Synthesis filter bank optimization in two-dimensional separablesubband coding systems

Subband coding is a popular and well established technique used in visual communications, such as image and video transmission. In the absence of quantization and transmission errors, the analysis and synthesis filters in a subband coding scheme can be designed to obtain perfect reconstruction of the input signal, but this is no longer the optimal solution in the presence of quantization of the subband coefficients. We presuppose the use of a two-dimensional (2-D) separable subband scheme and we address the problem of designing, for a given analysis filter bank and assuming uniform quantization of the subband coefficients, the set of row and column synthesis filters that minimize the mean squared reconstruction error at the output of the subband system. Since the corresponding optimization problem is inherently nonlinear, we propose a suboptimal solution that extends a one-dimensional (l-D) optimal filter design procedure, already presented in the literature, to a 2-D separable synthesis filter bank. The separable 2-D extension is not trivial, since the processing in one direction, e.g., the rows, alters the statistics of the signals for the design of the filters in the other direction, e.g., the columns. To further simplify the filter design, we propose to model the input image as a 2-D separable Markov process plus an additive white component. Several design examples using both synthetic signals and real world images are presented, showing that the filters designed using the proposed technique can give a significant gain with respect to the perfect reconstruction solution, especially when the dither technique is used for quantization. The simulation results also show that the proposed image model can be conveniently used in the synthesis filter design procedure.     

Optimal construction of subband coders using Lloyd-Max quantizers

A new method is presented for the analysis of the effects of Lloyd-Max quantization in subband filterbanks and for the optimal design of such filterbanks. A rigorous statistical model of a vector Lloyd-Max quantizer is established first, consisting of a linear time-invariant filter followed by additive noise uncorrelated/with the input. On the basis of this model, an expression for this variance of the error of a subband coder using Lloyd-Max quantizers is explicitly determined. Given analysis filters that statistically separate the subbands, it is shown that this variance is minimized if the synthesis filters are chosen, which mould achieve perfect reconstruction in lossless coding. The globally optimum of such a filterbank, minimizing the coder error variance, is further obtained by proper choice of its analysis filters. An alternative design method is also evaluated and optimized. In this, the errors correlated with the signal are set to zero, leaving a random error residue uncorrelated with the signal. This design method is optimized by choosing the analysis filters so as to minimize the random error variance. The results are evaluated experimentally in the realistic setting of a logarithmically split subband image coding scheme.     

Optimal signal reconstruction in noisy filter bank systems:multirate Kalman synthesis filtering approach

A multirate Kalman synthesis filter is proposed in this paper to replace the conventional synthesis filters in a noisy filter bank system to achieve optimal reconstruction of the input signal. Based on an equivalent block representation of subband signals, a state-space model is introduced for an M-band filter bank system with subband noises. The composite effect of the input signal, analysis filter bank, decimators, and interpolators is represented by a multirate state-space model. The input signal is embedded in the state vector, and the corrupting noises in subband paths are generally considered as additive noises. Hence, the signal reconstruction problem in the M-band filter bank systems with subband noises becomes a state estimation procedure in the resultant multirate state-space model. The multirate Kalman filtering algorithm is then derived according to the multirate state-space model to achieve optimal signal reconstruction in noisy filter bank systems. Based on the optimal state estimation theory, the proposed multirate Kalman synthesis filter provides the minimum-variance reconstruction of the input signal. Two numerical examples are also included. The simulation results indicate that the performance improvement of signal reconstruction in noisy filter bank systems is remarkable     

Two-dimensional filter bank design for optimal reconstruction usinglimited subband information

In this correspondence, we propose design techniques for analysis and synthesis filters of 2-D perfect reconstruction filter banks (PRFB's) that perform optimal reconstruction when a reduced number of subband signals is used. Based on the minimization of the squared error between the original signal and some low-resolution representation of it, the 2-D filters are optimally adjusted to the statistics of the input images so that most of the signal's energy is concentrated in the first few subband components. This property makes the optimal PRFB's efficient for image compression and pattern representations at lower resolutions for classification purposes. By extending recently introduced ideas from frequency domain principal component analysis to two dimensions, we present results for general 2-D discrete nonstationary and stationary second-order processes, showing that the optimal filters are nonseparable. Particular attention is paid to separable random fields, proving that only the first and last filters of the optimal PRFB are separable in this case. Simulation results that illustrate the theoretical achievements are presented.     

