Design of Multi-Dimensional Filter Banks

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1 Design of Multi-Dimensional Filter Banks Mikhail K. Tchobanou Moscow Power Engineering Institute (Technical University) Department of Electrical Physics 13 Krasnokazarmennaya st., Moscow, RUSSIA multi-dimensional filter banks, polynomial ap- Keywords: proach. Abstract The problem of the design of multi-dimensional filter banks with prescribed properties is considered. The most important requirements are perfect reconstruction property, linear phase property, optimization for a given class of signals, maximum number of vanishing moments and others. A survey of different approaches to the problem of filter banks design is presented. The most promising ones are polynomial approaches that allow to find the result directly in analytical form. 1 Introduction In recent years, increasingly more attention is being paid to multi-dimensional (M-D) digital filters. This is motivated by the growing demand for processing and compression of still two-dimensional (2-D) images and video (3-D) signals in telecommunications and multimedia technology. Another areas of application of M-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 solution of partial differential equations (PDE), which describe a wide variety of physical systems. In image enhancement and restoration applications, M- D filters (including adaptive filters) are also preferred. They can exploit the statistical correlation between all of the neighboring pixels in the images and track any spatial variations in statistics of the image signal. The application of filter banks (FB) to data compression, which is known as subband coding and transmultiplexing, has been studied as an effective coding scheme in audio and visual communications [1]. The analysis/synthesis multirate FBs find applications in another DSP systems, like spectrum analyzers, signal scramblers. Recently, M-D multirate signal processing has increasingly been used in sampling format conversion and in many other applications of digital video processing. Multirate FBs are usually composed of both analysis and synthesis banks. The analysis bank decomposes a signal into different frequency subbands, and the synthesis bank reconstructs the original signal from the subband signals. If the reconstructed signal is identical to the original one, except for delay and scaling, then the analysis/synthesis system is said to be a perfect reconstruction (PR) FB. Linear phase (LP) is a desired quality in many applications of multirate FB and wavelets for several reasons: applying filters with LP results in minimum phase distortion. This is important when the intraresolution level relation is to be exploited in multiresolution applications, as is done for instance in zero tree coding, where the correlation between adjacent resolution levels is used to improve compression; it is commonly accepted that the human visual system is more tolerant to symmetric distortions than it is to asymmetric ones; using symmetric filters allows one to use a symmetric periodical extension of finite length signals, which minimizes the discontinuity at the boundaries without enlarging the support size of the signal. Hence LP and PR properties of FBs are quite significant for the subband coding of images. The traditional approach in FBs design is to use a fixed transform for all signals to be processed. Although this may suffice for fixed classes of signals, being well suited in some sense to that fixed transform, it is a restriction dealing with arbitrary classes of signals with unknown or time-varying characteristics. This has motivated an alternative aprroach

2 that is adaptive in its representation and more robust in dealing with a large class of signals with unknown characteristics. This leads to partly signal optimized and partly adaptive FBs. Finite impulse response (FIR) analysis and synthesis filters are usually employed in the design of PR FB, since stability for such system is always guaranteed. 2 Fundamental Relations We consider a M-D subband coding scheme. Denote the decimation matrix V R k k. Then det V is the ratio of sampling densities of the original signal and the subsampled signal. For critically sampled signals the number of channels is equal to det V. The matrix V generates the lattice Γ = { λ i u i λ i Z}, where u i are columns k of i=1 V. One will denote z q = z q zq k k, with z = (z 1,..., z k ), q = (q 1,..., q k ). One can associate with V the unit cell UC(V ) = k { λ i u i λ i [0,1)}. The analysis filters E i (z) and the i=1 synthesis F i (z) filters can be decomposed as E i (z) = m 1 j=1 E ij (z V )z cj, F i (z) = m 1 j=1 F ij (z V )z cj where c j UC(V ), m = det V. In terms of polyphase matrices [2] E 0,0... E 0,(m 1) E = , E (m 1),0... E (m 1),(m 1) F = F 0,(m 1)... F (m 1),(m 1) F 0,0... F (m 1),0 PR and LP properties of FBs can be written as: PR property. If, E(z)F(z) = I z N, (1) where N Z k +, then the reconstructed signal y(n) is identical to the original signal x(n), except for a delay. LP property. If each FB, defined over the lattice Γ has symmetric (or anti-symmetric) unit sample response h( x c) = Sh(x 1 2c), x Γ, S {±1}, c Γ, then E(z) = εe(z 1 )z M J, (2) where ε - diagonal matrix of ±1, J - skew identity matrix, M Z k +. 