embedded image coding using zerotrees of wavelet coefficients the embedded zerotree wavelet algorithm ezw is a simple yet remarkably effective image compression algorithm having the property that the bits in the bit stream are generated in order of importance yielding a fully embedded code the embedded code represents a sequence of binary decisions that distinguish an image from the null image using an embedded coding algorithm an encoder can terminate the encoding at any point thereby allowing a target rate or target distortion metric to be met exactly also given a bit stream the decoder can cease decoding at any point in the bit stream and still produce exactly the same image that would have been encoded at the bit rate corresponding to the truncated bit stream in addition to producing a fully embedded bit stream the ezw consistently produces compression results that are competitive with virtually all known compression algorithms on standard test images yet this performance is achieved with a technique that requires absolutely no training no pre stored tables or codebooks and requires no prior knowledge of the image source the ezw algorithm is based on four key concepts 1 a discrete wavelet transform or hierarchical subband decomposition 2 prediction of the absence of significant information across scales by exploiting the self similarity inherent in images 3 entropy coded successive approximation quantization and 4 universal lossless data compression which is achieved via adaptive arithmetic coding

entropy constrained vector quantization an iterative descent algorithm based on a lagrangian formulation for designing vector quantizers having minimum distortion subject to an entropy constraint is discussed these entropy constrained vector quantizers ecvqs can be used in tandem with variable rate noiseless coding systems to provide locally optimal variable rate block source coding with respect to a fidelity criterion experiments on sampled speech and on synthetic sources with memory indicate that for waveform coding at low rates about 1 bit sample under the squared error distortion measure about 1 6 db improvement in the signal to noise ratio can be expected over the best scalar and lattice quantizers when block entropy coded with block length 4 even greater gains are made over other forms of entropy coded vector quantizers for pattern recognition it is shown that the ecvq algorithm is a generalization of the k means and related algorithms for estimating cluster means in that the ecvq algorithm estimates the prior cluster probabilities as well experiments on multivariate gaussian distributions show that for clustering problems involving classes with widely different priors the ecvq outperforms the k means algorithm in both likelihood and probability of error

extensions of the poisson summation formula the classical poisson summation formula 1 1 and the corresponding distributional formula 1 2 have found extensive applications in various scientific fields however they are not universally valid for instance if x is a smooth function the left hand side of 1 1 is generally divergent even when both sides of 1 1 converge absolutely they may do so to different numbers indeed in example 3 we are faced with the embarrassing situation where the series on the left hand side of 1 1 converges for re s 1 while that on the right hand side converges only for re s 0 our aim is to extend formulas 1 1 and 1 2 with the help of some new results in distributional theory for instance the evaluation of the distribution with zero mean as given by 3 1 at a test function x yields the relation k x dx both the series and the integral in this expression are generally divergent the concept of the cesaro limit is then used to obtain the finite difference of these two terms thereafter we extend the analysis to higher dimensions various innovative examples are presented to illustrate these concepts

the design of joint source and channel trellis waveform coders the generalized lloyd algorithm is applied to the design of joint source and channel trellis waveform coders to encode discrete time continuous amplitude stationary and ergodic sources operating over discrete memoryless noisy channels experimental results are provided for independent and autoregressive gaussian sources binary symmetric channels and absolute error and squared error distortion measures performance of the joint codes is compared with the tandem combination of a trellis source code and a trellis channel code on the independent gaussian source using the squared error distortion measure operating over an additive white gaussian noise channel it is observed that the jointly optimized codes achieve performance close to or better than that of separately optimized tandem codes of the same constraint length performance improvement via a predictive joint source and channel trellis code is demonstrated for the autoregressive gaussian source using the squared error distortion measure

image coding using wavelet transform a scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed this method involves two steps first a wavelet transform used in order to obtain a set of biorthogonal subclasses of images the original image is decomposed at different scales using a pyramidal algorithm architecture the decomposition is along the vertical and horizontal directions and maintains constant the number of pixels required to describe the image second according to shannon s rate distortion theory the wavelet coefficients are vector quantized using a multiresolution codebook to encode the wavelet coefficients a noise shaping bit allocation procedure which assumes that details at high resolution are less visible to the human eye is proposed in order to allow the receiver to recognize a picture as quickly as possible at minimum cost a progressive transmission scheme is presented it is shown that the wavelet transform is particularly well adapted to progressive transmission

optimal bit allocation via the generalized bfos algorithm p h westerink et al 1988 developed an optimal bit allocation algorithm that is simplified when all operational distortion rate functions which are referred to as quantizer functions are convex this algorithm is restated using the generalized breiman friedman olshen and stone bfos algorithm a recently developed technique for variable rate vector quantizer design for both cases of convex and nonconvex quantizer functions qfs and its complexity is analyzed the use of the generalized bfos algorithm for optimal bit allocation is analysed it is shown that if each source has a convex quantizer function then the complexity of the algorithm is low

adaptive entropy coded subband coding of image sequences a new approach to image sequence coding based on variable rate entropy constrained subband coding ecsbc is described the corresponding practical implementation of the ecsbc scheme for fixed rate channels is developed by extending recent adaptive entropy coded aec quantization techniques although the entropy constrained design of subband coding systems provides improve coding efficiency compared to level constrained design approaches the resulting coding system generates variable rate outputs which must be buffered before fixed rate transmissions in this case the finite buffer however large that interfaces the encoder and the channel will eventually overflow or underflow resulting in a catastrophic loss of encoder decoder synchronism with an associated large amount of distortion a buffer adaptive arithmetic coded implementation of the ecsbc scheme called adaptive entropy coded subband coding ecsbc aec is described to completely eliminate the associated encoder buffer overflow underflow problems even with a very small encoder buffer

a stabilization algorithm for multichannel multidimensional linear prediction of imagery the authors have investigated the stability problems observed in multichannel multidimensional linear predictive modeling of images it is known that based on a positive definite autocorrelation matrix singular values of the matrix h delta sub i 1 herm delta sub i 1 must lie inside the unit circle for a stable solution where delta sub i 1 is the normalized partial correlation matrix and herm denotes the hermitian operator the authors have developed a two step stabilization method to obtain stabilized linear prediction coefficients for short term analysis windows formed digitized images the authors have modified the multichannel levinson recursion algorithm to include this stability procedure they have tested the algorithm on numerous images commonly used in image coding and the results are very impressive

