perfect reconstruction versus mmse filter banks in source coding classically the filter banks fbs used in source coding schemes have been chosen to possess the perfect reconstruction pr property or to be maximally selective quadrature mirror filters qmfs this paper puts this choice back into question and solves the problem of minimizing the reconstruction distortion which in the most general case is the sum of two terms a first one due to the non pr property of the fb and the other being due to signal quantization in the subbands the resulting filter banks are called minimum mean square error mmse filter banks several quantization noise models are considered first under the classical white noise assumption the optimal positive bit rate allocation in any filter bank possibly nonorthogonal is expressed analytically and an efficient optimization method of the mmse filter banks is derived then it is shown that while in a pr fb the improvement brought by an accurate noise model over the classical white noise one is noticeable it is not the case for the mmse fb the optimization of the synthesis filters is also performed for two measures of the bit rate the classical one which is defined for uniform scalar quantization and the order one entropy measure finally the comparison of rate distortion curves where the distortion is minimized for a given bit rate budget enables us to quantify the snr improvement brought by mmse solutions

nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r sup n new results on multidimensional filter banks and their connection to multidimensional nonseparable wavelets are presented among the topics discussed are sampling in multiple dimensions multidimensional perfect reconstruction filter banks the two channel case in multiple dimensions the synthesis of multidimensional filter banks and the design of compactly supported wavelets

filters for distortion free two band multirate filter banks in this paper conditions are given for a two band multi rate filter bank to be alias free and to have a unity frequency response it is shown that the class of quadrature mirror filters qmf s that satisfies these conditions is quite limited a class of filters which does satisfy these conditions is given and a simple procedure for designing filters from this class is presented with an example

scalar quantization error analysis for image subband coding using qmfs the authors analyze the coding errors due to quantization by explicitly incorporating a mathematical model for a lloyd max quantizer into a quadrature mirror filter qmf splitting and reconstruction scheme p h westerink et al 1988 this approach explicitly incorporates quantization errors into a qmf system by means of a quantizer model this makes it possible to discriminate between different types of coding errors such as the aliasing error other errors that can be distinguished are a qmf design error a signal error and a random error which is uncorrelated with the original image both a mean squared error calculation and a subjective judgment of the coding errors show that the aliasing errors can be neglected for filter lengths of 12 taps or more the signal error determines the sharpness of the reconstructed image while the random error is most visible in the flat areas

nonuniform filter bank design with noises this work solves the signal reconstruction problem involving nonuniform filter bank systems with rational decimation factors and noise three main nonuniform filter bank systems i e filter block decimator fbd structure upsampler filter downsampler ufd structure and tree structure are included in this study according to different operating conditions two different signal reconstruction problems for nonuniform filter bank systems with noise under the unknown but identifiable input signal model and the unknown input signal model are discussed respectively at the first stage a unified block state space model for different nonuniform filter bank systems with noise is developed then by incorporating the identified input signal model with this unified state space model and appropriate choice of the augmented state vector the signal reconstruction problem is reduced to an equivalent state estimation problem for resulting augmented systems if the input signal is identifiable if the input signal is lacking in modeling the signal reconstruction is discussed from the minimax estimation point of view two state estimation techniques involving robust kalman filtering and h sub spl infin filtering are employed respectively to treat the signal reconstruction problem of nonuniform filter bank systems according to different a priori knowledge of the input signal finally several numerical examples are presented to illustrate the proposed algorithms and exhibit the performances

regular wavelets a discrete time approach regularity is a new filter property brought by wavelet theory for perfect reconstruction octave band filter banks tools for investigating its role in coding applications are provided first discrete time interpretations and optimal estimates of regularity are reviewed then a simple design procedure for paraunitary fir filter banks with optimal trade off between frequency selectivity and regularity is given finally the obtained filters are used to measure the effect of regularity versus frequency selectivity in a still image compression scheme with optimized rate distortion in this case regularity is shown to be more relevant than frequency selectivity especially for short filters

statistical design of analysis synthesis systems with quantization a statistical model is used for the optimal design of analysis synthesis systems that include quantization of the signals in separate bands two types of quantization approaches are analyzed fine quantization modelled by additive noise and matrix quantization with proper modification the model applies to other types of quantization approaches assuming that the system operates as a waveform coder two error measures are used a generalization of the usual statistical mean square error mse to time varying systems and the mse between the outputs of the analysis of the original and reconstructed signals the measures are shown to be equivalent a method that is applicable to the design of optimal synthesis filters given the analysis filters is presented for fine quantization an iterative algorithm for the design of an optimal analysis synthesis system is presented together with its convergence properties if no quantization is applied the results obtained with the presented method coincide with previously reported results

subband coding systems incorporating quantizer models a new method for dealing with the effects of quantization in a subband system is proposed it uses the gain plus additive noise linear model for the lloyd max quantizer based on this it is demonstrated how by an appropriate choice of synthesis filters one can cancel all signal dependent errors at the output of the system the only remaining error is random in nature and not correlated with the input signal we therefore have a tradeoff between the error being only random or having signal dependent components as well since the error variances in both cases are comparable as a result of having only a random error it is possible to reduce this error using for example a noise removal technique the result is then extended to the case where the input is a multidimensional signal and arbitrary sampling lattices are used as well as to the qmf alias cancellation case to demonstrate the validity of the proposed approach two types of experiments on images are carried out in a toy example it is shown that using noise removal could be beneficial for a more realistic coding scheme however it is demonstrated that even in the case when the model is no longer valid when some of the subbands are discarded the output error is still much less correlated with the input signal as opposed to the commonly used subband system while visually the reconstructed images look very similar

