A theory is outlined whereby it is possible to design a M x N channel two-dimensional quadrature mirror filter bank which has perfect reconstruction property. Such a property ensures freedom from aliasing, amplitude distortion, and phase distortion. The method is based on a simple property of certain transfer matrices, namely the losslessness property

Vaidyanathan, P. P.

English

Caltech Authors - Main

916 
Perfect Reconstruction QMF Banks for 
Two-Dimensional Applications 
P. P. VAIDYANATHAN 
Abstracr -A theory is outlined whereby it is possible to design a M X N 
channel two-dimensional quadrature mirror filter hank which has perfect 
reconstruction property. Such a property ensures freedom from aliasing, 
amplitude distortion, and phase distortion. The method is based on a 
simple property of certain transfer matrices, namely the losslessness prop- 
erty. 
I. INTRODUCTION 
The application of subband coding techniques for images has 
recently received considerable attention [l]-[4]. Vetterli [l] has 
extended the idea of quadrature mirror filtering [5] for the 
two-dimensional case: The usefulness of such techniques has been 
well demonstrated in recent contributions by Woods and O’Neil 
PI. 
The purpose of this correspondence is to introduce certain 
methods whereby two-dimensional quadrature mirror filter 
(QMF) banks can be designed with complete freedom from all 
linear distortions (namely, aliasing, amplitude, and phase distor- 
tions [S]). Our results here are extensions of the one-dimensional 
results in [7]; the presentation here can, however, be understood 
in a self-contained manner, as the developments do not depend 
on those in [7]. A two-dimensional QMF band with MN channels 
[l]-[4] is basically a parallel interconnection of MN branches of 
the form shown in Fig. 1. The branch labeled (m, n) takes the 
input signal x( n,, n,),-forms a subband signal by~passing through 
the two-dimensional filter H,,,, (z, , z2) (analysis filters), and deci- 
mates the filtered signal by the factor (M, N), where M and N 
are the decimation ratios in the horizontal and vertical directions, 
respectively. At the synthesis end, the signal is interpolated, 
filtered by the synthesis filters F,,(zi, z2) (to remove the images 
[6]), and then recombined. The operations of the decimators are 
described as r,,(n,, nz) = v,,( Mn,, Nn,). The interpolators are 
described by tmn(nl,n2) = r,,(n,/M, n,/N) if n, and n2 are 
multiples of ‘M and N, respectively, and fmn(nl, ns) = 0 other- 
wise. In the transform domain, the decimator and interpolator 
are described, respectively, by the input-output relations 
R,,(zl,z2) = (~/MN)~~~~~~‘V,,(Z~‘~~~,Z:‘~W~) (where 
W, = e-“‘jl”, etc.), and Tm,(zl, z2) = Rmn(ziM, z,"). Accord- 
ingly, the reconstructed output signal n( n,, n2) (which is the sum 
of the MN signals p,,,,( n,, n2) in Fig. 1) is related to the input 
M-lN-1 
. c c H,,(z,w~,z,w~)F,,(z,,z2). (1) 
m=O n=O 
The terms in (l), except those corresponding to k = I = 0, repre- 
sent the effect of aliasing. If aliasing is somehow canceled by 
appropriate choice of the synthesis filters Fmn(zl, z2), we are left 
Manuscript received February 26,1987. This work was supported in part by 
the National Science Foundation Grant DC1 8552579, and in part by Caltech’s 
programs in Advanced Technology Grant sponsored by Aerojet General, 
General Motors, GTE, and TRW. 
The author is with the Department of Electrical Engineering, California 
Institute of Technology, Pasadena, CA 91125. 
IEEE Log Number 8714985. 
Fig. 1. The (m. n)th branch of the M X N band 2-D QMF ba& 
Fig. 2. The (m, n)th branch of the “conceptual” structure. 
with the distortion function 
If T( zi, zz) is a linear-phase (FIR) function, there is no phase 
distortion; if, on the other hand, it is an all-pass (IIR) function, 
there is no amplitude distortion. Finally, if T( zi, z2) is a pure 
delay, i.e., T(z,, z2) = cz; Liz; ‘2, then we have perfect recon- 
struction, i.e., 
IEEE TRANSACTIONS 0~ CIRCUITS AND s~s~EMs,voL. as-34, NO. &AUGUST 1,987 
In order to understand how aliasing can be completely canceled, 
