context conditioning and run length coding for hybrid embedded progressive image coding an analysis of spatial context conditioning and run length coding in embedded progressive wavelet based image coding is presented the analysis shows that run length coding of certain context subsequences is superior to pure entropy coding both in terms of coding performance and of execution speed based on these considerations a novel intuitive context conditioning scheme using a spatial distance model to describe the statistics of significant coefficients is proposed the results for the proposed coding scheme are competitive to the best coding schemes found in the literature

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

asymptotical statistics of misspecified hidden markov models this paper deals with the problem of modeling data generated by an ergodic stochastic process as the output of a hidden markov model hmm more specifically we consider the problem of fitting a parametric family of hmm with continuous output to an ergodic stochastic process with continuous values which does not necessarily belong to the family in this context we derive the main asymptotic results almost sure consistency of the maximum likelihood estimator asymptotic normality of the estimation error and the exact rates of almost sure convergence

analyticity of entropy rate of hidden markov chains with continuous alphabet we first prove that under certain mild assumptions the entropy rate of a hidden markov chain observed when passing a finite state stationary markov chain through a discrete time continuous output channel is analytic with respect to the input markov chain parameters we then further prove under strengthened assumptions on the channel that the entropy rate is jointly analytic as a function of both the input markov chain parameters and the channel parameters in particular the main theorems establish the analyticity of the entropy rate for two representative channels 1 cauchy and 2 gaussian the analyticity results obtained are expected to be helpful in computation estimation of entropy rate of hidden markov chains and capacity of finite state channels with continuous output alphabet

asymptotic normality of the maximum likelihood estimator for general hidden markov models hidden markov models hmms have during the last decade become a widespread tool for modeling sequences of dependent random variables inference for such models is usually based on the maximum likelihood estimator mle and consistency of the mle for general hmms was recently proved by leroux in this paper we show that under mild conditions the mle is also asymptotically normal and prove that the observed information matrix is a consistent estimator of the fisher information

image coding based on a morphological representation of wavelet data an experimental study of the statistical properties of wavelet coefficients of image data is presented as well as the design of two different morphology based image coding algorithms that make use of these statistics a salient feature of the proposed methods is that by a simple change of quantizers the same basic algorithm yields high performance embedded or fixed rate coders another important feature is that the shape information of morphological sets used in this coder is encoded implicitly by the values of wavelet coefficients thus avoiding the use of explicit and rate expensive shape descriptors these proposed algorithms while achieving nearly the same objective performance of state of the art zerotree based methods are able to produce reconstructions of a somewhat superior perceptual quality due to a property of joint compression and noise reduction they exhibit

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

an embedded still image coder with rate distortion optimization it is well known that the fixed rate coder achieves optimality when all coefficients are coded with the same rate distortion r d slope in this paper we show that the performance of the embedded coder can be optimized in a rate distortion sense by coding the coefficients with decreasing r d slope we denote such a coding strategy as rate distortion optimized embedding rde rde allocates the available coding bits first to the coefficient with the steepest r d slope i e the largest distortion decrease per coding bit the resultant coding bitstream can be truncated at any point and still maintain an optimal r d performance to avoid the overhead of coding order transmission we use the expected r d slope which can be calculated from the coded bits and is available in both the encoder and the decoder with the probability estimation table of the qm coder the calculation of the r d slope can be just a lookup table operation experimental results show that the rate distortion optimization significantly improves the coding efficiency in a wide bit rate range

a generalized network formulation of the pairwise comparison consensus ranking model one of the best known and most widely referenced models for representing ordinal preferences is that due to kemeny and snell kemeny j g l j snell 1962 preference ranking an axiomatic approach mathematical models in the social sciences glnn new york 9 23 this model is designed to accommodate pairwise comparison data with an l1 norm used to measure voter disagreement while this model possesses many of the necessary properties for a social choice function solution procedures developed to date have been capable of handling only small problems due to the difficulty of modelling the transitivity requirements of an optimal consensus ranking this paper shows how the consensus formation problem for strict linear orderings can be modelled as a generalized network since efficient computer codes already exist for handling this special structure this approach will permit the solution of much larger problems than has been the case previously

consistent and asymptotically normal parameter estimates for hidden markov models hidden markov models are today widespread for modeling of various phenomena it has recently been shown by leroux that the maximum likelihood estimate mle of the parameters of a such a model is consistent and local asymptotic normality has been proved by bickel and ritov in this paper we propose a new clans of estimates which are consistent asymptoti cally normal and almost as good an the mle

