mismatch in high rate entropy constrained vector quantization bucklew s 1984 high rate vector quantizer mismatch result is extended from fixed rate coding to variable rate coding using a lagrangian formulation it is shown that if an asymptotically high rate optimal sequence of variable rate codes is designed for a k dimensional probability density function pdf g and then applied to another pdf f for which f g is bounded then the resulting mismatch or loss of performance from the optimal possible is given by the relative entropy or kullback leibler 1968 divergence i f spl par g it is also shown that under the same assumptions an asymptotically optimal code sequence for g can be converted to an asymptotically optimal code sequence for a mismatched source f by modifying only the lossless component of the code applications to quantizer design using uniform and gaussian densities are described including a high rate analog to the shannon rate distortion result of sakrison 1975 and lapidoth 1997 showing that the gaussian is the worst case for lossy compression of a source with known covariance by coupling the mismatch result with composite quantizers the worst case properties of uniform and gaussian densities are extended to conditionally uniform and gaussian densities which provides a lloyd clustering algorithm for fitting mixtures to general densities

distortion bounds for vector quantizers with finite codebook size upper and lower bounds are presented for the distortion of the optimal n point vector quantizer applied to k dimensional signals under certain smoothness conditions on the source distribution the bounds are shown to hold for each and every value of n the codebook size these results extend bounds derived in the high resolution limit which assume that the number of code vectors is arbitrarily large two approaches to the upper bound are presented the first constructive construction achieves the correct asymptotic rate of convergence as well as the correct dependence on the source density although leading to an inferior value for the constant the second construction based on a random coding argument is shown to additionally achieve a value of the constant which is much closer to the best known result derived within the asymptotic theory lower bound results derived in the correspondence are again shown to possess the correct asymptotic form and yield a constant which is almost indistinguishable from the best value achieved in the asymptotic regime finally application of the results to the problem of source coding yields upper bounds on the distortion rate function for a wide class of processes

reduced storage vq via secondary quantization this paper introduces methods for reducing the table storage required for encoding and decoding with unstructured vector quantization uvq or tree structured vector quantization tsvq specifically a low storage secondary quantizer is used to compress the code vectors and test vectors of the primary quantizer the relative advantages of uniform and nonuniform secondary quantization are investigated a linde buzo gray lbg like algorithm that optimizes the primary uvq codebook for a given secondary codebook and another that jointly optimizes both primary and secondary codebooks are presented in comparison to conventional methods it is found that significant storage reduction is possible typically a factor of two to three with little loss of signal to noise ratio snr moreover when reducing dimension is considered as another method of reducing storage it is found that the best strategy is a combination of both the method of secondary quantization is also applied to tsvq to reduce the table storage required for both encoding and decoding it is shown that by exploiting the correlation among the test vectors in the tree both encoder and decoder storage can be significantly reduced with little loss of snr by a factor of about four or two relative to the conventional method of storing test vectors or test hyperplanes

fourier transform of self affine measures abstract suppose f is a self affine set on r d d 2 which is not a singleton associated to affine contractions f j a j b j a j gl d r b j r d j a for some finite a we prove that if the group generated by the matrices a j j a forms a proximal and totally irreducible subgroup of gl d r then any self affine measure p j f j p j 1 0 p j 1 j a on f is a rajchman measure the fourier transform 0 as as an application this shows that self affine sets with proximal and totally irreducible linear parts are sets of rectangular multiplicity for multiple trigonometric series moreover if the zariski closure of is connected real split lie group in the zariski topology then has a power decay at infinity hence is l p improving for all 1 p and f has positive fourier dimension in dimension d 2 3 the irreducibility of and non compactness of the image of in pgl d r is enough for power decay of the proof is based on quantitative renewal theorems for random walks on the sphere s d 1

quantization methods for equal gain transmission with finite rate feedback we consider the design and analysis of quantizers for equal gain transmission egt systems with finite rate feedback based communication in flat fading multiple input single output miso systems egt is a beamforming technique that maximizes the miso channel capacity when there is an equal power per antenna constraint at the transmitter and requires the feedback of t 1 phase angles when there are t antennas at the transmitter in this paper we contrast two popular approaches for quantizing the phase angles vector quantization vq and scalar quantization sq on the vq side using the capacity loss with respect to egt with perfect channel information at transmitter as performance metric we develop a criterion for designing the beamforming codebook for quantized egt q egt we also propose an iterative algorithm based on the well known generalized lloyd algorithm for computing the beamforming vector codebook on the analytical side we study the performance of q egt and derive closed form expressions for the performance in terms of capacity loss and outage probability in the case of i i d rayleigh flat fading channels on the sq side assuming uniform scalar quantization and i i d rayleigh flat fading channels we derive the high resolution performance of quantized egt and contrast the performance with that of vq we find that although both vq and sq achieve the same rate of convergence to the capacity with perfect feedback as the number of feedback bits b increases there exists a fixed gap between the two

variance mismatched fixed rate scalar quantization of laplacian sources the paper investigates the mean squared error mse distortion of a variance mismatched fixed rate scalar quantizer that is optimal or asymptotically optimal in the minimum mse sense for a laplacian density with standard deviation q but is applied to another with standard deviation p a nitadori like formula is discovered for an optimal quantizer when p q 1 2 also it is shown that the distortion expressions derived rigorously for asymptotically optimal quantile quantizers essentially confirm the heuristically obtained previous result that the distortion decreases as 1 n3 for the heavy mismatch of 3 2 as lnn n2 for the critical mismatch of 3 2 and as 1 n2 for the mild mismatch of 1 and the resulting distortion decreases as in n n2 1 p

alphabet constrained vector quantization alphabet constrained rate distortion theory is extended to coding of sources with memory two different cases are considered when only the size of the codebook is constrained and when the codevector values are also held fixed for both cases nth order constrained alphabet rate distortion functions are defined and a convergent algorithm for their evaluation is presented specific simulations using ar 1 sources show that performance near the rate distortion bound is possible using a reproduction alphabet consisting of a small number of codevectors it is also shown that the additional constraint of holding the codevector values fixed does not degrade performance of the coder in relation to the size only constrained case this observation motivates the development of a fixed codebook vector quantizer called the alphabet and entropy constrained vector quantizer the performance of which is comparable to the entropy constrained vector quantizer a number of examples using an ar 1 and a speech source are presented to corroborate the theory

