified signal theory7 2.1 Reciprocal lattice and quantum...............................7 2.2 Integral and convolution..................................8 2.3 Periodic signals.......................................8 2.4 Period and fundamental cells................................9 2.5 Fourier transform...................................... 11 2.6 Sampling and aliasing.................................... 11 2.6.1 Sampling resolution................................ 11 A Applications 12 A.1 Hexagonal sampling of images............................... 12 A.2 Electrons in a crystal.................................... 15 A.2.1 The case of finite crystal.............................. 16 A.3 Spatio-temporal sampling.................................. 17 B Brief summary about quotient groups 19 1 C Proofs 20 C.1 Density of Z(a;b) ..................................... 22 1 What is a lattice? The usual model for a one-dimensional discrete-time signal is a function defined on the set Z(T) = fnT : n 2 Zg (1) representing the set of sampling instants. Value T in (1) is the sampling interval, has the dimension of time and it is the reciprocal of the sampling frequency Fc = 1=T. Set (1) formalizes the intuitive idea of “set of regularly spaced points on a line.” In order to extend this idea to a d-dimensional space, a natural approach is to sample regularly along every direction of the d-dimensional space (e.g., along rows and columns for a 2-dimensional signal like an image) obtaining d Z (T1;:::;Td) = f[n1T1;n2T2;:::;ndTd] : n1;n2;:::nd 2 Zg (2) = Z(T1) × Z(T2) × ··· × Z(Td) However, while in the 1-dimensional case sets like (1) are the only possibility for the domain of a 1-dimensional signal, in multiple dimensions sets of type (2) are not the only choice. Indeed, when moving to more than one dimension there are many other possibilities for a “regularly spaced collection of points.” Few examples for the two-dimensional case can be seen in Fig.1. In order to find a suitable mathematical description for sets like the ones shown in Fig.1, observe that a characteristic shared by the sets shown in Fig.1 is that they can be generated by “walking” along the shown vectors by an integer number of steps. For example, the point marked with a cross in Fig.1b can be reached by moving 2 steps forward along b2 and 1 step backward (that is, −1 steps. ) along b1. This observation suggests that the sets shown in Fig.1 can be written as L(b1;b2) := fn1b1 + n2b2 : n1;n2 2 Zg (3) The d-dimensional counterpart of (3) is, quite naturally, L(b1;b2;:::;bd) = fn1b1 + n2b2 + ··· + ndbd : n1;n2;:::;nd 2 Zg (4) Definition 1. Set L(b1;b2;:::;bd) in (4) is called the d-dimensional lattice with basis fb1;b2;:::;bdg. Notation (4) is fairly cumbersome. It is more convenient to use a notation that is more compact and also dimension-independent. This can be obtained by considering the vectors bi as column vectors of t matrix M = [b1;b2;:::;bd] and collecting n1;:::;nd in column vector n = [n1;n2;:::;nd] . Equation (4) becomes d L(M) = fMn : n 2 Z g (5) We will mostly use notation (5), resorting to the more explicit (4) only when necessary. 2 (a) (b) (c) (d) Figure 1: Examples of 2-dimensional lattices (a) Orthogonal (b) Quincunx (c) and (d) two generic lattices 3 Figure 2: Three different bases for the quincunx lattice Remark 1.1 (Lattices vs. vector spaces) Equations (4) and (5) suggest that a lattice generated by vectors b1;b2;:::;bd is similar to the vector space generated by the same set of vectors, but with the constraint that the coefficients must be integer. This seemly small difference has a big impact in the properties of L(M). d For example, suppose B = fb1;b2;:::;bdg is a set of d linearly independent vectors of R . If one d + + adds to B a vector c 2 R to obtain B = B [fcg, the space generated by B coincides, clearly, with the space generated by B. That is, adding extra vectors to a basis (in the case of vector spaces) does not change anything. With lattices things are much more complex. For example, working for simplicity with the one dimensional case, it is fairly easy to check that L(1=7;1=4) = L(1=28) (6) that is, the set obtained by taking integer linear combinations of 1=7 and 1=4 is the set of multiplies of 1=28. The proof is a (really ) easy exercise. More (much more) tricky is to prove that set L(2;p) ⊂ R (7) is a dense subset of R, that is, for every x 2 R and it is possible to find u 2 L(2;p) arbitrarily near to x (that is, such that jx − uj ≤ e with e > 0 arbitrarily small). See Property6 in AppendixC. 