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Hexagonal QMF Banks and Wavelets Hexagonal QMF Banks and Wavelets A chapter within Time-Frequency and Wavelet Transforms in Biomedical Engineering, M. Akay (Editor), New York, NY: IEEE Press, 1997. Sergio Schuler and Andrew Laine Computer and Information Science and Engineering Department UniversityofFlorida Gainesville, FL 32611 2 Hexagonal QMF Banks and Wavelets Sergio Schuler and Andrew Laine Department of Computer and Information Science and Engineering University of Florida Gainesville, FL 32611 Introduction In this chapter weshalllay bare the theory and implementation details of hexagonal sampling systems and hexagonal quadrature mirror lters (HQMF). Hexagonal sampling systems are of particular interest b ecause they exhibit the tightest packing of all regular two-dimensional sampling systems and for a circularly band-limited waveform, hexagonal sampling requires 13.4 p ercentfewer samples than rectangular sampling [1]. In addition, hexagonal sampling systems also lead to nonseparable quadrature mirror lters in which all basis functions are lo calized in space, spatial frequency and orientation [2]. This chapter is organized in two sections. Section I describ es the theoretical asp ects of hexagonal sampling systems while Section I I covers imp ortant implementation details. I. Hexagonal sampling system This section presents the theoretical foundation of hexagonal sampling systems and hexagonal quadrature mirror lters. Most of this material has app eared elsewhere in [1], [3], [4], [5], [2], [6] but is describ ed here under a uni ed notation for completeness. In addition, it will provide continuity and a foundation for the original material that follows in Section I I. The rest of the section is organized as follows. Section I-A covers the general formulation of a hexagonal sampling system. Section I-B intro duces up-sampling and down-sampling in hexagonal sampling systems. Section I-C reviews the theory of hexagonal quadrature mirror lters. Section I-D describ es redundant analysis/synthesis lter banks in hexagonal systems. Finally, Section I-E covers the formulation of the discrete Fourier transform in hexagonal sampling systems. 3 A. Hexagonal systems Let x (t)=x (t ;t ) b e a 2-D analog waveform, then a sampling op eration in 2-D can a a 1 2 b e represented by a lattice formed by taking all integer linear combinations of a set of two T T linearly indep endentvectors v =[v v ] and v =[v v ] . Using vector notation we 1 11 21 2 12 22 T can represent the lattice as the set of all vectors t =[t t ] generated by 1 2 t = Vn; (1) T where n =[n n ] is an integer-valued vector and V =[v v ]isa2 2 matrix, known 1 2 1 2 as the sampling matrix. Because v and v are chosen to b e linearly indep endent, the 1 2 determinantof V is nonzero. Note that V is not unique for a given sampling pattern and that two matrices representing the same sampling pro cess are related by a linear transformation represented by a unimo dular matrix [7]. Sampling an analog signal x (t) on the lattice de ned by (1) pro duces the discrete signal a x(n)= x (Vn): a Figure 1(a) shows a hexagonal sampling lattice de ned by the pair of sampling vectors 2 3 2 3 2T T 1 1 6 7 6 7 = and = ; (2) 4 5 4 5 v v 1 2 0 T 2 p 3 1 and T = . The lattice is hexagonal since each sample lo cation has exactly where T = 2 1 2 2 p six nearest neighb ors when T = T 3. 2 1 Let the Fourier transform of x (t) b e de ned by a Z +1 T x (t)exp(j t)dt; X ( )= a a 1 T where =[ ] . Similarly, let the Fourier transform of the sequence x(n) b e de ned 1 2 as X T X (! )= x(n)exp(j ! n); (3) n T where ! =[! ! ] . Mersereau [4] showed that the sp ectrum of the sequence x(n) and 1 2 the sp ectrum of the signal x (t) are related by a X 1 T X (! )= X (V (! 