Computations of Discrete Gabor Coefficients. Sigang Qiu

Transcription

1 !fie<?r ~oz ~~ G\.~t')M4'ffJ Characterizations of Gabor-Gram Matrices and Efficient Computations of Discrete Gabor Coefficients Sigang Qiu Department of Mathematics, University of Connecticut V-9, 196 Auditorium Road, Storrs, CT Tel: (203) Fax: (203) ABSTRACT The motivation of this paper is to give a direct way for determining the crucial Gabor coefficients. Based on the characterized Gabor-Gram matrix structures, we are able to propose fast and direct computational algorithms. We derive also a new way for dual Gabor atoms. 1 Introduction and Preliminaries The theory of Gabor transform is concerned with the main problem of representing a given signal into a series of building blocks (the Gabor elementary functions) and obtained from a basic one (Gabor atom or Gabor wavelet) by appropriate time-frequency shifts along a TF-Iattice. The importance of the Gabor transform is that Gabor coefficients reveal the frequency localization of a signal (or image) very well. Since the Gabor elementary functions are not orthogonal to each other, it is usually very complicated and difficult to determine the Gabor coefficients. By using a so-called bi-orthogonal function,,1 many practical algorithms have been presented in the recent studies on the subject3,7-11,18,19 mostly via the Zak transform and matrix decomposition methods. However, this, plays only an auxiliary role. Generally, we are only required to determine the Gabor coefficients, and do not need to know what,looks like. Our main motivation in this paper is to give a directly iterative method for the computation of Gabor coefficients. In the following, we fix some notations and introduce the concept of Gabor-Gram matrices. In Section 2, we present some equivalent formulations of the bi-orthogonality. Section 3 is the main part of this paper. We present structures of Gabor-Gram matrices and conjugate gradient (CG)-algorithms for calculations of Gabor coefficients and dual gabor atoms.

2 1.1 Notations In order to summarize the results, we have to fix some notations. Essentially, we will keep using the natations in our paper.14 For the self-containedness, we recall some of them. We consider signals as N-periodic row-vectors in ~N. The inner product of two signals is given as (m, y) = L~=-;/ x(k)y(k). We identify linear mapping from the right: m 1-+ m * A. "*,, is used for matrix multiplication. For a matrix A, we use A'to denote the conjugated and transposed matrix of A. We use F and F-1 to denote the discrete Fourier transform (DFT) and the inverse discrete Fourier transform (IDFT), respectively. They are performed by the efficient fast Fourier transform (FFT) and the inverse fast Fourier transform (IFFT). For a signal x E ~N, F(m) = m * FN; and F-1(m) = m * FN, where FN =" (e~) is the Fourier matrix of order N. We also use a: to denote F(x). NxN For a given Gabor atom 9 and lattice constants (a, b), we call (ii, b) := (~, If) the dual lattice constants of (a, b). A discrete Gabor representation of a signal x E ~N with respect to a Gabor triple (g, a, b) is of the form X 0.-1 b-1 = L: L: cnmgnm, n=om=o (1) where gnm = MmbTnag. {Cn,m}n,m are called Gabor coefficients. 1.2 Gabor-Gram Matrix Given a Gabor triple (g,a,b), we use GAB(g,a,b) to denote the Gabor basic matrix with the following block form. 9 Mbg GAB(g,a,b) = M(b-1)bg Tag MbTag M(b_1)bTag (2) MmbTnag where M(b_l)bTOi-1)ag MmbTnag = Mmb (g(o - na), g(l- na),..., g(n - 1- na»