adaptive mesh and algorithm refinement using direct simulation monte carlo adaptive mesh and algorithm refinement amar embeds a particle method within a continuum method at the finest level of an adaptive mesh refinement amr hierarchy the coupling between the particle region and the overlaying continuum grid is algorithmically equivalent to that between the fine and coarse levels of amr direct simulation monte carlo dsmc is used as the particle algorithm embedded within a godunov type compressible navier stokes solver several examples are presented and compared with purely continuum calculations

high rate vector quantization for detection we investigate high rate quantization for various detection and reconstruction loss criteria a new distortion measure is introduced which accounts for global loss in best attainable binary hypothesis testing performance the distortion criterion is related to the area under the receiver operating characteristic roc curve specifically motivated by sanov s theorem we define a performance curve as the trajectory of the pair of optimal asymptotic type i and type ii error rates of the most powerful neyman pearson test of the hypotheses the distortion measure is then defined as the difference between the area under the curve auc of the optimal pre encoded hypothesis test and the auc of the optimal post encoded hypothesis test as compared to many previously introduced distortion measures for decision making this distortion measure has the advantage of being independent of any detection thresholds or priors on the hypotheses which are generally difficult to specify in the code design process a high resolution zador gersho type of analysis is applied to characterize the point density and the inertial profile associated with the optimal high rate vector quantizer the analysis applies to a restricted class of high rate quantizers that have bounded cells with vanishing volumes the optimal point density is used to specify a lloyd type algorithm which allocates its finest resolution to regions where the gradient of the pre encoded likelihood ratio has greatest magnitude

high resolution quantization theory and the vector quantizer advantage the authors consider how much performance advantage a fixed dimensional vector quantizer can gain over a scalar quantizer they collect several results from high resolution or asymptotic in rate quantization theory and use them to identify source and system characteristics that contribute to the vector quantizer advantage one well known advantage is due to improvement in the space filling properties of polytopes as the dimension increases others depend on the source s memory and marginal density shape the advantages are used to gain insight into product transform lattice predictive pyramid and universal quantizers although numerical prediction consistently overestimated gains in low rate 1 bit sample experiments the theoretical insights may be useful even at these rates

quantization on the grassmann manifold the problem of quantization in an euclidean space with unitary constraints can be formulated as an unconstrained problem on a grassmann manifold such constraints arise in areas such as wireless communication with multiple antennas at the transmitter and at the receiver due to the constraints the distortion rate analysis developed for euclidean spaces cannot be applied directly this paper extends gersho s asymptotic large rate small distortion distortion bounds to the case when the source is distributed on the complex grassmann manifold the special structure of the grassmann manifold and the distortion measures defined on it differentiate this problem from the traditional vector quantization in euclidean spaces

non uniform convergence of finite volume schemes for riemann problems of ideal magnetohydrodynamics this paper presents riemann test problems for ideal magnetohydrodynamics mhd finite volume schemes the test problems place emphasis on the hyperbolic irregularities of the ideal mhd system namely the occurrence of intermediate shocks and non unique solutions we investigate numerical solutions for the test problems obtained by several commonly used methods roe hlle central scheme all methods turned out to show a non uniform convergence behavior which may be paraphrased as pseudo convergence initially the methods show convergence towards a wrong solution with irregular wave patterns only after heavy grid refinement the methods switch to converge to the true solution this behavior is most pronounced whenever the solution should be unique but its phase space trajectory lies in the vicinity of that of a non unique solution we show detailed grid convergence studies and empirical error analysis the results may be related to similar results for a 2 2 model system and to time dependent investigations of other authors the non uniform convergence is expected to be present for any diffusive finite volume method and question the reliability of coarse grid ideal mhd solutions also in higher space dimensions

a method for incorporating gauss lasw into electromagnetic pic codes abstract an algorithm is presented which solves a modified set of maxwell s equations none of them elliptic in an electromagnetic pic type code the algorithm prevents large violations of gauss law by diffusing away numerical errors arising from the particle to grid apportionment method there are no restrictions on the manner in which charge and current densities are determined from the particle distribution in particular point particles and simple linear interpolation are used

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

asymptotically optimal model estimation for quantization using high rate theory approximations we introduce flexible practical quantizers based on possibly non gaussian models in both the constrained resolution cr and the constrained entropy cases we derive model estimation criteria optimizing asymptotic with increasing rate quantizer performance we show that in the cr case the optimal criterion is different from the maximum likelihood criterion commonly used for that purpose and introduce a new criterion that we call constrained resolution minimum description length cr mdl we apply these principles to the generalized gaussian scaled mixture model which is accurate for many real world signals we provide an explanation of the reason why the cr mdl improves quantization performance in the cr case and show that cr mdl can compensate for a possible mismatch between model and data distribution thus this criterion is of a great interest for practical applications our experiments apply the new quantization method to controllable artificial data and to the commonly used modulated lapped transform representation of audio signals we show that both the cr mdl criterion and a non gaussian modeling have significant advantages

time domain differential feedback for massive miso ofdm systems in correlated channels massive multiple input multiple output mimo orthogonal frequency division multiplexing ofdm is a promising technology for next generation wireless communications however when channel state information csi at the transmitter is obtained using channel feedback the benefits of this system are severely limited by the tradeoff between downlink capacity and feedback overhead to solve this problem we propose a time domain differential feedback scheme for massive multiple input single output miso ofdm systems the proposed scheme exploits channel correlations in time frequency and space domains simultaneously by considering differential channel impulse response cir to simplify the codebook design and to reduce codeword search complexity we partition and then quantize the differential cir using a number of subcodebooks because the total feedback bits are shared by the subcodebooks we further optimize the bit allocation for them to minimize the total quantization error for this we analyze the quantization error of the proposed scheme and then use the analysis results for the bit allocation optimization in simulations the proposed scheme efficiently exploits all correlations and achieves significant spectral efficiency gain compared to conventional feedback schemes