2 k)); (4) a jdet V j k 4 v Ω 1 t1, n1 1 u1 v2 u2 n 2 t Ω 2 2 (a) (b) Fig. 1. (a) A hexagonal sampling lattice in the spatial domain. (b) Hexagonal sampling lattice in the frequency domain. T 1 T where k is an integer-valued vector and V denotes (V ) . Alternatively,we can de ne the Fourier transform of the sequence x(n)as X T X ( ) = x(n) exp(j Vn) V n T = X (V ); (5) then Equation (4) may b e written as X 1 X ( )= X ( Uk); (6) V a jdet V j k where T U =2 V : (7) Thus, Equation (6) can b e interpreted as a p erio dic extension of X ( ) with p erio dicity a T T vectors u =[u u ] and u =[u u ] , where U =[u u ]. The set of all vectors 1 11 21 2 12 22 1 2 generated by = Un de nes a lattice in the frequency domain known as the mo dulation or recipro cal lattice. Thus, the sp ectrum of a sequence x(n) can b e viewed as the convolution of the sp ectrum of x (t) with a mo dulation lattice de ned by U. a Figure 1(b) shows the recipro cal lattice corresp onding to the sampling vectors de ned in Equation (2), that is the lattice de ned by the pair of mo dulation vectors 3 3 2 2 0 7 7 6 6 T 1 and : = = 5 5 4 4 u u 2 1 2 T T 2 2 5 Λ Λ K 1 23n1 1 2 3 n1 1 1 2 2 3 3 n2 n2 (a) (b) Fig. 2. (a) Integer sampling lattice. (b) Sampling sublattice. B. Up-sampling and down-sampling in hexagonal systems Let denote the integer lattice de ned by the set of integer vectors n, and let K denote the sampling sublattice generated by the subsampling matrix K, that is the set of integer vectors m such that m = Kn. Note that in order to prop erly de ne a sublattice of , a subsampling matrix must b e nonsingular with integer-valued entries. In general, a sublattice of is called separable if it can b e represented by a diagonal matrix K, otherwise it is called nonseparable. Figure 2 shows an integer sampling lattice and a sampling sublattice , for the separable subsampling matrix K 3 2 2 0 7 6 K = : (8) 5 4 0 2 With and de ned in this way,we can view the op erations of up-sampling and K down-sampling as describ ed b elow. The pro cess of up-sampling maps a signal on to a new signal that is nonzero only at p oints on the sampling sublattice . The output of an up-sampler is related to the input K by 8 > 1 1 < x(K n); if K n 2 , y (n)= > : 0; otherwise. It is easy to show [5] that the Fourier transform relates the output and input of an up-sampler by T Y (! )=X (K ! ); 6 K (a) 0 1 23n1 0 1 23n1 x x x x 0 x 0 00 10 20 x30 00 10 1 1 x x x 0 0 0 0 01 11 21 x31 2 2 x x x x 0 x 0 02 12 22 x32 01 11 3 3 x x x 0 0 0 0 03 13 23 x33 n2 n2 (b) Fig. 3. (a) Up-sample op erator. (b) Mapping of samples under up-sampling: (left) input signal, (right) output signal. where X (! )isde nedby Equation (3). Figure 3 shows the blo ck diagram of an up-sampler and the pro cess of up-sampling for the subsampling matrix de ned by Equation (8). The pro cess of down-sampling maps p oints on the sublattice to according to K y (n)= x(Kn); (9) and discards all other p oints. The Fourier transform relation b etween the output and input of a down-sampler can b e derived byintro ducing the concept of a sampling function s (n) asso ciated with the K sampling matrix K [5], that is, 8 > < 1; if n 2 , K s (n)= K > : 0; otherwise. Since s (n) can b e interpreted as a p erio dic sequence, with p erio dicity matrix K,i.e. K s (n)=s (n + Km), it may b e expressed as a Fourier series K K jdet Kj1 X 1 T 1 s (n)= exp(j 2 k K n); (10) K l jdet Kj l=0 T T where eachofthejdet Kj vectors k =[k k ] is asso ciated with one of the cosets of K . l l l 1 2 Notice that a coset of a sublattice is de ned as the set of p oints obtained by shifting K the entire sublattice byaninteger shift vector k. There are exactly jdet Kj distinct cosets 7 of , and their union is the integer lattice . Each shift vector k asso ciated with a K l certain coset is known as a coset vector. For example, one choice for the k given the l sampling sublattice de ned by Equation (8) is 2 3 2 3 2 3 2 3 0 1 0 1 6 7 6 7 6 7 6 7 = ; = ; = ; and = : (11) 4 5 4 5 4 5 4 5 k k k k 0 1 2 3 0 0 1 1 Let w (n)=x(n)s (n); (12) K then it is easy to see from (9) that a down-sampled signal y (n) can b e written as y (n)=w (Kn); since w (n) equals x(n)on .
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