Optimal Filter Bank Research Articles

Two-dimensional FIR signal adapted filter banks: Optimality and design

The effect of different subsampling matrices with the same sampling density, on the optimality of two-dimensional filter banks is studied in this paper. It is seen that, in the case of two channel filter banks, there can be three distinct filter banks for an input power spectral density, with each one optimum for the corresponding subsampling matrix. For most of the natural images, the quincunx filter bank is optimum. In addition to this, a suboptimal solution that extends one-dimensional (1D) signal adapted filter bank design procedure, to two-dimensional (2D) nonseparable and separable signal adapted filter banks, using iterative greedy algorithm, is proposed. The nonseparable filter bank is designed for both four channel and two channel quincunx filter banks. A cascade structure is used for the factorization of 2D paraunitary polyphase matrix in the nonseparable case and a 2D separable FIR PU polyphase matrix is used in the separable case. The simulated FIR responses match the ideal responses very closely.

Optimal wavelet filter design for remote sensing image compression

A new approach for designing the Biorthogonal Wavelet Filter Bank (BWFB) for the purpose of image compression is presented in this letter. The approach is decomposed into two steps. First, an optimal filter bank is designed in theoretical sense based on Vaidyanathan’s coding gain criterion in SubBand Coding (SBC) system. Then the above filter bank is optimized based on the criterion of Peak Signal-to-Noise Ratio (PSNR) in JPEG2000 image compression system, resulting in a BWFB in practical application sense. With the approach, a series of BWFB for a specific class of applications related to image compression, such as remote sensing images, can be fast designed. Here, new 5/3 BWFB and 9/7 BWFB are presented based on the above approach for the remote sensing image compression applications. Experiments show that the two filter banks are equally performed with respect to CDF 9/7 and LT 5/3 filter in JPEG2000 standard; at the same time, the coefficients and the lifting parameters of the lifting scheme are all rational, which bring the computational advantage, and the case for VLSI implementation.

Subcanceller: adaptive generalised sidelobe canceller with optimum subband decomposition

This paper presents a subband beamformer with the generalised sidelobe canceller (GSC) as the underlying structure. This new beamformer, referred to as the subcanceller, determines the statistically optimum filter bank succeeding the blocking matrix of the GSC based on the minimum mean-square error criterion. As a consequence, the signals passing through the lower branch of the GSC are decomposed into more appropriate subband components by such filter banks according to the input data statistics to enhance the interference suppression capability. The determination of the filter bank coefficients, however, calls for computationally demanding nonlinear optimisation. To alleviate the computational overhead, an iterative procedure is also proposed, which solves the Rayleigh quotient in each iteration. Furthermore, a detailed performance analysis of the proposed beamformer is conducted, which includes the development expression of the output signal to interference-plus-noise ratio and its implications, and the analysis of the convergence characteristic. Furnished simulations show that the new scheme yields superior interference suppression performance with a faster convergence rate compared with previous studies.

Minimum Variance Multi-User Detection with Optimum Subband Decomposition over Multipath Channels

This paper presents a linear constrained minimum variance multiuser detection (MUD) scheme for DS-CDMA systems, which makes full use of the available spreading sequences of the users as well as the relevant channel information of the incoming rays in the construction of the constraint matrix. To further enhance the performance, a statistical optimum filter bank in combination with the developed minimum variance MUD with the partitioned linear interference canceller (PLIC) as the underlying structure is also addressed. The determination of the filter bank coefficients, however, calls for computationally demanding nonlinear programming. To alleviate the computational overhead, an iterative procedure is also proposed, which solves the Rayleigh quotient in each iteration. Furthermore, the expressions of the output signal to interference plus noise ratio (SINR) are also determined to provide further insights into the proposed approach. Conducted simulations validate the new scheme.

