The Structure of the Gabor Matrix and Ecient Numerical Algorithms for Discrete Gabor Expansions Sigang Qiu and Hans G. Feichtinger Department of Mathematics, University of Vienna Strudlhofgasse 4, A-1090, Wien, Austria Email:
[email protected]
[email protected]
ABSTRACT The standard way to obtain suitable coecients for the (non-orthogonal) Gabor expansion of a general signal for a given Gabor atom g and a pair of lattice constants in the (discrete) time/frequency plane, requires to compute the dual Gabor window function g~ rst. In this paper, we present an explicit description of the sparsity, the block and banded structure of the Gabor frame matrix G . On this basis ecient algorithms are developed for computing g~ by solving the linear equation g~ 3 G = g with the conjugate-gradients method. Using the dual Gabor wavelet, a fast Gabor reconstruction algorithm with very low computational complexity is proposed.
Keywords:
Gabor expansion (synthesis), Gabor (frame) matrix, Gabor basic matrix, Gabor (frame) triple, dual Gabor atom.
1
Introduction
As Gabor9 originally suggested the fundamental problem of (non-orthogonal) Gabor expansion consists in expanding arbitrary signals into a series of elementary terms which are time-frequency shifted copies (along a certain TF-lattice) of a given \Gabor atom (Gabor wavelet or Gabor window function)". Nowadays arbitrary signals g are discussed as possible Gabor atoms, not only Gauss-type bump functions. It is well-known that the main diculty involved in the Gabor expansion is to select a suitable family of Gabor coecients. Many scientist have studied the problem of Gabor transforms theoretically, using the Zak transform. We refer to the latest survey paper3,21 in which the authors have presented analytic formulas for Gabor coecients and the dual Gabor window function (or bi-orthogonal window function) which has been rst introduced by Bastiaans1,2 for the case of the Gaussian function. Since a suitable set of Gabor coecients can be determined as the short-time Fourier transform of the expanded signal with respect to the dual Gabor window, it is crucial to obtain the dual Gabor window function eciently. Unfortunately, the determination of the dual Gabor window function - according to the literature - is considered computationally expensive. When the Gabor atom is of Gaussian shape, Bastiaans1 has obtained an analytic solution in the critically sampled case. With the bi-orthogonality relation developed by Wexler and Raz,19 the determination of the dual Gabor window function is reduced to a readily computable linear system in certain cases. In the paper,21 the author presented a fast algorithm for Gabor coecients based on block-Toeplitz matrix multiplication in the critically sampled case (ab = N ). The implementations presented in13 and14 gave some discrete versions for the in nite case. In the present paper, we take both the critical and oversampling cases into our consideration. Based on the
careful description of the Gabor-matrix structure, we are developing ecient ways of determining the dual Gabor atom, which can servere as dual Gabor window function, for the nite (i.e. discrete and periodic) case. The new approach allows to deal eciently with arbitrary Gabor atoms, which need not be of Gaussian shape, and both in the critically sampled and oversampled settings (ab < N ). It can be seen as a successful attempt to continue the lines of investigations of previous papers, starting with the papers of Bastiaans,1,2 and later.3,6,8,11{13,17,19{22 Arrangement: Section 2 contains notations and basic facts about Gabor transforms. Section 3 is the main part and includes the mathematical description of the structure of the Gabor-matrix and practical algorithms. Proofs of the results presented here can be found in .15 In section 4 we present some numerical results and give an outlook on further results to be discussed in subsequent papers. Concerning the basic facts about frames, we refer to.5,10
2
Preliminaries
x
We treat signals (or atoms) = (x0 ; x1 ; :::; xN 01) 2 C I N ; as complex row vectors in CI N or actually as functions on the cyclic group ZZ N , or equivalently as N -periodic sequences over the integers ZZ . The k-th coordinate of is usually expressed either as xk or x(k): Thus we have (j + kN ) = (j ); for j; k 2 ZZ: We use jj jj to denote the norm of 2 CI N , i.e., jj jj = (PNj =001 jxj j2 )1=2: The inner product of two signals ; 2 CI N is h ; i = PNj =001 xj yj :
x
x xy
of
x
Definition 2.1 (rotation
x with rotation number a if
x
x
x
xy
rot). For any signal x 2 CI N ; the signal y 2 CI N is said to be the cyclic rotation
y = (xN 0a ; xN 0a ; :::; xN 0 ; x ; :::; xN 0a0 ) : We usually denote this rotated vector y as rot(x; a): rot(1 ; a) is called the rotation operator with rotation +1
1
0
1
time-translation of periodic signals we will use also for convenience
number a: Since it corresponds to ordinary the symbol Ta for the cyclic rotation operator.