On reconstruction methods for processing finite-length signals withparaunitary filter banks

New expressions are developed for the perfect reconstruction of the boundary regions of a finite-length signal after subband processing. The time-invariant filter bank is required to be uniform and paraunitary, using FIR filters regardless of phase or symmetry. They accommodate a linear boundary extension in the analysis section, and avoid periodic extensions or storage of extended subband signals. The reconstruction methods are based on the formulation of linear systems that are built as a function of the filters     

The design of 2-D nonseparable directional perfect reconstruction filter banks

In this paper, we develop a directional 2-D nonseparable filter bank that can perfectly reconstruct the downsampled subband signals. The filter bank represents two powerful image and video processing tools: directional subband decomposition and perfect reconstruction. The directional filter banks consist of (1) the input signal and the subband signals modulation, (2) diamond shape prefilter, and (3) four different parallelogram shape prefilters. This paper addresses the design and implementation of a two-band filter bank that is proved to be able to provide perfect reconstruction of the downsampled subband signals. Finally, we use a conventional 1-D half-band filter as a prototype and then apply the McClellan transform for the specific 2-D diamond shape and parallelogram shape subfilters. This method is extremely simple in designing the analysis/synthesis subfilters for the filter bank.     

Construction of optimal subband coders using optimized and optimalquantizers

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.     

基于模数混合子带划分的信号重构建模分析与优化

为了应对宽带阻塞式干扰,通信干扰对消系统通常应用基于子带划分的干扰对消技术来提升宽带干扰抑制能力,同时为了兼顾通信性能,将子带信号重构恢复为通信信号,保障通信质量。该文采用模拟电路与数字处理相结合的方式,搭建了子带划分与信号重构框架,通过模拟滤波器实现第1级宽滤波,减小信号处理带宽,利用数字滤波器完成第2级窄滤波,进一步提高信号信噪比。针对通信信号在子带划分过程中,存在跨子带重构失真的问题,建立了子带划分与信号重构系统时频域模型,分析子带间幅相不一致对信号重构的影响。为了解决重构信号的幅相不一致问题,提出了相位校准方法与滤波器幅频优化方法,从幅度和相位两方面同时保证跨子带信号的近似无失真重构。仿真与实验结果表明,该文所设计的滤波器幅频响应具有良好的重构精度,并且解决了跨子带信号重构的相位失真问题,有效降低了重构通信信号的误码率。     

Flexible design of multidimensional perfect reconstruction FIR2-band filters using transformations of variables

An approach to designing multidimensional linear-phase FIR diamond subband filters having the perfect reconstruction property is presented. It is based on a transformation of variables technique and is equivalent to the generalized McClellan transformation. Methods for designing a whole class of transformation are given. The approach consists of two parts; design of the transformation and design of the 1-D filters. The use of Lagrange halfband filters to design the 1-D filters is discussed. The modification of a particular Lagrange halfband filter which gives a pair of simple 1-D filters that are almost similar to each other in their frequency characteristics but still form a perfect reconstruction pair is presented. The design technique is extended to other types of two-channel sampling lattice and subband shapes, in particular, the parallelogram and the diagonally quadrant subband cases. Several numerical design examples are presented to illustrate the flexibility of the design method.     

Efficient design of oversampled NPR GDFT filterbanks

We propose a flexible, efficient design technique for the prototype filter of an oversampled near perfect reconstruction (NPR) generalized discrete Fourier transform (GDFT) filterbank. Such filterbanks have several desirable properties for subband processing systems that are sensitive to aliasing, such as subband adaptive filters. The design criteria for the prototype filter are explicit bounds (derived herein) on the aliased components in the subbands and the output, the distortion induced by the filterbank, and the imaged subband errors in the output. It is shown that the design of an optimal prototype filter can be transformed into a convex optimization problem, which can be efficiently solved. The proposed design technique provides an efficient and effective tool for exploring many of the inherent tradeoffs in the design of the prototype filter, including the tradeoff between aliasing in the subbands and the distortion induced by the filterbank. We calculate several examples of these tradeoffs and demonstrate that the proposed method can generate filters with significantly better performance than filters obtained using current design methods.     

