exact subband image decomposition and reconstruction in discrete space and discrete frequency...

SIGNAL PROCESSING:

Ill4AGE COMMUNICATION

Signal Processing: image Communication 7 (1995) 249-257

Exact subband image decomposition and reconstruction in discrete space and discrete frequency domains

Rtmy Prost*, Chaouki Diab, Robert Goutte

lnstitut National des Sciences Appliqkes De Lyon, Luboratoire de Traitement du Signal et Ultrasons - CREATiS - URA CNRS 1216, B&t. 502, 20 avenue Albert Einstein, 69621 Villeurbanne Cedex, France

Received 17 August 1994

Abstract

This work extends previously reported work on image multi-subband decomposition/reconstruction using DFT. The purpose of the method proposed here is to achieve exact decomposition and reconstruction using ideal band-pass filters implemented in the DFT domain without any excess data.

1. Introduction

Image subband coding consists of splitting the image into some subimages corresponding to a set of oriented frequency bands. Each subimage is then quantized and coded using an appropriate coder matched to its statistics. The classical approach uses Quadrature Mirror Filters (QMF’s) implemented in the spatial domain by finite impulse response (FIR) filtering [9]. A four subband decomposition of an A4 x N image using an FIR filter of length L results in (M + L - 1) x (N + L - 1) pixels. To preserve the total amount of data, the decomposed images are truncated to M x N pixels. Thus, a boundary error is induced in the recon-

*Corresponding author. Tel.: + (33) 72.43.82.27. Fax: + (33) 72.43.85.26. E-mail: [email protected].

strutted image. This error prevents the coding of the image by blocks and then deprives the tech- nique of the advantages of an adaptivity of the coding process to the local statistical changes of the image. In [6, S] some signal extension methods are proposed to reduce these border effects. These extension methods are still a partial remedy.

Perfect band-splitting requires ideal band-pass filters. Unfortunately such filters have an infinite impulse response which prevents implementation in the spatial domain. In [2] an implementation of these filters in the frequency domain using the Dis- crete Fourier Transform (DFT) has been proposed. In this method, there is no border effect in the reconstructed image because the process is equivalent to a circular convolution. The DFT introduces cyclic boundary effects in the subbands but this phenomenon does not appear in the reconstructed image. Perfect decomposition and reconstruction of an M x N image in four subbands requires

0923-5965/95/$9.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0923-5965(95)00029-l

250 R. Prod et al. / Signal Processing: Image Communication 7 (1995) 249-257

M + N real values corresponding to the imaginary part of the DFT of each row and column of the original image at a quarter of the Nyquist frequency, because the ideal filters used have complex impulse responses. This is an excess amount of data.

In this paper a derivation based on this latter method is proposed to eliminate the excess data. This work is based on equivalent equations of the very well known conditions for exact reconstruction [S] for the case of finite extent signals and the discrete frequency domain. These conditions are elaborated upon in Section 2 for a two-band decomposition/reconstruction scheme for a one dimensional problem. The method is presented in Section 3 and extended to multiband decomposition and reconstruction for the two dimensional case in Sections 4 and 5. Finally, computational complexity is evaluated.

2. Exact decomposition/reconstruction conditions in the frequency domain

To reduce the complexity of the presentation we consider first the one dimensional case. Here the image is reduced to a signal of N samples. The basic component of subband coder is the two-band analysis and reconstruction system (Fig. 1). The discrete input signal sz (whose z-transform is S(z)) is ana- lysed by a half-band low-pass filter H,(z) and a half-band high-pass filter G,(z). Then the filtered signals are decimated a factor of two.