universal multiple description scalar quantization analysis and design this paper introduces a new high rate analysis of the multiple description scalar quantizer mdsq with balanced descriptions the analysis provides insight into the structure of the mdsq suggesting the nonoptimality of uniform central quantizer cell lengths as well as a method to approximate optimal cell lengths for both level constrained and entropy constrained mdsq new upper bounds on the granular distortion for sources with smooth probability density functions pdfs are derived under the mean squared error measure which are 0 4 db lower than previous results based on the insights a universal multiple description scalar quantizer umdsq is proposed which at high rate can achieve nearly the same performance as the fully optimized entropy constrained mdsq ecmdsq without requiring extensive training the proposed umdsq has only two control parameters and a continuum of tradeoff points between the central and side distortions can be achieved as the two parameters are varied

asymptotic entropy constrained performance of tessellating and universal randomized lattice quantization two results are given first using a result of csiszar 1973 the asymptotic i e high resolution low distortion performance for entropy constrained tessellating vector quantization heuristically derived by gersho 1979 is proven for all sources with finite differential entropy this implies using gersho s conjecture and zador s formula that tessellating vector quantizers are asymptotically optimal for this broad class of sources and generalizes a rigorous result of gish and pierce 1968 from the scalar to the vector case second the asymptotic performance is established for zamir and feder s 1992 randomized lattice quantization with the only assumption that the source has finite differential entropy it is proven that the low distortion performance of the zamir feder universal vector quantizer is asympotically the same as that of the deterministic lattice quantizer

finding the number of clusters in a dataset one of the most difficult problems in cluster analysis is identifying the number of groups in a dataset most previously suggested approaches to this problem are either somewhat ad hoc or require parametric assumptions and complicated calculations in this article we develop a simple yet powerful nonparametric method for choosing the number of clusters based on distortion a quantity that measures the average distance per dimension between each observation and its closest cluster center our technique is computationally efficient and straightforward to implement we demonstrate empirically its effectiveness not only for choosing the number of clusters but also for identifying underlying structure on a wide range of simulated and real world datasets in addition we give a rigorous theoretical justification for the method based on information theoretic ideas specifically results from the subfield of electrical engineering known as rate distortion theory allow us to describe the behavior of the dist

high rate vector quantization for the neyman pearson detection of correlated processes this paper investigates the effect of quantization on the performance of the neyman pearson test it is assumed that a sensing unit observes samples of a correlated stationary ergodic multivariate process each sample is passed through an n point quantizer and transmitted to a decision device which performs a binary hypothesis test for any false alarm level it is shown that the miss probability of the neyman pearson test converges to zero exponentially as the number of samples tends to infinity assuming that the observed process satisfies certain mixing conditions the main contribution of this paper is to provide a compact closed form expression of the error exponent in the high rate regime i e when the number n of quantization levels tends to infinity generalizing previous results of gupta and hero to the case of nonindependent observations if d represents the dimension of one sample it is proved that the error exponent converges at rate n2 d to the one obtained in the absence of quantization as an application relevant high rate quantization strategies which lead to a large error exponent are determined numerical results indicate that the proposed quantization rule can yield better performance than existing ones in terms of detection error

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

hybrid block amr in cartesian and curvilinear coordinates mhd applications abstract we present a novel hybrid block adaptive scheme for use in solving sets of near conservation laws in general orthogonal coordinate systems the adaptive mesh refinement amr scheme is block based i e individual grids have a pre fixed number of grid cells and is implemented for any dimensionality d its hybrid character relaxes the common approach where a block that needs refinement triggers 2 d subblocks when grids are refined by a factor of 2 this introduces incomplete families in the grid hierarchy but approaches the optimal fit to developing flow features inherent in the original patch based amr strategy our hybrid block amr approach is compared with a patch based amr one both exploiting openmp parallelism the implementation is able to handle general curvilinear coordinates for which restriction and prolongation formulae are presented along with boundary treatments at singular boundaries demonstrative examples cover hydro and magnetohydrodynamic mhd applications including tests on a 2d polar grid a 2 5d spherical and a 3d cylindrical configuration the applications cover classical up to relativistic mhd simulations of particular relevance for astrophysical magnetized jet and stellar wind studies

do optimal entropy constrained quantizers have a finite or infinite number of codewords an entropy constrained quantizer q is optimal if it minimizes the expected distortion d q subject to a constraint on the output entropy h q we use the lagrangian formulation to show the existence and study the structure of optimal entropy constrained quantizers that achieve a point on the lower convex hull of the operational distortion rate function d sub h r inf sub q d q h q spl les r in general an optimal entropy constrained quantizer may have a countably infinite number of codewords our main results show that if the tail of the source distribution is sufficiently light resp heavy with respect to the distortion measure the lagrangian optimal entropy constrained quantizer has a finite resp infinite number of codewords in particular for the squared error distortion measure if the tail of the source distribution is lighter than the tail of a gaussian distribution then the lagrangian optimal quantizer has only a finite number of codewords while if the tail is heavier than that of the gaussian the lagrangian optimal quantizer has an infinite number of codewords

optimality of klt for high rate transform coding of gaussian vector scale mixtures application to reconstruction estimation and classification the karhunen loeacuteve transform klt is known to be optimal for high rate transform coding of gaussian vectors for both fixed rate and variable rate encoding the klt is also known to be suboptimal for some non gaussian models this paper proves high rate optimality of the klt for variable rate encoding of a broad class of non gaussian vectors gaussian vector scale mixtures gvsm which extend the gaussian scale mixture gsm model of natural signals a key concavity property of the scalar gsm same as the scalar gvsm is derived to complete the proof optimality holds under a broad class of quadratic criteria which include mean squared error mse as well as generalized f divergence loss in estimation and binary classification systems finally the theory is illustrated using two applications signal estimation in multiplicative noise and joint optimization of classification reconstruction systems

a lagrangian formulation of zador s entropy constrained quantization theorem zador s 1963 1966 classic result for the asymptotic high rate behavior of entropy constrained vector quantization is recast in a lagrangian form which better matches the lloyd algorithm used to optimize such quantizers the equivalence of the two formulations is shown and the result is proved for source distributions that are absolutely continuous with respect to the lebesgue measure which satisfy an entropy condition thereby generalizing the conditions stated by zador under which the result holds

