Orthonormal and biorthonormal filter banks as convolvers, andconvolutional coding gain

Convolution theorems for filter bank transformers are introduced. Both uniform and nonuniform decimation ratios are considered, and orthonormal as well as biorthonormal cases are addressed. All the theorems are such that the original convolution reduces to a sum of shorter, decoupled convolutions in the subbands. That is, there is no need to have cross convolution between subbands. For the orthonormal case, expressions for optimal bit allocation and the optimized coding gain are derived. The contribution to coding gain comes partly from the nonuniformity of the signal spectrum and partly from nonuniformity of the filter spectrum. With one of the convolved sequences taken to be the unit pulse function,,e coding gain expressions reduce to those for traditional subband and transform coding. The filter-bank convolver has about the same computational complexity as a traditional convolver, if the analysis bank has small complexity compared to the convolution itself     

On the optimality of nonunitary filter banks in subband coders

This paper investigates the energy compaction capabilities of nonunitary filter banks in subband coding. It is shown that nonunitary filter banks have larger coding gain than unitary filter banks because of the possibility of performing half-whitening in each channel. For long filter unit pulse responses, optimization of subband coding gain for stationary input signals results in a filter bank decomposition, where each channel works as an optimal open-loop DPCM system. We derive a formula giving the optimal filter response for each channel as a function of the input power spectral density (PSD). For shorter filter bank responses, good gain is obtained by suboptimal half-whitening responses, but the impact on the theoretical coding gain is still highly significant. Image coding examples demonstrate that better performance is achieved using nonunitary filter banks when the input images are in correspondence with the signal model.     

Multiplicative filtering in the fractional Fourier domain

In this article, we investigate the multiplicative filtering in the fractional Fourier transform (FRFT) domain based on the generalized convolution theorem which states that the convolution of two signals in time domain results in simple multiplication of their FRFTs in the FRFT domain. In order to efficiently implement multiplicative filtering, we express the generalized convolution structure by the conventional convolution operation. Utilizing the generalized convolution structure, we convert the multiplicative filtering in the FRFT domain easily to the time domain. Based on the model of multiplicative filtering in the FRFT domain, a practical method is proposed to achieve the multiplicative filtering through convolution in the time domain. This method can be realized by classical Fast Fourier transform (FFT) and has the same capability compared with the method achieved in the FRFT domain. As convolution can be performed by FFT, this method is more useful from practical engineering perspective.     

Analysis of linear prediction, coding, and spectral estimation fromsubbands

Linear prediction schemes make a prediction xˆ_i of a data sample x_i using p previous samples. It has been shown by Woods and O'Neil (1986) as well as Pearlman (1991) that as the order of prediction p→∞, there is no gain to be obtained by coding subband samples. This paper deals with the less well understood theory of finite-order prediction and optimal coding from subbands which are generated by ideal (brickwall) filtering of a stationary Gaussian source. We first prove that pth-order prediction from subbands is superior to pth-order prediction in the fullband, when p is finite. This fact adduces that optimal vector p-tuple coding in the subbands is shown to offer quantifiable gains over optimal fullband p-tuple coding, again when p is finite. The properties of subband spectra are analyzed using the spectral flatness measure. These results are used to prove that subband DPCM provides a coding gain over fullband DPCM, for finite orders of prediction. In addition, the proofs provide means of quantifying the subband advantages in linear prediction, optimal coding, and DPCM coding in the form of gain formulas. Subband decomposition of a source is shown to result in a whitening of the composite subband spectrum. This implies that, for any stationary source, a pth-order prediction error filter (PEF) can be found that is better than the pth-order PEF obtained by solving the Yule-Walker equations resulting from the fullband data. We demonstrate the existence of such a “super-optimal” PEF and provide algorithmic approaches to obtaining this PEF. The equivalence of linear prediction and AR spectral estimation is then exploited to show theoretically, and with simulations, that AR spectral estimation from subbands offers a gain over fullband AR spectral estimation     

Unequal error protection for MPEG-2 video transmission over wireless channels

This paper proposes an unequal error protection (UEP) method for MPEG-2 video transmission. Since the source and channel coders are normally concatenated, if the channel is noisy, more bits are allocated to channel coding and fewer to source coding. The situation is reversed when the channel conditions are more benign. Most of the joint source channel coding (JSCC) methods assume that the video source is subband coded, the bit error sensitivity of the source code can be modeled, and the bit allocations for different subband channels will be calculated. The UEP applied to different subbands is the rate compatible punctured convolution channel coder. However, the MPEG-2 coding is not a subband coding, the bit error sensitivity function for the coded video can no longer be applied. Here, we develop a different method to find the rate-distortion functions for JSCC of the MPEG-2 video. In the experiments, we show that the end-to-end distortion of our UEP method is smaller than the equal error protection method for the same total bit-rate.     