At the receiver these signals are upsampled and filtered for interpolation by a low-pass filter H,(z)

and a high-pass filter G,(z), respectively, and re- combined to give the reconstructed signal f(z) (Fig. 1). The reconstructed signal contains two components:

%) = CH,(z)H,(z) + G,(z) G,(z)1 S(z)

+ [H,(z) Hat - z) + G,(z) G( - z) 1 St - 4. (1)

The first term corresponds to the weighted original signal and the second is an aliasing term. It follows that the exact reconstruction conditions are

W)K(z) + G,(z) G,(z) = 1, (2)

H,(z) H,( - z) + G,(z) G,( - z) = 0. (3)

For the case of finite extent signals we can write Eqs. (2) and (3) in the discrete frequency domain (the DFT domain) by substituting z by exp(i2zk/N) and - z by exp[(i2n/N)(k + IN)] (i = 0). These

equations become

&X0WJ + ~~(ga)~z7r) = 17

fork=O,...,N-1, (4a)

F;+,,,(UF,N(hJ + @+:,,&a)FkN(S*) = 0,

fork=O, . . ..N-1. (4’4

where F:(x) denotes the component k of the N-sample DFT (denoted N-DFT) of the discrete signal x of length N. The impulse response of the filters N,, G,, H, and G,, are h,, ga, h, and gr, respectively.

Fig. 1. Two-band analysis/reconstruction subband coder/decoder scheme.

R. Prost et al. / SignaI Processing: Image Communication 7 (1995) 249-257 251

3. Half-band ideal filters for error-free reconstruction

The idea is to use half-band ideal filters implemented in the frequency domain by the Discrete Fourier Transform (DFT). To obtain orthogonal subbands we search for two ideal filters defined by

CML) = Mr,), G%7,) = m&)1

~ I [l,O] k=O, . . ..$N- 1

@a) = [[O,l] k=fN+ 1, . . . . $N,

and for k = N/4

To obtain filters with real impulse responses h,, h,, gB, gr we use the following conjugate relationship:

F:(*)=(F;_k(.))*, k= 1, . . . . $N-1, (6)

where . applies for all impulse response filters. Clearly from the filter definition (Sa), the perfect

reconstruction conditions (4a) and (4b) are satisfied for k = 0, . . . ,N/4-1 and k=N/4+1, . . . . N/2 - 1.

The choice of (h,, ga) and (h,, gr) equal to (1,O) and (0, 1) in (5b), satisfies the reconstruction conditions, but leads to complex impulse response filters. To overcome this difficulty we have chosen complex coefficients which match Eqs. (4a) and (4b). Then, by incorporating (5b) into (4a) and (4b) we obtain the following equations:

ah,% + a,#,, = l/2,

bhr% + bgraga = 0,

bhrba + bgrbga = - l/Z

ahr bha + agrbga = 0.

(74

G’W

(7c)

(7d)

The system of equations (7) is overdetermined. To solve it we use the following additional constraint of mirror analysis filters (equal magnitude at one quarter of the sampling frequency):

ai& +- bi, = ai. + b,, 2 ea2, 0<a2d1. (8)

A sufficient condition for this is

Fr(h,) = {Ff(g,)}*, for k = N/4,

then

(9)

a@ = aha and b,, = - bha. Pa)

By incorporating (9a) into the system of equations (7) we obtain

@,r = a,, and bbr = - bgr, UW

%a(ahr + agr) = l/2 and bha(bhr - bg,) = - l/2,

WW

Combining the above relationships leads to

1

uha = 4a,,, Wa)

and

bha= - f. hr

(1 lb)

By incorporating (1 la) and (11 b) into (8) the latter equation becomes

(12)

The equation remains overdetermined. To solve this equation, we impose the following arbitrary constraint:

ahr = bhre a,. (13)

Then from (lla) and (lib)

aha = - bha &a,, (14)

from (lOa)

bgr= -bhr= -c1, and a,,r=ahr=a,,

and from (9a)

bea= -bha=a, and aga=aha=@,.

From the definition of filters (5b), we can write

tFt(k), G(s,)l = [a,(1 - 0, a,(1 + $1, CFkN(k), Gkh)l = Cd + i), 4 - 01,

for k = $N. (15)

252 R. Prost et al. 1 Signal Processing: Image Communication 7 (1995) 249-257

From (lla), (13) and (14)

1

c(“=40(,. (16a)

and from (12) and (13)

1 &@. UW

Solution 1. The magnitude of the filters is equal to 1 at the point N/4 (a2 = 1). From (16b) it follows that

a, = l/(2$) and, from (16a), a, = ,/2/2.