modeling of multichannel time series and extrapolation of matrix valued autocorrelation sequences this paper deals with multichannel time series and extends the autocorrelation method for fitting the time series from the nondegenerate case to the general case we shall show that the ar model obtained by solving the normal equations is always stable even in the degenerate case the multichannel levinson algorithm for solving the normal equations is extended from the nondegenerate case to the general case in the nondegenerate case the first n 1 values of the auto correlation sequence of the obtained ar model of order n are identical with those of the original data the next considered is the extrapolation problem of an autocorrelation sequence that is given data r 0 r 1 r n it is to construct a model whose first n 1 values of the autocorrelation sequence are identical with these data we shall show that even in the degenerate case when the given data r 0 r 1 r n are taken from the autocorrelation sequence of a purely nondeterministic time series they are identical with the first n 1 values of the autocorrelation sequence of the ar model of order n

subband coding of images subband coding has become quite popular for the source encoding of speech this paper presents a simple yet efficient extension of this concept to the source coding of images we specify the constraints for a set of two dimensional quadrature mirror filters qmf s for a particular frequency domain partition and show that these constraints are satisfied by a separable combination of one dimensional qmf s bits are then optimally allocated among the subbands to minimize the mean squared error for dpcm coding of the subbands also an adaptive technique is developed to allocate the bits within each subband by means of a local variance mask optimum quantization is employed with quantizers matched to the laplacian distribution subband coded images are presented along with their signal to noise ratios snr s the snr performance of the subband coder is compared to that of the adaptive discrete cosine transform dct vector quantization and differential vector quantization for bit rates of 0 67 1 0 and 2 0 bits per pixel for 256 256 monochrome images the adaptive subband coder has the best snr performance

multiple local optima in vector quantizers corresp two results are presented on vector quantizers meeting necessary conditions for optimality first a simple generalization of well known centroid and moment properties of the squared error distortion measure to a weighted quadratic distortion measure with an input dependent weighting is presented the second result is an application of the squared error special case of the first result to a simulation study of the design of 1 bit per sample two and three dimensional quantizers for a memoryless gaussian source using the generalized lloyd technique the existence of multiple distinct local optima is demonstrated thereby showing that sufficient conditions for unique local optima do not exist for this simple common case it is also shown that at least three dimensions are required for a vector quantizer to outperform a scalar quantizer for this source

improving the rate distortion performance of dpcm using multirate processing with application in low rate image coding differential pulse code modulation dpcm is able to code highly correlated sources efficiently at high bit rates but not at low bit rate regions motivated by the rate distortion theory a simple modified dpcm codec using multirate processing is proposed a low pass filter is used to limit the source signal spectrum according to the optimal mapping scheme bit rate reduction is achieved by reducing the number of transmitted samples through downsampling a wiener filter is appended at the decoder for further noise reduction simulation results agree with theoretical analysis and show that for a first order autoregressive ar 1 process the proposed codec makes substantial improvements over the classic dpcm and performs close to the rate distortion bound the proposed system is then implemented as a low rate predictive image coder in addition to apply two dimensional dpcm adaptive entropy coding is used to exploit variation in the local statistics of the prediction error image simulation results show that the system is able to outperform jpeg at low rates and has subjective quality comparable to that of jpeg2000

trellis excitation speech coding at low bit rates trellis coded vector quantization tcvq and code excited linear prediction celp coding are combined to form an efficient low bit rate speech coding system the resulting system uses a trellis search to select the synthesis filter excitation sequence and is referred to as trellis excitation coding tec simulations are performed for encoding rates of 6 4 and 8 kbps informal listening tests indicate that the 8 kbps tec system has quality roughly between that of 6 bit and 7 bit spl mu law pcm with spl mu 255 the 6 4 kbps tec system provides speech quality between 5 bit and 6 bit spl mu law pcm a subjective comparison with vector sum excited linear prediction vselp indicates that the 8 kbps tec and the vselp reconstructed speech is about equally preferable

gradient algorithms for designing predictive vector quantizers a predictive vector quantizer pvq is a vector extension of a predictive quantizer it consists of two parts a conventional memoryless vector quantizer vq and a vector predictor two gradient algorithms for designing a pvq are developed in this paper the steepest descent sd algorithm and the stochastic gradient sg algorithm both have the property of improving the quantizer and the predictor in the sense of minimizing the distortion as measured by the average mean squared error the differences between the two design approaches are the period and the step size used in each iteration to update the codebook and predictor the sg algorithm updates once for each input training vector and uses a small step size while the sd updates only once for a long period possibly one pass over the entire training sequence and uses a relatively large step size code designs and tests are simulated for both gauss markov sources and for sampled speech waveforms and the results are compared to codes designed using techniques that attempt to optimize only the quantizer for the predictor and not vice versa

predictive vector quantizer design using deterministic annealing a new approach is proposed for predictive vector quantizer pvq design which is inherently probabilistic and is based on ideas from information theory and analogies to statistical physics the approach effectively resolves three longstanding fundamental shortcomings of standard pvq design the first complication is due to the pvq prediction loop which has a detrimental impact on the convergence and the stability of the design procedure the second shortcoming is due to the piecewise constant nature of the quantizer function which makes it difficult to optimize the predictor with respect to the overall reconstruction error finally a shortcoming inherited from standard vq design is the tendency of the design algorithm to terminate at a locally rather than the globally optimal solution we propose a new pvq design approach that embeds our previous asymptotic closed loop acl approach within a deterministic annealing da framework the overall da acl method profits from its two main components in a complementary way acl is used to overcome the first difficulty and offers the means for stable quantizer design as it provides an open loop design platform yet allows the pvq design algorithm to asymptotically converge to optimization of the closed loop performance objective da simultaneously mitigates or eliminates the remaining design shortcomings its probabilistic framework replaces hard quantization with a differentiable expected cost function that can be jointly optimized for the predictor and quantizer parameters and its annealing schedule allows the avoidance of many poor local optima substantial performance gains over traditional methods have been achieved in the simulations