Subband coding of images using asymmetrical filter banks

We address the problem of the choice of subband filters in the context of image coding. The ringing effects that occur in subband-based compression schemes are the major unpleasant distortions. A new set of two-band filter banks suitable for image coding applications is presented. The basic properties of these filters are linear phase, perfect reconstruction, asymmetric length, and maximum regularity. The better overall performances compared to the classical QMF subband filters are explained. The asymmetry of the filter lengths results in a better compaction of the energy, especially in the highpass subbands. Moreover, the quantization error is reduced due to the short lowpass synthesis filter. The undesirable ringing effect is considerably reduced due to the good step response of the synthesis lowpass filter. The proposed design takes into account the statistics of natural images and the effect of quantization errors in the reconstructed images, which explains the better coding performance.     

调制混合滤波器组的宽带模拟信号采样研究

对于超宽带模拟信号,很难用单个模拟数字转换器（ADC）直接进行采样。该文提出了一种新的并行调制混合滤波器组结构用于实现超宽带模拟信号的采样,首先,将每一路宽带模拟输入信号进行余弦调制,并用相同的低通模拟滤波器均匀分割输入信号的带宽;然后,采用相同的ADC将子带信号数字化;各路子带信号通过上采样器后用数字综合滤波器综合得到原宽带模拟输入信号的数字重构。综合滤波器采用总体最小二乘准则下的特征值滤波器设计方法得到。本文所提出的系统结构不需要使用高速的采样保持电路,降低了系统实现的难度,并且设计的系统具有与其它混合滤波器组相近的重构性能。仿真结果表明了本方法的有效性。      

A Mapping-Based Design for Nonsubsampled Hourglass Filter Banks in Arbitrary Dimensions

Multidimensional hourglass filter banks decompose the frequency spectrum of input signals into hourglass-shaped directional subbands, each aligned with one of the frequency axes. The directionality of the spectral partitioning makes these filter banks useful in separating the directional information in multidimensional signals. Despite the existence of various design techniques proposed for the 2-D case, to our best knowledge, the design of hourglass filter banks in 3-D and higher dimensions with finite impulse response (FIR) filters and perfect reconstruction has not been previously reported. In this paper, we propose a novel mapping-based design for the hourglass filter banks in arbitrary dimensions, featuring perfect reconstruction, FIR filters, efficient implementation using lifting/ladder structures, and a near-tight frame construction. The effectiveness of the proposed mapping- based design depends on the study of a set of conditions on the frequency supports of the mapping kernels. These conditions ensure that we can still get good frequency responses when the component filters used are nonideal. Among all feasible choices, we then propose an optimal specification for the mapping kernels, which leads to the simplest passband shapes and involves the fewest number of frequency variables. Finally, we illustrate the proposed techniques by a design example in 3-D, and an application in video denoising.     

High compression image coding using an adaptive morphologicalsubband decomposition

A morphological subband decomposition with perfect reconstruction is proposed. Critical subsampling is achieved. The reconstructed images using this decomposition do not suffer from any ringing effect. In order to avoid poor texture representation by the morphological filters an adaptive subband decomposition is introduced. It chooses linear filters on textured regions and morphological filters otherwise. A simple and efficient texture detection criterion is proposed and applied to the adaptive decomposition. Comparisons to other coding techniques such as JPEG and linear subband coding show that the proposed scheme performs significantly better both in terms of PSNR and visual quality     

Design of efficient M-band coders with linear-phase andperfect-reconstruction properties

This paper presents a new design technique for obtaining M-band orthogonal coders where M=2ⁱ. The structures obtained using the proposed technique have the perfect reconstruction property. Furthermore, all filters that constitute the subband coder are linear-phase FIR-type filters. In contrast with conventional design techniques that attempt to find a unitary alias-component matrix in the frequency domain, we carry out the design in the time domain, based on time-domain orthonormality constraints that the filters must satisfy. The M-band design problem is reduced to the problem of finding a suitable lowpass filter h₀(n). Once a suitable lowpass filter is found, the remaining (M-1) filters of the coder are obtained through the use of shuffling operators on the lowpass filter. This approach leads to a set of filters that use the same numerical coefficient values in different shift positions, allowing very efficient numerical implementation of the subband coder. In addition, by imposing further constraints on the lowpass branch impulse response h₀(n), we are able to construct continuous bases of M-channel wavelets with good regularity properties. Design examples are presented for four-, eight-, and 16-band coders, along with examples of continuous wavelet bases that they generate     