and how perfect reconstruction can be eventually attained, let us 
first consider the conceptual structure of Fig. 2, which we shall 
eventually relate to Fig. 1. Assume that we have a parallel 
connection of MN branches of the form shown in Fig. 2 (with 
O<m<M-landO<ngN-l)andletj(ni,n,)denotethe 
output (w*tich is the sum of the MN signals qmn(nl, nz)). The 
quantity Y( zi , z2) is then equal to 
=1 
-(M-l),;(N-1) M-l N-1 
MN c c x(~lcfMY~) k=O I=0 
M-lN-1 
. c c Smn(Zy,z:)kvp%p. (4) 
m=O n=O 
The signal j( n,, ni) is thus free from aliasing if (and only if) all 
the terms in (4) are zero except those corresponding to k = I = 0. 
Accordingly, the system is alias-free if and only if Smn(zl, z2) is 
independent of m and n. Letting S,, (zi, z2) = S( zi, z2) for all 
m, n , we then have 
P( Zl ) z,)/X( Zl , z2) = Z;(“-l)Z;(N-l)S( zy, z?). (5) 
This result enables us to construct various types of QMF banks 
as shown below. 
II. PERFECTRECONSTRUCTION QMF BANKS 
If S( zi, z2) in (5) is a pure delay, we have perfect reconstruc- 
tion. Now consider Fig. 3(a), where we have inserted two MN X 
MN transfer matrices E(z,, z2) and R(z,, z2) into our concep- 
tual circuit of Fig. 2. (The figure is drawn for the case of 
M = N = 2, for convenience.) If the matrices are such that 
R( zi, z2) = E-‘( zi, z,), then we still satisfy the perfect recon- 
struction property (i.e., (5) holds with S( zi, z2)). If the transfer 
matrices are now moved past the decimators and interpolators 
(according to standard rules [6]), this results in the equivalent 
009%4094/87/0800-0976$01.00 01987 IEEE 
EEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-34, NO. 8, AUGUST 1987 
Fig. 3. (a) Inserting transfer matrices E(r,,rz) and R(z1,z2) into the con- 
ceptual structure. (b) Moving the decimators and interpolators to obtain an 
equivalent structure. 
Fig. 4. A method for forcing E(zl.zZ) to be unitary. Here, the transfer 
matrix of the cascaded lattice structure is taken to be E( zl, Q). 
Fig. 5. All-pass-band uniform DFT QMF bank. Here, the 2X2 DFT and 
IDFT blocks represent two-dimensional DFT and IDFT on 2 x 2 arrays. 
structure of Fig. 3(b). (Even though the rules in [6] are for the 
1-D case, it is easily verified that similar results hold for the 2-D 
case.) By comparison with the QMF bank of Fig. 1, we see that 
Fig. 3(b) represents a standard QMF bank with 
M-l N-l 
H,,(zI,z~) = c c E~+~M,~+~M(z~~,z~)z;~z;” (6) 
m-o n=O 
M-l N-l 
Frs(z1,4 = c c %+nM,r+sM(+2N) 
m-o n=O 
.Z;(M-l-m)Z;(N-l-n) 
. (7) 
Thus, the transfer function HrS(zl, zs) has polyphase compo- 
nents [l]-[4] given by Ejj(zl, zs), where i = r + sM, j = m + n&l. 
Given a set of analysis filters, we can therefore find the matrix 
E(z,, z2) and then invert it to obtain R ( zl, z2) so that the 
synthesis filters for perfect reconstruction can be found. This 
approach, however, usually gives rise to synthesis filters of very 
high order which, in addition, are not guaranteed to be stable. A 
second approach is to restrict H,,,, ( zl, z2) so that E( ej”l, &) is 
unitary for all wi, w2.1 Under this condition, if we choose 
R (zi, z2) to be equal to ET( z; ‘, z; ‘), then we still have a perfect 
reconstruction system. If the analysis filters are FIR, then the 
entries of E( zi, zs) are FIR, and, hence, R ( zl, z2) is FIR, with 
entries of the same length. However, R(z,, zs) is typically non- 
‘Such matrices E(r,, z2) are said to be lossless, analogous to the one-dimen- 
sional case [7]. 
causal; if this is undesirable, appropriate amounts of delays can 
be inserted without destroying the perfect reconstruction prop- 
erty. 
In summary, if the analysis filters are FIR and such that the 
polyphase component matrix is unitary for z1 = @I, z2 = &‘z, 
then it is easy to find a set of FIR synthesis filters of the same 
length (as the analysis filters), leading to perfect reconstruction. 
In order to force such unitariness, we have to structurally con- 
strain E(z,, z2). For example, with M = N = 2, Fig. 4 shows a 
way of doing this, where the matrices Y are orthogonal. The 