multiplication free vector quantization using l sub 1 distortion measure and its variants the author considers vector quantization that uses the l sub 1 distortion measure for its implementation a gradient based approach for codebook design that does not require any multiplications or median computation is proposed convergence of this method is proved rigorously under very mild conditions simulation examples comparing the performance of this technique with the lbg algorithm show that the gradient based method in spite of its simplicity produces codebooks with average distortions that are comparable to the lbg algorithm the codebook design algorithm is then extended to a distortion measure that has piecewise linear characteristics once again by appropriate selection of the parameters of the distortion measure the encoding as well as the codebook design can be implemented with zero multiplications the author applies the techniques in predictive vector quantization of images and demonstrates the viability of multiplication free predictive vector quantization of image data

trellis coded quantization of memoryless and gauss markov sources trellis coded quantization tcq is developed and applied to the encoding of memoryless and gauss markov sources the theoretical justification for the approach is alphabet constrained rate distortion theory which is a dual to the channel capacity argument that motivates trellis coded modulation tcm the authors adopt the notions of signal set expansion set partitioning and branch labeling of tcm but modify the techniques to account for the source distribution to design tcq coders of low complexity with excellent mean squared error mse performance for a memoryless uniform source tcq provides an mse within 0 21 db of the distortion rate bound at all positive integral rates the performance is superior to that promised by the coefficient of quantization for all of the best lattices known in dimensions 24 or less for a memoryless gaussian source the tcq performance at rates of 0 5 1 and 2 b sample is superior to all previous results the authors found in the literature the encoding complexity of tcq is very modest tcq is incorporated into a predictive coding structure for the encoding of gauss markov sources simulation results for first second and third order gauss markov sources are presented

borel transform resurgence and the density of states lattice schrodinger operators with exponential disorder the authors study the averaged green function of a tight binding model with exponentially distributed disorder as an explicit example for a resurgent function in the sense of ecalle 1981 as an application one can find the exact asymptotic decay of the density of states and the large order behaviour of the coefficients of the perturbation expansion

fast quantizing and decoding and algorithms for lattice quantizers and codes for each of the lattices a n n geq 1 d n n geq 2 e 6 e 7 e 8 and their duals a very fast algorithm is given for finding the closest lattice point to an arbitrary point if these lattices are used for vector quantizing of uniformly distributed data the algorithm finds the minimum distortion lattice point if the lattices are used as codes for a gaussian channel the algorithm performs maximum likelihood decoding

vector quantization of speech and speech like waveforms an algorithm for the design of vector quantizers that are locally optimum in the sense of minimizing an average quantitative distortion measure is used to design 1 and 2 bit sample vector quantizers for both real sampled speech and a simulated speech like auto regressive random process both weighted and unweighted squared error distortion measures are considered several comparisons are made and discussed based on the average distortions of the vector quantization schemes the results for the simulated speech are compared to mathematical performance bounds from information theory to provide an indication of how nearly globally optimal vector quantization is for such highly correlated sources a comparison of the results for the real speech and the simulated speech provides a quantitative measure of the accuracy of such models and hence of the applicability of information theory bounds and code designs based on probabilistic models the signal to quantization noise ratios of vector quantizers designed to minimize squared error distortion are compared to those of several popular speech waveform coding systems of similar rates

wavelets and filter banks theory and design the wavelet transform is compared with the more classical short time fourier transform approach to signal analysis then the relations between wavelets filter banks and multiresolution signal processing are explored a brief review is given of perfect reconstruction filter banks which can be used both for computing the discrete wavelet transform and for deriving continuous wavelet bases provided that the filters meet a constraint known as regularity given a low pass filter necessary and sufficient conditions for the existence of a complementary high pass filter that will permit perfect reconstruction are derived the perfect reconstruction condition is posed as a bezout identity and it is shown how it is possible to find all higher degree complementary filters based on an analogy with the theory of diophantine equations an alternative approach based on the theory of continued fractions is also given these results are used to design highly regular filter banks which generate biorthogonal continuous wavelet bases with symmetries

classified vector quantization of images vector quantization vq provides many attractive features for image coding with high compression ratios however initial studies of image coding with vq have revealed several difficulties most notably edge degradation and high computational complexity we address these two problems and propose a new coding method classified vector quantization cvq which is based on a composite source model blocks with distinct perceptual features such as edges are generated from different subsources i e belong to different classes in cvq a classifier determines the class for each block and the block is then coded with a vector quantizer designed specifically for that class we obtain better perceptual quality with significantly lower complexity with cvq when compared to ordinary vq we demonstrate with cvq visual quality which is comparable to that produced by existing coders of similar complexity for rates in the range 0 6 1 0 bits pixel

regularization pseudofunction and hadamard finite part abstract first we discuss and correlate the various types of regularizations available in the literature for the singular function h x x k where k is an integer and h x is the heaviside function then we present the corresponding regularization for the function r k where r is the radial distance in r n thereafter we express the recently discovered distributional derivatives of this function in terms of pseudofunctional language commonly used in the coulomb gravitational and interparticle potentials where the function 1 r plays a fundamental role

efficient codebooks for vector quantization image compression with an adaptive tree search algorithm this paper discusses some algorithms to be used for the generation of an efficient and robust codebook for vector quantization vq some of the algorithms reduce the required codebook size by 4 or even 8 b to achieve the same level of performance as some of the popular techniques this helps in greatly reducing the complexity of codebook generation and encoding we also present a new adaptive tree search algorithm which improves the performance of any product vq structure our results show an improvement of nearly 3 db over the fixed rate search algorithm at a bit rate of 0 75 b pixel

a distributional theory for asymptotic expansions the purpose of this chapter is to present the distributional theory of asymptotic expansions for functions of one variable this chapter and the next where the multidimensional expansions are studied are the central part of the book

on vector transformation it has been shown that a vector transform is a good tool for digital filtering and image coding this paper addresses some of the issues related to vector transforms first a fundamental difference between vector transforms and conventional transforms is examined second two approaches to constructing vector transforms are discussed finally an application of vector transformation to an efficient implementation of 2 d nonseparable digital filters is proposed

generalized cost measure based address predictive vector quantization address predictive vector quantization apvq exploits the interblock dependency by jointly encoding the addresses of the codewords associated with spatially close blocks it profiles the same image quality as memoryless vq for a much lesser bit rate br and the same computational complexity in the generalized cost measure based apvq the two steps of the encoding process namely vq and predictive address encoding are carried out jointly by minimizing a generalized cost measure which takes into account both the br and the distortion computer simulations show that a significant improvement can be obtained with respect to apvq in terms of both br and distortion compared with memoryless vq a bit rate reduction of almost 60 is obtained for the same image quality