space frequency quantization for wavelet image coding a new class of image coding algorithms coupling standard scalar quantization of frequency coefficients with tree structured quantization related to spatial structures has attracted wide attention because its good performance appears to confirm the promised efficiencies of hierarchical representation this paper addresses the problem of how spatial quantization modes and standard scalar quantization can be applied in a jointly optimal fashion in an image coder we consider zerotree quantization zeroing out tree structured sets of wavelet coefficients and the simplest form of scalar quantization a single common uniform scalar quantizer applied to all nonzeroed coefficients and we formalize the problem of optimizing their joint application we develop an image coding algorithm for solving the resulting optimization problem despite the basic form of the two quantizers considered the resulting algorithm demonstrates coding performance that is competitive often outperforming the very best coding algorithms in the literature

gradient convergence in gradient methods with errors we consider the gradient method x t 1 x t g t s t w t where s t is a descent direction of a function f rn to re and w t is a deterministic or stochastic error we assume that gr f is lipschitz continuous that the stepsize g t diminishes to 0 and that s t and w t satisfy standard conditions we show that either f x t to infty or f x t converges to a finite value and gr f x t to0 with probability 1 in the stochastic case and in doing so we remove various boundedness conditions that are assumed in existing results such as boundedness from below of f boundedness of gr f x t or boundedness of xt

analyticity of entropy rates of continuous state hidden markov models the analyticity of the entropy and relative entropy rates of continuous state hidden markov models is studied here using the analytic continuation principle and the stability properties of the optimal filter the analyticity of these rates is established for analytically parameterized models the obtained results hold under relatively mild conditions and cover several useful classes of hidden markov models these results are relevant for several theoretically and practically important problems arising in statistical inference system identification and information theory

asymptotic properties of the maximum likelihood estimator in autoregressive models with markov regime an autoregressive process with markov regime is an autoregressive process for which the regression function at each time point is given by a nonobservable markov chain in this paper we consider the asymptotic properties of the maximum likelihood estimator in a possibly nonstationary process of this kind for which the hidden state space is compact but not necessarily finite consistency and asymptotic normality are shown to follow from uniform exponential forgetting of the initial distribution for the hidden markov chain conditional on the observations

image coding using wavelet transforms and entropy constrained trellis coded quantization the discrete wavelet transform has recently emerged as a powerful technique for decomposing images into various multi resolution approximations multi resolution decomposition schemes have proven to be very effective for high quality low bit rate image coding in this work we investigate the use of entropy constrained trellis coded quantization ectcq for encoding the wavelet coefficients of both monochrome and color images ectcq is known as an effective scheme for quantizing memoryless sources with low to moderate complexity the ectcq approach to data compression has led to some of the most effective source codes found to date for memoryless sources performance comparisons are made using the classical quadrature mirror filter bank of johnston and nine tap spline filters that were built from biorthogonal wavelet bases we conclude that the encoded images obtained from the system employing nine tap spline filters are marginally superior although at the expense of additional computational burden excellent peak signal to noise ratios are obtained for encoding monochrome and color versions of the 512 spl times 512 lenna image comparisons with other results from the literature reveal that the proposed wavelet coder is quite competitive

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

the o d e method for convergence of stochastic approximation and reinforcement learning it is shown here that stability of the stochastic approximation algorithm is implied by the asymptotic stability of the origin for an associated ode this in turn implies convergence of the algorithm several specific classes of algorithms are considered as applications it is found that the results provide i a simpler derivation of known results for reinforcement learning algorithms ii a proof for the first time that a class of asynchronous stochastic approximation algorithms are convergent without using any a priori assumption of stability iii a proof for the first time that asynchronous adaptive critic and q learning algorithms are convergent for the average cost optimal control problem

recursive algorithms for estimation of hidden markov models and autoregressive models with markov regime this paper is concerned with recursive algorithms for the estimation of hidden markov models hmms and autoregressive ar models under the markov regime convergence and rate of convergence results are derived acceleration of convergence by averaging of the iterates and the observations are treated finally constant step size tracking algorithms are presented and examined