Vector space framework for unification of one- and multidimensionalfilter bank theory

A number of results in filter bank theory can be viewed using vector space notations. This simplifies the proofs of many important results. In this paper, we first introduce the framework of vector space, and then use this framework to derive some known and some new filter bank results as well. For example, the relation among the Hermitian image property, orthonormality, and the perfect reconstruction (PR) property is well-known for the case of one-dimensional (1-D) analysis/synthesis filter banks. We can prove the same result in a more general vector space setting. This vector space framework has the advantage that even the most general filter banks, namely, multidimensional nonuniform filter banks with rational decimation matrices, become a special case. Many results in 1-D filter bank theory are hence extended to the multidimensional case, with some algebraic manipulations of integer matrices. Some examples are: the equivalence of biorthonormality and the PR property, the interchangeability of analysis and synthesis filters, the connection between analysis/synthesis filter banks and synthesis/analysis transmultiplexers, etc. Furthermore, we obtain the subband convolution scheme by starting from the generalized Parseval's relation in vector space. Several theoretical results of wavelet transform can also be derived using this framework. In particular, we derive the wavelet convolution theorem     

Subband coding of images using vector quantization

A novel two-dimensional subband coding technique is presented that can be applied to images as well as speech. A frequency-band decomposition of the image is carried out by means of 2D separable quadrature mirror filters, which split the image spectrum into 16 equal-rate subbands. These 16 parallel subband signals are regarded as a 16-dimensional vector source and coded as such using vector quantization. In the asymptotic case of high bit rates, a theoretical analysis yields that a lower bound to the gain is attainable by choosing this approach over scalar quantization of each subband with an optimal bit allocation. It is shown that vector quantization in this scheme has several advantages over coding the subbands separately. Experimental results are given, and it is shown the scheme has a performance that is comparable to that of more complex coding techniques     

Wavelet packets and spatial adaptive intraband coding of images

In this work, we present a coding scheme based on a rate-distortion optimum wavelet packets decomposition and on an adaptive coding procedure that exploits spatial non-stationarity within each subband. We show, by means of a generalization of the concept of coding gain to the case of non-stationary signals, that it may be convenient to perform subband decomposition optimization in conjunction with intraband optimal bit allocation. In our implementation, each subband is partitioned into blocks of coefficients that are coded using a geometric vector quantizer with a rate determined on the basis of spatially local statistical characteristics. The proposed scheme appears to be simpler than other wavelet packets-based schemes presented in the literature and achieves good results in terms of both compression and visual quality.     

The coding gain of multiplexed wavelet transforms

Evangelista (1994, 1993) introduced the multiplexed wavelet transform (MWT) and pointed out its potential applications to the analysis, synthesis, processing and coding of pseudo-periodic signals such as voiced speech and music. Coders based on the MWT have been shown to outperform the conventional subband coders when a reliable pitch parameter can be extracted from data. In this paper, we investigate the effects of uniform quantization of the MWT coefficients. We compare the performance of the new coders with that of WT, block-DCT, and KLT coders in terms of the coding gain achieved when optimal bit allocation schemes are adopted     

Low-complexity subband encoding for HDTV images

The transmission of high-definition television (HDTV) signals on available digital networks and satellites requires the adoption of sophisticated compression techniques to limit the bit rate requirements and to provide high-quality and reliable service to customers. For processing and transmission of image signals, a low-complexity codec without visible degradation is desired. A low-complexity intraframe subband image coding algorithm is developed. The low band is DPCM encoded and the high bands are PCM encoded. An efficient entropy coder is designed which reduces the overall bit rate significantly. It is shown that high-quality HDTV images can be obtained at as low a bit rate as 45 Mb/s or less with a very low-complexity encoder. For dividing the image into subbands, a new class of quadrature mirror filters (QMFs) called generalized quadrature mirror filters (GQMFs) is used for filtering. Performance is also evaluated by using short kernel filters (SKFs), which are easy to implement and require very few computations     