Minimum Mean-Squared Error Reconstruction for Generalized Undersampling of Cyclostationary Processes

The generalized sampling problem is considered using a filter bank model with a stochastic process framework. A signal reconstruction solution minimizing the time-averaged mean-squared error is found using a filter bank optimization technique for cyclostationary signals. The use of a stochastic model approach to find a minimized error solution rather than a perfect reconstruction solution allows consideration of sampling problems that deterministic approaches cannot solve. For instance, a particular sampling density is not required, allowing reconstruction optimization in cases of undersampling. An extension is also presented for optimal reconstruction in the presence of additive noise and interference. The result is a generalized sampling model which can be used for an extensive variety of scenarios, combining aspects of undersampling, multiple-input multiple-output (MIMO) sampling, and periodic nonuniform sampling

On the spectral factor ambiguity of FIR energy compaction filter Banks

This paper focuses on the design of signal-adapted finite-impulse response (FIR) paraunitary (PU) filter banks optimized for energy compaction (EC). The design of such filter banks has been shown in the literature to consist of the design of an optimal FIR compaction filter followed by an appropriate Karhunen-Loe/spl grave/ve transform (KLT). Despite this elegant construction, EC optimal filter banks have been shown to perform worse than common nonadapted filter banks for coding gain, contrary to intuition. Here, it is shown that this phenomenon is most likely due to the nonuniqueness of the compaction filter in terms of its spectral factors. This nonuniqueness results in a finite set of EC optimal filter banks. By choosing the spectral factor yielding the largest coding gain, it is shown that the resulting filter bank behaves more and more like the infinite-order principal components filter bank (PCFB) in terms of numerous objectives such as coding gain, multiresolution, noise reduction with zeroth-order Wiener filters in the subbands, and power minimization for discrete multitone (DMT)-type nonredundant transmultiplexers.

Uniform FIR filterbank optimization with group delay specifications

We characterize the delay properties of uniformly modulated finite impulse response (FIR) filterbanks. Conditions on the permissable delay are developed for this class of filterbanks when the perfect reconstruction condition is relaxed. Accurate linear approximations for the phase and the group delay of the total filterbank are derived. These approximations allow linear phase or group delay constraints to be introduced in the filter optimization problem. A tractable quadratic optimization problem for the design of optimal analysis and synthesis filter prototypes is proposed. The problem involves the minimization of the aliasing distortion while constraining group delay and amplitude distortion. Thus, a new algorithm is presented to solve this optimization problem for the analysis and synthesis filterbanks simultaneously. Numerical examples are presented that confirm the theoretical results and verify that the approximations used are highly accurate.

영상압축을 위한 최적의 서브밴드 양자화 필터 설계

본 논문은 다차원 서브밴드 필터 뱅크에서 양자화 효과에 대한 분석과 그에 따른 최적의 필터 설계 방법에 대한 이론을 제안한다. 기존의 서브밴드 필터뱅크에 대한 연구에서 양자화 과정에 대한 고려는 매우 미미하였다. 본 연구에서는 확률 밀도에 최적화된 양자화기가 이득에 무상관성 잡음이 더하여진 비선형 형태로 모델링 되어 서브밴드 구조에 결합된다. 분석/합성 필터 뱅크에 다위상 분리 방법을 적용하여 양자화 과정에 의한 출력에서의 MSE가 유도되며 이 에러의 최소화 과정을 통하여 최적의 필터 설계 방법이 개발된다. 또한 5점형 구조의 표본화 격자를 이용한 최적의 비분리 파라유니터리와 이중직교 필터의 설계와 이를 이용한 영상 데이터의 압축 및 복원 과정을 통하여 성능 분석을 한다. 시뮬레이션 결과 제안한 설계방법에 따른 필터가 기존의 단순한 PR 서브밴드 필터보다 영상의 압축 및 복원에서 10에서 20퍼센트 정도의 MSE 감소를 보여주었고 낮은 비트율에서 특히 우수한 성능을 나타내었다. This paper provides a rigorous theory for analysis of quantization effects and optimum filter bank design in quantized multidimensional subband filter banks. Even though subband filter design has been a hot topic for last decades, a few results have been reported on the subband filter with a quantizer. Each pdf-optimized quantizer is modeled by a nonlinear gain-plus-additive uncorrelated noise and embedded into the subband structure. Using polyphase decomposition of the analysis/synthesis filter banks, we derive the exact expression for the output mean square quantization error. Based on the minimization of the output mean square error, the technique for optimal filter design methodology is developed. Numerical design examples for optimum nonseparable paraunitary and biorthogonal filter banks are presented with a quincunx subsampling lattice. Through the simulation, <TEX>$10\~20\;\%$</TEX> decreases in MSE have been observed compared with subband filter with no quantizers especially for low bit rate cases.