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 have been widely used mainly for subband coding of speech, still images and video [4, 2, 5]. The underlying theory progressed from cancellation of aliasing to building systems achieving exact reconstruction of the signal in one or multiple dimensions, from the two-channel orthogonal banks to general multichannel systems. For implementation reasons all of these efforts were concentrated on filters with rational transfer functions. Multi-Dimensional Filter Banks and Wavelet Bases. Most of the developments were focused on 1-D signals, and the M-D case was handled via the tensor product. Some of the more recent efforts are concentrated on the true M-D case. Here it means that both nonseparable sampling and filtering are allowed. Although the true M-D approach suffers from higher computational complexity, it offers some important advantages: a) using nonseparable filters leads to more degrees of freedom in design, and consequently better filters; b) nonseparable sampling opens a possibility to have schemes better adapted to the human visual system. While in one dimension sampling by N can be performed in only one way, in two or more dimensions this is not so. M-D sampling is represented by the lattice Γ which can be separable or nonseparable. In [6] it was given the analysis of possible combinations separable/nonseparable sampling/filters/polyphase components. Nonseparable operations, which include separable ones as a special case, offer more flexibility and better performance. Moreover, they are required in some applications [7], for example in the conversion between progressive and interlaced video signals [8]. The use of subband coding for HDTV was first proposed and strongly advocated in [9]. Independent of this approach, the theory of wavelets was developed by a group of applied mathematicians (Daubechies, Mallat, Meyer et al). Later it became clear that FBs and wavelets are closely related. Discrete wavelet transforms are a subset of digital multirate FBs. Signal Representations. Signal optimization of M-D FBs requires a model for the class of signals to be processed. Natural images are believed to be realizations of non-gaussian processes and there is no stochastic model yet that incorporates the diversity of complex scenes with edges and textures. Large class of images have bounded total variation f T V = f (x) dx < +. More restricted classes of images, such as homogeneous textures, are better represented by Markov random fields over sparse representations. A linear approximation of f from M inner products f, g m is an orthogonal projection on a space V M generated from the first M vectors of B = {g m } m N.

3 The maximum approximation error over a signal set or class S is M 1 ɛ(s, M) = sup f f, g m g m 2 f S = sup + f S m=m m=0 f, g m 2. Using the concept of M-width introduced by Kolmogorov [10], one can prove that for a ball S BV of bounded variation functions, the most rapid decay in a basis B is ɛ(s BV, M) M 1, see [11]. To improve this result, a more adaptive representation was constructed by projecting f over M basis vectors selected depending upon f: f M = m I M f, g m g m. A nonlinear wavelet approximation keeps the M wavelet coefficients of largest amplitude. The impact of the wavelet bases comes from the fact that they are unconditional bases of a large family of smoothness spaces (Besov spaces) and are optimal for nonlinear approximations in balls of these spaces. It was shown, that when M increases, the asymptotic decay of ɛ(s BV, M) = O(M 2 ) is faster than any linear approximation using M parameters, which decays at most like M Applied Design Methods Structural Methods. The design is based on optimizing the 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 due to the lack of a factorization theorem. State-Space Representation. Another major design technique for M-D FBs is applying of the state-space representations [14, 15]. Unfortunately, the set of transfer matrices, which can be factored in cascade form, represents only a subset inside the complete class of such matrices. Optimization Methods. The design is based on optimizing the filters coefficients (with respect to some performance index) under the constraints of orthogonality (or other properties) [16, 17]. An important issue remains the problem of finding an initial point for optimization procedure, which satisfies given constraints. The arbitrary choice of initial points may not result in the effective results. This solution is an approximation of the exact result and has several disadvantages. Transformation of Variables. There are various techniques proposed by different researchers ([6, 18]) that are related to the McClellan transformation. Transformation-based designs have the disadvantage that the shape of the frequency response of the filters is determined uniquely from the subsampling matrix. Polynomial Approach 1. Bernstein Polynomials Taking advantage of the multivariate polynomials (like Bernstein polynomials or other types) allows to obtain filters with maximum number of vanishing moments and with other good properties [19]. The theory and techniques are derived for the design of FBs in which the filter impulse responses, the scaling functions and wavelets have rectangular