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

side match and overlap match vector quantizers for images a class of vector quantizers with memory that are known as finite state vector quantizers fsvqs in the image coding framework is investigated two fsvq designs namely side match vector quantizers smvqs and overlap match vector quantizers omvqs are introduced these designs take advantage of the 2 d spatial contiguity of pixel vectors as well as the high spatial correlation of pixels in typical gray level images smvq and omvq try to minimize the granular noise that causes visible pixel block boundaries in ordinary vq for 512 by 512 gray level images smvq and omvq can achieve communication quality reproduction at an average of 1 2 b pixel per image frame and acceptable quality reproduction because block boundaries are less visible the perceived improvement in quality over ordinary vq is even greater owing to the structure of smvq and omvq simple variable length noiseless codes can achieve as much as 60 bit rate reduction over fixed length noiseless codes

the inverse of a borel summable function let b1 be the borel transform of f f 0 0 by means of an integral equation similar to that proposed by t hooft the whys of subnuclear physics edited by a zichichi plenum new york 1979 an explicit representation of the borel transform b2 of 1 f as a series in convolution powers of b1 and its derivative is given it is proved that the singularities of b1 b2 on the circle of convergence coincide and have the same strength when approaching these singularities on a path inside their common analyticity domain but possibly outside the circle of convergence of their power series representation borel circle

wavelet filter evaluation for image compression choice of filter bank in wavelet compression is a critical issue that affects image quality as well as system design although regularity is sometimes used in filter evaluation its success at predicting compression performance is only partial a more reliable evaluation can be obtained by considering an l level synthesis analysis system as a single input single output linear shift variant system with a response that varies according to the input location module 2 sup l 2 sup l by characterizing a filter bank according to its impulse response and step response in addition to regularity we obtain reliable and relevant for image coding filter evaluation metrics using this approach we have evaluated all possible reasonably short less than 36 taps in the synthesis analysis pair minimum order biorthogonal wavelet filter banks of this group of over 4300 candidate filter banks we have selected and present here the filters best suited to image compression while some of these filters have been published previously others are new and have properties that make them attractive in system design

optimal nonlinear interpolative vector quantization a process by which a reduced dimensionality feature vector can be extracted from a high dimensionality signal vector and then vector quantized with lower complexity than direct quantization of the signal vector is discussed in this procedure a receiver must estimate or interpolate the signal vector from the quantized features the task of recovering a high dimensional signal vector from a reduced dimensionality feature vector can be viewed as a generalized form of interpolation or prediction a way in which optimal nonlinear interpolation can be achieved with negligible complexity eliminating the need for ad hoc linear or nonlinear interpolation techniques is presented the range of applicability of nonlinear interpolative vector quantization is illustrated with examples in which optimal nonlinear estimation from quantized data is needed for efficient signal compression

asymptotics of the cut discontinuity and large order behaviour from the instanton singularity the case of lattice schrodinger operators with exponential disorder we discuss the relation between the singularity structure of the borel transform the asymptotics of the cut discontinuity and the large order behaviour of perturbation theory in an explicit example a tight binding model with exponential disorder we show how to obtain the first instanton singularity from a cluster expansion in the borel variable and as an application we determine the exact decay of the density of states ase the method opens some perspectives for similar problems arising from different models of mathematical physics

adaptive entropy coded predictive vector quantization of images the authors consider 2 d predictive vector quantization pvq of images subject to an entropy constraint and demonstrate the substantial performance improvements over existing unconstrained approaches they describe a simple adaptive buffer instrumented implementation of this 2 d entropy coded pvq scheme which can accommodate the associated variable length entropy coding while completely eliminating buffer overflow underflow problems at the expense of only a slight degradation in performance this scheme called 2 d pvq aecq adaptive entropy coded quantization is shown to result in excellent rate distortion performance and impressive quality reconstructions of real world images indeed the real world coding results shown demonstrate little distortion at rates as low as 0 5 b pixel

power series quantization for noisy channels a recently proposed method for transmission of correlated sources under noise free conditions power series quantization psq uses a separate linear or nonlinear predictor for each quantizer region and has shown to increase performance compared to several common quantization schemes for sources with memory in this paper it is shown how to apply psq for transmission of a source with memory over a noisy channel a channel optimized psq copsq encoder and codebook optimization algorithms are derived the suggested scheme is shown to increase performance compared with previous state of the art methods

adaptive entropy coded subband coding of images the authors describe a design approach called 2 d entropy constrained subband coding ecsbc based upon recently developed 2 d entropy constrained vector quantization ecvq schemes the output indexes of the embedded quantizers are further compressed by use of noiseless entropy coding schemes such as huffman or arithmetic codes resulting in variable rate outputs depending upon the specific configurations of the ecvq and the ecpvq over the subbands many different types of sbc schemes can be derived within the generic 2 d ecsbc framework among these the authors concentrate on three representative types of 2 d ecsbc schemes and provide relative performance evaluations they also describe an adaptive buffer instrumented version of 2 d ecsbc called 2 d ecsbc aec for use with fixed rate channels which completely eliminates buffer overflow underflow problems this adaptive scheme achieves performance quite close to the corresponding ideal 2 d ecsbc system