3 for m = 0.1,..., b - 1; n = 0,1,..., a - 1, is an N-elements row vector and Ml is the modulation operator. Remark: 1. It is clear that the row vectors of GAB(g.,b) form the Gabor family {gnm}n,m' 2. It is easy to see that GAB(g,,b) and the Gabor basic matrix GAB(g, a, b) introduced in the previous papers12 14 are exactly the same, except for the different orders of the row vectors. Both the row vectors of GAB(g..b) and GAB(g, a, b) form the Gabor family {gnm}' 3. It is not difficult to check that [GAB(g,,b)]' * [GAB(g,.b)] = [GAB(g, a, b)]' * [GAB(g, a, b)]. (3) Therefore, the Gabor matrix corresponding to (g, a, b) can be formulated by Eq. (3) either with GAB(g, a, b) or with GAB(g.,b)' That is why we also call GAB(g..b) a Gabor basic matrix. We use the Gabor basic matrix GAB(g..b) to define the Gabor-Gram matrix. DEFINITION 1.1 (GABOR-GRAM MATRIX). For a Gabor triple (g,a,b), we call the N x N (N = ~:) matrix GM(g..b) = [GAB(g.,b)] * [GAB(g.,bl' the Gabor-Gram matrix associated to (g, a, b). We generally call GM(9i,92,.b) = [GAB(g!,,b)] * [GAB(92.,bl' the Gabor-Gram matrix associated to Gabor triples (g1, a, b) and (g2,a,b). In addition, we use the following rotation operator 0". DEFINITION 1.2. For a given nu.mber r which divides N, we define 0"(', r) as the following rotation operator on (CN. where y = (y(j))f:~1 is given by (j : (CN -+ (CN; X = (X(j));=~1 -+ y, Y ( J') = { x(j + -1) r - 1) if 1=0.. N for J = kr + 1 wzth k = 0, 1,..., and 1 = 0,1,..., r #0 r Definition 1.2 depends only on the coordinates of the given vector x. 0"(" r) is well-defined for both row and column vectors. It is not difficult to check that the inverse of o{, r) is defined as follows. where y = (y(j))f=~l is given by (j-1 : (f}n -+ (CN; X = (x(j));=~1 -+ y, Y (')=={ J x(j-r+l) + 1) if l=r-1 1 # r - 1 for j = kr + 1 with k = 0,1,..., 1'f - 1 and l = 0, 1,..., r - 1.

4 For a matrix A E q;nxq column vector of A. and a number r which divides N, we define o-(a, r) by applying 0-(., r) to each Similarly, we have no difficulty to see that 0--1(A, r) is obtained by applying 0--1 to each column vector of A. 2 Matrix-Formulation of the Bi-orthogonality Using the introduced Gabor basic matrix, we are able to state the Gabor bi-orthogonality18 and the equivalent conditions in matrix forms. THEOREM 1. Given a Gabor triple (g, a, b) and another signal h, the following statements are equivalent: where lr denotes the r x r identity matrix, for rein. ab GM(g,h,b,ii) = GM(h,g,b,ii) = N * lab, [ ]' ab h * GAB(g,b,ii) = N * eab, where er E q;r denotes the 'Unit vector with the first entry being ]' ab 9 * [GAB(h,b,ii) = N * eab GM(h.,U,a, H b) = GM(. 9, h.,a, - b) = N*lab. h.* [GAB(g,Q.,b) ]' = ab * eab. 9 * [GAB(h.,ii,b)]' = ab * eab Moreover, (g, a, b) generates a Gabor frame if and only if there exists a signal h such that one of the above conditions is satisfied. For the higher dimensional cases, if Gabor functions are separable, we have no difficulty to state the results similar to Theorem 1. The last three conditions are based on the fact that (g, a, b) generates a frame if and only if (9, b, a) generates a frame.12