Remark:
By the N -periodicity of the signal, we usually choose the rotation number a satisfying 0 a < N: The same notation can be stated for the column vectors.
rotm
Definition 2.2 (matrix rotation ). Given any n by m matrix B and a positive number a(usually, 0 a n). the rotation of B with rotation number a is de ned as
rotm(B; a) = (rot(B ; a); rot(B ; a); :::; rot(Bm ; a)); where Bl is the l-th column vector of B; l = 1; 2; :::; m: rotm(1; a) is called the matrix rotation operator with 1
2
rotation number a:
The
modulation operator (or frequency translation) is given by
Ml (x0 ; x1; :::; xN 01 ) := x0 ; x1!l ; x2!2l ; :::; xN 01 !(N 01)l for l 2 ZZ: where ! = e02i=N : It is easy to check that
x
x
x
x
FFT(Tk ( )) = M0k (FFT( )) and FFT(Mk ( )) = Tk (FFT( )): Definition 2.3 (Gabor triple ).
a signal of length N . We also call g a
Gabor triple
(1)
We call (g; a; b) a if a and b be divisors of N and g is The constants a and b are usually called and
Gabor atom.
the time
frequency gaps and form a pair of lattice constants. We also use ~a = N=a and ~b = N=b to denote the dual lattice constants associated to (a; b) with respect to N: In this paper we will consider only the cases that (a; b) satis es that a 1 b N which we call critical and oversampling cases. A Gabor expansion with respect to the Gabor triple (g; a; b) is a series expansion of the form
x=
~ a01 X ~ b01 X
n=0 m=0
cnm gnm
Gabor elementary functions.
where the building blocks gnm de ned by gnm(k) = Mmb Tna g(k) are called We call fgnm g a or .18 The coecients fcnm g are referred as . The S associated to (g; a; b) is de ned as
Gabor family Weyl-Heisenberg wavelet system Gabor coecients Gabor operator Sx =
~ a~X 01 X b01
hx; gnm ignm
(2)
n=0 m=0
Gabor frame triple
Definition 2.4 (Gabor frame triple). We say that Gabor triple (g; a; b) is the provided the linear span of the Gabor family fgnm g is all of C I N : In this case, we call g a Gabor frame atom.
It is well known that a Gabor triple is a Gabor frame triple if and only if the associated Gabor operator
S is invertible CI N , which is usually described as the frame property of of the Gabor family. Naturally we call the corresponding frame operator a g~ = S 01 g:
Gabor frame operator (cf.
10
or5 ). The
dual Gabor atom is de ned as
Because we view signals as row vectors we prefer to identify linear mappings (operators) on the signal space with matrix multiplication from the right (which reads in MATLAB notation as follows): x 7! x 3 A. The adjoint operator (in terms of the Euclidean scalar product on the N -dimensional space C I N ) is thus represented by matrix 0 multiplication from the right by the transposed and conjugated matrix A : For an arbitrary Gabor triple (g; a; b); we use GAB(g; a; b) to denote the following M 2 N matrix (M = a~ 1 ~b) which we call the
Gabor basic matrix.
0 B B B B B B B B B B B GAB(g; a; b) = B B B B B B B B B B B @
g Ta g .. .
T(~a01)ag Mb g Mb Ta g .. .
Mb T(~a01)a g .. .
Mmb Tna g .. .
M(~b01)bT(~a01)ag
1 C C C C C C C C C C C C C C C C C C C C C C A
In our notation, we have the following matrix representation of the Gabor operator
S x = x 3 G;
(3)
where
G = [GAB(g; a; b)]0 3 [GAB(g; a; b)]:
Gabor frame matrix if G is invertible. Generally, for any two Gabor triples (gk ; a; b) for k = 1; 2; we Gabor matrix G as
We call G de ne the associated
G = [GAB(g1; a; b)]0 3 [GAB(g2 ; a; b)]:
Since g~ = S 01 g = g 3 G01, we can show the following remark.
