Design of Multi-Dimensional Filter Banks

Mikhail K. TchobanouMoscow Power Engineering Institute

(Technical University)Department of Electrical Physics

13 Krasnokazarmennaya st., 105835Moscow, RUSSIAcmk2@orc.ru

Keywords: multi-dimensional filter banks, polynomial ap-proach.

Abstract

The problem of the design of multi-dimensional filter bankswith prescribed properties is considered. The most impor-tant requirements are perfect reconstruction property, linearphase property, optimization for a given class of signals,maximum number of vanishing moments and others. A sur-vey of different approaches to the problem of filter banks’ de-sign is presented. The most promising ones are polynomialapproaches that allow to find the result directly in analyticalform.

1 Introduction

In recent years, increasingly more attention is being paid tomulti-dimensional (M-D) digital filters . This is motivatedby the growing demand forprocessingandcompressionofstill two-dimensional (2-D) images and video (3-D) signalsin telecommunicationsandmultimedia technology. Anotherareas of application ofM-D digital filters are:

• medical image processing,

• seismic signal processing,

• high-definition TV (HDTV),

• scanning rate converters,

• PAL decoders and digital video codecs ,

• exploiting of M-D digital filtering methods for the solu-tion of partial differential equations (PDE), which de-scribe a wide variety of physical systems.

In image enhancementand restoration applications, M-D filters (includingadaptivefilters) are also preferred. Theycan exploit the statistical correlation between all of the neigh-boring pixels in the images and track any spatial variations instatistics of the image signal.

The application offilter banks (FB) to data compression,which is known assubband codingandtransmultiplexing ,has been studied as an effective coding scheme in audio andvisual communications[1]. Theanalysis/synthesis multirateFBsfind applications in another DSP systems, likespectrumanalyzers, signal scramblers. Recently, M-D multirate sig-nal processing has increasingly been used insampling formatconversionand in many other applications ofdigital videoprocessing.

Multirate FBs are usually composed of both analysis andsynthesis banks. The analysis bank decomposes a signal intodifferent frequency subbands, and the synthesis bank recon-structs the original signal from the subband signals. If thereconstructed signal is identical to the original one, exceptfor delay and scaling, then the analysis/synthesis system issaid to be aperfect reconstruction (PR)FB.

Linear phase (LP) is a desired quality in many applica-tions of multirate FB andwaveletsfor several reasons:

• applying filters with LP results in minimum phase dis-tortion. This is important when the intraresolution levelrelation is to be exploited in multiresolution applica-tions, as is done for instance in zero tree coding, wherethe correlation between adjacent resolution levels isused to improve compression;

• it is commonly accepted that the human visual systemis more tolerant to symmetric distortions than it is toasymmetric ones;

• using symmetric filters allows one to use a symmetricperiodical extension of finite length signals, which min-imizes the discontinuity at the boundaries without en-larging the support size of the signal.

HenceLP andPR properties of FBs are quite significantfor the subband coding of images.

The traditional approach in FBs design is to use a fixedtransform for all signals to be processed. Although this maysuffice for fixed classes of signals, being well suited in somesense to that fixed transform, it is a restriction dealing witharbitrary classes of signals with unknown or time-varyingcharacteristics. This has motivated an alternative aprroach

that is adaptive in its representation and more robust in deal-ing with a large class of signals with unknown characteris-tics. This leads to partlysignal optimizedand partlyadap-tive FBs.

Finite impulse response (FIR) analysis and synthesis filtersare usually employed in the design of PR FB, since stabilityfor such system is always guaranteed.

2 Fundamental Relations

We consider a M-D subband coding scheme. Denote thedecimation matrixV ∈ Rk×k. Thendet V is the ratio ofsampling densities of the original signal and the subsampledsignal. For critically sampled signals the number of chan-nels is equal to|det V |. The matrixV generates the lat-

tice Γ = {k

∑

i=1

λiui | λi ∈ Z} , whereui are columns of

V . One will denotezq = zq11 ...zqkk , with z = (z1, ..., zk),

q = (q1, ..., qk).One can associate withV the unit cell UC(V ) =

{k

∑

i=1

λiui | λi ∈ [0,1)}. The analysis filtersEi(z) and the

synthesisFi(z) filters can be decomposed as

Ei(z) =m−1∑

j=1

Eij(zV )z−cj , Fi(z) =m−1∑

j=1

Fij(zV )zcj

wherecj ∈ UC(V ), m = |det V |.In terms of polyphase matrices[2]

E =

E0,0 . . . E0,(m−1). . . . . . . . .