Mixed H2/H∞ filtering design inmultirate transmultiplexer systems: LMI approach

A mixed H₂/H_∞ filter design is proposed for multirate transmultiplexer systems with dispersive channel and additive noise. First, a multirate state-space representation is introduced for the transmultiplexer with the consideration of channel dispersion. Then, the problem of signal reconstruction can be regarded as a state estimation problem. In order to design an efficient separating filterbank for a transmultiplexer system with uncertain input signal and additive noise, the H_∞ filter is employed for robust signal reconstruction. The H₂ filter design is considered to be a suboptimal approach to achieve the optimal signal reconstruction in transmultiplexer system under unitary noise power. Finally, a mixed H₂/H_∞ filter is proposed to achieve a better signal reconstruction performance in transmultiplexer systems. These design problems can be transformed to solving the eigenvalue problems (EVP) under some linear matrix inequality (LMI) constraint. The LMI Matlab toolbox can be applied to efficiently solve the EVP by convex optimization technique     

A delayless subband adaptive filter architecture

Some adaptive signal processing applications, such as wideband active noise control and acoustic echo cancellation, involve adaptive filters with hundreds of taps. The computational burden associated with these long adaptive filters precludes their use for many low-cost applications. In addition, adaptive filters with many taps may also suffer from slow convergence, especially if the reference signal spectrum has a large dynamic range. Subband techniques have been previously developed for adaptive filters to solve these problems. However, the conventional approach is ruled out for many applications because delay is introduced into the signal path. The paper presents a new type of subband adaptive filter architecture in which the adaptive weights are computed in subbands, but collectively transformed into an equivalent set of wideband filter coefficients. In this manner, signal path delay is avoided while retaining the computational and convergence speed advantages of subband processing. An additional benefit accrues through a significant reduction of aliasing effects. An example of the general technique is presented for a 32-subband design using a polyphase FFT implementation. For this example, the number of multiplies required are only about one-third that of a conventional full band design with zero delay, and only slightly greater than that of a conventional subband design with 16 ms delay     

基于多尺度Wiener滤波器的分形噪声滤波

针对淹没在1/f噪声中的有用信号恢复问题,本文提出了一套基于双正交小波变换与Wiener滤波的多尺度滤波算法,并设计出多尺度Wiener滤波器.首先,利用双正交小波变换将带有1/f噪声的信号分解成多尺度的子带信号,通过小波变换对1/f噪声的白化作用,消除了1/f噪声的非平稳性、自相似性和长程相关性.其次,在小波域内,利用Wiener滤波,实现了噪声和有用信号的分离,估计出了各子带中的有用信号.最后,利用双正交小波的精确重构性,较好地恢复出淹没在1/f噪声中的有用信号.仿真实验表明,该滤波器能有效的抑制分形噪声,显著地提高信噪比.     

Cyclostationary modeling, analysis, and optimal compensation ofquantization errors in subband codecs

The paper is concerned with the analysis and modeling of the effects of quantization of subband signals in subband codecs. Using cyclostationary representations, the authors derive equations for the autocorrelation and power spectral density (PSD) of the reconstructed signal y(n) in terms of the analysis/synthesis filters, the PSD of the input, and the pdf-optimized quantizer model. Formulas for the mean-square error (MSE) and for compaction gain are obtained in terms of these parameters. The authors constrain the filter bank to be perfect reconstruction (PR) (but not necessarily paraunitary) in the absence of quantization and transmission errors. These formulas set the stage for filter optimization (maximization of compaction gain and minimization of MSE) subject to PR and bit constraints. Optimal filters are designed, optimal compensation is performed, and the theoretical results are confirmed with simulations. The floating-point quantizer wherein only the mantissa is uniformly quantized is also analyzed and compared with the fixed point, pdf-optimized filter bank. For high bit rates, their performance is comparable     

A Low-Complexity Delayless Selective Subband Adaptive Filtering Algorithm

Adaptive filters of significant order, requiring high computational complexity, are necessary in many applications such as acoustic echo cancellation and wideband active noise control. Successful approaches to lessen the computational complexity of such filters are subband methods, and partial updating schemes where only a part of the filter is updated at each instant. To avoid the time delay introduced by the subband-splitting, delayless structures which reconstructs a fullband filter, producing delayless output, from the adaptive subband filters have been proposed. This paper proposes a delayless subband adaptive filter partial updating scheme, where the general idea is to only update the most misadjusted subband filter(s). Analysis in terms of mean square deviation is presented and shows that the fullband filter convergence speed is significantly increased, even for flat spectrum signals, as compared to traditional periodic subband filter update with the same computational complexity. Echo cancellation simulations with an artificial system to verify the analysis, using both flat spectrum signals and speech, is also presented, as well as offline calculations using signals from a real system.