5 3 Gabor-Gram Matrix Characterization We present observations on the Gabor-Gram matrix structures in the first part. This is the basis for algorithms proposed in the second part of this section. 3.1 Gabor-Gram Matrix Structure THEOREM 2. Let us write the Gabor-Gram-matrix GM as the following block-matrix form GM = [B1,B2,, BN], (4) where Bk for k = 1,2,..., IV is the k-th column vector of GM. Set B = [Bl,Bb+1,..., B(ii-l)bH], (5) then the IV x IV matrix GM is completely determined by the IV x a small block matrix B via cyclic rotations operator 0'(" b). p'recisely, if we write then B(s) = [Bs,Bb+S'..., B(ii-l)b+S], for s = 2,...,b, B(s) = (7s-1(B, b). 3.2 Algorithms Gabor-Gram Matrix-Vector (GGMV) Multiplication In this section, we are presenting a useful Gabor-Gram matrix-vector (GGMV') multiplication: For any vector Xc E q:n, we show that Yc = Xc * GM can be obtained by the Gabor-Gram block matrix B only. ALGORITHM 1. Under the assumptions of Theorem 2, let Xc be an IV-entries vector. Then Yc = Xc * GM can be computed by Xc and the Gabor-Gram block matrix B only. Yc = (YcU»f:o is given by Yc(j) = O'-k(xc,b) * B(s), where j = sb + k for k = 0, 1,..., b - 1, s = 0,1,...,a-I, and B(s) for k = 0,1,...,a-I is the s + 1-th column vector of B. Proof. Using the notations of Theorem 2, for j = sb + k with k = 0,1,..., b - 1 and s = 0,1,..., a, Theorem 2 implies that B 81+1<:+1 = O'k (B 8;;+1)' Thus, We finished the proof. 0 Yc(j) = Yc(sb + k) = Xc * B sbh = Xc * O'k(B sbh) = Xc * (7k (B(s»'

6 3.2.2 Gabor Coefficients For an arbitrary Gabor triple (g, a, b), the following CG-algorithm gives a general way for computing the Gabor coefficients. The algorithm holds for oversampling (ab < N), critical sampling (ab = N) and undersampling (ab > N) cases. We do not assume that (g, a, b) generates a frame. In this case, we determine the best approximations by linear combinations of Gabor familif1s, lfi:j""-, ':l~,,{_} ALGORITHM 2 (GABOR COEFFICIENTS). For a signal x E (CN, the Gabor coeficients C = {cnm} with respect to (g, a, b) are determined via the following procedu.res: 1st step: Caculate Xc = X * [GAB(g,a,b)]" 2nd step: Compu.te the Gabor-Gram block matrix B. 3rd step: Obtain the Gabor coefficients C = {cn,m} by solving the linear system Xc = C * GM via the CG-method.6 Remark: Algorithm 2 is based on the fact that if we can get the solution for C from the linear system C * GM = Xc, i.e., C[GAB(g,a,b)J * [GAB(g,a,b)l' = x * GAB(g,a,b)' then it is easy to see that C * [GAB(g,a,b)] * S = x * S. where S = [GAB(g,a,b)l' * [GAB(g,a,b)] is the associated Gabor matrix. If (g, a, b) generates a Gabor frame, since S is invertible, we deduce that x = C * [GAB(g,a,b)J. This is the same as Gabor representation Eq. (1). For technical details, we refer to the subsequent papers. The 1st step and 2nd step are performed efficiently by the short time Fourier transform (STFT). Although GM is generally not invertible (e.g., for the oversampling cases), we show that the standard CG-method6 can be applied to the 3rd step for the undersampled cases. to determine the Gabor coefficients. Algorithm 2 works fast The matrix-vector multiplications involved in the 3rd step are performed by the efficient GGMV-multiplication with the precalculated Gabor-Gram block B Dual Gabor Atom If a given Gabor triple (g,a,b) generates a frame, by Theorem 1, there exists a Gabor signal g (we call it a dual Gabor atom such that 9 * {GAB(g,b,ii)l' = ~ eab. This implies that --1 g = {g * GAB( g, -b,a-) * GM } * [GAB( 9, b-,a-)]