Performance evaluation of subband coding and optimization of its filter coefficients

In this paper, two analytical methods for evaluating the coding efficiency of subband coding are proposed, and optimization of filter coefficients of the perfect reconstruction FIR filter banks is considered, based on a new performance measure called unified coding gain. First, matrix representation of the subband coding in the time domain is considered, and conventional subband filter banks are classified into orthogonal ones such as the QMF and nonorthogonal ones such as the SSKF. For the orthogonal filter banks, the coding gain shown by Jayant and Noll is introduced, and their theoretical performance evaluation is carried out. However, this first method cannot be applied to nonorthogonal filter banks any longer because the coding gain is defined on the assumption of filter orthogonality. Therefore, an optimum bit allocation problem for subband coding is considered, and the unified coding gain, which can be applied to arbitrary subband filter banks, is derived as a new performance measure to take the place of the coding gain. This second method enables us to estimate the coding efficiency of arbitrary transform techniques as well as the subband approaches, and its result suggests that the SSKF(5 × 3) outperforms the QMF as long as the number of subbands is not too large, even though its filter length is much shorter. This result encourages us to find filter coefficients that maximize the unified coding gain according to filter length. In addition, new perfect reconstruction FIR filter banks which have not only low computational complexity but also good energy compaction properties are presented.     

Theory of rate-distortion-optimal, constrainedfilterbanks-application to IIR and FIR biorthogonal designs

We design filterbanks that are best matched to input signal statistics in M-channel subband coders, using a rate-distortion criterion. Previous research has shown that unconstrained-length, paraunitary filterbanks optimized under various energy compaction criteria are principal-component filterbanks that satisfy two fundamental properties: total decorrelation and spectral majorization. In this paper, we first demonstrate that the two properties above are not specific to the paraunitary case but are satisfied for a much broader class of design constraints. Our results apply to a broad class of rate-distortion criteria, including the conventional coding gain criterion as a special case. A consequence of these properties is that optimal perfect-reconstruction (PR) filterbanks take the form of the cascade of principal-component filterbanks and a bank of pre- and post-conditioning filters. The proof uses variational techniques and is applicable to a variety of constrained design problems. In the second part of this paper, we apply the theory above to practical filterbank design problems. We give analytical expressions for optimal IIR biorthogonal filterbanks; our analysis validates a conjecture by several researchers. We then derive the asymptotic limit of optimal FIR biorthogonal filterbanks as filter length tends to infinity. The performance loss due to FIR constraints is quantified theoretically and experimentally. The optimal filters are quite different from traditional filters. Finally, a sensitivity analysis is presented     

Subband image coding using watershed and watercourse lines of thewavelet transform

Reports progress in primitive-based image coding using nonorthogonal dyadic wavelets. A 3D isotropic wavelet is used to approximate the difference-of-Gaussians (D-o-G) operator. Convolution of the image with dilated versions of the wavelet produces three band-pass signals that approximate multiscale smoothed second derivatives. An additional convolution of the image with a Gaussian-shaped low-pass wavelet creates a fourth subband signal that preserves low-frequency information not described by the three band-pass signals. The authors show that the original image can be recovered from the watershed and watercourse lines of the three band-pass signals plus the lowpass subband signal. By thresholding the watershed/watercourse representation, subsampling the low-pass subband, and using edge post emphasis, the authors achieve data reduction with little loss of fidelity. Further compression of the watersheds and watercourses is achieved by chain coding their shapes and predictive coding their amplitudes prior to lossless arithmetic coding. Results are presented for grey-level test images at data rates between 0.1 and 0.3 b/pixel.     