on the asymptotic distribution of the errors in vector quantization in a recent paper lee and neuhoff see ibid vol 42 no 2 p446 60 1996 found an asymptotic formula for the distribution of the length of the errors produced by a vector quantizer with many quantization points this distribution depends on the source probability density the quantizer point density and the quantizer shape profile the latter characterizes the shapes of the quantization cells as a function of position the purpose of this paper is to give a rigorous derivation of this formula by identifying precise conditions under which it is shown that if a sequence of vector quantizers with a given dimension and an increasing number of points has specific point densities and specific shape profiles converging to a model point density and a model shape profile respectively then the distribution of the length of the quantization error suitably normalized converges to the aforementioned formula with the model point density and the model shape profile substituted

vector quantization by companding a union of z lattices an encoding scheme is presented based on random coding theory in conjunction with companding techniques this combination provides an easy design and a low storage quantizer the groundwork of the study is gaussian sources the motivation is the attractive properties of gaussian and gaussian mixtures gms and their applications in data compression here codebooks are generated by companding a union of randomly translated cubic lattices the compander functions operate scalar wise providing an affordable encoding complexity that is rate independent for such a codebook an optimal search and the indexing of the codevectors are addressed performance close to finite dimensional random coding is achieved while the complexity is not far from scalar coding the structure of the codebook balances between complexity and performance in comparison to some other methods the advantage of the proposed scheme is more distinct for correlated sources

lagrangian vector quantization with combined entropy and codebook size constraints in this paper the lagrangian formulation of variable rate vector quantization is extended to quantization with simultaneous constraints on entropy and codebook size including variable and fixed rate quantization as special cases the formulation leads to a lloyd quantizer design algorithm and generalizations of gersho s approximations characterizing optimal performance for asymptotically large rate a variation of gersho s approach is shown to yield rigorous results partially characterizing the asymptotically optimal performance

analysis of multiple antenna systems with finite rate feedback using high resolution quantization theory this paper considers the development of a general framework for the analysis of transmit beamforming methods in multiple antenna systems with finite rate feedback inspired by the results of classical high resolution quantization theory the problem of finite rate quantized communication system is formulated as a general fixed rate vector quantization problem with side information available at the encoder or the quantizer but unavailable at the decoder the framework of the quantization problem is sufficiently general to include quantization schemes with general non mean squared distortion functions and constrained source vectors asymptotic distortion analysis of the proposed general quantization problem is provided by extending the vector version of the bennett s integral specifically tight lower and upper bounds of the average asymptotic distortion are proposed sufficient conditions for the achievability of the distortion bounds are also provided and related to corresponding classical fixed rate quantization problems the proposed general methodology provides a powerful analytical tool to study a wide range of finite rate feedback systems to illustrate the utility of the framework we consider the analysis of a finite rate feedback multiple input single output miso beamforming system over independent and identically distributed i i d rayleigh flat fading channels numerical and simulation results are presented that further confirm the accuracy of the analytical results

transform coding with integer to integer transforms a new interpretation of transform coding is developed that downplays quantization and emphasizes entropy coding allowing a comparison of entropy coding methods with different memory requirements with conventional transform coding based on computing karhunen loeve transform coefficients and then quantizing them vector entropy coding can be replaced by scalar entropy coding without an increase in rate thus the transform coding advantage is a reduction in memory requirements for entropy coding this paper develops a transform coding technique where the source samples are first scalar quantized and then transformed with an integer to integer approximation to a nonorthogonal linear transform among the possible advantages is to reduce the memory requirement further than conventional transform coding by using a single common scalar entropy codebook for all components the analysis shows that for high rate coding of a gaussian source this reduction in memory requirements comes without any degradation of rate distortion performance

absolute continuity of non homogeneous self similar measures abstract we prove that self similar measures on the real line are absolutely continuous for almost all parameters in the super critical region in particular confirming a conjecture of s m ngai and y wang while recently there has been much progress in understanding absolute continuity for homogeneous self similar measures this is the first improvement over the classical transversality method in the general non homogeneous case in the course of the proof we establish new results on the dimension and fourier decay of a class of random self similar measures

multidimensional optical transport based on optimized vector quantization inspired signal constellation design an optimized vector quantization inspired signal constellation design ovq scd suitable for multidimensional optical transport is proposed in which signal constellation radii transformation function is optimized and near uniform distribution of points is achieved the proposed ovq scd is used in a tandem with a hybrid multidimensional coded modulation scheme employing slepian sequences as electrical discrete time basis functions orthogonal prolate spheroidal wave functions as impulse responses of optical filters in orthogonal division multiplexing and spatial modes as optical continuous time basis functions it has been shown that the proposed multidimensional coded modulation schemes based on ovq scds outperform corresponding counterparts and can be used to enable beyond 10 pb s serial optical transport over spatial division multiplexing sdm fibers as well as beyond 1 pb s transport over smfs the proposed ovq scd based hybrid multidimensional coded modulation scheme can simultaneously solve the problems related to the limited bandwidth of information infrastructure high energy consumption and heterogeneity of network segments while enabling elastic and dynamic bandwidth allocation

multidimensional companding quantization of the gaussian source multidimensional companding quantization is analyzed theoretically for the case of high resolution and mean squared error distortion the optimality of choosing the expander function to be the inverse of the compressor function is established first heuristic derivations of the point density and moment of inertia of companding are given which combined result in the distortion formula by bucklew further the interaction between the lattice quantizer lq and compressor function is studied optimality is achieved with a second moment optimal lq shaped by a compressor dependent linear transform high rate theory for a radial compander and spherically symmetric sources is reviewed the result is used to evaluate the performance on an independent and identically distributed i i d gaussian source the radial compander clearly outperforms both a product scalar quantizer and a spherical quantizer for dimensions higher than two radial companding is also generalized to the correlated gaussian source finally a comparison of theory and practical companders is made in a gaussian framework

entropy density and mismatch in high rate scalar quantization with renyi entropy constraint properties of scalar quantization with th power distortion and constrained renyi entropy of order are investigated for an asymptotically high rate optimal sequence of quantizers the contribution to the renyi entropy due to source values in a fixed interval is identified in terms of the entropy density of the quantizer sequence this extends results related to the well known point density concept in optimal fixed rate quantization a dual of the entropy density result quantifies the distortion contribution of a given interval to the overall distortion the distortion loss resulting from a mismatch of source densities in the design of an asymptotically optimal sequence of quantizers is also determined this extends bucklew s fixed rate and gray et al s variable rate mismatch results to general values of the entropy order parameter

