FIR/IIR Exact Reconstruction Filter Banks with Applications to Subband Coding of Images

Chai W. Kim and Rashid Ansaxit University of Pennsylvania, Philadelphia, PA 19104

'Bellcore, Morristown, NJ 07960

Abstract A class of two-channel analysis/synthesis filter banks which are useful in subband coding are described. The proce- dure is particularly useful in hierarchical coding where a low resolution embedded signal is obtained by using sepa- rable halfband filters. The filter bank can be composed of finiteduration impulse response or infinite-duration impulse response filters. The basic structure is simple and one can use well-known classes of filters as prototypes in a flexible manner to obtain the exact reconstruction solutions. Even if the filter coefficients are quantized for a low bit represen- tation, the exact reconstruction property is preserved.

1. Introduction Multiresolution representation of signals is being increas- ingly used in applications such as computer vision, signal analysis, image and video coding. In image and video cod- ing, it provides a signal partition that is useful for embed- ded or hierarchical coders which allow interworking between coders operating with signals of different resolution and quality. The basic idea in multiresolution representation is to decompose the signal into a hierarchy of sub-signals of different resolutions. Each sub-signal carries information of different perceptual importance and accordingly wn be coded individually according to the sub-signal character- istics. Several techniques have been used for creating an image resolution hierarchy, and among the most popular ones are pyramidal (11 and subband [2] decomposition. A pyramidal decomposition consists of the generation of a low resolution approximation of the input signal and a residual high frequency signal. The low resolution signal is obtained by low-pass filtering and then downsampling the original signal by a factor of 2x2 and the low frequency signal is interpolated, typically after coding and local decoding in an encoder. The interpolated signal is subtracted from the original to yield the high frequency signal which is then sep- arately coded. The lowpass filter used in decimation can be flexibly chosen. However in practice often a class of one- dimensional (1-D) filters called half-band filters [3] are used in a separable manner to filter an image vertically and hor- izontally to create the low resolution signal. The aliasing in the low resolution signal is considered acceptable in this case. In subband coding [4, 5, 61, quadrature mirror fil- ters or exact reconstruction filters are used. In this case the aliasing in the low resolution signal is larger than in the case of half-band filters, which is a disadvantage. How- ever one advantage in subband coding is that the number of samples in the input signal and the suitably decimated sub- band signals are chosen to be equal in practice whereas after one stage of pyramidal decomposition of images we end up with a sampling redundancy of 25%. A question that we

91-645128/92$03.00 0 1992 EEE

might ask is whether it is possible to design subband ex- act reconstruction filter banks for this application in which the lowpass analysis filter is composed of halfband filters and the remaining filters have frequency responses that are "good" approximations to the desired responses. Moreover it is desirable that such a structure provide the flexibility such that by a proper choice of the filter parameters one can improve the response as desired. This paper proposes such a structure. A solution in which the lowpass filter is a half- band filter was recently proposed [7]. But in this case the solution is restrictive. The corresponding highpass filter has a non-linear phase response, and it has a wide bandwidth causing a large aliasing after downsampling.

We begin by examining the conditions of exact recon- struction in a 1-D two-channel analysis/synthesis filter bank pair. Since we are considering separable filters here, we can directly extend this to the case of two-dimensional (2-D) filter banks. We then impose, the requirement that the low- pass filter be a halfband filter. This filter could have either a finiteduration impulse response (FIR) or infinite-duration impulse response (IIR). A class of solutions of the remain- ing filters is then derived. One can employ linear-phase FIR filters and approximately linear phase IIR filters with a right-sided impulse. response. A procedure for design will be described. The basic structure is simple and one can use well-known classes of filters as prototypes in the class of so- lutions. Filters with simple coefficients can be used. A very attractive feature of the structure is such that even with a quantization of coefficients the exact recostruction condition is satisfied. The problem of two-band exact recostruction is described in section 2 and the solutions are examined in sec- tion 3. Some examples of exact reconstruction solutions are presented in section 4 followed by a discussion of the results in section 5 .

2. 2-band Exact Reconstruction A two channel analysis/synthesis filter is shown as in Figure 1. The analysis section decomposes a given input signal z(n) into a set of subband signals yo(n) and y l ( n ) by processing the input signal with a lowpass filter &(z ) and a highpass filter H l ( r ) . These filtered signals are then decimated by a factor of 2. Let q(n) for i = 0 , l be the downsampled versions of yi(n). The Ztransformof q(n) can be expressed as

1 '

2 m=O

where W = eJ" = -1. At the synthesis side, the signals xi(.) are interpolated by a factor of 2. The upsampled versions w;(n) of z;(n) (i.e. V;.(z)= X;(z')) are then filtered with G i ( z ) where Go(%) and G l ( z ) are the lowpass filter and the highpass filter, respectively. The output signals of these filters are added together to reconstruct the original signal.