Symmetric extension methods for M-channel linear-phaseperfect-reconstruction filter banks

The symmetric extension method has been shown to he an efficient way for subband processing of finite-length sequences. This paper presents an extension of this method to general linear-phase perfect-reconstruction filter banks. We derive constraints on the length and symmetry polarity of the permissible filter banks and propose a new design algorithm. In the algorithm, different symmetric sequences are formulated in a unified form based on the circular-symmetry framework. The length constraints in symmetrically extending the input sequence and windowing the subband sequences are investigated. The effect of shifting the input sequence is included. When the algorithm is applied to equal-length filter banks, we explicitly show that symmetric extension methods can always be constructed to replace the circular convolution approach     

Oversampled filter banks as error correcting codes: theory and impulse noise correction

Oversampled filter banks (OFBs) provide an overcomplete representation of their input signal. This paper describes how OFBs can be considered as error-correcting codes acting on real or complex sequences, very much like classical binary convolutional codes act on binary sequences. The structured redundancy introduced by OFBs in subband signals can be used to increase robustness to noise. In this paper, we define the notions of code subspace, syndrome, and parity-check polynomial matrix for OFBs. Furthermore, we derive generic expressions for projection-based decoding, suitable for the case when a simple second-order model completely characterizes the noise incurred by subband signals. We also develop a nonlinear hypotheses-test based decoding algorithm for the case when the noise in subbands is constituted by a Gaussian background noise and impulsive errors (a model that adequately describes the action of both quantization noise and transmission errors). Simulation results show that the algorithm effectively removes the effect of impulsive errors occurring with a probability of 10/sup -3/.     

Computation of the coding gain for subband coders

A procedure to evaluate the coding gain for 2-D subband systems is explicitly presented. The technique operates in the signal domain and requires the knowledge of the input process auto-correlation function. Both the case of uniform subband and pyramid decomposition are considered. In the case of a separable input process spectrum, the evaluation can be performed by considering appropriately defined 1-D systems, thus, making the procedure very convenient in terms of computational complexity. Using a model that has been recently derived for difference images in motion-compensated image sequence coders, we compare the performance of several filter banks and transform coders in terms of coding gain and asymptotic rate-distortion figures. The results for intraframe and interframe coding show that uniform subband coders can have a performance superior to that of transform coders. Pyramidal schemes appear to have a slightly worse performance     

Codage audio haute qualité (20 Hz-15 kHz) en sous bandes à débit réduit (64 kbit/s) et à faible retard (5 ms)

In this paper, we present a new method for high quality audio coding at low delay and low bit rate for telecommunications applications such as audioconfe-rence or videoconference. The developped coder is adapted to code generic audio signals at a bit rate of 64 kbit/s with a delay close to 5 ms in the 20-15000 Hz bandwidth. The method is based on speech coding as well as audio coding concepts. The coder combines subband decomposition of the input signal and LD-CELP techniques. We introduce in this structure of coding a psychoacoustic model which allows to allocate an optimal bit rate on each subband according to perceptual properties of the human hearing. In order to satisfy the bit rate requirement of the psychoacoustic model and to reduce the complexity of such a coding algorithm, we suggested a new method of vector quantization based on lattice quantization. This method allows to quantify the residual signal in the LD-CELP coder and avoid the complexity of the full search. Objective and subjective tests have been made on a test set of audio signals which is a critical sub-set used by ISO. Formal tests showed that the quality of the proposed coder is comparable to the best implementation of the MPEG-1, Layer II, but our solution has the advantage of reaching a very low delay (5 ms).     

Theory of optimal orthonormal subband coders

The theory of the orthogonal transform coder and methods for its optimal design have been known for a long time. We derive a set of necessary and sufficient conditions for the coding-gain optimality of an orthonormal subband coder for given input statistics. We also show how these conditions can be satisfied by the construction of a sequence of optimal compaction filters one at a time. Several theoretical properties of optimal compaction filters and optimal subband coders are then derived, especially pertaining to behavior as the number of subbands increases. Significant theoretical differences between optimum subband coders, transform coders, and predictive coders are summarized. Finally, conditions are presented under which optimal orthonormal subband coders yield as much coding gain as biorthogonal ones for a fixed number of subbands     

分布式子带自适应滤波算法

为了解决分布式最小均方算法在输入信号相关性较高时收敛速度较慢、分布式仿射投影算法计算复杂度较高等问题,本文提出了两种分布式子带自适应滤波算法,即递增式和扩散式子带自适应滤波算法.分布式子带自适应滤波算法将节点信号进行子带分割来降低信号的相关性,从而加快收敛速度.由于用于子带分割的滤波器组中包含了抽取单元,所以分布式子带自适应滤波算法和对应的分布式最小均方算法的计算复杂度相近.仿真结果表明,与分布式最小均方算法相比,分布式子带自适应滤波算法具有更好的收敛性能.