new results in binary multiple descriptions an encoder whose input is a binary equiprobable memoryless source produces one output of rate r 1 and another of rate r 2 let d 1 d 2 and d 0 respectively denote the average error frequencies with which the source data can be reproduced on the basis of the encoder output of rate r l only the encoder output of rate r 2 only and both encoder outputs the two descriptions problem is to determine the region r of all quintuples r 1 r 2 d 1 d 2 d 0 that are achievable in thc usual shannon sense let r d 1 d log 2 d 1 d log 2 1 d denote the error frequency rate distortion function of the source the no excess rate case prevails when r 1 r 2 r d 0 and the excess rate case when r 1 r 2 r d 0 denote the section of r at r 1 r 2 d 0 by d r 1 r 2 d 0 d 1 d 2 r 1 r 2 d 1 d 2 d 0 in r in the no excess rate case we show that a portion of the boundary of d r 1 r 2 d 0 coincides with the curve frac 1 2 d 1 2d 0 frac 1 2 d 2 2d 0 frac 1 2 1 2d 0 2 this curve is an extension of witsenhausen s hyperbola bound to the case d 0 0 it follows that the projection of r onto the d 1 d 2 plane at fixed d 0 consists of all d 1 geq d 0 and d 2 geq d 0 that lie on or above this hyperbola in the excess rate case we show by counterexample that the achievable region of el gamal and cover is not tight

high rate vector quantization for noisy channels with applications to wideband speech spectrum compression this paper considers the high rate performance of source coding for noisy discrete symmetric channels with random index assignment ia accurate analytical models are developed to characterize the expected distortion performance of vector quantization vq for a large class of distortion measures it is shown that when the point density is continuous the distortion can be approximated as the sum of the source quantization distortion and the channel error induced distortion expressions are also derived for the continuous point density that minimizes the expected distortion next for the case of mean squared error distortion a more accurate analytical model for the distortion is derived by allowing the point density to have a singular component the extent of the singularity is also characterized these results provide analytical models for the expected distortion performance of both conventional vq as well as for channel optimized vq as a practical example compression of the linear predictive coding parameters in the wideband speech spectrum is considered with the log spectral distortion as performance metric the theory is able to correctly predict the channel error rate that is permissible for operation at a particular level of distortion

lattice and trellis quantization with lattice and trellis bounded codebooks high rate theory for memoryless sources high rate lattice and trellis quantizers for nonuniform sources are introduced and analyzed the performance of these quantizers is determined by two separable quantities the granular gain and the boundary gain which are determined by the shapes of the granular cells and of the support region respectively the granular gain and boundary gain are the duals of shaping and coding gain in data transmission applications using this duality it is shown for gaussian sources that the ultimate achievable boundary gain with high rate lattice bounded lattice codebooks is the same as the ultimate gain that can be obtained from variable rate entropy coding it is observed that if lattice codebooks can achieve the ultimate granular gain of 0 255 b per dimension then lattice bounded lattice codebooks can approach the rate distortion limit the performance of lattice quantizers is compared to that of optimum vector quantizers

vanishing distortion and shrinking cells we establish an asymptotic connection between vanishing rth power distortion and shrinking cell diameters for vector quantizers with convex cells as a consequence a number of shrinking cell conditions may be easily verified by showing that the quantizers in question have distortion tending to zero this also plays an important role in the asymptotic analysis of a common greedy growing scheme for tree structured vector quantizers

on the minimum average distortion of quantizers with index dependent distortion measures in many applications one is interested in optimally deploying a network of unidentical agents to a certain area of interest examples include heterogeneous sensor networks and cellular networks such deployment problems can equivalently be formulated as quantizer design problems where different distortion measures should be associated with different quantization indices in this paper we consider the case where the distortion measure is the r th power r geq 1 of the distance between the reproduction point and the source sample weighted by a factor that varies from one index to another for a uniform distribution of source samples we determine the corresponding optimal scalar quantizers and their average distortions for nonuniform distributions and vector quantization we provide a high resolution analysis of the minimum possible average distortion applications to sensor and cellular networks are also discussed together with numerical design examples

multiple description wavelet based image coding we consider the problem of coding images for transmission over error prone channels the impairments we target are transient channel shutdowns as would occur in a packet network when a packet is lost or in a wireless system during a deep fade when data is delivered it is assumed to be error free but some of the data may never reach the receiver the proposed algorithms are based on a combination of multiple description scalar quantizers with techniques successfully applied to the construction of some of the most efficient subband coders a given image is encoded into multiple independent packets of roughly equal length when packets are lost the quality of the approximation computed at the receiver depends only on the number of packets received but does not depend on exactly which packets are actually received when compared with previously reported results on the performance of robust image coders based on multiple descriptions on standard test images our coders attain similar psnr values using typically about 50 60 of the bit rate required by these other state of the art coders while at the same time providing significantly more freedom in the mechanism for allocation of redundancy among descriptions

the shannon lower bound is asymptotically tight the shannon lower bound is one of the few lower bounds on the rate distortion function that holds for a large class of sources in this paper which considers exclusively norm based difference distortion measures it is demonstrated that its gap to the rate distortion function vanishes as the allowed distortion tends to zero for all sources having finite differential entropy and whose integer part has finite entropy conversely it is demonstrated that if the integer part of the source has infinite entropy then its rate distortion function is infinite for every finite distortion level thus the shannon lower bound provides an asymptotically tight bound on the rate distortion function if and only if the integer part of the source has finite entropy

a rigorous approach to high resolution entropy constrained vector quantization the nonnegativity of relative entropy implies that the differential entropy of a random vector x with probability density function pdf f is upper bounded by mathsf e left log g text x right for any arbitrary pdf g using this inequality with a cleverly chosen g we derive a lower bound on the asymptotic excess rate of entropy constrained vector quantization for d dimensional sources and r th power distortion where the asymptotic excess rate is defined as the difference between the smallest output entropy of a vector quantizer satisfying the distortion constraint and the rate distortion function in the limit as the distortion tends to zero specialized to the one dimensional case this lower bound coincides with the asymptotic excess rate achieved by a uniform quantizer thereby recovering the result by gish and pierce that uniform quantizers are asymptotically optimal as the allowed distortion tends to zero furthermore in the one dimensional case the derivation of the lower bound reveals a necessary condition for a sequence of quantizers to be asymptotically optimal this condition implies that any sequence of asymptotically optimal almost regular quantizers must converge to a uniform quantizer as the distortion tends to zero while the obtained lower bound itself is not novel to the best of our knowledge we present the first rigorous derivation that follows the direct approach by gish and pierce without resorting to heuristic high resolution approximations commonly found in the quantization literature furthermore our derivation holds for all d dimensional sources having finite differential entropy and whose integer part has finite entropy in contrast to gish and pierce we do not require additional constraints on the continuity or decay of the source pdf