analyticity convergence and convergence rate of recursive maximum likelihood estimation in hidden markov models this paper considers the asymptotic properties of the recursive maximum likelihood estimator for hidden markov models the paper is focused on the analytic properties of the asymptotic log likelihood and on the point convergence and convergence rate of the recursive maximum likelihood estimator using the principle of analytic continuation the analyticity of the asymptotic log likelihood is shown for analytically parameterized hidden markov models relying on this fact and some results from differential geometry lojasiewicz inequality the almost sure point convergence of the recursive maximum likelihood algorithm is demonstrated and relatively tight bounds on the convergence rate are derived as opposed to the existing result on the asymptotic behavior of maximum likelihood estimation in hidden markov models the results of this paper are obtained without assuming that the log likelihood function has an isolated maximum at which the hessian is strictly negative definite

mutual information based analysis of jpeg2000 contexts context based arithmetic coding has been widely adopted in image and video compression and is a key component of the new jpeg2000 image compression standard in this paper the contexts used in jpeg2000 are analyzed using the mutual information which is closely related to the compression performance we first show that when combining the contexts the mutual information between the contexts and the encoded data will decrease unless the conditional probability distributions of the combined contexts are the same given i the initial number of contexts and f the final desired number of contexts there are s i f possible context classification schemes where s i f is called the stirling number of the second kind the optimal classification scheme is the one that gives the maximum mutual information instead of using an exhaustive search the optimal classification scheme can be obtained through a modified generalized lloyd algorithm with the relative entropy as the distortion metric for binary arithmetic coding the search complexity can be reduced by using dynamic programming our experimental results show that the jpeg2000 contexts capture the correlations among the wavelet coefficients very well at the same time the number of contexts used as part of the standard can be reduced without loss in the coding performance

comparison of different methods of classification in subband coding of images this paper investigates various classification techniques applied to subband coding of images as a way of exploiting the nonstationary nature of image subbands the advantages of subband classification are characterized in a rate distortion framework in terms of classification gain and overall subband classification gain two algorithms maximum classification gain and equal mean normalized standard deviation classification which allow unequal number of blocks in each class are presented the dependence between the classification maps from different subbands is exploited either directly while encoding the classification maps or indirectly by constraining the classification maps the trade off between the classification gain and the amount of side information is explored coding results for a subband image coder based on classification are presented the simulation results demonstrate the value of classification in subband coding

analyticity of entropy rate of hidden markov chains we prove that under mild positivity assumptions the entropy rate of a hidden markov chain varies analytically as a function of the underlying markov chain parameters a general principle to determine the domain of analyticity is stated an example is given to estimate the radius of convergence for the entropy rate we then show that the positivity assumptions can be relaxed and examples are given for the relaxed conditions we study a special class of hidden markov chains in more detail binary hidden markov chains with an unambiguous symbol and we give necessary and sufficient conditions for analyticity of the entropy rate for this case finally we show that under the positivity assumptions the hidden markov chain itself varies analytically in a strong sense as a function of the underlying markov chain parameters

recursive maximum likelihood algorithm for dependent observations a recursive maximum likelihood algorithm rml is proposed that can be used when both the observations and the hidden data have continuous values and are statistically dependent between different time samples the algorithm recursively approximates the probability density functions of the observed and hidden data by analytically computing the integrals with respect to the state variables where the parameters are updated using gradient steps a full convergence proof is given based on the ordinary differential equation approach which shows that the algorithm converges to a local minimum of the kullback leibler divergence between the true and the estimated parametric probability density functions a result that is useful even for a miss specified parametric model compared to other rml algorithms proposed in the literature this contribution extends the state space model and provides a theoretical analysis in a nontrivial statistical model that was not analyzed so far we further extend the rml analysis to constrained parameter estimation problems two examples including nonlinear state space models are given to highlight this contribution

significance linked connected component analysis for wavelet image coding the success in wavelet image coding is mainly attributed to a recognition of the importance of data organization and representation there have been several very competitive wavelet coders developed namely shapiro s 1993 embedded zerotree wavelets ezw servetto et al s 1995 morphological representation of wavelet data mrwd and said and pearlman s see ieee trans circuits syst video technol vol 6 p 245 50 1996 set partitioning in hierarchical trees spiht we develop a novel wavelet image coder called significance linked connected component analysis slcca of wavelet coefficients that extends mrwd by exploiting both within subband clustering of significant coefficients and cross subband dependency in significant fields extensive computer experiments on both natural and texture images show convincingly that the proposed slcca outperforms ezw mrwd and spiht for example for the barbara image at 0 25 b pixel slcca outperforms ezw mrwd and spiht by 1 41 db 0 32 db and 0 60 db in psnr respectively it is also observed that slcca works extremely well for images with a large portion of texture for eight typical 256 spl times 256 grayscale texture images compressed at 0 40 b pixel slcca outperforms spiht by 0 16 db 0 63 db in psnr this performance is achieved without using any optimal bit allocation procedure thus both the encoding and decoding procedures are fast