Remarks
For a Gabor frame triple (g; a; b),
I = [GAB(~g ; a; b)]0 3 [GAB(g; a; b)]
(4)
where I is the N 2 N identity matrix. This leads directly to the following Gabor synthesis formula:
x = fx 3 [GAB(~g; a; b)]0g 3 [GAB(g; a; b)]
x
where C := 3 [GAB(~g; a; b)]0 is a suitable family of Gabor coecients. Using the natural double index set to P label the elements of C as cnm we obtain the typical Gabor synthesis formula as = n;m cnm gnm :
3
x
The Description of the Gabor Frame Matrix Structure
In this section, we describe the block and banded structure of the Gabor matrix in Theorems 1, 2. Theorem 1 (Banded Structure). Let Gl = GAB(gl ; a; b) be M 2 N Gabor basic matrices associated to Gabor triples (gl ; a; b) for l = 1; 2: Then the matrix product G = G02 3 G1 has the following banded structure: the non-zero entries of G are only located in the main-diagonal and k-th sub-diagonals of G for k = 0; 6~b; 62~b; : : :; 6(b 0 1)~b: Moreover, the general entry Gk+1;j +1 of the Gabor matrix G can be formulated as
Gk+1;j +1 = for k; j = 0; 1; : : :; N
~ Pa~01 b n=0 Tnag1 (k)Tna g2(j ) if jk 0 j j is divided by ~b 0 otherwise
(5)
0 1:
Theorem 2 (Block Structure).
matrix of G: Then the k-th N
Under the assumptions of Theorem 1, let B be the rst N
2 a block matrix Bk of G can be obtained from B by matrix rotation: Bk = rotm (B; (k 0 1)a) ; for k = 2; 3; : : :; ~a:
2 a block
For any l = 0; 6~b; 62~b; : : :; 6(b 0 1)~b; the l-th subdiagonal of G described in Theorem 1 is an a-periodic vector of length N 0 jlj~b: Corollary 1.
Corollary 2. Assume that (g; a; b) is a Gabor frame triple, then the associated Gabor frame matrix G is a positive de nite matrix with all the entries in the main diagonal being strictly positive.
We present next some algorithms based on the described structures. The rst one can be used to eciently build the Gabor matrix. The second one is used to obtain the image of an arbitrary signal under the Gabor frame operator. The third one is a special case of Algorithm 2 and can be seen as Gabor reconstruction method.
Algorithm 1 (Gabor Matrix and Dual Gabor Atom). Let (g1; a; b) and (g2 ; a; b) be two Gabor triples , and G = [GAB(g1 ; a; b)]0 3 [GAB(g2 ; a; b)] be the associated Gabor matrix. Then we can set up G using only ba(~a + 1) = ba + nb multiplications and some proper rotations as follows:
1st step:
We calculate the rst N 2 a block matrix B: As a consequence of Theorem 1 and Theorem 2, there are only ba nonzeros entries of B: The general (k + 1; j + 1)-th nonzero entry of B for j = 0; 1; :::; a 0 1 and k = p 1 ~b + j for p = 0; 1; :::; b 0 1 can be formulated as
Bk+1;j +1 = ~b
a~01 X n=0
Tnag1 (k)Tna g2(j )
2nd step: We build the full Gabor matrix from B using the formula G = [B; rotm(B; a); :::;
rotm(B; (~a 0 1)a)]
In particular, for a given Gabor frame triple (g; a; b); the associated Gabor frame matrix G = [GAB(g; a; b)]0 3 [GAB(g; a; b)] can be obtained via the above two steps.
3rd step:
dual Gabor atom
The g~ can be determined eciently by solving the linear equation g~ 3 G = g with sparse-matrix technology and the conjugate-gradients method using Algorithm 2.
The following algorithm is very ecient for calculating the image of an arbitrary signal under the Gabor operator. Using Algorithm 2, the dual Gabor atom can be solved eciently by means of the conjugate-gradients method. Algorithm 2.
suppose that
x
With all the assumptions in Algorithm 1, for an arbitrary signal = (x(0); x(1); :::; x(N 0 1)) ; a 0 1g; we have 0 + ra; where j0 2 f0; 1; : : : ; a 0 1g and r 2 f0; 1; : : :; ~
y = x 3 G; then for any j = j y(j ) = ~b
b01 X p=0
x(j + p~b)f
a~01 X n=0
Tna g1 (p~b + j0 )Tna g2(j0 )g
(6)
An important special case of Algorithm 2 is the following Gabor reconstruction algorithm. Algorithm 3 (Gabor reconstruction).