transfer matrix of the cascaded lattice structure in Fig. 4 is itself 
taken to be E( zl, zz). If the coefficients of these matrices are 
optimized such that the analysis filters (given by (6)) have “good” 
stopbands, then a good perfect reconstruction .system is auto- 
matically ensured. 
Comments on Separability 
If a perfect reconstruction system (which is of course free from 
aliasing) is designed in the above manner, H,,,,( zl, z2) does not 
have to be separable. This can be demonstrated as follows for the 
M = N = 2 case, by taking the special case where E(z,, z2) is a 
constant matrix E with the first row equal to c, c, c,2c, where 
c = l/c. The remaining rows of E can always be chosen such 
that E is orthogonal. With this choice, we have Hoo(zl, zz) = c(1 
+z;1+z;1+2z;1 -l z2 ), which is nonseparable, and at the same 
time we have the perfect reconstruction property (in particular, 
aliasing has been eliminated). 
978 IEEETRANSACTIONS ON CIRCUITSAND SYSTEMS,VOL.CAS-34,NO,8,AUGUSTl987 
III. ALLPASS-BASED QMF BANKS 
One disadvantage with FIR, QMF banks is the requirement of 
high filter order for a given performance (such as stopband 
attenuation). It is possible to use IIR filters H,,(z,, z,), 
F,,,(z,, z2) in such a way that aliasing is canceled, and such that 
Z’(zl, z2) in (2) is allpass. This leaves behind some phase distor- 
tion, which can subsequently be compensated for by cascading 
with allpass equalizers. If such a scheme is practicable in a 
certain application, then the results of Section I offer a method 
for doing this. Fig. 5 demonstrates such a scheme for M = N = 2, 
where A,,, ( zl, z2) and B,,,,(z,, z2) are all-pass functions such 
that S(zl,z,)=A,,(z,,zZ)Bmn(~1,~2) is independent of m,n. 
As a result, the structure is equivalent to a parallel combination 
of the branches in Fig. 2 with S,,( zl, z2) independent of m, n. 
Aliasing is therefore absent, and T(z,, z2) given by the RHS 
of (5) is allpass. Notice that the analysis and synthesis banks 
correspond to the uniform-DFT type. For example, with 
M=N=2, we have Hlo(~l,z,)=H,(-zl,z,),Hol(z,,z,)= 
%a(~,, - zz>, H,,(z,,z,) = f&o = (- ~1, - ~2). 
IV. CONCLUSIONS 
The design of transfer functions H,,, ( zl, z2) such that E( zl, z2) 
remains unitary should be done by computer-aided optimization 
by formulating an objective function which reflects the filtering 
quality of H,,,,(zl, z2). Since the objective function to be mini- 
mized is a nonlinear function, and since the responses are two- 
dimensional, the design time may ultimately limit the sharpness 
and stopband attenuation of the analysis filters. For the design of 
the all-pass-based structure in Fig. 5, a similar computer-aided 
technique is required. But the IIR scheme has fewer parameters 
to optimize as compared to the perfect reconstruction FIR struc- 
ture. Some of these design problems are currently under study. 
ACKNOWLEDGMENT 
The author is thankful to a reviewer for making certain im- 
portant and useful remarks about the separability issue. Also, 
interesting discussions with V. C. Liu, graduate student at Caltech, 
are gratefully acknowledged. 
WI 
PI 
[31 
[41 
151 
161 
[71 
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REFERENCES 
M. Vetterli, “Multi-dimensional sub-band coding: Some theory and al- 
gorithms,” Signal Processing, vol. 6, pp. 97-112, Apr. 1984. 
J. W. Woods and S. D. O’Neil, “Subband coding of images,” IEEE 
Truns. Acourt., Speech, Signal Process., vol. 34, pp. 1278-1288, Oct. 
1986. 
A. V. Brandt, “Subband coding of videoconference signals using quadra- 
ture mirror filt&s,” in Proc. IASTED Conf. on Applied Signal Processing 
and Digital Filtering (Paris), June 1985. 
G. Wackersreuther, “On two-dimensional polyphase filter banks,” IEEE 
Trans. Acourt., Speech, Signul Process., vol. 34, pp. 192-199, Feb. 1986. 
A. Croisier, D. Esteban, and C. Galand, “Perfect channel splitting by use 
of interpolation/decimation/tree decomposition techniques,” presented 
at Int. Conf. on Info. Sci. and Sys., Patras, Greece, 1976. 
R. E. Crochiere and L. R. Rabiner, Multirute Digital Signal Processing. 