support. The filters designed by using the Bernstein polynomial have LP property and an arbitrarily number of vanishing moments [19]. Polynomial Approach 2. Lifting Technique Recently the lifting technique emerged providing a new angle for studying FBs and wavelets constructions [20]. The basic idea behind lifting is a simple relationship between all FBs that share the same lowpass (or the same highpass) filter. Lifting also leads to a ladder-type FB implementation. A general method, based on lifting, for building FBs and wavelets in M-D case, for any lattice and any number of vanishing moments is presented in [20]. Polynomial Approach 3. Matrix Completion Methods. The problem which is solved by applying of matrix completion method is to construct a M-D FB, given a subset, or one of the filters of the FB [21, 22]. It was shown, that for FBs with three or more channels the set of all solutions can be characterized. Any rectangular unimodular matrix can always be made a submatrix of a square unimodular matrix, allowing analysis of the non-critically subsampled FB [22, 21]. Signal optimized filter banks. One of the widely used design methods of signal-adapted filters uses energy compaction as the adaptation criterion [23, 24]. The use of energy compaction filters is motivated by applications in signal analysis, denoising, compression and progressive transmission. The goal is to design the FB so that good L 2 - approximations to the signal of interest can be computed from a reduced number of channels. The idea is to design the components of the FB so that the energy in the first channel is maximized. Then, use the remaining degrees of freedom to maximize the energy in the second channel, and repeat this operation succesively for all remaining channels. It was shown, that the theoretic coding gain (TCG) of a uniform PU analysis/synthesis system is maximized if and only if the FB is principal component FB [24]. Principal component FBs depend on the averaged autocorrelation matrix of the signal. For nonstationary signals, as almost all images and video signals are, the filters with simple analytic forms cannot be found. The main drawback is that such kind of FBs is highly signal dependent. Another design method finds best FBs in an operational rate-distortion (R/D) sense [25, 26, 27]. This treatment is based on best-basis design in which FBs, subband tree structure and quantizers are chosen to optimize R/D performance. It should be mentioned, that the methods developed are not directly applicable to the design of signal-adapted M- D FIR FBs due to the lack of a spectral factorization theorem for M-D polynomials (for example, in [28, 29] the result was obtained for 1-D case and was based on factorization theorem). Numerical solutions were found for some constrained designs [24] for the case of separable and nonseparable lattices (and separable FBs [30]) and the solution for unconstrained-length filters was a trivial extension of known

4 1-D results. Optimization Criteria. In lossy image coding small reconstruction errors introduced by FBs do not necessary lead to significant coding quality reduction. Thus the PR property (and possibly other properties and requirements) may be relaxed in designing FBs and the overall coding performance may be improved. Such design problem can be formulated as an optimization problem whose objectives may include several performance criteria of the overall image coder, such as: coding gain; minimizing first-order entropy; minimizing the mean-squared error between the original and reconstructed signals; stopband and passband energies (or ripples, or flatness, or transition bandwidth) of the low-pass filter; Figure 1: Low-pass filter, N=5. 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 = ( ). The PR condition can be written as E 0 (z 1, z 2 )E 1 ( z 1, z 2 ) E 1 (z 1, z 2 )E 0 ( z 1, z 2 ) = = z 2k1 1 1 z 2k2 2 where k 1 and k 2 are arbitrary. The Bernstein polynomials may be applied in order to obtain the necessary FBs. The Bernstein operator, acting on a function f(x, y) with the [0, 1] [0, 1] is defined for 2-D case as where B N f(x, y) = b N i,j(x, y) = ( N i N N f i=0 j=0 ( i N, j ) b N N i,j(x, y), )( ) N x i (1 x) N i y j (1 y) N j. j In this case it was found the analysis filter [19] E 0 (z 1, z 2 ) = 1 2 4N N N i i=0 j=0 g i,j ( N i )( ) N ( 1) i+j j (1 z 1 1 )2i (1 + z 1 1 )2(N i) (1 z 1 2 )2j (1 + z 1 2 )2(N j), Figure 2: High-pass filter, N=5. As usual, only PR property is insufficient. It is desirable by iteration of a M-channel FB to obtain limit functions - scaling function and M 1 wavelets. The scaling and wavelet functions satisfy the equations: φ(x) = det(v ) k ψ i (x) = det(v ) k i = 1... M 1 e 0 (k)φ(v x k), e 1 (k)φ(v x k), 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 differentiable. The properties of biorthogonal FBs (when E 0 is half-band) depend on the number of vanishing moments for a given support size. The examples of low-pass and high-pass analysis filters for different values of N are given in Fig.1-4. The lowpass analysis filter E 0 will be a half-band filter and will have a zero of order 2N + 1 at ω 1 = ω 2 = π and all derivatives up to 2N + 1-st at the origin ω 1 = ω 2 = 0 will be also zero. with g i,j chosen according to the given FB s properties.