synthesis filter bank optimization in two dimensional separable subband 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

environmental variability in a stochastic epidemic model abstract in this paper we investigate the stochastic dynamics of a simple epidemic model incorporating the mean reverting ornstein uhlenbeck process analytically and numerically we define two threshold parameters the stochastic demographic reproduction number r d s and the stochastic basic reproduction number r 0 s to utilize in identifying the stochastic extinction and persistence of the disease we find that the stochastic disease dynamics can be determined by the environment fluctuations which measured by the intensity of volatility and the speed of reversion the larger intensity of volatility or the smaller speed of reversion can suppress the outbreak of the disease the smaller intensity of volatility or the the higher speed of reversion can enhance the outbreak of the disease furthermore via numerical simulations we find that the stochastic model has an endemic stationary distribution which leads to the stochastic persistence of the disease our results show that mean reverting process is a well established way of introducing stochastic environmental noise into biologically realistic population dynamic models

optimal pyramidal decomposition for progressive multidimensional signal coding using optimal quantizers optimal hierarchical coding is sought for progressive or scalable multidimensional signal transmission by minimizing the variance of the error difference between the original image and its lower resolution renditions the optimal according to the above criterion pyramidal coders are determined for images quantized using the optimal vector lloyd max quantizers a rigorous general statistical model of a vector lloyd max quantizer is used consisting of a linear time invariant filter followed by additive noise uncorrelated with the input given arbitrary analysis filters the optimal synthesis filters are found the optimal analysis filters are subsequently determined leading to formulas for globally optimal structures for pyramidal multidimensional signal decompositions these structures produce replicas of the original image which at lower resolutions retain as much similarity to the original as possible this is highly useful for the progressive coding of two or three dimensional 2 d or 3 d images needed in applications such as fast browsing through image databases furthermore the minimization of the variance of the error image leads to minimization of the variance of the quantization noise for this image and hence to its optimally efficient compression experimental results illustrate the implementation and performance of the optimal pyramids in application for the coding of still 2 d images

optimal pyramidal and subband decompositions for hierarchical coding of noisy and quantized images optimal hierarchical coding is sought for progressive or scalable image transmission by minimizing the variance of the error difference between the original image and its lower resolution renditions the optimal according to the above criterion pyramidal and subband image coders are determined for images subject to corruption by quantization or transmission noise given arbitrary analysis filters and assuming adequate knowledge of the noise statistics optimal synthesis filters are found the optimal analysis filters are subsequently determined leading to formulas for globally optimal structures for pyramidal and subband image decompositions experimental results illustrate the implementation and performance of the optimal coders

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

bit allocation for dependent quantization with applications to multiresolution and mpeg video coders we address the problem of efficient bit allocation in a dependent coding environment while optimal bit allocation for independently coded signal blocks has been studied in the literature we extend these techniques to the more general temporally and spatially dependent coding scenarios of particular interest are the topical mpeg video coder and multiresolution coders our approach uses an operational rate distortion r d framework for arbitrary quantizer sets we show how a certain monotonicity property of the dependent r d curves can be exploited in formulating fast ways to obtain optimal and near optimal solutions we illustrate the application of this property in specifying intelligent pruning conditions to eliminate suboptimal operating points for the mpeg allocation problem for which we also point out fast nearly optimal heuristics additionally we formulate an efficient allocation strategy for multiresolution coders using the spatial pyramid coder as an example we then extend this analysis to a spatio temporal 3 d pyramidal coding scheme we tackle the compatibility problem of optimizing full resolution quality while simultaneously catering to subresolution bit rate or quality constraints we show how to obtain fast solutions that provide nearly optimal typically within 0 3 db full resolution quality while providing much better performance for the subresolution layer typically 2 3 db better than the full resolution optimal solution

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

three dimensional subband coding of video we describe and show the results of video coding based on a three dimensional 3 d spatio temporal subband decomposition the results include a 1 mbps coder based on a new adaptive differential pulse code modulation scheme adpcm and adaptive bit allocation this rate is useful for video storage on cd rom coding results are also shown for a 384 kbps rate that are based on adpcm for the lowest frequency band and a new form of vector quantization geometric vector quantization gvq for the data in the higher frequency bands gvq takes advantage of the inherent structure and sparseness of the data in the higher bands results are also shown for a 128 kbps coder that is based on an unbalanced tree structured vector quantizer utsvq for the lowest frequency band and gvq for the higher frequency bands the results are competitive with traditional video coding techniques and provide the motivation for investigating the 3 d subband framework for different coding schemes and various applications

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

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

a multiscale relaxation algorithm for snr maximization in nonorthogonal subband coding develops a technique for improving the applicability of complete nonorthogonal multiresolution transforms to image coding as is well known the l sup 2 norm of the quantization errors is not preserved by nonorthogonal transforms so the l sup 2 reconstruction error may be unacceptably large however given the quantizers and synthesis filters the authors show that this artifact can be eliminated by formulating the coding problem as that of minimizing the l sup 2 reconstruction error over the set of possible encoded images with this new formulation the coding problem becomes a high dimensional discrete optimization problem and features a coupling between the redundancy removing and quantization operations a practical solution to the optimization problem is presented in the form of a multiscale relaxation algorithm using inter and intrascale quantization noise feedback filters bounds on the coding gain over the standard coding technique are derived a simple extension of the algorithm allows for the use of a weighted l sup 2 error criterion and deadband non mmse quantizers experiments using biorthogonal spline filter banks demonstrate appreciable snr gains over the standard coding technique and comparable visual improvements