E(m−1),0 . . . E(m−1),(m−1)

,

F =

F0,(m−1) . . . F(m−1),(m−1). . . . . . . . .F0,0 . . . F(m−1),0

,

PR andLP properties of FBs can be written as:

• PR property. If

E(z)F(z) = I · zN, (1)

whereN ∈ Zk+ , then the reconstructed signaly(n) isidentical to the original signalx(n), except for a delay.

• LP property. If each FB, defined over the latticeΓhas symmetric (or anti-symmetric) unit sample responseh(−x + 12c) = Sh(x−

12c), ∀x ∈ Γ, S ∈ {±1}, c ∈

Γ, then

E(z) = εE(z−1)zMJ, (2)

whereε - diagonal matrix of±1, J - skew identity ma-trix, M ∈ Zk+.

3 Principal results

3.1 Main Theoretical Approaches

Since the introduction of digital multirate filter banks (FBs)for the compression of speech signals in 1976[3], they havebeen widely used mainly for subband coding of speech, stillimages and video[4, 2, 5]. The underlying theory pro-gressed from cancellation of aliasing to building systemsachieving exact reconstruction of the signal in one or mul-tiple dimensions, from the two-channel orthogonal banks togeneral multichannel systems. For implementation reasonsall of these efforts were concentrated on filters with rationaltransfer functions.

Multi-Dimensional Filter Banks and Wavelet Bases.Most of the developments were focused on 1-D signals, andthe M-D case was handled via the tensor product. Some ofthe more recent efforts are concentrated on the ”true” M-Dcase. Here it means that both nonseparable sampling and fil-tering are allowed.

Although the true M-D approach suffers from higher com-putational complexity, it offers some important advantages:a) using nonseparable filters leads to more degrees of free-dom in design, and consequently better filters; b) nonsep-arable sampling opens a possibility to have schemes betteradapted to the human visual system.

While in one dimension sampling byN can be performedin only one way, in two or more dimensions this is not so.M-D sampling is represented by the latticeΓ which can beseparable or nonseparable. In[6] it was given the anal-ysis of possible combinations separable/nonseparable sam-pling/filters/polyphase components.

Nonseparable operations, which include separable ones asa special case, offer more flexibility and better performance.Moreover, they are required in some applications[7], for ex-ample in the conversion between progressive and interlacedvideo signals[8]. The use of subband coding for HDTV wasfirst proposed and strongly advocated in[9].

Independent of this approach, the theory of wavelets wasdeveloped by a group of applied mathematicians (Daube-chies, Mallat, Meyer et al). Later it became clear that FBsand wavelets are closely related. Discrete wavelet transformsare a subset of digital multirate FBs.

Signal Representations.Signal optimization of M-D FBsrequires a model for the class of signals to be processed. Nat-ural images are believed to be realizations of non-Gaussianprocesses and there is no stochastic model yet that incorpo-rates the diversity of complex scenes with edges and textures.Large class of images havebounded total variation

‖f‖TV =∫

|f ′(x)|dx < +∞.

More restricted classes of images, such as homogeneous tex-tures, are better represented by Markov random fields oversparse representations.

A linear approximation off from M inner products〈f, gm〉 is an orthogonal projection on a spaceVM gener-ated from thefirst M vectors ofB = {gm}m∈N.

The maximum approximation error over a signal set orclassS is

(S,M) = supf∈S

‖f −M−1∑

m=0

〈f, gm〉 gm‖2

= supf∈S

+∞∑

m=M

| 〈f, gm〉 |2.

Using the concept of M-width introduced by Kolmogorov[10], one can prove that for a ballSBV of bounded variationfunctions, the most rapid decay in a basisB is (SBV,M) ∼M−1, see [11].

To improve this result, a more adaptive representation wasconstructed by projectingf over M basis vectors selecteddepending uponf : fM =

∑

m∈IM 〈f, gm〉 gm. A nonlinearwavelet approximation keeps theM wavelet coefficients oflargest amplitude. The impact of the wavelet bases comesfrom the fact that they are unconditional bases of a largefamily of smoothness spaces (Besov spaces) and are opti-mal for nonlinear approximations in balls of these spaces. Itwas shown, that whenM increases, the asymptotic decay of(SBV, M) = O(M−2) is faster than any linear approxima-tion usingM parameters, which decays at most likeM−1.