Solution 2. The magnitude of the reconstruction filters is equal to that of the analysis ones: a, = a,

and, from (16a), a, = a, = 4.

4. Ideal filter bank for error-free reconstruction

A number B of orthogonal subbands can be easily obtained using an ideal band-pass filter bank (h,[b], h,[b] for analysis and reconstruction, respectively, for b = 0, . . . , B - 1) generated by the same approach as in Section 3. We will consider filters of equal bandwidth: N/B = Q (where Q is an integer). To avoid complexity of presentation we consider only the practical case where B is a power of two: B = 2’ J E N. Then

We recall that so is the input to the filter bank. Let ai be the output signal of the filter h,[b]. Its N-DFT is Ft(ai). By decimation of ui by a factor B: 1, we obtain the signal abJ whose N/B-DFT is obtained from Ft(aE) using the following equation:

F:‘B(a;J) = ; B$l F,“,&z,O), I-O

which is derived in Appendix A.

5. Implementation of the subband analysis and reconstruction process in the frequency domain

(18)

We first consider the decomposition process of a 1D signal s,” (n = 0, . . . , N - 1) into two subbands using the solution (15). This process requires four steps:

(i) N-DFT of so. (ii) As the filter transfer function is zero or one

for most of the frequency components (Eq. (5a)) the filtering step is reduced to a simple component selection except at the frequencies N/4 and 3N/4 where the transfer function has complex values.

(iii) The subsampling step is done using Eq. (A.5) for the two N-DFTs Fc(a’), Ff(d’). It can be noted that by applying (AS) at the frequency point

:

k=O, . ...;- 1 if b = 0,

Vf(kCbl), F~(kCbl)) g (1, I), for k = bQ + 1, ... ,tb + ‘)Q 2

-----1 if l<b<B-2, 2

k=bQ+l E 2

3 .. 2 2

if b= B- 1,

4 (aa(l - i), a,(1 + i), for k = v if 0 < b < B - 2,

a(~~(1 + i), a,(1 - i)), for k = F if 1 < b < B - 1,

4 (0, O), otherwise, for k d g,

= ([F~_,(h,[b])]*, [Fz_,(h,[b])]*), for k = 5 + 1, . . . , N - 1. (17)

R. Prod et al. / Signal Processing: Image Communication 7 (1995) 249-257 253

N/4, we obtain

F$:(a- ‘) = + [F&&O) + &v&0)1

= 5 [ff,( 1 - i) F&4(so)

+ cc,(l + i)Fgh(s”)l

= a, Re[(l - i) F&&‘)]

= cc,{ReCJ’&+(s”)l + Im[f’&~(s”)l).

(19)

Similar equations apply to d- 1 to give

Fgj:(d- ‘) = a,(Re[F&+(s’)] - Im[F&(s0)]}.(20)

(iv) two ~-IDFTs are applied to FL/2(u-1) and

F:‘2 (d- ‘).

We conclude that steps (ii) and (iii) can be merged. We will now consider the decomposition process

into B subbands. All the previously described steps are applied; however, Eq. (18) is used instead of (AS). For the transition points of the filter transfer functions (17) we obtain equations similar to (19) or

(20),

F:%b J, = $ [Re(Ft+ .(s”)}

+ ( - l)b+2k’QIm{F,“,x(s0))],

fork=0 ork=z,

with the following conditions:

For b even, X = bQ/2 and

ifk=Oandb=OthenY=l, else Y = 2a,. For b odd,

Q if k = T then X = (b - l)Q 2

and Y = 2a,,

if k = 0 then X = (b :I)’ and if b = B - I

then Y = 1, else Y = 2a,.

(21)

Fig. 2 illustrates the decomposition of a signal into B = 4 subbands. For a 2D signal we first decompose rows into B horizontal subbands and then we decompose each column of these subbands into B vertical subbands. As a result we obtain B2 regularly spaced subbands. Fig. 3 illustrates the decomposition into 64 subband images of Lena (512 x 512 pixels). Without quantization the reconstructed image is exactly the original one.