Under the assumptions of Algorithm 1 and 2, suppose that
g1 = g and g2 = g~ is the dual Gabor atom associated with (g; a; b); then the signal y derived from formula (6) is exactly the reconstructed signal of x:
Remarks
(A) Complexity Analysis:
direct
Algorithm 3 does actually provide a Gabor synthesis algorithm using the bi-orthogonal function which has given an answer to the problem of Orr.11 It needs only a total count of operations T1 = 4Nb + ab: The derived Gabor-synthesis algorithms in11 for the critical sampling case only requires T2 = N (b + log2 N ) total count of operations, where is a positive constant being less than 1. We will give a more detailed complexity analysis in .15
(B) Total Nonzero Entries:
The theorems and algorithms derived above imply that the total number of the non-zero entries of the Gabor frame matrix is at most ~a 1 (b 1 a) = N 1 b: Thus the sparsity of the matrix is the better the smaller b is, for xed N . Figure 1 gives a clear picture of the sparsity structure of Gabor matrices. We can see from the plots that the total number of the non-zero elements of the Gabor matrix is = N 1 b: The smaller the frequency gap b is, the more sparsity we have in the Gabor matrix, and the more ecient is sparse matrix multiplication.
nz
When the given time gap a is small compared to the frequency gap b; we sometimes apply the following
commutation relation in order to take advantage of the better sparsity (and therefore reduced FLOPs) on the
Fourier transform side.
Theorem 3 (Commutation Relation). The Gabor triple (g; a; b) is a Gabor frame triple if and only if (^g; b; a) is a Gabor frame triple, where g^ = FFT(g): Moreover, we have
SF 1 FFT = N 1 FFT 1 ST : that is,
(7)
SF (FFT(x)) = N 1 FFT (ST (x)) for x 2 CI N :
where ST and SF are the frame operators associated with the Gabor frame generated by (g; a; b) and (^g; b; a) respectively.
For the case that the time gap a is less than the frequency gap b; with the help of the commutation relation, we usually compute the dual Gabor atom fast via the following corollary. Corollary 3. Suppose that (g; a; b) is a Gabor frame triple with length N; then g ~ can be obtained from FFT(g) via the following formula g~ = N 1 IFFT(FFT(g) 3 G0F 1):
where GF denotes the Gabor frame matrix associated with the Gabor frame triple (FFT(g); b; a); and FFT(g) 3 G0F 1 is exactly the dual Gabor atom corresponding to (FFT(g); b; a):
Based on the above results we can obtain the following series of corollaries. Corollary 4.
aX ~ 01 n=0
The Gabor triple (g; a; b) generates a tight Gabor frame if and only if
Tna g(k~b + j )Tna g(j ) =
(
jjgjj2 if k = 0 a
0
otherwise
for j = 0; 1; :::; a 0 1; k = 0; 1; :::; b 0 1:
Corollary 5. Suppose that (g; a; b) is real-valued Gabor triple (i.e., g is real-valued signal), then the associated Gabor matrix is matrix.
real
Corollary 6.
A Gabor frame atom g is real if and only if the associated dual Gabor atom is real.
