multiple description trellis coded quantization we present a construction of multiple description trellis coded quantizers we use the tensor product of trellises to build a trellis which is applicable to multiple description coding the problems of index assignment and set partitioning for the resulting trellis are considered the viterbi algorithm provides the best path for encoding and the design procedure utilizes a generalized lloyd algorithm the encoding process simultaneously generates all the transmitted sequences furthermore the complexity of the scheme is almost independent of the rate the quantizer provides remarkable performance with little encoding complexity

lagrangian optimization of two description scalar quantizers in this paper we study the problem of optimal design of balanced two description fixed rate scalar quantizer 2dsq under the constraint of convex codecells using a graph based approach to model the problem we show that the minimum expected distortion of the 2dsq is a convex function of the number of codecells in the side quantizers this property allows the problem to be solved by lagrangian minimization for which the optimal lagrangian multiplier exists given a trial multiplier we exploit a monotonicity of the objective function and develop a simple and fast dynamic programming technique to solve the parameterized problem to further improve the algorithm efficiency we propose an rd guided search strategy to find the optimal lagrangian multiplier in our experiments on distributions of interest for signal compression applications the proposed algorithm improves the speed of the fastest algorithm so far by a factor of o k logk where k is the number of codecells in each side quantizer we also assess the impact on the optimality of the convex codecell constraint using a published performance analysis of 2dsq at high rates we show that asymptotically this constraint does not preclude optimality for l 2 distortion measure when channels have a higher than 0 12 loss rate

the boundary of the irreducible components for invariant subspace varieties given partitions alpha beta gamma the short exact sequences begin aligned 0 longrightarrow n alpha longrightarrow n beta longrightarrow n gamma longrightarrow 0 end aligned of nilpotent linear operators of jordan types alpha beta gamma respectively define a constructible subset mathbb v alpha gamma beta of an affine variety geometrically the varieties mathbb v alpha gamma beta are of particular interest as they occur naturally and since they typically consist of several irreducible components in fact each littlewood richardson tableau vargamma of shape alpha beta gamma contributes one irreducible component overline mathbb v vargamma we consider the partial order vargamma le mathsf boundary widetilde vargamma on lr tableaux which is the transitive closure of the relation given by mathbb v widetilde vargamma cap overline mathbb v vargamma ne emptyset in this paper we compare the boundary relation with partial orders given by algebraic combinatorial and geometric conditions it is known that in the case where the parts of alpha are at most two all those partial orders are equivalent we prove that those partial orders are also equivalent in the case where beta setminus gamma is a horizontal and vertical strip moreover we discuss how the orders differ in general

gaussian multiple description coding with low density generator matrix codes it is shown that the coding problem for an arbitrary point on the dominant face of an l description el gamal cover egc region can be converted to that for a vertex of a k description egc region for some k 2l 1 where the latter problem can be solved via successive coding a practical scheme is proposed for the quadratic gaussian case by reducing each step in successive coding to a gaussian quantization operation and implementing such an operation using low density generator matrix codes the effectiveness of this scheme is verified through extensive simulation experiments

dithered lattice based quantizers for multiple descriptions multiple description md source coding is aimed at achieving graceful degradation in reconstruction with respect to losing portions of the code with the cost of some redundancy we examine md schemes which use entropy coded dithered lattice quantizers ecdq we propose two techniques one based on successive refinement sr and the other a dithered and periodic version of the mid scalar quantizer mdsq with distributed cells proposed by vaishampayan 1993 1994 similarly to the single description case both techniques are universal in nature and are equivalent to additive noise channels this allows one to derive analytical expressions for the rate distortion performance for general sources and to compare them to the optimal rate regions at both high and low resolutions among other results we establish that while the dithered mdsq scheme loses only the space filling loss of the scalar lattice at any resolution the sr based scheme loses an additional 0 5 bit at any lattice dimension possible improvements such as refinement time sharing and dependent dithering are discussed