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

series the convergence of which should be interpreted in the sense of l schwartz s distributions often in practical questions divergent multiple series arise and are handled by heuristic ad hoc techniques taking up the discussion in connection with siam problem 92 11 we show how to treat such series in a mathematically rigorous way by the use of the weak convergence concept of distribution theory

predictive multistage vector quantizer design using asymptotic closed loop optimization this correspondence builds on the asymptotic closed loop approach to predictive vector quantizer design khalil et al 2001 and extends it to the design of predictive multistage vector quantizers for low bit rate video coding the design approach resolves longstanding shortcomings in particular design stability and empty cell problems simulation results show substantial gains over traditional design approaches

borel summability in the disorder parameter of the averaged green s function for gaussian disorder in this note we prove borel summability in the disorder parameter of the averaged green s function y of tight binding models h v delta v with gaussian disorder d lambda v 2 pi gamma 1 2 exp v 2 2 gamma dv for 0 and fixed large e using this we can reconstruct the density of states e from the borel sums of y with ime 0 and ime 0

vector quantization of image subbands a survey subband and wavelet decompositions are powerful tools in image coding because of their decorrelating effects on image pixels the concentration of energy in a few coefficients their multirate multiresolution framework and their frequency splitting which allows for efficient coding matched to the statistics of each frequency band and to the characteristics of the human visual system vector quantization vq provides a means of converting the decomposed signal into bits in a manner that takes advantage of remaining inter and intraband correlation as well as of the more flexible partitions of higher dimensional vector spaces since 1988 a growing body of research has examined the use of vq for subband wavelet transform coefficients we present a survey of these methods

the asymptotic closed loop approach to predictive vector quantizer design with application in video coding the basic vector quantization vq technique employed in video coding belongs to the category of predictive vector quantization pvq as it involves quantization of the motion compensated frame prediction error it is well known that the design of pvq suffers from fundamental difficulties due to the prediction loop which have an impact on the convergence and the stability of the design procedure we propose an approach to pvq design that enjoys the stability of open loop design while it ensures ultimate optimization of the closed loop system the method is derived for general predictive quantization and we demonstrate it on video compression at low bit rates where it provides substantial improvement over standard open and closed loop design techniques further the approach outperforms standard dct based video coding

geometric source coding and vector quantization a geometric formulation is presented for source coding and vector quantizer design motivated by the asymptotic equipartition principle the authors consider two broad classes of source codes and vector quantizers elliptical codes and quantizers based on the gaussian density function and pyramid codes and quantizers based on the laplacian density function elliptical and weighted pyramid vector quantizers are developed by selecting codewords as points in a lattice that lie on or near a specified ellipse or pyramid the combination of geometric structure and lattice basis allows simple encoding and decoding algorithms

predictive residual vector quantization image coding this paper presents a new vector quantization technique called predictive residual vector quantization prvq it combines the concepts of predictive vector quantization pvq and residual vector quantization rvq to implement a high performance vq scheme with low search complexity the proposed prvq consists of a vector predictor designed by a multilayer perceptron and an rvq that is designed by a multilayer competitive neural network a major task in our proposed prvq design is the joint optimization of the vector predictor and the rvq codebooks in order to achieve this a new design based on the neural network learning algorithm is introduced this technique is basically a nonlinear constrained optimization where each constituent component of the prvq scheme is optimized by minimizing an appropriate stage error function with a constraint on the overall error this technique makes use of a lagrangian formulation and iteratively solves a lagrangian error function to obtain a locally optimal solution this approach is then compared to a jointly designed and a closed loop design approach in the jointly designed approach the predictor and quantizers are jointly optimized by minimizing only the overall error in the closed loop design however a predictor is first implemented then the stage quantizers are optimized for this predictor in a stage by stage fashion simulation results show that the proposed prvq scheme outperforms the equivalent rvq operating at the same bit rate and the unconstrained vq by 2 and 1 7 db respectively furthermore the proposed prvq outperforms the pvq in the rate distortion sense with significantly lower codebook search complexity

image coding using adaptive recursive interpolative dpcm a predictive image coder having minimal decoder complexity is presented the image coder utilizes recursive interpolative dpcm in conjunction with adaptive classification entropy constrained trellis coded quantization and optimal rate allocation to obtain signal to noise ratios snrs in the range of those provided by the most advanced transform coders

on adaptive vector transform quantization for speech coding adaptive vector transform quantization avtq as a coding system is discussed the optimal bit assignment is derived based on vector quantization asymptotic theory for different pdfs probability density functions of the transform coefficients strategies for shaping the quantization noise spectrum and for adapting the bit assignment to the changes in the speech statistics are discussed a good estimate of the efficiency of any coding system is given by the system coding gain over scalar pcm pulse code modulation based on the optimal bit allocation the coding gain of the vector transform quantization vtq system operating on a stationary input signal is derived the vtq coding gain demonstrates a significant advantage of vector quantization over scalar quantization within the framework of transform coding system simulation results are presented for a first order gauss markov process and for typical speech waveforms the results of fixed and adaptive systems are compared for speech input also the avtq results are compared to known scalar speech coding systems

image compression by vector quantization of multiresolution decompositions abstract an image compression scheme is introduced which involves a multiresolution decomposition derived from the wavelet transform the use of this transformation in image coding is motivated by its similarity to the processing which occurs at the cortical level of the visual system and by the fact that subband coding in general allows the coder to be better matched to the signal statistics in this work each subband is coded separately by vector quantization this work differs from previous wavelet transform coding schemes involving vector quantization in that the quantization parameters for each subband are selected by a nonlinear optimization procedure which involves constraints on the overall bit rate and the encoding complexity the multiplicative visual model is reviewed and a motivation for cascading this system with the wavelet transform to give a more complete description of human vision is provided empirical results are presented which demostrate the effectiveness of the technique for compressing monochromatic images signal to noise ratio measurements are shown which demostrate the system performance for various selections of bit rate and complexity

on the structure of vector quantizers vector quantization is intrinsically superior to predictive coding transform coding and other suboptimal and em ad hoc procedures since it achieves optimal rate distortion performance subject only to a constraint on memory or block length of the observable signal segment being encoded the key limitation of existing techniques is the very large randomly generated code books which must be stored and the computational complexity of the associated encoding procedures the quantization operation is decomposed into its rudimentary structural components this leads to a simple and elegant approach to derive analytical properties of optimal quantizers some useful properties of quantizers and algorithmic approaches are given which are relevant to the complexity of both storage and processing in the encoding operation highly disordered quantizers which have been designed using a clustering algorithm are considered finally lattice quantizers are examined which circumvent the need for a code book by using a highly structured code based on lattices the code vectors are algorithmically generated in a simple manner rather than stored in a code book and fast algorithms perform the encoding algorithm with negligible complexity