non-negative
Corollary 7. For any Gabor triple (g; a; 1) the associated Gabor matrix is always a diagonal matrix, or equivalently, the associated Gabor operator is a non-negative pointwise multiplication operator. A sucient condition for (g; a; 1) to constitute Gabor frame is that there are at least a consecutive nonzero coordinates of g:
With the help of the Commutation Relation Theorem, we can also deduce the following corollary. Corollary 8. A necessary condition for Gabor triple (g; a; b) to generate Gabor frame is that that there are at least a nonzero coordinates of g and there are at least b nonzero coordinates of the Fourier transform FFT(g) of g: A sucient condition for (g; 1; b) to constitute Gabor frame is that there are at least b consecutive nonzero coordinates in FFT(g):
Remarks
As a special case for the lattice constants (a; b) = (1; 1); by Corollary 7, we can easily deduce Lemma 5 presented in the previous paper.8
4
Numerical Results and Further Remarks
In this section, we present some experimental results and make some remarks on further results. We will also give comparisons between the approach presented in this paper (we will call it \the new approach" ) and the approach obtained in.8,17 We will discuss the details in subsequent papers. All the numerical experiments were carried out using MATLAB 4.0 on a SUN EPC Workstation. (1). With the calculated dual Gabor atom g~ corresponding to (g; a; b); we can easily obtain the inverse of the associated Gabor frame matrix G by Algorithm 1 and the formula G01 = [GAB(~g; a; b)]0 3 [GAB(~g; a; b)]: (2). The implementation based on the conjugate-gradients method reaches numerical precision after at most
ab number of iterations. Furthermore, it is also applicable for any Gabor triple (g; a; b) (even if it is not Gabor
frame triple) which provides an ecient algorithm for computing the best approximation of any signal from the Gabor space.22 Using the (generalized) dual Gabor atom,16 we have actually derived an e ective numerical way for testing if a given Gabor triple (g; a; b) generates a frame or not. In fact, writing g~ for the calculated (generalized) dual Gabor atom,16 then (g; a; b) is a Gabor frame triple if and only if the Gabor matrix G = GAB(g; a; b)]0 3 GAB(~g ; a; b) is the N 2 N identity matrix. Since G can be completely determined by the the rst N 2 a block matrix B , we only need to test if B is equal to BI or not, where BI = (e1; :::; ea ) ; ek for k = 1; :::; a is the k-th \canonical" column vector of length N : ek = (0; :::; 0; 1; 0; :::; 0)0 : We refer to 15 for more details. | {z } | {z } k01
N 0k
(3). Figure 2 shows the Gabor atom and the associated dual one. The signal length is N = 360 and the lattice constants (a; b) = (20; 12): Table 1 gives us the obvious comparison between the approach (we call it \the old approach" for simplicity) developed in8 and the new approach. The old approach is based on the conjugate-gradients method and the short-time Fourier transform. Methods New Approach Old Approach
Time required(seconds) FLOPs required 2.167 69, 761 130.5 89,521,311
Table 1: Comparison between the new approach and the old approach for computing the dual Gabor atom associated to (g; a; b) shown in Figure 2 with the signal length N = 360 and the lattice constants (a; b) = (20; 12): (4). The new approach based on the sparse-matrix technology and the conjugate-gradients idea combined with Algorithm 2 pushes the maximal signal length that can be handled for numerical calculations to new orders. This point is obviously important for practical purposes. Figure 3 shows the typical Gaussian shape of Gabor atom and the associated dual Gabor atom where the signal length N = 2; 520 and the lattice constants are taken (a; b) = (56; 15): It took only about 8.3406 seconds and 366, 717 FLOPs for computing the associated dual Gabor atom. The reconstruction error for a random is around 1.1098e-11 which can be seen as error-free reconstruction.
x
(5). The new approach can be used to work with Gabor atoms of arbitrary shapes. Table 2 shows some of our experimental results for computing the associated dual Gabor atoms with the given Gabor atom shown in Figure 4 and di erent lattice constants. From the reconstruction errors, the reconstruction can be regarded as error-free reconstruction. The experimental results corresponding to the lattice constants (a; b) = (63; 40) is the critical sampling. With Theorem 3, we did the experiments only on the cases that the frequency gap b is less than or equal to the time gap a: (6). The experiments show that the MATLAB implementation presented here is much more ecient and faster than the recent complex methods obtained by D. F. Stewart, Lee C. Potter and S. C. Ahalt,17 where the
Time gap a 8 9 10 12 14 15 18 20 21 24 28 30 35 36 40 42 45 56 60 63
Frequency gap b 3 4 5 6 7 8 9 10 12 14 15 18 20 21 24 28 30 35 36 40
Time required (seconds) 2.5119 2.9258 3.3608 3.8459 4.1795 4.8227 5.1048 5.8098 6.7068 7.7741 8.0404 9.7974 10.6406 11.0059 12.9224 15.0518 16.2724 19.1514 19.6585 22.1683
FLOPs Reconstruction required error 77469 1.3267e-14 97578 1.4585e-13 111719 1.2550e-14 134,872 3.7255e-14 152,119 2.0053e-14 178,866 3.0434e-13 199,351 2.6250e-14 229,336 2.8263e-14 287,050 1.5658e-13 351,244 3.7342e-14 381,905 9.0525e-14 499,900 5.2249e-14 584,302 9.4007e-14 625,241 6.6831e-14 773,524 4.3654e-13 989,684 1.2606e-13 1,107,876 5.0337e-13 1,429,217 1.4152e-12 1,503,352 7.5835e-12 1,800,498 1.8528e-12
Table 2: Time, FLOPs required for the computation of the dual Gabor atom associated to a xed Gabor atom shown in Figure 4 and di erent lattice constants listed in the table. The last column gives the reconstruction errors for a typical random signal of length N = 2; 520.