asymptotic distribution of the errors in scalar and vector quantizers high rate or asymptotic quantization theory has found formulas for the average squared length more generally the qth moment of the length of the error produced by various scalar and vector quantizers with many quantization points in contrast this paper finds an asymptotic formula for the probability density of the length of the error and in certain special cases for the probability density of the multidimensional error vector itself the latter can be used to analyze the distortion of two stage vector quantization the former permits one to learn about the point density and cell shapes of a quantizer from a histogram of quantization error lengths histograms of the error lengths in simulations agree well with the derived formulas also presented are a number of properties of the error density including the relationship between the error density the point density and the cell shapes the fact that its qth moment equals bennett s integral a formula for the average distortion of a scalar or vector quantizer and the fact that for stationary sources the marginals of the multidimensional error density of an optimal vector quantizer with large dimension are approximately i i d gaussian

high resolution scalar quantization with renyi entropy constraint we consider optimal scalar quantization with rth power distortion and constrained renyi entropy of order for sources with absolutely continuous distributions the high rate asymptotics of the quantizer distortion has long been known for 0 fixed rate quantization and 1 entropy constrained quantization these results have recently been extended to quantization with renyi entropy constraint of order r 1 here we consider the more challenging case 0 0 1 and for a large class of absolutely continuous source distributions we determine the sharp asymptotics of the optimal quantization distortion the achievability proof is based on finding asymptotically optimal quantizers via the companding approach and is thus constructive

bennett s integral for vector quantizers this paper extends bennett s 1948 integral from scalar to vector quantizers giving a simple formula that expresses the rth power distortion of a many point vector quantizer in terms of the number of points point density function inertial profile and the distribution of the source the inertial profile specifies the normalized moment of inertia of quantization cells as a function of location the extension is formulated in terms of a sequence of quantizers whose point density and inertial profile approach known functions as the number of points increase precise conditions are given for the convergence of distortion suitably normalized to bennett s integral previous extensions did not include the inertial profile and consequently provided only bounds or applied only to quantizers with congruent cells such as lattice and optimal quantizers the new version of bennett s integral provides a framework for the analysis of suboptimal structured vector quantizers it is shown how the loss in performance of such quantizers relative to optimal unstructured ones can be decomposed into point density and cell shape losses as examples these losses are computed for product quantizers and used to gain further understanding of the performance of scalar quantizers applied to stationary memoryless sources and of transform codes applied to gaussian sources with memory it is shown that the short coming of such quantizers is that they must compromise between point density and cell shapes

sequential linear interpolation of multidimensional functions we introduce a new approach that we call sequential linear interpolation sli for approximating multidimensional nonlinear functions the sli is a partially separable grid structure that allows us to allocate more grid points to the regions where the function to be interpolated is more nonlinear this approach reduces the mean squared error mse between the original and approximated function while retaining much of the computational advantage of the conventional uniform grid interpolation to obtain the optimal grid point placement for the sli structure we appeal to an asymptotic analysis similar to the asymptotic vector quantization vq theory in the asymptotic analysis we assume that the number of interpolation grid points is large and the function to be interpolated is smooth closed form expressions for the mse of the interpolation are obtained from the asymptotic analysis these expressions are used to guide us in designing the optimal sli structure for cases where the assumptions underlying the asymptotic theory are not satisfied we develop a postprocessing technique to improve the mse performance of the sli structure the sli technique is applied to the problem of color printer characterization where a highly nonlinear multidimensional function must be efficiently approximated our experimental results show that the appropriately designed sli structure can greatly improve the mse performance over the conventional uniform grid

lagrangian empirical design of variable rate vector quantizers consistency and convergence rates the lagrangian formulation of variable rate vector quantization is known to yield useful necessary conditions for quantizer optimality and generalized lloyd algorithms for quantizer design the lagrangian formulation is demonstrated to provide a convenient framework for analyzing the empirical design of variable rate vector quantizers in particular the consistency of empirical design based on minimizing the lagrangian performance over a stationary and ergodic training sequence is shown for sources with finite second moment the finite sample performance is also studied for independent training data and sources with bounded support

on self similar sets with overlaps and inverse theorems for entropy we study the dimension of self similar sets and measures on the line we show that if the dimension is less than the generic bound of minf1 sg where s is the similarity dimension then there are superexponentially close cylinders at all small enough scales this is a step towards the conjecture that such a dimension drop implies exact overlaps and conrms it when the generating similarities have algebraic coecients as applications we prove furstenberg s conjecture on projections of the one dimensional sierpinski gasket and achieve some progress on the bernoulli convolutions problem and more generally on problems about parametric families of self similar measures the key tool is an inverse theorem on the structure of pairs of probability measures whose mean entropy at scale 2 n has only a small amount of growth under convolution

a novel approach of divergence free reconstruction for adaptive mesh refinement a simple novel approach to maintain the divergence free condition with adaptive mesh refinement is presented this new approach uses reconstructions on the coarse faces together with the divergence free condition to reconstruct the field values on the internal fine faces it does not construct a global interpolation polynomial over a whole coarse cell therefore it can be easily applied to any refinement ratio it is implemented via a directionally splitting approach so that it can be applied to any kind of grids and in any dimensions implementation is presented in the cartesian cylindrical and spherical geometries it is shown by several 2d magneto hydrodynamic simulations that such a method can keep the divergence free error of magnetic fields at the round off level

two results on the asymptotic performance of quantizers a necessary and sufficient condition is presented for the normalized asymptotic r th power distortion of a mismatched multidimensional quantizer to converge to a certain optimum constant known as bennett s integral the dimensionality of our results is allowed to approach infinity in order to make some universal source coding comparisons to quantization theory