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

optimal quantizer design for noisy channels an approach to combined source channel coding we present an analysis of the zero memory quantization of memoryless sources when the quantizer output is to be encoded and transmitted across a noisy channel necessary conditions for the joint optimization of the quantizer and the encoder decoder pair are presented and an iterative algorithm for obtaining a locally optimum system is developed the performance of this locally optimal system obtained for the class of generalized gaussian distributions and the binary symmetric channel is compared against the optimum performance theoretically attainable using rate distortion theoretic arguments as well as against the performance of lloyd max quantizers encoded using the natural binary code and the folded binary code it is shown that this optimal design could result in substantial performance improvements the performance improvements are more noticeable at high bit rates and for broad tailed densities

index assignment for multichannel communication under failure we consider the problem of constructing multiple description scalar quantizers and describing the achievable rate distortion tuples in that setting we model this as a combinatorial optimization problem of number arrangements in a matrix this approach gives a general technique for deriving lower bounds on the distortion at given channel rates this technique is constructive thus allowing an algorithm that gives an upper bound for the case of two communication channels with equal rates the bounds coincide thus giving the precise lowest achievable distortion at fixed rates the bounds are within a small constant for a higher number of channels to the best of our knowledge this is the first result involving systems with more than two communication channels

on the design of symmetric entropy constrained multiple description scalar quantizer with linear joint decoders this paper addresses the design of symmetric entropy constrained multiple description scalar quantizers ec mdsq with linear joint decoders i e where some of the decoders compute the reconstruction by averaging the reconstructions of individual descriptions thus the use of linear decoders reduces the space complexity at the decoder since only a subset of the codebooks needs to be stored the proposed design algorithm locally minimizes the lagrangian which is a weighted sum of the expected distortion and of the side quantizers rates the algorithm is inspired by the ec mdsq design algorithm proposed by vaishampayan and domaszewicz and it is adapted from two to k descriptions differently from the aforementioned work the optimization of the reconstruction values can no longer be performed separately at the decoder optimization step interestingly we show that the problem is a convex quadratic optimization problem which can be efficiently solved moreover the generalization of the encoder optimization step from two to k descriptions increases drastically the amount of computations we show how to exploit the special form of the cost function conferred by the linear joint decoders to significantly reduce the time complexity at this step we compare the performance of the proposed design with multiple description lattice vector quantizers mdlvq and with the multiple description scheme based on successive refinement and unequal erasure protection uep our experiments show that the proposed approach outperforms mdlvq with dimension 1 quantization as expected additionally when more codebooks are added our scheme even beats mdlvq with quantization dimension approaching infty for rates sufficiently high furthermore the proposed approach is also superior to uep with dimension 1 quantization when the rates are low

multiple description coding with randomly and uniformly offset quantizers in this paper two multiple description coding schemes are developed based on prediction induced randomly offset quantizers and unequal deadzone induced near uniformly offset quantizers respectively in both schemes each description encodes one source subset with a small quantization stepsize and other subsets are predictively coded with a large quantization stepsize in the first method due to predictive coding the quantization bins that a coefficient belongs to in different descriptions are randomly overlapped the optimal reconstruction is obtained by finding the intersection of all received bins in the second method joint dequantization is also used but near uniform offsets are created among different low rate quantizers by quantizing the predictions and by employing unequal deadzones by generalizing the recently developed random quantization theory the closed form expression of the expected distortion is obtained for the first method and a lower bound is obtained for the second method the schemes are then applied to lapped transform based multiple description image coding the closed form expressions enable the optimization of the lapped transform an iterative algorithm is also developed to facilitate the optimization theoretical analyzes and image coding results show that both schemes achieve better performance than other methods in this category

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

asymptotical analysis of several multiple description scenarios with l 3 descriptions this work compares theoretically the performance of several representative practical multiple description md frameworks with l 3 symmetric descriptions the first scenario is the classic unequal erasure protection uep scheme using a successively refinable code src and reed solomon codes the second scenario is an improvement upon uep by applying domain partitioning and permuted reed solomon codes the third scenario uses a finer partitioning and erasure correction via repetition codes additionally the md lattice vector quantizer and another recent md scheme are considered in the comparison the aforementioned md schemes are compared in terms of the expected squared error asymptotically achievable as the rate r of a description approaches assuming independent description losses our analysis reveals that the improvement of the second scenario upon the first one when r can reach up to 1 68 db but it approaches 0 as the description loss rate p goes to 0 and l approaches additionally we find that the first two schemes outperform the third one with an unbounded gain when r as p 0 or l further we show that the first three scenarios achieve unbounded improvements over the other two as r and p 0 on the other hand we point out that some of the results of our asymptotic analysis rely on strong assumptions and therefore an experimental validation is needed before applying them to practical situations