1.1 Change of basis and sub-lattices In the one-dimensional case the sampling interval T > 0 is uniquely determined by the corresponding lattice L(T) = Z(T). In multiple dimensions this is not true anymore since a lattice can be generated by many different bases. See, for example, Fig.2 that shows three different bases for the quincunx lattice. The set of bases that generate the same lattice has an easy characterization. d×d Property 1. Two matrices M;N 2 R are two bases for the same lattice (i.e., L(M) = L(N)) if and only if d×d M = NH; H 2 Z ;jdetHj = 1 (8) Property1 is a corollary of the following property. 4 Property 2. Lattice L(M) is a sub-lattice of L(N) (that is, L(M) ⊆ L(N) if and only if d×d M = NH; H 2 Z (9) Remark 1.2 Note the similarity of Property2 with the one-dimensional case where Z(U) is a subset of Z(T) if and only if U is a multiple of T, that is, U = NT, N 2 Z. The proofs of Property1 and Property2 are in AppendixC, Proof C.1 and Proof C.2, respectively. 1.1.1 Change of basis and elementary transformations There are three simple elementary transformations that can be done on a basis M in order to obtain an equivalent one; they are 1. Multiply a column by −1 2. Swap two columns 3. Add to column i column j multiplied by an integer. It is (really ) easy to verify that the three operations above correspond to right-multiplying M by a matrix H with jdetHj = 1. Therefore, by doing the operations above we do not change the generated lattice. Moreover, it is possible to prove that every matrix H with jdetHj = 1 can be obtained by using the elementary operations above. Remark 1.3 The proof of the latter result is definitively not trivial. A possibility is to use the Hermite normal forma) of an integer matrix to prove that every matrix H with jdetHj = 1 can be reduced to the identity matrix by elementary transformations. It is less trivial than it seems since we can multiply only by integers and cannot do divisions. ahttps://en.wikipedia.org/wiki/Hermite_normal_form 1.2 Rational Lattices It is often convenient to suppose that the lattice of interest is a rational lattice. Definition 2. Lattice L(M) is said to be rational if for every r = 1;:::;d of M, the elements of row r are in rational ratio. More precisely, L(M) is rational if [M]r;c2 8r;c1;c2 2 f1;:::;dg [M]r;c1 = 0 _ 2 Q (10) [M]r;c1 where _ denotes, as usual, logical OR. 5 Remark 1.4 Definition2 seems to depend on the specific basis of the lattice. However, it is easy to show that if d×d H 2 Z , jdetHj = 1, then M satisfies (10) if and only if MH does. In other words, if a basis of L(M) satisfies (10), then all the bases of L(M) satisfy it; if a basis of L(M) does not satisfies (10), then no basis of L(M) satisfy it. The class of rational lattices is very large, but not every lattice is a rational lattice; for example, the lattice of basis 2 p 3 1 2 4 p 5 (11) p 3 p is clearly non rational since 1= 2 62 Q. However, it is also clear that in practical applications it is quite unlikely to find lattices with non rational bases. Therefore, the hypothesis of rational lattice is quite cheap, since we can expect that no case of practical interest will be left out. Rational lattices enjoy the following property Property 3. Every basis M of a rational lattice L(M) can be written as M = DK (12) d×d d×d with D = diag(D1;:::;Dd) 2 R a diagonal matrix and K 2 Z . Moreover, there is another basis d×d N of the same lattice L(M) (that is, N = MH, H 2 Z , jdetHj = 1) such that N = DKb (13) with Kb in lower triangular (actually Kb can be made in Hermite normal form1). The proof is in AppendixC, Proof C.3. Informally, Property3 can be rephrased that if L(M) is rational, one can choose the units of measure along the d axis in order to have a lattice with integer coordinates.
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