best wavelet packet bases in a rate distortion sense a fast rate distortion r d optimal scheme for coding adaptive trees whose individual nodes spawn descendents forming a disjoint and complete basis cover for the space spanned by their parent nodes is presented the scheme guarantees operation on the convex hull of the operational r d curve and uses a fast dynamic programing pruning algorithm to markedly reduce computational complexity applications for this coding technique include r coefman et al s yale univ 1990 generalized multiresolution wavelet packet decomposition iterative subband coders and quadtree structures applications to image processing involving wavelet packets as well as discrete cosine transform dct quadtrees are presented

image coding using vector quantization a review a review of vector quantization techniques used for encoding digital images is presented first the concept of vector quantization is introduced then its application to digital images is explained spatial predictive transform hybrid binary and subband vector quantizers are reviewed the emphasis is on the usefulness of the vector quantization when it is combined with conventional image coding techniques or when it is used in different domains

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

product code vector quantizers for waveform and voice coding memory and computation requirements imply fundamental limitations on the quality that can be achieved in vector quantization systems used for speech waveform coding and linear predictive voice coding lpc one approach to reducing storage and computation requirements is to organize the set of reproduction vectors as the cartesian product of a vector codebook describing the shape of each reproduction vector and a scalar codebook describing the gain or energy such shape gain vector quantizers can be applied both to waveform coding using a quadratic error distortion measure and to voice coding using an itakura saito distortion measure in each case the minimum distortion reproduction vector can be found by first selecting a shape code word and then based on that choice selecting a gain codeword several algorithms are presented for the design of shape gain vector quantizers based on a traning sequence of data or a probabilistic model the algorithms are used to design shape gain vector quantizers for both the waveform coding and voice coding application the quantizers are simulated and their performance is compared to that of previously reported vector quantization systems

analyticity of the density of states and replica method for random schrodinger operators on a lattice we analyze the density of states and some aspects of the replica method for anderson s tight binding model on a lattice of arbitrary dimension with diagonal disorder we give heuristic arguments for the conjectures that the classical value of the exponent of the localization length is 1 2 and that the upper critical dimension d c loc is bounded by 4 d c loc 6

next state functions for finite state vector quantization in this paper the finite state vector quantization scheme called dynamic finite state vector quantization dfsvq is investigated with regard to its subcodebook construction in dfsvq each input block is encoded by a small codebook called subcodehook which is created from a much larger codebook called supercodebook each subcodebook is constructed by selecting using a reordering procedure a set of appropriate codevectors from the supercodebook the performance of the dfsvq depends on this reordering procedure therefore several reordering procedures are introduced and their performances are evaluated in this paper the reordering procedures that are investigated are based on the conditional histogram of the codevectors index prediction vector prediction nearest neighbor design and frequency usage of the code vectors the performances of the reordering procedures are evaluated by comparing their hit ratios the number of blocks encoded by the subcodebook and their computational complexity experimental results are presented and it is found that the reordering procedure based on the vector prediction performs the best when compared with the other reordering procedures

vector quantizers with direct sum codebooks the use of direct sum codebooks to minimize the memory requirements of vector quantizers is investigated assuming arbitrary fixed partitions necessary conditions for minimum distortion codebooks are derived first for scalar codebooks assuming mean squared error distortion and then for vector codebooks and a broader class of distortion measures an iterative procedure is described for designing locally optimal direct sum codebooks both optimal and computationally efficient suboptimal encoding schemes are considered it is shown that although an optimal encoding can be implemented by a sequential encoder the complexity of implementing optimal stagewise partitions generally exceeds the complexity of an exhaustive search of the direct sum codebook it is also shown that sequential nearest neighbor encoders can be extremely inefficient the m search method is explored as one method of improving the effectiveness of suboptimal sequential encoders representative results for simulated direct sum quantizers are presented

dynamic finite state vector quantization of digital images a vector quantization vq scheme with finite memory called dynamic finite state vector quantization dfsvq is presented the encoder consists of a large codebook so called super codebook where for each input vector a fixed number of its codevectors are chosen to generate a much smaller codebook sub codebook this sub codebook represents the best matching codevectors that could be found in the super codebook for encoding the current input vector the choice for the codevectors in the sub codebook is based on the information obtained from the previously encoded blocks where directional conditional block probability histogram matrices are used in the selection of the codevectors the index of the best matching codevector in the sub codebook is transmitted to the receiver an adaptive dfsvq scheme is also proposed in which when encoding an input vector first the sub codebook is searched for a matching codevector to satisfy a pre specified waveform distortion if such a codevector is not found in tile current sub codebook then the whole super codebook is checked for a better match if a better match is found then a signaling flag along with the corresponding index of the codevector is transmitted to the receiver both the dfsvq encoder and its adaptive version are implemented experimental results for several monochrome images with a super codebook size of 256 or 512 and different sub codebook sizes are presented

ell_ 2 optimized predictive image coding with ell_ infty bound in many scientific medical and defense applications of image video compression an l error bound is required however pure l optimized image coding colloquially known as near lossless image coding is prone to structured errors such as contours and speckles if the bit rate is not sufficiently high moreover most of the previous l based image coding methods suffer from poor rate control in contrast the l2 error metric aims for average fidelity and hence preserves the subtlety of smooth waveforms better than the l error metric and it offers fine granularity in rate control but pure l2 based image coding methods e g jpeg 2000 cannot bound individual errors as the l based methods can this paper presents a new compression approach to retain the benefits and circumvent the pitfalls of the two error metrics a common approach of near lossless image coding is to embed into a dpcm prediction loop a uniform scalar quantizer of residual errors the said uniform scalar quantizer is replaced in the proposed new approach by a set of context based l2 optimized quantizers the optimization criterion is to minimize a weighted sum of the l2 distortion and the entropy while maintaining a strict l error bound the resulting method obtains good rate distortion performance in both l2 and l metrics and also increases the rate granularity compared with jpeg 2000 the new method not only guarantees lower l error for all bit rates but also it achieves higher psnr for relatively high bit rates