5 Figure 4: High-pass filter, N=10. 5 Conclusion Figure 3: Low-pass filter, N=10. The analysis of the design methods shows that the polynomial approach is the most promising one in the design of M-D FBs with prescribed properties [31]. These properties are PR, LP and the optimality with respect to a class of signals. Matrix completion method can be applied for this purpose. In the case of LP PR FBs, since the LP property dictates certain symmetries [32], the construction of the PR FB may be obtained via an application of Suslin s theorem [33]. The remaining degrees of freedom can be used for signal optimization of FBs characteristics. To answer this question Gröbner basis techniques can be used and results from the theory of rings, modules and algebraic K-theory can be applied. Bernstein polynomials and lifting technique can be also applied. At present this approach has led only to two- and four-channel biorthogonal FBs and has been applied only for the case of two and three dimensions. The techniques which were already derived should be extended and generalized for the case of arbitrary number of channels and dimensions. These constructions should succeed many advantages of lifting, like custom-design, in-place computation, integerto-integer transforms, and speed, which are very important for implementation purposes. As usual, the optimization is undertaken in accordance with a prescribed parametric signal class. A mathematical theory, developed in [34], demonstrated that the use of L 1 -norm for the error measures is usually more appropriate with the properties of the human visual system. It was also shown that the smoothness of the image in certain functional spaces (Besov spaces) has a direct impact on the quality of the compressed image for a given bit rate. Markov random fields provide a general framework to construct processes with sparse interactions over appropriate representations. Knowledge of how to optimize these components, and analyzing the properties of such Markov random fields over functional spaces are also problems of future investigations. Acknowledgments This work was partly done during the stay as a visiting researcher in NTNU, Trondheim, Norway. References [1] T.A. Ramstad, S.O. Aase, and J.H. Husøy. Subband Compression of Images Principles and Examples. ELSEVIER Science Publishers BV, North Holland, [2] P. P. Vaidyanathan. Multirate Systems and Filter Banks. Prentice Hall, Englewood Cliffs, [3] A. Croisier, D. Esteban, and C. Galaud. Perfect channel splitting by use of interpolation/decimation/tree decomposition techniques. In Intern. Confer. Inform. Sci. Syst., pages , Greece, [4] A. Tekalp. Digital video processing. Prentice Hall, [5] J. W. Woods. Subband Image Coding. Kluwer Academic Publishers, 3300 AH Dortrecht, The Netherlands, [6] J. Kovačević and M. Vetterli. Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for R n. IEEE Trans. Inform. Th., special issue on Wavelet Transforms and Multiresolution Signal Analysis, 38(2): , March [7] T. Chen and P. P. Vaidyanathan. Recent developments in multidimensional multirate systems. IEEE Trans. Circ., Syst. for Video Technol., 3(2): , April [8] E. Dubois. The sampling and reconstruction of timevarying imagery with application to video systems. Proc. IEEE, 73(4): , April [9] W. Schreiber. Channel-compatible 6-MHz HDTV distribution systems. SMPTE J., pages 5 13, January 1989.

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