Optimized Multichannel Filter Bank with Flat Frequency Response for Texture Segmentation

Previous approaches to texture analysis and segmentation use multichannel filtering by applying a set of filters in the frequency domain or a set of masks in the spatial domain. This paper presents two new texture segmentation algorithms based on multichannel filtering in conjunction with neural networks for feature extraction and segmentation. The features extracted by Gabor filters have been applied for image segmentation and analysis. Suitable choices of filter parameters and filter bank coverage in the frequency domain to optimize the filters are discussed. Here we introduce two methods to optimize Gabor filter bank. First, a Gabor filter bank with a flat response is implemented and the optimal feature dimension is extracted by competitive networks. Second, a subset of Gabor filter bank is selected to compose the best discriminative filters, so that each filter in this small set can discriminate a pair of textures in a given image. In both approaches, multilayer perceptrons are employed to segment the extracted features. The comparisons of segmentation results generated using the proposed methods and previous research using Gabor, discrete cosine transform (DCT), and Laws filters are presented. Finally, the segmentation results generated by applying the optimized filter banks to textured images are presented and discussed.

Optimum filter design for quantized multidimensional subband filter bank

This paper provides a rigorous theory for analysis of quantization effects and optimum filter bank design in quantized multidimensional subband filter banks. Each pdf-optimized quantizer is modeled by a nonlinear gain-plus-additive uncorrelated noise and embedded into the subband structure. Using polyphase decomposition of the analysis/synthesis filter banks, we derive the exact expression for the output mean square quantization error. Based on the minimization of the output mean square error, the technique for optimal filter design methodology is developed. Numerical design examples for optimum nonseparable paraunitary and biorthogonal filter banks are presented with a quincunx subsampling lattice.

DCT-based general structure for linear-phase paraunitary filterbanks

In this paper, we present a general structure for linear phase paraunitary filterbanks (LPPUFB) via both time-domain pre-processing and frequency-domain post-processing of the discrete cosine transform (DCT). The proposed structure not only covers existing frameworks with either pre-processing or post-processing of the DCT but enables the design of DCT-based LPPUFB with both partial-block overlapping and variable-length filters as well. Some high-performance and low-complexity transforms can be obtained by fine-tuning the structure parameters. Furthermore, a DCT-oriented initialization method is developed to improve the convergence and to simplify,the parameterization in the filterbank optimization process. The application in image compression is demonstrated.

Algebraic theory of optimal filterbanks

We introduce an optimality theory for finite impulse response (FIR) filterbanks using a general algebraic point of view. We consider an admissible set /spl Lscr/ of FIR filterbanks and use scalability as the main notion based on which performance of the elements in /spl Lscr/are compared. We show that quantification of scalability leads naturally to a partial ordering on the set /spl Lscr/. An optimal solution is, therefore, represented by the greatest element in /spl Lscr/. It turns out that a greatest element does not necessarily exist in /spl Lscr/. Hence, one has to settle with one of the maximal elements that exist in /spl Lscr/. We provide a systematic way of finding a maximal element by embedding the partial ordering at hand in a total ordering. This is done by using a special class of order-preserving functions known as Schur-convex. There is, however, a price to pay for achieving a total ordering: there are infinitely many possible choices for Schur-convex functions, and the optimal solution specified in /spl Lscr/ depends on this (subjective) choice. An interesting aspect of the presented algebraic theory is that the connection between several concepts, namely, principal component filterbanks (PCFBs), filterbanks with maximum coding gain, and filterbanks with good scalability, is clearly revealed. We show that these are simply associated with different extremal elements of the partial ordering induced on /spl Lscr/ by scalability.