high resolution source coding for non difference distortion measures multidimensional companding entropy coded vector quantization is studied using high resolution multidimensional companding over a class of nondifference distortion measures for distortion measures which are locally quadratic a rigorous derivation of the asymptotic distortion and entropy coded rate of multidimensional companders is given along with conditions for the optimal choice of the compressor function this optimum compressor when it exists depends on the distortion measure but not on the source distribution the rate distortion performance of the companding scheme is studied using an asymptotic expression for the rate distortion function which parallels the shannon lower bound for difference distortion measures it is proved that the high resolution performance of the scheme is arbitrarily close to the rate distortion limit for large quantizer dimensions if the compressor function and the lattice quantizer used in the companding scheme are optimal extending an analogous statement for entropy coded lattice quantization and mse distortion the companding approach is applied to obtain a high resolution quantizing scheme for noisy sources

adaptive learning vector quantization for online parametric estimation this paper addresses the problem of parameter estimation in a quantized and online setting a sensing unit collects random vector valued samples from the environment these samples are quantized and transmitted to a central processor which generates an online estimate of the unknown parameter this paper provides a closed form expression of the excess mean square error mse caused by quantization in the high rate regime i e when the number of quantization levels is supposed to be large next we determine the quantizers which mitigate the excess mse the optimal quantization rule unfortunately depends on the unknown parameter to circumvent this issue we introduce a novel adaptive learning vector quantization scheme which allows to simultaneously estimate the parameter of interest and select an efficient quantizer

on source transmission over mimo channels with limited feedback the problem of source channel coding over a multiple antenna mimo channel with quantized channel state information at the transmitter csit is considered upper bounds on the distortion exponents achieved with partial csit under a long term power constraint are developed it is shown that the distortion exponent with perfect csit grows unbounded as the ratio between the channel and source bandwidth increases while the exponent achieved with any feedback link of fixed finite resolution is bounded above by a polynomial of the product between the number of transmit and number of receive antennas the resolution of the feedback link should grow with the bandwidth ratio to make the distortion exponent scale as fast as that in the perfect csit case we show that in order to achieve the optimal scaling the csit feedback resolution must grow logarithmically with the bandwidth ratio for mimo channels and faster than linear for the single input single output channel the achievable distortion exponent of some hybrid schemes with heavily quantized feedback is also derived the results demonstrate that dramatic performance improvement over the case of no csit can be achieved by combining simple schemes with a very coarse csit feedback

enhanced multistage vector quantization by joint codebook design multistage vector quantization msvq can achieve very low encoding and storage complexity in comparison to unstructured vector quantization however the conventional stage by stage design of the codebooks in msvq is suboptimal with respect to the overall performance measure the authors introduce an algorithm for the joint design of the stage codebooks to optimize the overall performance the performance improvement although modest is achieved with no effect on encoding or storage complexity and only a slight increase in design effort

three dimensional adaptive mesh refinement for hyperbolic conservation laws a local adaptive mesh refinement algorithm for solving hyperbolic systems of conservation laws in three space dimensions is described the method is based on the use of local grid patches superimpo

a simple class of asymptotically optimal quantizers a simple class of quantizers is introduced which are asymptotically optimal as the number of quantization levels increases to infinity with respect to a mean r th power absolute error distortion measure these asymptotically optimal quantizers are very easy to compute their performance is evaluated for several distributions and compares favorably with the performance of the optimal quantizers in all cases for which the latter have been computed in addition their asymptotic robustness is studied under location scale and shape mismatch for several families of distributions

comparison of some flux corrected transport and total variation diminishing numerical schemes for hydrodynamic and magnetohydrodynamic problems two versions of flux corrected transport and two versions of total variation diminishing schemes are tested for several one and two dimensional hydrodynamic and magnetohydrodynamic problems two of the schemes ydfct and tvdlf are tested extensively for the first time the results give an insight into the limitations of the methods their relative strengths and weaknesses some subtle points of the algorithms and the effects of selecting different options for certain methods are emphasised

optimal companding vector quantization for circularly symmetric sources in this paper it is shown that optimal z 2 lattice vector quantization can be implemented using radial companding technique we derive the optimal vector compressor function for radial compander of memoryless gaussian source this result is obtained by taking into consideration the source geometry and by establishing the relation between the volumes and the point densities at the compressor input and compressor output we also derive the linearized model the piecewise linear compander its performance closely approaches that of optimal vector quantization for example for r 8bits dimension and l 16 regions the difference between corresponding distortions is about 0 037db while the asymptotic performances are identical

suboptimality of the karhunen loeve transform for transform coding we examine the performance of the karhunen loeve transform klt for transform coding applications the klt has long been viewed as the best available block transform for a system that orthogonally transforms a vector source scalar quantizes the components of the transformed vector using optimal bit allocation and then inverse transforms the vector this paper treats fixed rate and variable rate transform codes of non gaussian sources the fixed rate approach uses an optimal fixed rate scalar quantizer to describe the transform coefficients the variable rate approach uses a uniform scalar quantizer followed by an optimal entropy code and each quantized component is encoded separately earlier work shows that for the variable rate case there exist sources on which the klt is not unique and the optimal quantization and coding stage matched to a worst klt yields performance as much as 1 5 db worse than the optimal quantization and coding stage matched to a best klt in this paper we strengthen that result to show that in both the fixed rate and the variable rate coding frameworks there exist sources for which the performance penalty for using a worst klt can be made arbitrarily large further we demonstrate in both frameworks that there exist sources for which even a best klt gives suboptimal performance finally we show that even for vector sources where the klt yields independent coefficients the klt can be suboptimal for fixed rate coding

scalar hyperbolic pde simulations and coupling strategies we report on grid adaptive multi dimensional simulations for hyperbolic pdes with a deliberate focus on the analytically tractable scalar case motivated by recent efforts towards multi physics simulation strategies we investigate a variety of coupling strategies for numerically solving hyperbolic partial differential equations pdes we use adaptive mesh refinement in combination with shock capturing spatio temporal discretizations and first present accuracy and validation tests on both smooth and shock dominated evolutions to investigate the feasibility of coupling means for multi physics simulations we then introduce new reference tests where spatially different flux prescriptions require coupling strategies across the domains where local advection burgers or nonconvex behavior is imposed for these nonlinear single scalar equations we can illustrate and analytically explain the evolutions obtained when handling cases where fluxes differ in spatially non overlapping regions we discuss both conservative and non conservative ways of coupling across the region boundaries when coupling scalar conservation laws where the characteristic speeds change discontinuously across the regional boundaries these two strategies yield differing but mathematically sound solution behaviors their relevance for multi physics simulations where one couples systems of hyperbolic pdes is discussed