Gabor frame matrix (2) 0 50
n = 360, (a, b) = (5, 12)
n = 360, (a, b) = (12, 5)
Gabor frame matrix (1) 0 50 100 150 200 250 300 350 0
100 150 200 250 300
100
200 nz = 1800
350 0
300
0 50
100 150 200 250 300 350 0
200 nz = 4320
300
Gabor frame matrix (4)
0 50
n = 360, (a, b) = (4, 8)
n = 360, (a, b) = (8, 4)
Gabor frame matrix (3)
100
100 150 200 250 300
100
200 nz = 1440
300
350 0
100
200 nz = 2880
300
Figure 1: Gabor matrix structure for xed signal length N = 360 and four di erent pairs of lattice constants (a; b) = (12; 5); (5; 12); (8; 4); (4; 8):
Gabor atom 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -200
x 10
-150
-100
-50
-2
0 50 Signal length n = 360
100
150
200
100
150
200
Dual Gabor atom
10 8
Lattice size (a, b) = (20,12)
6 4 2 0 -2 -4 -6 -8 -200
-150
-100
-50
0 50 Signal length n = 360
Figure 2: Gabor atom and the associated dual one, signal length N = 360; and lattice constants (a; b) = (20; 12):
Gabor atom 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -1500
x 10
-1000
-500
-2
0 Signal length n = 2520
500
1000
1500
500
1000
1500
Dual Gabor atom
2.5 2
Lattice size (a, b) = (56,15)
1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -1500
-1000
-500
0 Signal length n = 2520
Figure 3: Gabor atom of large signal length N = 2; 520 and the associated dual with lattice constants (a; b) = (56; 15):
Gabor atom 1.5
1
0.5
0
-0.5
-1
-1.5 -1500
-1000
-500
0 Signal length n = 2520
500
1000
1500
Figure 4: Gabor atom of large signal length N = 2; 520:
Gabor atom 2.5
2
1.5
1
0.5
0 -150
x 10
-100
-2
-50
0 Signal length n = 288
gamma1
x 10
6
6
4
4
2
2
0
0
-2
-2
-4
-6 -200
50
-2
100
150
gamma2
-4
-100
0
100
200
-6 -200
-100
0
100
200
Figure 5: Gabor atom of signal length N = 288 with lattice constants (a; b) = (24; 12) and the associated dual Gabor atoms gamma1 and gamma2 calculated from and the new approach and Stewarts' method, respectively. The absolutely error between gamma1 and gamma2 is around 4.8627e-15.
Methods New Approach Stewart's Approach
Time required(seconds) FLOPs required 0.9834 32,302 3.681 1,175,750
Table 3: Comparison between the new approach and the approach developed by D. F. Stewart, Lee C. Potter and S. C. Ahalt. The Gabor atom and the calculated dual Gabor atoms with two methods are shown in Figure 5. The signal length N = 288 and the lattice constants (a; b) = (24; 12):
authors considered only the critical sampling case. Table 3 gives the comparison between the complex approach developed in17 and the new approach presented in this paper for computing the dual Gabor atom associated to the Gabor atom shown in Figure 5. (7). Based on the investigations concerning Gabor expansions with oversampling we have also developed ecient implementations for the problem of best approximation by undersampled Gabor families. We refer to.16 (8). For the two dimensional case, if the Gabor atom is given as 2D separable signal, we have proved that the associated dual Gabor atom is also 2D separable and thus the problem of 2D Gabor expansions can be reduced to the 1D case in a natural way. We refer to15 for details.