Xi(.) = - Hi(W"z'I2)X(Wmz'I2), (1)

221

The reconstructed signal 2(n) is then the sum of the output signals of Gi(z) with its 2-transform shown as

1 ' 1 X ( Z ) = 5 X(Wmz) C Hi(W"z)Gi(Z), (2)

m=O i=O

where &Hi(-z)Gi(z) is known as the aliasing term and Hi(z)Gi(z) as the system response. For cancelling the

aliasing, there are a number of ways of choosing the relations between the filters such as

(3)

GI( 2) = (-1)'-'azkHo( - z ) , (4)

1 H ~ ( z ) = (-l)'-k-~-kGo(-~), a

where I , k E 2 and a E R. Here, 2 and 72 denote the set of integers and real numbers, respectively. For exact reconstruction we consider the requirement that

IfHi(z)Gi(z) = 22-"O, (5) i=O

1

CHi(-z)Gi(z) = 0, (6) i=O

where n,, E 2.

3. Procedure for Solutions With the choices of Go(z) and Gl(z) as shown in (3) and (4), the analysis/synthesis filter bank is guaranteed to be free of aliasing (i.e., (6) is always true). In order to obtain exact reconstruction, we must have a proper relationship between Ho(z) and GO(%) so that (5) is satisfied. It can be shown that for a given Ho(z) there is no unique solutions which satisfy (5) . Here, we will focus on I = 0, k = 0, a = 2 and no = 1. With these values of I, k, CI and no, we can add (5) and (6) to get

Go(%) [Ho(z) + Ho(-z)] = 2z-l - Gl(z) [HI(z) + HI(-%)]. (7)

Substituting from (3) and (4) into (7), we have

Go(z) (Ho(z) + Ho(-z)] = 22-' [l + zH0(-z)B(z2)] , (8) where B(z2) = Hl(z) +Hl(-z). Let us now consider a case where the lowpass analysis filter Ho(z) is a halfband filter. The transfer function of a halfband filter can be expressed

Ho(z) = 5 [1+ zA(z2)] . (9)

(10)

1 as

Substituting (9) into (8), we have

GO(t) = 2z-' + [l - zA(z')] B(z').

Thus, the exact reconstruction conditions in (5) and (6) are always satisfied when the subfilters are given as in (3), (4), (9) and (10) regardless of the parameter values of A(z2) and B(z2) .

Let us now examine proper choices of A ( z Z ) and B(zz) . It is clear from (9) that the frequency response of A(%) should approximate

A(&") M e-'"/', 0 5 w 5 2wm (11)

for a good lowpass response of Ho(z) with an approximately zero phase in w E [0, wm] where w, < 7r/2. Thus, from (10)

and ( l l ) , we notice that the frequency response of B(z) should also approximate

B(ej") M 0 5 w 5 2wpl (12) for a good lowpass response of Go(z) where wp, < 1712. The proper choices of A(%) and B(z) are the transfer func- tions whose impulse responses are non-causal. When the causality is required for Ho(z) , Hl(z), Go(z) and G,(z), we constrain A ( z ) and B ( z ) to be

A(z ) = zNaAc( t ) , (13) B ( z ) = zNbBc(z), (14)

where A,(z) and B,(z) are the causal and stable transfer functions (i.e., poles are within the unit circle for IIR cases) so that the causality can be achieved by inserting proper amounts of delays. When A ( z ) and B(z) are FIR filters, they can always be expressed as the forms shown in (13) and (14). The commutative structures of a (causal) exact reconstruction system are shown in Figure 2.

The solutions of linear phase FIR filters which approxi- mate (11) and (12) can be more explicitly expressed as

A(z ) = + zi) , Nb

i=O B ( z ) = bi (z-~-' + zi) . (16)

For IIR filters with right-sided impulse responses and ratio- nal transfer functions, linear phase solutions do not exist. However, approximately linear solutions can be obtained numerically. IIR filters which consist of allpass sections [a] was shown to be suitable for approximating the linear phase response, and they are in the forms of (13) and (14) where A,(z) and B,(z) are causal allpass sections with their respec- tive orders of N,, + 1 and Na + 1. Instead of approximating A(z ) and B ( z ) individually, We can achieve better lowpass and highpass characteristics by minizing the suitablely cho- sen objective function.

Even if H&) is not confined to a halfband filter, we can obtain non unique solutions of Go(%) which satisfy the exact reconstruction conditions. For a given Ho(z), choose Go(z) to be

Then from (3), (4) and (17), the equation (7) can be written Go(z) = Fo(z) + F1(t2)Ho(-z) . (17)

as FO(Z)HO(Z) - F o ( - ~ ) H o ( - z ) = 22-l. (18)

Thus, it can easily be seen that exact reconstruction can be obtained regardless of Fl(z') if Fo(z) and Ho(z) satisfy

Fo(z)Ho(z) = 2-' + f(z2), (19) where f(z') is a function of z'.

4. Examples Here we will consider some solutions by choosing the filters A ( z Z ) and B(zZ) derived from well known filters. First we consider linear phase FIR filters where A(%') and B(z2) are obtained from Lagrange halfband filters. The coefficients of the filters are tabulated in Table 1. Magnitude reponses obtained from these filters are shown in Figure 3 - 5, where L3, L7 and L11 denote 3rd, 7'h and 1lth order Lagrange filters, respectively.