Optimal filter banks for multiple description coding: analysis and synthesis

Multiple description (MD) coding is a source coding technique for information transmission over unreliable networks. In MD coding, the coder generates several different descriptions of the same signal and the decoder can produce a useful reconstruction of the source with any received subset of these descriptions. In this paper, we study the problem of MD coding of stationary Gaussian sources with memory. First, we compute an approximate MD rate distortion region for these sources, which we prove to be asymptotically tight at high rates. This region generalizes the MD rate distortion region of El Gamal and Cover (1982), and Ozarow (1980) for memoryless Gaussian sources. Then, we develop an algorithm for the design of optimal two-channel biorthogonal filter banks for MD coding of Gaussian sources. We show that optimal filters are obtained by allocating the redundancy over frequency with a reverse "water-filling" strategy. Finally, we present experimental results which show the effectiveness of our filter banks in the low complexity, low rate regime.

A direct approach to H/sub 2/ optimal deconvolution of periodic digital channels

This paper studies the H/sub 2/ optimal deconvolution problem for periodic finite impulse response (FIR) and infinite impulse response (IIR) channels. It shows that the H/sub 2/ norm of a periodic filter can be directly quantified in terms of periodic system matrices and linear matrix inequalities (LMIs) without resorting to the commonly used lifting technique. The optimal signal reconstruction problem is then formulated as an optimization problem subject to a set of matrix inequality constraints. Under this framework, the optimization of both the FIR and IIR periodic deconvolution filters can be made convex, solved using the interior point method, and computed by using the Matlab LMI Toolbox. The robust deconvolution problem for periodic FIR and IIR channels with polytopic uncertainties are further formulated and solved, also by convex optimization and the LMIs. Compared with the lifting approach to the design of periodic filters, the proposed approach is simpler yet more powerful in dealing with multiobjective deconvolution problems and channel uncertainties, especially for IIR deconvolution filter design. The obtained solutions are applied to the design of an optimal filterbank yielding satisfactory performance.

Wavelet networks for face processing.

Wavelet networks (WNs) were introduced in 1992 as a combination of artificial neural radial basis function (RBF) networks and wavelet decomposition. Since then, however, WNs have received only a little attention. We believe that the potential of WNs has been generally underestimated. WNs have the advantage that the wavelet coefficients are directly related to the image data through the wavelet transform. In addition, the parameters of the wavelets in the WNs are subject to optimization, which results in a direct relation between the represented function and the optimized wavelets, leading to considerable data reduction (thus making subsequent algorithms much more efficient) as well as to wavelets that can be used as an optimized filter bank. In our study we analyze some WN properties and highlight their advantages for object representation purposes. We then present a series of results of experiments in which we used WNs for face tracking. We exploit the efficiency that is due to data reduction for face recognition and face-pose estimation by applying the optimized-filter-bank principle of the WNs.

Two-channel adaptive biorthogonal filterbanks via lifting

In the high bit-rate data compression, the filterbanks that match the statistical properties of inputs have been widely researched. For orthonormal uniform filterbanks and biorthogonal filterbanks without order constraints, the design problems have been solved on the whole. However, for biorthogonal FIR filterbanks, an efficient method associated with an algorithm for the design still needs to be developed. In this paper, two-channel adaptive biorthogonal filterbanks via lifting scheme are investigated. Starting from an arbitrary filterbank (orthonormal or biorthogonal), two-channel biorthogonal filterbanks are designed by the adaptive lifting scheme, which is a general form of our previous work. Two adaptive lifting algorithms are developed, namely, the sequential adaptive lifting and the joint adaptive lifting. Further, it is proved that starting from a linear-phase filterbank, the optimal filterbank by the sequential adaptive lifting preserves the linear-phase property. Experimental results show that the subband coding gains can be markedly improved when an appropriate initial filterbank is chosen.