It can be noted that, in Fig. 3, the cyclic boundary effect has a lower magnitude when the analysis filters of Solution 2 are used. Clearly, this solution is more suitable for quantization in an image com- pression system. This cyclic boundary effect can be completely eliminated by using a symmetric extension of the image [l, 71. This approach is equivalent to the use of the DCT instead of the DFT in the proposed scheme (see [3]). The main drawbacks of using the DCT approach are the nonspatial-invari- ant property of the DCT domain filtering and the nonequivalence to spatial-decimation [4] of the merging process of Section 5 (step iii).

6. Computational complexity

In this section we will evaluate both the number of multiplications and additions per pixel needed to decompose an M x N image into B2 = 22J subbands using the frequency domain implementation of the proposed algorithm.

The image decomposition into B2 regularly spaced subbands constrains the number of rows (M) and columns (N) to be of the form No2J and Mo2’, respectively. Then it is appropriate to use the radix-2 FFT algorithm (Tukey-Cooley) for DFT and IDFT computations. However, this algorithm requires the additional constraint that MO and No be powers of 2. The evaluation of the DFTs of two real sequences of N samples in one pass through a single complex N-FFT needs 2(Nlog2 N - 3N + 4) real multiplications and (3N log,N + 2N) real additions. The computation of the N-IDFTs of two complex sequences (which are the DFTs of two real sequences) in one pass through a single complex N-IFFT needs 2(N log,N - 2N + 4) real multiplications and (3N log, N + 2N) real additions. The decomposition

254 R. Prost et aI. / Signal Processing: Image Communication 7 (1995) 249-257

Fig. 3.

a mag

to the

R. Prost et al. / Signal Processing: Image Communication 7 (I995) 249-257 255

Illustration of the decomposition, into 64 subbands, of Lena-512 x 512: (a) using filters of solution 1 (a2 = I), the filters ha

nitude equal to one at a quarter of the sampling frequency;(b) using filters of solution 2 (a2 = l/2), The analysis filters are identic

reconstruction filters.

ve :a1

256 R. Prost et al. / Signal Processing: Image Communication 7 (1995) 249-257

of M rows into B subbands requires YN-FFTs and y #-IFFTs. It follows that the number of real multiplications is

P, =M (Nlog,N-3N+4)

and that of real additions is

Al=MN ;log,;+2 . 1 The multiplication (respectively addition) count

for the decomposition of the N columns into B subbands is obtained from PI (respectively A,) where we exchange M and N:

P2 = N (M log, M - 3 M + 4) [

. Az=MN

The total real multiplication and addition counts are

P = PI + P2

=2MN[log,y-5]+4(B+l)(M+N),

A=Al+Az=MN MN

310gzB+4 1 Then the number p of real multiplications per pixel is

p = P/MN

= 2 log, y-5)+4(8+1)(~+~),

and the number a of real additions per pixel is

MN a=A/MN=310gzB+4.

For a square image (M = N):

p=2(log,~-5)+8~,

NZ a = 3 log, B + 4.

For example, taking N = 1024 and B = 4 we obtain p = 26 and a = 58. The equivalent decomposition using a QMF of length 2n0 = 32 [S) requires 2n010g, B = 64 real multiplications per pixel and 64 real additions per pixel.

The hardware implementation of the decomposition/reconstruction system should integrate exist- ing FFT chips and an additional quite simple ad- dressing logical circuit that reorganizes data to do the one stage multi-band decomposition or reconstruction process.

7. Conclusions

The method proposed here achieves the exact decomposition and reconstruction using ideal band-pass filters implemented in the DFT domain without any excess data, as was the case for the method of the previous paper [2]. In addition, this method allows one stage multi-band decomposition/reconstruction at a computational cost lower than in the case of standard FIR filter schemes (QMF, CQF, . ..).

Acknowledgements

The authors would like to thank the reviewers for their helpful comments.