analysis and further results on adaptive entropy coded quantization buffer underflow and overflow problems associated with entropy coding are completely eliminated by effectively imposing reflecting walls at the buffer endpoints synchronous operation of the aecq adaptive entropy coded quantizer encoder and decoder is examined in detail and it is shown that synchronous operation is easily achieved without side information a method is developed to explicitly solve for the buffer state probability distribution and the resulting average distortion when memoryless buffer state feedback is used as well as when the source is stationary and memoryless this method is then used as a tool in the design of low distortion aecq systems with particular attention given to developing source scale invariant distortion performance it is shown that the introduction of reflecting buffer walls in a properly designed aecq system results in a very small rate distortion performance penalty and that the resulting aecq system can be an extremely simple and effective solution to the stationary memoryless source coding problem for a wide range of source types operation with nonstationary sources is also examined

vector adaptive vector quantization with application to speech coding a vector adaptive vector quantization vavq scheme which may be viewed as a generalization of gain adaptive vector quantization is described the proposed scheme adjusts each component of the encoding signal vector according to a statistical estimate of the signal characteristics the vavq scheme can cope with large input dynamic ranges it can be used in either the time domain or the transform domain and exhibits approximately 4 db improvement in segmental signal to noise ratio snr over the fixed vq

a greedy tree growing algorithm for the design of variable rate vector quantizers image compression a technique for directly designing a variable rate tree structured vector quantizer by growing the tree one node at a time rather than one layer at time is presented the technique is a natural extension of a tree growing method for decision trees when the tree is pruned with a generalized algorithm for optimally pruning trees improvement is measured in the signal to noise ratio at high rates over pruning a fixed rate tree structured vector quantizer of the same initial rate the growing algorithm can be interpreted as a constrained inverse of the pruning algorithm

adaptive entropy coded pruned tree structured predictive vector quantization of images simpler versions of a previously introduced adaptive entropy coded predictive vector quantization pvq scheme where the embedded entropy constrained vector quantizer ecvq is replaced by a pruned tree structured vq ptsvq are described the resulting encoding scheme is shown to result in drastically reduced complexity at only a small cost in performance coding results for selected real world images are given

trellis coded vector quantization trellis coded quantization is generalized to allow a vector reproduction alphabet three encoding structures are described several encoder design rules are presented and two design algorithms are developed it is shown that for a stationary ergodic vector source if the optimized trellis coded vector quantization reproduction process is jointly stationary and ergodic with the source then the quantization noise is zero mean and of a variance equal to the difference between the source variance and the variance of the reproduction sequence several examples illustrate the encoder design procedure and performance

optimal pruning with applications to tree structured source coding and modeling an algorithm introduced by l breiman et al 1984 in the context of classification and regression trees is reinterpreted and extended to cover a variety of applications in source coding and modeling in which trees are involved these include variable rate and minimum entropy tree structured vector quantization minimum expected cost decision trees variable order markov modeling optimum bit allocation and computer graphics and image processing using quadtrees a concentration on the first of these and a detailed analysis of variable rate tree structured vector quantization are provided it is found that variable rate tree structured vector quantization outperforms not only the fixed rate variety but also full search vector quantization the successive approximation character of variable rate tree structured vector quantization permits it to degrade gracefully if the rate is reduced at the encoder this has applications to the problem of buffer overflow

trellis coded quantization designed for noisy channels trellis coded quantization tcq of memoryless sources is developed for transmission over a binary symmetric channel the optimized tcq coder can achieve essentially the same performance as ayanoglu and gray s 1987 unconstrained trellis coding optimized for the binary symmetric channel but with a much lower implementation complexity for transmission rates above 1 b sample in most cases the optimized tcq coder also provides larger signal to noise ratio than farvardin and vaishampayan s 1991 channel optimized vector quantization algorithms are developed for the joint design of trellis coded quantization modulation tcq tcm the jointly designed tcq tcm system outperforms the straightforward cascade of separately designed tcq and tcm systems the improvement is most significant at low channel signal to noise ratio for a first order gauss markov source the predictive tcq tcm performance can exceed that of optimum pulse amplitude modulation

design of vector quantizer for image compression using self organizing feature map and surface fitting we propose a new scheme of designing a vector quantizer for image compression first a set of codevectors is generated using the self organizing feature map algorithm then the set of blocks associated with each code vector is modeled by a cubic surface for better perceptual fidelity of the reconstructed images mean removed vectors from a set of training images is used for the construction of a generic codebook further huffman coding of the indices generated by the encoder and the difference coded mean values of the blocks are used to achieve better compression ratio we proposed two indices for quantitative assessment of the psychovisual quality blocking effect of the reconstructed image our experiments on several training and test images demonstrate that the proposed scheme can produce reconstructed images of good quality while achieving compression at low bit rates

vector quantization for entropy coding of image subbands vector quantization for entropy coding of image subbands is investigated rate distortion curves are computed with mean square error as a distortion criterion the authors show that full search entropy constrained vector quantization of image subbands results in the best performance but is computationally expensive lattice quantizers yield a coding efficiency almost indistinguishable from optimum full search entropy constrained vector quantization orthogonal lattice quantizers were found to perform almost as well as lattice quantizers derived from dense sphere packings an optimum bit allocation rule based on a lagrange multiplier formulation is applied to subband coding coding results are shown for a still image

vector quantization with complexity costs vector quantization is a data compression method by which a set of data points is encoded by a reduced set of reference vectors the codebook a vector quantization strategy is discussed that jointly optimizes distortion errors and the codebook complexity thereby determining the size of the codebook a maximum entropy estimation of the cost function yields an optimal number of reference vectors their positions and their assignment probabilities the dependence of the codebook density on the data density for different complexity functions is investigated in the limit of asymptotic quantization levels how different complexity measures influence the efficiency of vector quantizers is studied for the task of image compression the wavelet coefficients of gray level images are quantized and the reconstruction error is measured the approach establishes a unifying framework for different quantization methods like k means clustering and its fuzzy version entropy constrained vector quantization or topological feature maps and competitive neural networks