5
Conclusion
With the established Gabor-matrix structure, we have developed a very ecient Gabor transform in the nite discrete setting with the following big advantages: The experimental results show clearly that the implementations for computing the dual Gabor atom based on the observations described in this note work very eciently. The algorithms can also be used to compute the inverse of the Gabor frame matrix eciently. The algorithms can be used for Gabor frame atoms of large signal length. If the given Gabor triple does not generate frame, using the implementation based on the conjugate-gradients method, we can obtain the best approximation by linear combinations of Gabor family. We usually can get a good reconstruction with the calculated dual Gabor atom. The experimental results presented in the above section show that the Gabor reconstruction with the dual Gabor atom can be seen as practically error-free reconstruction.
(1). Eciency:
(2). (3). Wide applicability:
(4). Error-free reconstruction:
6
Acknowledgements
The authors acknowledge partial support from the project PH 08784 at the Austrian Science foundation FWF. We are thankful to Mr. D. Stewart for providing us the MATLAB functions related to the latest work.17
7 [1]
REFERENCES
M. J. Bastiaans, Gabor's signal expansion of a signal into Gaussian elementary signals, Proc. IEEE, 538{ 539, 1980.
[2] [3] [4] [5] [6]
A sampling theorem for the complex spectrogram, and Gabor's expansion of a signal in Gaussian elementary signals, Optical Engineering, 20(4), 594{598, 1981. M. J. Bastiaans, Gabor's Signal Expansion and Its Relation to Sampling of the Sliding-Window Spectrum, M. J. Bastiaans,
Advanced Topics in Shannon Sampling and Interpolation Theory, edited by Robert J. Marks II, 1{35.
J. J. Benedetto, Frame decompositions, sampling, and uncertainty principle inequalities, WAVELETS Mathematics and Application, Edited by J. Benedetto and M. Frazier, 247{304, 1993.
I. Daubechies, Ten Lectures 61, Philadelphia, 1992.
on Wavelets, CBMS-NSF regional conference series in applied mathematics
T. Ebrahimi and M. Kunt, Image Compression by Gabor Expansion, Optical Engineering, 30(7), 873{880, 1991.
[7]
H. G. Feichtinger and K. Grochenig, Theory and Practice of Irregular Sampling, WAVELETS Mathematics and Application, Edited by J. Benedetto and M. Frazier, 303{363, 1993.
[8]
H. G. Feichtinger and O. Christensen, Communication, SPIE, Nov. 1993.
[9] [10] [11]
New ecient methods for Gabor analysis, Proc. Conf. Visual
D. Gabor, Theory of Communication, Journal of Electrical Engineering, 93(1946), 429{441.
C.E. Heil and D.F. Walnut, Continuous and Discrete Wavelet Transforms, SIAM Reviews, 31(4), 628{666, 1989.
R. S. Orr, The Order of Computation for cessing, 41(2), 122{130, 1993.
Finite Discrete Gabor Transform, IEEE Trans. on Signal Pro-
[12]
M. Porat and Y. Y. Zeevi, The generalized Gabor scheme of image representation in biological and machine vision, IEEE Trans. Pattern Anal. Machine Intell., 10, 452{468, 1988.
[13]
S. Qian and D. Chen, 1993.
[14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
Discrete Gabor Transform,
IEEE Trans. on Signal Processing, 41(7), 2429{2438,
S. Qiu, H. G. Feichtinger and T. Strohmer, Inexpensive ence in San Diego, July, 1994.
S. Qiu and H. G. Feichtinger, Discrete IEEE Trans. on Signal Processing.
Gabor Expansions, accepted for the SPIE confer-
Gabor Structure and Optimal Representation, to be submitted to
S. Qiu, On the Best Approximation by Undersampled Gabor Families, submitted to IEEE Trans. on Signal Processing. D. Stewart, L. Potter and S. C. Ahalt, Computationally Attractive Real Gabor Transform, Preprint.
R. Tolimieri and R. S. Orr, Poisson Summation, the Ambiguity Function and the Theory of WeylHeisenberg Frames, Preprint. J. Wexler and S. Raz, Discrete Gabor Expansion, Signal Processing, 21(1990), 207{220. J. Yao, Complete Gabor Transformation for Signal Representation, IEEE Trans. on Image Processing,
2(2), 152{159, 1993.
J. Yao, Computations for the Generalized
Gabor Transforms and Bi-orthogonal Functions, Preprint. M. Zibulski and Y. Y. Zeevi, Matrix Algebra Approach to Gabor-Scheme Analysis, Preprint. Z. Zlatev, Computational Methods for General Sparse Matrices, KLUWER ACADEMIC PUBLISHERS, 1991.