derivatives of entropy rate in special families of hidden markov chains consider a hidden markov chain obtained as the observation process of an ordinary markov chain corrupted by noise recently zuk et al showed how in principle one can explicitly compute the derivatives of the entropy rate of at extreme values of the noise namely they showed that the derivatives of standard upper approximations to the entropy rate actually stabilize at an explicit finite time we generalize this result to a natural class of hidden markov chains called black holes we also discuss in depth special cases of binary markov chains observed in binary symmetric noise and give an abstract formula for the first derivative in terms of a measure on the simplex due to blackwell

multiscale image segmentation using wavelet domain hidden markov models we introduce a new image texture segmentation algorithm hmtseg based on wavelets and the hidden markov tree hmt model the hmt is a tree structured probabilistic graph that captures the statistical properties of the coefficients of the wavelet transform since the hmt is particularly well suited to images containing singularities edges and ridges it provides a good classifier for distinguishing between textures utilizing the inherent tree structure of the wavelet hmt and its fast training and likelihood computation algorithms we perform texture classification at a range of different scales we then fuse these multiscale classifications using a bayesian probabilistic graph to obtain reliable final segmentations since hmtseg works on the wavelet transform of the image it can directly segment wavelet compressed images without the need for decompression into the space domain we demonstrate the performance of hmtseg with synthetic aerial photo and document image segmentations

on the optimality of symbol by symbol filtering and denoising we consider the problem of optimally recovering a finite alphabet discrete time stochastic process x sub t from its noise corrupted observation process z sub t in general the optimal estimate of x sub t will depend on all the components of z sub t on which it can be based we characterize nontrivial situations i e beyond the case where x sub t z sub t are independent for which optimum performance is attained using symbol by symbol operations a k a singlet decoding meaning that the optimum estimate of x sub t depends solely on z sub t for the case where x sub t is a stationary binary markov process corrupted by a memoryless channel we characterize the necessary and sufficient condition for optimality of symbol by symbol operations both for the filtering problem where the estimate of x sub t is allowed to depend only on z sub t sub t spl les t and the denoising problem where the estimate of x sub t is allowed dependence on the entire noisy process it is then illustrated how our approach which consists of characterizing the support of the conditional distribution of the noise free symbol given the observations can be used for characterizing the entropy rate of the binary markov process corrupted by the binary symmetric channel bsc in various asymptotic regimes for general noise free processes not necessarily markov general noise processes not necessarily memoryless and general index sets random fields we obtain an easily verifiable sufficient condition for the optimality of symbol by symbol operations and illustrate its use in a few special cases for example for binary processes corrupted by a bsc we establish under mild conditions the existence of a spl delta sup 0 such that the say what you see scheme is optimal provided the channel crossover probability is less than spl delta sup finally we show how for the case of a memoryless channel the large deviations ld performance of a symbol by symbol filter is easy to obtain thus characterizing the ld behavior of the optimal schemes when these are singlet decoders and constituting the only known cases where such explicit characterization is available

a randomized algorithm for the capacity of finite state channels inspired by ideas from the field of stochastic approximation we propose a randomized algorithm to compute the capacity of a finite state channel with a markovian input when the mutual information rate of the channel is concave with respect to the chosen parameterization the proposed algorithm proves to be convergent to the capacity of the channel almost surely with the derived convergence rate we also discuss the convergence behavior of the algorithm without the concavity assumption

consistent estimation of the filtering and marginal smoothing distributions in nonparametric hidden markov models in this paper we consider the filtering and smoothing recursions in nonparametric finite state space hidden markov models hmms when the parameters of the model are unknown and replaced by estimators we provide an explicit and time uniform control of the filtering and smoothing errors in total variation norm as a function of the parameter estimation errors we prove that the risk for the filtering and smoothing errors may be uniformly upper bounded by the mathrm l 1 risk of the estimators it has been proved very recently that statistical inference for finite state space nonparametric hmms is possible we study how the recent spectral methods developed in the parametric setting may be extended to the nonparametric framework and we give explicit upper bounds for the mathrm l 2 risk of the nonparametric spectral estimators in the case where the observation space is compact this provides explicit rates for the filtering and smoothing errors in total variation norm the performance of the spectral method is assessed with simulated data for both the estimation of the nonparametric conditional distribution of the observations and the estimation of the marginal smoothing distributions