7 ab --1 = {Neab * GM } * [GAB(g,b,a)1 where GM = [GAB(g,b,a)]' * [GAB(g,b,ei)] is the Gabor-Gram matrix corresponding to (g, b, ii). Using the CG-algorithm, we solve the linear equation tl * GM = ~e. Then we obtain h = tl * GAB(g,a,b) by STFT. If (g, a, b) does not generate a frame, we will use 9 = {~eab *pinv(gm)} * [GAB(g,b,ei)] to get the generalized dual Gabor atom (GDGA),13 where pinv(gm) is the pseudo-inverse of GM. 4 Further Results In this section we report some of the further results obtained after our submission of the proposal. For more detailed information, we refer to the subsequent papers. THEOREM 3 (16). Under the assumptions of Theorem 2, if we write GM = (Uk,I)NxN as the following block form, D2 1 D2 2. D2 a GM=.'.' ( Dei,l Dl,l Dei,2 Dl,2... Dei,a D1,ei 1 (6) where Dp,q = (U(P-l)b+l:,(q-l)b+l) b,b then Dp,q are circulant matrices of order b. for p, q = 1,2,...,a. THEOREM 4 (16). Gabor-Gram matrices are simultaneously unitarily similar to banded matrices. Precisely, under the assumptions of Theorem 3, GM = u' * G M * U is a banded sparse matrix with all the possible nonzero entries being in k-th (for k = 0, ±(b - 1),..., ±(b - l)(a - 1») diagonals, where U = Ia 0 Fb, Ia is the a x a identity matrix and Fb is the Fourier matrix of order b. We call GM the unitary equivalent Gabor-Gram matrix (UEGM) associated with Gabor triple (g, a, b). Based on Theorem 3 and Theorem 4, a new fast direct way of determining the Gabor coefficients has been proposed.16 5 Numerical Illustrations We illustrate some of the numerical results done with MATLAB at a SUN4 workstation. Figure 1 illustrates a Gabor atom of signal length N = 560 and the associated dual Gabor atoms calculated by the stated in Section and the under-oversampling relationship established in.is Case (0) is a oversampling

8 a where we take lattice constants pair (a, b) = (28,16). Case (U) is undersampling with lattice constants (a, b) = (40,35). Figure 2 illustrates a Chirp signal and the Gabor coefficients corresponding to Case (0) obtained by Algorithm 2. The reconstruction (relative) error is in order of Figure 3 is for the Case (U). It shows a signal from the Gabor space and the Gabor coefficients. reconstruction (relative) error is also in order of The Figure 4 shows the explicit sparsity of the unitary equivalent Gabor-Gram matrix corresponding to three cases. The case (01) and (02) are oversampling with lattice constants pair (a,b) = (6,6), (8,4). Case (u) is undersampling with (a, b) = (8,8). The signal length N is Conclusion We have introduced concepts of basic Gabor matrix and Gabor-Gram matrices. Simply equivalent conditions for the bi-orthogonality have been formulated. We have developed the structural properties of the Gabor-Gram matrices. A direct algorithm based on the CG-procedure has been presented for the determination of the Gabor coefficients. The algorithm is applied for undersampling as well as critical and oversampling cases. It works even if a given Gabor triple (g, a, b) does not generate a frame. In this case, we are able to use the calculated Gabor coefficients to obtain the best approximation by linear combinations of Gabor families. A new algorithm is also proposed for the dual Gabor atom. In addition, we have summarized some of the latest new results. This enables us to give much efficient computational algorithms. 7 REFERENCES [1] M. J. Bastiaans, Gabor's signal expansion of a signal into Gaussian elementary signals, Proc. IEEE, ,1980. [2] M. J. Bastiaans, Gabor's Signal Expansion and Its Relation to Sampling of the Sliding- Window Spectrum, Advanced Topics in Shannon Sampling and Interpolation Theory (edited by R. J. Marks II), 1-35, [3] J. Benedetto, Gabor representations and wavelets, Commutative Harmonic Analysis (edited by D. Colella), Contemp. Math. 19, 9-27, [4] E. C. Boman, fast Algorithms for Toeplitz Equations, Ph.D thesis, University of Connecticut, [5] C. E. Heil and D. F. Walnut, Continuous and Discrete Wavelet Transforms, SIAM Review 31 (4), , [6] G. Golub and C. van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore and London, 1993.