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Next we consider equiripple linear phase FIR filters ob- tained the Parks-McCldan program. Here, we set A ( z ) = B(z). The coefficients of the filter are given in Table 2. The magnitude responses are shown in Figure 6.

Now we consider IIR solutions where the filters are ob- taied from approximately linear phase allpass IIR filters [8]. We also set A(z ) = B(z) for this example. The pole loca- tions are given in Table 3, and the magnitude responses are shown in Figure 7.

Finally we consider Hybrid IIR/FIR solutions where A(z) is an IIR filter and B(z ) is an FIR filter. Using A(z ) in Table 3 and B ( z ) in Table 2, we plotted the responses in Figure 8. For all these examples, the frequency responses can be further optimized by minimizing a suitably chosen objective function.

1

1 2 3 4 5

7

9 10

6

8

5. Results and Discussion

ai 0.316133803+00 -0.997250233-01 0.535115083-01

-0.322048783-01 0.197815823-01

0.677359853-02

0.163070763-02 -0.638495263-03

-0.118826123-01

-0.35497919E02

In this paper we have proposed an exact reconstruction filter bank in which the lowpass analysis filter is a halfband filter. We derive conditions for choosing the remaining filters such that we get an exact reconstruction filter bank. The struc- ture is such that we can quantize the coefficients for low bit representation and still maintain the exact reconstruc- tion condition. The filters A ( 2 ) and B ( 2 ) can be chosen according to some optimization criteria and then coarsely quantized for ease of implementation in applications such as image coding.

The technique is applied to the Lena Image and a sin- gle frame of the HDTV Kiel Harbour sequence. The low- band subsignal is coded using the discrete cosine transform. The three highband subsignals are quantized in the spatial domain then coded using Huffmann coding together with runlength coding of zeros. The results will be reported else- where. It is observed that the lowband subsignal has con- siderably lower alliasing compared with subband schemes commonly employed.

References [I] P. J. Burt and E. H. Adelson, “The Laplacian Pyramid

as a Compact Image Code,” IEEE Trans. Commlrn vol. COM-31, pp, 532-540, April 1983.

[2] M. Vetterli, “Multi-Dimensional Subband Coding: Some Theory and Algorithms,” Signal Processing, vol. 6, pp, 97-112, Feb. 1984.

[3] M. G. Bellanger, J. L. Daguet and G. P. Lepagnol, “In- terpolation, Extrapolation and Reduction of Compu- tation Speed in Digital Filters,” IEEE Trans. Acoust., Speech, Signal Processing, Vol. ASSP-22, No. 4, pp. 231-235, August 1974.

[4] J. W. Woods and S. D. O’Neill, “Subband Coding of Images,” IEEE Trans. Acoust., Speech, Signal Process- ing, vol. ASSP-34, pp. 1278-1288, Oct. 1986.

[5] H. Gharavi and A. Tabatabai, “Sub-band Coding of Digital Images Using Two-dimensional Quadrature Mirror Filtering,” P m . SPIE Conf. on Visual Comm. and Image Processing, vol. 707, pp. 51-61, Nov. 1986.

[6] J. W. Woods, Ed., Subband Image Coding, Kluwer Aca- demic Publishers, Norwell, MA, 1991.

[7] J . Katto and Y. Yasuda, “A New Structure of the Per- fect Reconstructuin Filter Banks for Sub-bnad Coding” Bans. IEICE, vol. E73, No. 10, Oct. 1990, pp. 16.16- 16.24.

[SI R. Ansari and B. Liu, “Efficient Sampling Rate Al- teration Using Recursive (IIR) Digital Filters,” IEEE Trans. Acoust., Speech, Signal Processing, Vol. ASSP- 31, No. 6, pp. 1366-1373, December 1983.

Table 1. Lagrange Filter Coefficients.

Filter Order

3 7 11

4 32

512

11 150 -25

Table 2. FIR Filter Coefficients Obtained from Parks-McClellan Program.

Table 3. Pole Locations of a 5-th Order Allpass Filter A&).

Figure 1: Two-band analysis/synthesis filter banks.

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Figure 5: Magnitude response of FIR filters with lowpass filter L11 and highpass filters using B(z2) derived from L3, L7 and L11.

.l:t-- /\ 1 ,-- -_ -Na-Nbl Q, E z-Na D1 E

Figure 2: Commutative structures of a causal exact recon- struction system (a) Analysis section (b) synthesis section.

w L1 I Figure 6: Magnitude response of FIR filters in Table 2.

I O , . . . . . , . . .

Figure 3: Magnitude response of FIR filters with lowpass filter L3 and highpass filters using B(z2) derived from L3, L7 and L11.

Figure 7: Magnitude response of IIR filters in Table 3.

I 01

IO, . . . . . . . . . ,

O0 L Figure 4: Magnitude response of FIR filters with lowpass filter L7 and highpass filters using B(z*) derived from L3, L7 and L11.

Figure 8: Magnitude response of Hybrid IIR/FIR filters.

230