asymptotic formulas for variance mismatched fixed rate scalar quantization of a gaussian source asymptotic formulas are derived for the mean squared error mse distortion and the snr of a variance mismatched fixed rate scalar quantizer designed mse optimally for a gaussian source it is discovered that in the case that the standard deviation ratio of the applied to density to the designed for density is greater than 3 2 the snr in db increases at large rate r as 9 03r 2 15 1 1 2 log10r plus a constant that depends only on

scalar versus vector quantization worst case analysis we study the potential merits of vector quantization and show that there can be an arbitrary discrepancy between the worst case rates required for scalar and vector quantization specifically we describe a random variable and a distortion measure where quantization of a single instance to within a given distortion requires an arbitrarily large number of bits in the worst case but quantization of multiple independent instances to within the same distortion requires an arbitrarily small number of bits per instance in the worst case we relate this discrepancy to expander graphs representation and cover numbers of set systems and a problem studied by slepian wolf and wyner 1973

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

distributed scalar quantization for computing high resolution analysis and extensions communication of quantized information is frequently followed by a computation we consider situations of distributed functional scalar quantization distributed scalar quantization of possibly correlated sources followed by centralized computation of a function under smoothness conditions on the sources and function companding scalar quantizer designs are developed to minimize mean squared error mse of the computed function as the quantizer resolution is allowed to grow striking improvements over quantizers designed without consideration of the function are possible and are larger in the entropy constrained setting than in the fixed rate setting as extensions to the basic analysis we characterize a large class of functions for which regular quantization suffices consider certain functions for which asymptotic optimality is achieved without arbitrarily fine quantization and allow limited collaboration between source encoders in the entropy constrained setting a single bit per sample communicated between encoders can have an arbitrarily large effect on functional distortion in contrast such communication has very little effect in the fixed rate setting

a new extension of weibull distribution properties and different methods of estimation abstract the weibull distribution has been generalized by many authors in recent years here we introduce a new generalization of the weibull distribution called alpha logarithmic transformed weibull distribution that provides better fits than some of its known generalizations the proposed distribution contains weibull exponential logarithmic transformed exponential and logarithmic transformed weibull distributions as special cases our main focus is the estimation from frequentist point of view of the unknown parameters along with some mathematical properties of the new model the proposed distribution accommodates monotonically increasing decreasing bathtub and unimodal and then bathtub shape hazard rates so it turns out to be quite flexible for analyzing non negative real life data we briefly describe different frequentist approaches namely maximum likelihood estimators percentile based estimators least squares estimators weighted least squares estimators maximum product of spacings estimators and compare them using extensive numerical simulations monte carlo simulations are performed to compare the performances of the proposed methods of estimation for both small and large samples the potentiality of the distribution is analyzed by means of two real data sets

a robust moving mesh finite volume method applied to 1d hyperbolic conservation laws from magnetohydrodynamics in this paper we describe a one dimensional adaptive moving mesh method and its application to hyperbolic conservation laws from magnetohydrodynamics mhd the method is robust because it employs automatic control of mesh adaptation when a new model is considered without manually set parameters adaptive meshes are a common tool for increasing the accuracy and reducing computational costs when solving time dependent partial differential equations pdes mesh points are moved towards locations where they are needed the most to obtain a time dependent adaptive mesh monitor functions are used to automatically monitor the importance of the various parts of the domain by assigning a weight value to each location based on the equidistribution principle all mesh points are distributed according to their assigned weights we use a sophisticated monitor function that tracks both small local phenomena as well as large shocks in the same solution the combination of the moving mesh method and a high resolution finite volume solver for hyperbolic pdes yields a serious gain in accuracy at relatively no extra costs the results of several numerical experiments including comparisons with h refinement are presented which cover many intriguing aspects typifying nonlinear magneto fluid dynamics with higher accuracy than often seen in similar publications

dimension theory of iterated function systems let si l i 1 be an iterated function system ifs on d with attractor k let denote the one sided full shift over the alphabet 1 l we define the projection entropy function h on the space of invariant measures on e associated with the coding map k and develop some basic ergodic properties about it this concept turns out to be crucial in the study of dimensional properties of invariant measures on k we show that for any conformal ifs respectively the direct product of finitely many conformal ifss without any separation condition the projection of an ergodic measure under is always exactly dimensional and its hausdorff dimension can be represented as the ratio of its projection entropy to its lyapunov exponent respectively the linear combination of projection entropies associated with several coding maps furthermore for any conformal ifs and certain affine ifss we prove a variational principle between the hausdorff dimension of the attractors and that of projections of ergodic measures 2008 wiley periodicals inc

design of multiple description scalar quantizers the design of scalar quantizers for communication systems that use diversity to overcome channel impairments is considered the design problem is posed as an optimization problem and necessary conditions for optimality are derived a design algorithm a generalization of s p lloyd s 1962 algorithm for quantizer design is developed unlike a single channel scalar quantizer the performance of a multiple description scalar quantizer is dependent on the index assignment the problem of index assignment is addressed good index assignments performance results and sample quantizer designs are presented for a memoryless gaussian source comparisons are made with rate distortion bounds for the multiple description problem

optimal entropy constrained scalar quantization of a uniform source optimal scalar quantization subject to an entropy constraint is studied for a wide class of difference distortion measures including rth power distortions with r 0 it is proved that if the source is uniformly distributed over an interval then for any entropy constraint r in nats an optimal quantizer has n e sup r interval cells such that n 1 cells have equal length d and one cell has length c spl les d the cell lengths are uniquely determined by the requirement that the entropy constraint is satisfied with equality based on this result a parametric representation of the minimum achievable distortion d sub h r as a function of the entropy constraint r is obtained for a uniform source the d sub h r curve turns out to be nonconvex in general moreover for the squared error distortion it is shown that d sub h r is a piecewise concave function and that a scalar quantizer achieving the lower convex hull of d sub h r exists only at rates r log n where n is a positive integer