bayesian tree structured image modeling using wavelet domain hidden markov models wavelet domain hidden markov models have proven to be useful tools for statistical signal and image processing the hidden markov tree hmt model captures the key features of the joint probability density of the wavelet coefficients of real world data one potential drawback to the hmt framework is the need for computationally expensive iterative training to fit an hmt model to a given data set e g using the expectation maximization algorithm we greatly simplify the hmt model by exploiting the inherent self similarity of real world images the simplified model specifies the hmt parameters with just nine meta parameters independent of the size of the image and the number of wavelet scales we also introduce a bayesian universal hmt uhmt that fixes these nine parameters the uhmt requires no training of any kind while extremely simple we show using a series of image estimation denoising experiments that these new models retain nearly all of the key image structure modeled by the full hmt finally we propose a fast shift invariant hmt estimation algorithm that outperforms other wavelet based estimators in the current literature both visually and in mean square error

on analytic properties of entropy rate entropy rate is a real valued functional on the space of discrete random sources for which it exists however it lacks existence proofs and or closed formulas even for classes of random sources which have intuitive parameterizations a good way to overcome this problem is to examine its analytic properties relative to some reasonable topology a canonical choice of a topology is that of the norm of total variation as it immediately arises with the idea of a discrete random source as a probability measure on sequence space it is shown that both upper and lower entropy rate hence entropy rate itself if it exists are lipschitzian relative to this topology which by well known facts is close to differentiability an application of this theorem leads to a simple and elementary proof of the existence of entropy rate of random sources with finite evolution dimension this class of sources encompasses arbitrary hidden markov sources and quantum random walks

image compression via joint statistical characterization in the wavelet domain we develop a probability model for natural images based on empirical observation of their statistics in the wavelet transform domain pairs of wavelet coefficients corresponding to basis functions at adjacent spatial locations orientations and scales are found to be non gaussian in both their marginal and joint statistical properties specifically their marginals are heavy tailed and although they are typically decorrelated their magnitudes are highly correlated we propose a markov model that explains these dependencies using a linear predictor for magnitude coupled with both multiplicative and additive uncertainties and show that it accounts for the statistics of a wide variety of images including photographic images graphical images and medical images in order to directly demonstrate the power of the model we construct an image coder called epwic embedded predictive wavelet image coder in which subband coefficients are encoded one bitplane at a time using a nonadaptive arithmetic encoder that utilizes conditional probabilities calculated from the model bitplanes are ordered using a greedy algorithm that considers the mse reduction per encoded bit the decoder uses the statistical model to predict coefficient values based on the bits it has received despite the simplicity of the model the rate distortion performance of the coder is roughly comparable to the best image coders in the literature

capacity of finite state channels based on lyapunov exponents of random matrices the finite state markov channel fsmc is a time varying channel having states that are characterized by a finite state markov chain these channels have infinite memory which complicates their capacity analysis we develop a new method to characterize the capacity of these channels based on lyapunov exponents specifically we show that the input output and conditional entropies for this channel are equivalent to the largest lyapunov exponents for a particular class of random matrix products we then show that the lyapunov exponents can be expressed as expectations with respect to the stationary distributions of a class of continuous state space markov chains this class of markov chains which is closely related to the prediction filter in hidden markov models is shown to be nonirreducible hence much of the standard theory for continuous state space markov chains cannot be applied to establish the existence and uniqueness of stationary distributions nor do we have direct access to a central limit theorem clt in order to address these shortcomings we utilize several results from the theory of random matrix products and lyapunov exponents the stationary distributions for this class of markov chains are shown to be unique and continuous functions of the input symbol probabilities provided that the input sequence has finite memory these properties allow us to express mutual information and channel capacity in terms of lyapunov exponents we then leverage this connection between entropy and lyapunov exponents to develop a rigorous theory for computing or approximating entropy and mutual information for finite state channels with dependent inputs we develop a method for directly computing entropy of finite state channels that does not rely on simulation and establish its convergence we also obtain a new asymptotically tight lower bound for entropy based on norms of random matrix products in addition we prove a new functional clt for sample entropy and apply this theorem to characterize the error in simulated estimates of entropy finally we present numerical examples of mutual information computation for intersymbol interference isi channels and observe the capacity benefits of adding memory to the input sequence for such channels