Appendix A. Decimation in the DFT domain

A.I. 2 : I decimation

The 2 : 1 decimation of x0 can be modelled by the two-step process:

(i) (xL)~ = 3 Cxll + ( - Vxfl, n=o, . . ..N- 1. (W

R. Prod et al. / Signal Processing: Image Communication 7 (1995) 249-257 257

Taking the N-DFT of (A.l) we obtain

QW:,) = 3 r_GYxO) + JT+N,ZbOn, k=O, . . ..N-1.

(ii) The 2: 1 decimated signal is (n = 0, . . . , N/2 - 1). It is obtained (n = 0, . . . ,N-l)by

X” -‘=(x;:&~, n=O ,..., N/2-1.

64.2) noted xi 1

from C-G 1 1”

(A.3)

In Fig. 1 the block 2 : 1 gives xi ’ at its output when x,” is applied at its input. The relationship between their DFTs can be easily obtained:

F;“(x-‘) = F;(x&), k = 0, . . . , N/2 - 1. (A.4)

From (A-2) we have

F;‘2(x- ‘) = $[Fk”(xO) + F;+N,2(X0)],

k = 0, . . . ,N/2 - 1. (A.5)

(AS) is used to evaluate the DFT of the decimated signal x0 by a factor 2: 1.

A.2. B: 1 decimation (B = 2J)

The decimation of x- 1 by a factor 2 : 1 produces x- *; the decimated signal x0 by a factor 4 : 1. From (AS) the $-DFT of x-~ is

F,“‘“(x_‘) = f [F;‘qx-l) + F;&,4(x-‘)],

k = 0, . . . ,N/4 - 1. (‘4.6)

By incorporating (AS) into (A.6), we obtain

F,“‘“(x-*) = $[F:(xO) + F,N,N,Z(XO) + F:+N,4(x")

+ F;+3N,4(~0)], k = 0, . . . , N/4 - 1.

(A.7)

By repeating this process J times, we obtain x-‘; the signal x0 decimated by a factor B: 1.

FfIB(x - J, = $ “z;-l FkN,[,v,&‘). I-O

References

64.8)

Cl1

PI

r31

II41

I?1

PI

c71

PI

c91

R.H. Bamberger, S.L. Eddins and V. Nuri, “Generalized symmetric extension for size-limited multirate filter banks”, IEEE Trans. Image Process., Vol. 3, No. 1, January 1994, pp. 82-86. C. Diab, R. Prost and R. Goutte, “Error-free image decomposition/reconstruction for subband coding schemes”, Signal Processing: Image Communication, Vol. 2, No. 1, May 1 1990, pp. 53-68. C. Diab, R. Prost and R. Goutte, “Exact image subband decomposition/reconstruction by DCT”, Signal Processing: Image Communication, Vol. 4, No. 6, November 1992, pp. 489-496. C. Diab, R. Prost and R. Goutte, “Subsampling versus folding in the DCT domain”, Submitted, June 1994. D. Esteban and C.R. Galand, “Application of quadrature mirror filters to split band voice coding schemes”, IEEE Proc. Internat. Conf: Acoust. Speech Signal Process., Hartford, CT, May 1977, pp. 191-195. G. Karlson and M. Vetterli, “Extension of finite length signals for sub-band coding”, Signal Processing, Vol. 17, No. 2, June 1989, pp. 161-168. H. Kiya, K. Nishikawa and M. Iwahashi, “A development of symmetric extension method for subband images coding”, IEEE Trans. Image Process., Vol. 3, No. 1, January 1994, pp. 78-81. M.J.T. Smith and S.L. Eddins, “Subband coding of images with octave band tree structures”, IEEE Proc. Internat. Co@ Acoust. Speech Signal Process. (ICASSP), Dallas, Texas, April 1989, pp. 1382-1385. J.W. Woods and S.D. O’Neil, “Subband coding of images”, IEEE Trans. Acoust. Speech Signal Process., Vol. ASSP-34, No. 5, 1986, pp. 1278-1288.

exact subband image decomposition and reconstruction in discrete space and discrete frequency...

Documents