Englewood Cliffs, NJ: Prentice-Hall, 1983. 
P. P. Vaidyanathan, “Theory and design of M-channel maximally deci- 
mated quadrature mirror filters with arbitrary M hating perfect- 
reconstruction property,” Tech. Rep., California Institute of Technology, 
Apr. 1986 (also to be published in IEEE Trans. ASSP, Apr. 1987). 
M. J. T. Smith and T. P. Bamwell III, “A unifying framework for 
analysis/synthesis systems based on maximally decimated filter banks,” 
in Proc. IEEE Int. Conf. on Acourt., Speech, Signal Process. (Tampa, 
FL), Mar. 1985, pp. 521-524. 
Split Vector Radix 2-D Fast Fourier Transform 
SOO-CHANGPEIANDJA-LINWU 
Abtrucl -The split vector radii is used to develop a 2-D fast Fourier 
transform (FFIJ algorithm; it is performed “in-place,” and requires no 
matrix transpose operation. This method greatly improves the conventional 
vector radix 2-D FFT; an overall saving of about 23 percent in complex 
multiplications for a typical 2048 X 2048 array could be obtained. 
I. INTRODUCTION 
The two-dimensional (2-D) discrete Fourier transform (DFT) 
is an important tool in digital image processing. Traditionally, 
the “row-column” method has been used to compute the 2-D fast 
Fourier transform (FFT) by taking 1-D FFT row-wise and col- 
umn-wise; however, the time-consuming matrix transpose is 
needed in this row-column transform. Rivard, Harris, and 
McClellan [l], [2] use the vector radix to perform the decimation 
in both rows and columns simultaneously. The 2-D DFT is 
decomposed successively into several smaller 2-D DFT’s until, 
ultimately, only trivial 2-D DFT’s need to be evaluated. This 
vector radix method saves 25 percent in complex multiplications, 
and also avoids the matrix transpose operation. 
The split radix has been proposed for 1-D FFT computations 
by Duhamel and Hollman [3], [4]. This algorithm has the ad- 
vantage of being performed “in-place” in an FFT-like structure, 
and requires the lowest number of multiplications and additions 
for length N = 2”. Recently, Sorensen, Heideman, and Burrus 
have developed an efficient Fortran program on computing the 
split radix FFT [5]. Also, this algorithm can be used to efficiently 
calculate the fast Hartley transform [6], [7]. 
In this paper, we use the split vector decomposition to develop 
a 2-D fast Fourier transform; it greatly improves the conven- 
tional vector radix 2-D FFT. An overall saving of about 23 
percent in complex multiplications for a typical 2048 X 2048 array 
could be obtained. 
II. SPLITVECTORRADIXALGORITHM 
The two-dimensional discrete Fourier transform is defined as 
N-l N-l 
X(k,,k,) = c c. X(nl,n,)W;‘k’W;Zk2 
n,=O n,=O 
where 
W, = exp( - jZn/N). 
In the conventional vector radix 2-D FFT, we decompose the 
indices-k,, k, simultaneously into four groups (see Fig. 1): 
X(2k,&) even-even 
X(2kl,2k, +1) even-odd 
X(2k, +1,2k,) odd-even 
X(2& +1,2k, +1) odd-odd. 
Manuscript received March 9, 1987. 
S.-C. Pei is with the Department of Electrical Engineering, National Taiwan 
University, Taipei, Taiwan, R.O.C. 
J.-L. Wu is with the Department of Electrical Engineering, Tatung Institute 
of Technology, Taipei, Taiwan, R.O.C. 
IEEE Log Number 8714983.' 
0098-4094/87/0800-0978$01.0001987IEEE 


Perfect reconstruction QMF banks for two-dimensional applications

Bamwell III, “A unifying framework for analysis/synthesis systems based on maximally decimated filter banks,” in

Multi-dimensional sub-band coding: Some theory and algorithms,”

On two-dimensional polyphase filter banks,”

Perfect channel splitting by use of interpolation/decimation/tree decomposition techniques,” presented at Int.

S.-C. Pei is with the Department of Electrical Engineering,

Subband coding of images,”

Subband coding of videoconference signals using quadrature mirror filt&s,” in

Theory and design of M-channel maximally decimated quadrature mirror filters with arbitrary M hating perfectreconstruction property,”

http://authors.library.caltech.edu/5896/1/VAIieeetcs87c.pdf

Perfect reconstruction QMF banks for two-dimensional applications

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