3.2 Applied Design Methods

Structural Methods. The design is based on optimizingthe cascade filter structures, which ensure design constraints(like orthogonality, LP or regularity) structurally[12, 6, 13].No complete cascade has been available in the M-D case dueto the lack of a factorization theorem.

State-Space Representation.Another major design tech-nique for M-D FBs is applying of the state-space represen-tations[14, 15]. Unfortunately, the set of transfer matrices,which can be factored in cascade form, represents only a sub-set inside the complete class of such matrices.

Optimization Methods. The design is based on optimiz-ing the filters’ coefficients (with respect to some performanceindex) under the constraints of orthogonality (or other prop-erties)[16, 17]. An important issue remains the problem offinding an initial point for optimization procedure, which sat-isfies given constraints. The arbitrary choice of initial pointsmay not result in the effective results. This solution is anapproximation of the exact result and has several disadvan-tages.

Transformation of Variables. There are various tech-niques proposed by different researchers ([6, 18]) that are re-lated to the McClellan transformation. Transformation-baseddesigns have the disadvantage that the shape of the frequencyresponse of the filters is determined uniquely from the sub-sampling matrix.

Polynomial Approach 1. Bernstein Polynomials Tak-ing advantage of the multivariate polynomials (like Bernsteinpolynomials or other types) allows to obtain filters with max-imum number of vanishing moments and with other good

properties[19]. The theory and techniques are derived forthe design of FBs in which the filter impulse responses, thescaling functions and wavelets have rectangular support. Thefilters designed by using the Bernstein polynomial have LPproperty and an arbitrarily number of vanishing moments[19].

Polynomial Approach 2. Lifting Technique Recently thelifting technique emerged providing a new angle for study-ing FBs and wavelets constructions[20]. The basic idea be-hind lifting is a simple relationship between all FBs that sharethe same lowpass (or the same highpass) filter. Lifting alsoleads to a ladder-type FB implementation. A general method,based on lifting, for building FBs and wavelets inM-D case,for any lattice and any number of vanishing moments is pre-sented in[20].

Polynomial Approach 3. Matrix Completion Methods.The problem which is solved by applying of matrix com-pletion method is to construct a M-D FB, given a subset, orone of the filters of the FB[21, 22]. It was shown, that forFBs with three or more channels the set of all solutions canbe characterized. Any rectangular unimodular matrix can al-ways be made a submatrix of a square unimodular matrix, al-lowing analysis of the non-critically subsampled FB[22, 21].

Signal optimized filter banks. One of the widely useddesign methods of signal-adapted filters uses energy com-paction as the adaptation criterion[23, 24]. The use of en-ergy compaction filters is motivated by applications in signalanalysis, denoising, compression and progressive transmis-sion.

The goal is to design the FB so that goodL2-approximations to the signal of interest can be computedfrom a reduced number of channels. The idea is to designthe components of the FB so that the energy in the first chan-nel is maximized. Then, use the remaining degrees of free-dom to maximize the energy in the second channel, and re-peat this operation succesively for all remaining channels. Itwas shown, that the theoretic coding gain (TCG) of a uni-form PU analysis/synthesis system is maximized if and onlyif the FB is principal component FB[24]. Principal compo-nent FBs depend on the averaged autocorrelation matrix ofthe signal. For nonstationary signals, as almost all imagesand video signals are, the filters with simple analytic formscannot be found. The main drawback is that such kind of FBsis highly signal dependent.

Another design method finds best FBs in an operationalrate-distortion (R/D) sense[25, 26, 27]. This treatment isbased on best-basis design in which FBs, subband tree struc-ture and quantizers are chosen to optimize R/D performance.

It should be mentioned, that the methods developed arenot directly applicable to the design of signal-adapted M-D FIR FBs due to the lack of a spectral factorization theo-rem for M-D polynomials (for example, in[28, 29] the resultwas obtained for 1-D case and was based on factorizationtheorem). Numerical solutions were found for some con-strained designs[24] for the case of separable and nonsep-arable lattices (and separable FBs[30]) and the solution forunconstrained-length filters was a trivial extension of known

1-D results.Optimization Criteria. In lossy image coding small re-

construction errors introduced by FBs do not necessary leadto significant coding quality reduction. Thus the PR property(and possibly other properties and requirements) may be re-laxed in designing FBs and the overall coding performancemay be improved. Such design problem can be formulated asan optimization problem whose objectives may include sev-eral performance criteria of the overall image coder, such as:

• coding gain;

• minimizing first-order entropy;

• minimizing the mean-squared error between the origi-nal and reconstructed signals;

• stopband and passband energies (or ripples, or flatness,or transition bandwidth) of the low-pass filter;

• channel bandwidth and power constraint.