wavelet transforms versus fourier transforms this note is a very basic introduction to wavelets it starts with an orthogonal basis of piecewise constant functions constructed by dilation and translation the wavelet transform maps each f x to its coefficients with respect to this basis the mathematics is simple and the transform is fast faster than the fast fourier transform which we briefly explain but approximation by piecewise constants is poor to improve this first wavelet we are led to dilation equations and their unusual solutions higher order wavelets are constructed and it is surprisingly quick to compute with them always indirectly and recursively

predictive vector quantization with intrablock prediction support region a predictive vector quantization pvq structure is proposed where the encoder uses a predictor based on an intrablock support region followed by a modified vector quantizer stage simulation results show that a modification on a previously published pvq system led to an improvement of 1 db in psnr for lenna

vector predictive coding of speech at 16 kbits s vector quantization in its simplest form may be regarded as a generalization of pcm independent quantization of each sample of a waveform to what might be called vector pcm where a block of consecutive samples a vector is simultaneously quantized as one unit in theory a performance arbitrarily close to the ultimate rate distortion limit is achievable with waveform vector quantization if the dimension of the vector k is large enough the main obstacle in effectively using vector quantization is complexity a vector quantizer of dimension k operating at a rate of r bits sample requires a number of computations on the order of k2 kr and a memory of the same order however a low dimensional vector quantizer dimensions 4 8 achieves a remarkable improvement over scalar quantization pcm consequently using the vector quantizer as a building block and imbedding it with other waveform data compression techniques may lead to the development of a new and powerful class of waveform coding systems this paper proposes and analyzes a waveform coding system adaptive vector predictive coding avpc in which a low dimensionality vector quantizer is used in an adaptive predictive coding scheme in the encoding process a locally generated prediction of the current input vector is subtracted from the current vector and the resulting error vector is coded by a vector quantizer each frame consisting of many vectors is classified into one of m statistical types this classification determines which one of m fixed predictors and of m vector quantizers will be used for encoding the current frame

gain adaptive vector quantization with application to speech coding the generalization of gain adaptation to vector quantization vq is explored in this paper and a comprehensive examination of alternative techniques is presented we introduce a class of adaptive vector quantizers that can dynamically adjust the gain or amplitude scale of code vectors according to the input signal level the encoder uses a gain estimator to determine a suitable normalization of each input vector prior to vq encoding the normalized vectors have reduced dynamic range and can then be more efficiently coded at the receiver the vq decoder output is multiplied by the estimated gain both forward and backward adaptation are considered and several different gain estimators are compared and evaluated gain adaptive vq can be used alone for vector pcm coding i e direct waveform vq or as a building block in other vector coding schemes the design algorithm for generating the appropriate gain normalized vq codebook is introduced when applied to speech coding gain adaptive vq achieves significant performance improvement over fixed vq with a negligible increase in complexity

a new dynamic finite state vector quantization algorithm for image compression the picture quality of conventional memory vector quantization techniques is limited by their supercodebooks this paper presents a new dynamic finite state vector quantization dfsvq algorithm which provides better quality than the best quality that the supercodebook can offer the new dfsvq exploits the global interblock correlation of image blocks instead of local correlation in conventional dfsvqs for an input block we search the closest block from the previously encoded data using the side match technique the closest block is then used as the prediction of the input block or used to generate a dynamic codebook the input block is encoded by the closest block dynamic codebook or supercodebook searching for the closest block from the previously encoded data is equivalent to expand the codevector space thus the picture quality achieved is not limited by the supercodebook experimental results reveal that the new dfsvq reduces bit rate significantly and provides better visual quality as compared to the basic vq and other dfsvqs

transform and hybrid transform dpcm coding of images using pyramid vector quantization the transform and hybrid transform dpcm methods of image coding are generalized to allow pyramid vector quantization of the transform coefficients an asymptotic mean squared error performance expression is derived for the pyramid vector quantizer and used to determine the optimum rate assignment for encoding the various transform coefficients coding simulations for two images at average rates of 0 5 1 bit pixel demonstrate a 1 3 db improvement in signal to noise ratio for the vector quantization approach in the hybrid coding with more modest improvements in signal to noise ratio in the transform coding however this improvement is quite noticeable in image quality particularly in reducing blockiness in the low bit rate encoded images

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

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

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

predictive vector quantization of images the purpose of this paper is to present new image coding schemes based on a predictive vector quantization pvq approach the predictive part of the encoder is used to partially remove redundancy and the vq part further removes the residual redundancy and selects good quantization levels for the global waveform two implementations of this coding approach have been devised namely sliding block pvq and block tree pvq simulations on real images show significant improvement over the conventional dpcm and tree codes using these new techniques the strong robustness property of these coding schemes is also experimentally demonstrated

robust adaptive buffer instrumented entropy coded quantization of stationary sources an investigation is conducted of the effects of various source statistics and buffer management algorithms for a buffer of fixed size and an algorithm that provides particularly robust rate distortion performance for a wide range of stationary sources is described the performance of this scheme which is referred to as the adaptive entropy coded quantizer aecq is compared to that of conventional adaptive pulse code modulation apcm operating at the same fixed channel transmission rate experimental results indicate that the aecq can provide considerably improved performance and more robust operation than apcm the relative sensitivity of the aecq to scale mismatch is directly related to the rather simple memoryless feedback control strategy employed this might be remedied by alternative choices of the memoryless feedback control function or by introducing memory the former is much easier to implement and or analyze than the latter

entropy constrained trellis coded quantization trellis coded quantization is generalized to allow noiseless coding of the trellis branch reproduction symbols an entropy constrained trellis coded quantization ectcq design algorithm is presented based on the generalized lloyd algorithm for trellis code design and the entropy constrained vector quantization design algorithm ectcq systems are designed for memoryless gaussian and laplacian sources and for first and second order gauss markov sources with an 8 state trellis the mean squared error ectcq encoding performance for memoryless sources with smooth densities is within about 0 5 db of the rate distortion function small additional improvements are achieved with larger trellises similar though not quite as good performance is achieved for the gauss markov sources by using the ectcq in a predictive coding structure