9 [7] 1. Daubechies, The wavelet transform, time-frequency localization, and signal an31ysis, IEEE Trans. Inform. Theory, 36, , [8] A. J. E. M. Janssen, Gabor representation of generalized functions, J. Math. Anal. Appl., 83, , [9] A. J. E. M. Janssen, Duality and biorthogonality for Weyl-Heisenberg frames, Journal of Fourier Analysis and Applications, 1(3), [10] R. S. Orr, Tbe Order of Computation for Finite Discrete Gabor Transform, IEEE Trans. on Signal Processing, 41(2), , [11] S. Qian and D. Chen, Discrete Gabor Transform, IEEE Trans. on Signal Processing, 41(7), , [12] S. Qiu and H. G. Feichtinger, Discrete Gabor Structure and Optimal Representation, to appear in IEEE Trans. on Signal Processing, October, [13] S. Qiu, Generalized Dual Gabor Atoms and Best Approximations by Gabor Family, submitted for publication. [14] S. Qiu, Superfast Computations of Dual and Tight Gabor Atoms, SPIE-Proceeding: San Diego'95. [15] S. Qiu, On the Best Approximation by Undersampled Gabor Families, submitted to IEEE Trans. on Signal Processing. [16] S. Qiu, Discrete Gabor Transforms: the Gabor-Gram Matrix Approacb, submitted for publication. [17] M. Unser, Fast Gabor-Like Windowed Fourier and Continuous Wavelet Transforms, IEEE Signal Processing Letters, 1(5), [18] J. Wexler and S. Raz, Discrete Gabor Expansion, Signal Processing, 21(1990), [19] M. Zibulski and Y. Y. Zeevi, Oversampling in tbe Gabor Scbeme, IEEE Trans. on Signal Processing, 41(8), , August, 1993.

10 I 1 Gabor 12 atom dual Gabor atom (0) C) 1 -o.sl O.st (0): (U): oversampling undersampling 2 0.5; generalized dual Gabor atom (U) Figure 1: Gabor atom and the a.gsociated dual Gabor atoms corresponding to a oversampling ca.geand a undersampling case, where lattice constants (a, b) are (28,16) and (40,35). The signal length N is 560. abor coefficients I (0) 20 Of- -11-, ,., v 400, , V V V 10 t ''> } ([l..'n~ <~ "-,.'" T.... Chirp signal --=--.:;::: \ ~ ~ A Figure 2: A Chirp signal and the Gabor coefficients for Case (0). The reconstruction relative error is in order of

11 1 signal from Gabor space o Gabor coefficients (U) Figure 3: A signal from the Gabor space and the Gabor coefficients. It is the undersampled case (U). The reconstruction relative error is in order of I 0.5 (01): (a, b) = (6, 6) o (02): (a, b) =.(8, 4) -0.5 (u): (a, b) = (8, 8) UEGM matrix (02) o ~ '....,, " ', 20 ~ ~ ' ~~ '~"'",., ~ ~ ~....., 40~.,. 60 ~~:::::~:::::::::::::::::::::::::::~::::::::::::::::::: o nz = ~_1 UEGM matrix (01) o nz = UEGM matrix (u) 01_. ~ ~ ~ ~ ~ _. _. _. _. a_. e a t -Cl. -_. - -_. e - -. a_. a_. _ _.. 20~ _.. e. a _. _. _ 30 te -. o nz = 216 Figure 4: The unitarily equivalent matrices structures of the Gabor-Gram matrices corresponding to three cases. (01) and (02) are oversampling and (u) is undersampling.