4 Bernstein Polynomials

It is assumed that the type of downsampling is quincuncial,which is the simplest nonseparable downsampling lattice[2].The quincunx sublattice is generated by

V =(

1 11 −1

)

.

The PR condition can be written as

E0(z1, z2)E1(−z1,−z2)− E1(z1, z2)E0(−z1,−z2) =

= z−2k1−11 z−2k22

wherek1 andk2 are arbitrary.The Bernstein polynomials may be applied in order to ob-

tain the necessary FBs. The Bernstein operator, acting on afunctionf(x, y) with the[0, 1]× [0, 1] is defined for 2-D caseas

BNf(x, y) =N

∑

i=0

N∑

j=0

f(

iN

,jN

)

bNi,j(x, y),

where

bNi,j(x, y) =(

Ni

)(

Nj

)

xi(1− x)N−iyj(1− y)N−j .

In this case it was found the analysis filter[19]

E0(z1, z2) =1

24N

N∑

i=0

N−i∑

j=0

gi,j

(

Ni

)(

Nj

)

(−1)i+j

(1− z−11 )2i(1 + z−11 )

2(N−i) · (1− z−12 )2j(1 + z−12 )

2(N−j),

with gi,j chosen according to the given FB’s properties.

Figure1: Low-pass filter, N=5.

Figure2: High-pass filter, N=5.

As usual, only PR property is insufficient. It is desirable byiteration of aM -channel FB to obtain limit functions - scal-ing function andM − 1 wavelets. The scaling and waveletfunctions satisfy the equations:

φ(x) = |det(V )|∑

k

e0(k)φ(V x− k),

ψi(x) = |det(V )|∑

k

e1(k)φ(V x− k),

i = 1 . . . M − 1

For the synthesis FB similar equations can be written.Not all PR FBs give rise to limit functions. Furthermore,

it is desirable to obtain smooth limit functions, possibly dif-ferentiable.

The properties of biorthogonal FBs (whenE0 is half-band)depend on the number of vanishing moments for a given sup-port size.The examples of low-pass and high-pass analysisfilters for different values of N are given in Fig.1-4. The low-pass analysis filterE0 will be a half-band filter and will havea zero of order2N + 1 at ω1 = ω2 = π and all derivativesup to2N + 1-st at the originω1 = ω2 = 0 will be also zero.

Figure3: Low-pass filter, N=10.

5 Conclusion

The analysis of the design methods shows that thepolyno-mial approach is the most promising one in thedesign ofM-D FBs with prescribed properties [31]. These proper-ties arePR, LP and theoptimality with respect to a class ofsignals.

Matrix completion method can be applied for this pur-pose. In the case of LP PR FBs, since the LP property dic-tates certain symmetries[32], the construction of the PR FBmay be obtained via an application of Suslin’s theorem[33].The remaining degrees of freedom can be used for signal op-timization of FBs characteristics. To answer this questionGröbner basis techniques can be used and results from thetheory of rings, modules and algebraic K-theory can be ap-plied.

Bernstein polynomialsandlifting technique can be alsoapplied. At present this approach has led only to two- andfour-channel biorthogonal FBs and has been applied onlyfor the case of two and three dimensions. The techniqueswhich were already derived should be extended and general-ized for the case of arbitrary number of channels and dimen-sions. These constructions should succeed many advantagesof lifting, like custom-design, in-place computation, integer-to-integer transforms, and speed, which are very importantfor implementation purposes.

As usual, theoptimization is undertaken in accordancewith a prescribed parametric signal class. A mathemati-cal theory, developed in[34], demonstrated that the use ofL1-norm for the error measures is usually more appropriatewith the properties of the human visual system. It was alsoshown that the smoothness of the image in certainfunctionalspaces(Besov spaces) has a direct impact on the quality ofthe compressed image for a given bit rate.Markov ran-dom fields provide a general framework to construct pro-cesses with sparse interactions over appropriate representa-tions. Knowledge of how to optimize these components, andanalyzing the properties of such Markov random fields overfunctional spaces are also problems of future investigations.

Figure4: High-pass filter, N=10.

Acknowledgments

This work was partly done during the stay as a visiting re-searcher in NTNU, Trondheim, Norway.

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