Construction of optimal subband coders using optimized and optimal quantizers

A method is presented for the optimization of arbitrary quantizers by use of a compensating postfilter. It is shown that the resulting optimized quantizers fit the model of a linear time-invariant filter followed by additive noise uncorrelated with the input which also characterizes the optimal (Lloyd-Max) quantizers. On the basis of this model, an expression for the variance of the error of a subband coder using optimized quantizers is explicitly determined. Given analysis filters which statistically separate the subbands, it is shown that this variance is minimized if these synthesis filters are chosen, which would achieve perfect reconstruction in lossless coding. The globally optimum filter bank, minimizing the coder error variance, is further obtained by proper choice of its analysis filters. A novel method for the determination of optimal bit allocation to subbands of the filter banks with optimized quantizers is also developed. The results are evaluated experimentally by comparison of the optimum uniformly split subband image coding scheme to classical logarithmically-split filter bank (wavelet) coding methods.

A Search for Submillisecond Pulsars

We have conducted a search of 19 southern Galactic globular clusters for submillisecond pulsars at 660 MHz with the Parkes 64 m radio telescope. To minimize dispersion smearing we used the CPSR baseband recorder, which samples the 20 MHz observing band at the Nyquist rate. By possessing a complete description of the signal we could synthesize an optimal filter bank in software, and in the case of globular clusters of known dispersion measure, much of the dispersion could be removed using coherent techniques. This allowed for very high time resolution (25.6 μs in most cases), making our searches in general sensitive to submillisecond pulsars with flux densities greater than about 3 mJy at 50 cm. No new pulsars were discovered, placing important constraints on the proportion of pulsars with very short spin periods in these clusters.

A Review of the Theory and Applications of Optimal Subband and Transform Coders

The problem of optimizing digital filter banks based on input statistics was perhaps first addressed nearly four decades ago by Huang and Schultheiss. These authors actually considered a special case, namely transform coder optimization. Many of the subband coder optimization problems considered in recent years have close similarities to this work, though there are fundamental differences as well. Filter banks are used today not only for signal compression, but have found applications in signal denoising and in digital communications. A recent result is that principal component filter banks (PCFBs) offer an optimal solution to many problems under certain theoretical assumptions. While this result is quite powerful and includes several earlier results as special cases, there still remain some open problems in the area of filter bank optimization. We first give a review of the older classical methods to place the ideas in the right perspective. We then review recent results on PCFBs. The generality of these results is demonstrated by showing an application in digital communications (the discrete multitone channel). We show, for example, that the PCFB minimizes transmitted power for a given probability of error and bit rate. Future directions and open problems are discussed as well.

Rate-distortion-optimal subband coding without perfect-reconstruction constraints

We investigate the design of subband coders without the traditional perfect-reconstruction constraint on the filters. The coder uses scalar quantizers, and its filters and bit allocation are designed to optimize a rate-distortion criterion. Using convexity analysis, we show that optimality can be achieved using filterbanks that are the cascade of a (paraunitary) principal component filterbank for the input spectral process and a set of pre and postfilters surrounding each quantizer. Analytical expressions for the pre and postfilters are then derived. An algorithm for computing the globally optimal filters and bit allocation is given. We also develop closed-form solutions for the special case of two-channel coders under an exponential rate-distortion model. Finally, we investigate a constrained-length version of the filter design problem, which is applicable to practical coding scenarios. While the optimal filterbanks are nearly perfect-reconstruction at high rates, we demonstrate an apparently surprising advantage of optimal FIR filterbanks; they significantly outperform optimal perfect-reconstruction FIR filterbanks at all bit rates.