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

invariant subspaces of nilpotent linear operators i let k be a field we consider triples v u t where v is a finite dimensional k space u a subspace of v and t v v a linear operator with t n 0 for some n and such that t u u thus t is a nilpotent operator on v and u is an invariant subspace with respect to t we will discuss the question whether it is possible to classify these triples these triples v u t are the objects of a category with the krullremak schmidt property thus it will be sufficient to deal with indecomposable triples obviously the classification problem depends on n and it will turn out that the decisive case is n 6 for n 6 we deal with what is called wild representation type so no complete classification can be expected for n 6 we will exhibit a complete description of all the indecomposable triples let k be a field and v a finite dimensional k vector space we consider a linear operator t v v which is nilpotent say satisfying t n 0 for some fixed natural number n and we are interested in the subspaces u of v which are t invariant this means that t u u to be more precise we are going to consider the corresponding triples v u t and ask for a classification of all the isomorphism classes in order to do so we consider these triples v u t as the objects of a category s n or sk n in case a reference to the base field

degenerations in a class of matrix varieties and prinjective modules abstract let k be an algebraically closed field and let i be a finite partially ordered set to any vector v n i we associate the irreducible variety mat v i with an action of the algebraic group g v i see section 6 the main aim of this paper is to give a description of degenerations of orbits of this action in terms of homomorphisms and extensions of prinjective ki modules

n channel asymmetric entropy constrained multiple description lattice vector quantization this paper is about the design and analysis of an index assignment ia based multiple description coding scheme for the n channel asymmetric case we use entropy constrained lattice vector quantization and restrict attention to simple reconstruction functions which are given by the inverse ia function when all descriptions are received or otherwise by a weighted average of the received descriptions we consider smooth sources with finite differential entropy rate and mse fidelity criterion as in previous designs our construction is based on nested lattices which are combined through a single ia function the results are exact under high resolution conditions and asymptotically as the nesting ratios of the lattices approach infinity for any n the design is asymptotically optimal within the class of ia based schemes moreover in the case of two descriptions and finite lattice vector dimensions greater than one the performance is strictly better than that of existing designs in the case of three descriptions we show that in the limit of large lattice vector dimensions points on the inner bound of pradhan can be achieved furthermore for three descriptions and finite lattice vector dimensions we show that the ia based approach yields in the symmetric case a smaller rate loss than the recently proposed source splitting approach

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

flexible symmetric multiple description lattice vector quantizer with l geq 3 descriptions in the previous work on multiple description lattice vector quantizers mdlvqs with l geq 3 descriptions once the central and side lattice codebooks are fixed the decoding quality is determined for all numbers k of received descriptions therefore it is not possible to achieve tradeoffs between the quality of reconstruction for different values of k 1 leq k leq l 1 this paper proposes a flexible mdlvq capable of overcoming the above drawback for this a different reconstruction method is employed and a heuristic index assignment ia algorithm which uses l 2 parameters to control the distortions for 2 leq k leq l 1 is developed experimental results show that the proposed mdlvq in addition to achieving the desired tradeoffs significantly outperforms the classic md scheme based on unequal erasure protection the second contribution of this paper is a structured ia for the case of l 3 and the derivation of the corresponding expressions of the distortions at high resolution the proposed ia has a simple mechanism for controlling the tradeoff between the reconstruction quality for k 1 2 the ia is able to achieve a wide range of distortion values while keeping the product of the distortions for k 1 2 the same as in the prior work

the rate distortion region for multiple descriptions without excess rate during recent years there has been strong interest in a certain source coding problem which some authors call the problem of multiple descriptions old and new wringing techniques enable us to establish a single letter characterization of the rate distortion region in the case of no excess rate for the joint description

optimality and suboptimality of multiple description vector quantization with a lattice codebook the asymptotic analysis of multiple description vector quantization mdvq with a lattice codebook for sources with smooth probability density functions pdfs is considered in this correspondence goyal et al 2002 observed that as the side distortion decreases and the central distortion correspondingly increases the quantizer cells farther away from the coarse lattice points shrink in a spatially periodic pattern in this correspondence two special classes of index assignments are used along strategic groupings of central quantizer cells to derive a straightforward asymptotic analysis which provides an analytical explanation for the aforementioned observation mdvq with a lattice codebook was shown earlier to be asymptotically optimal in high dimensions with a curious converging property that the side quantizers achieve the space filling advantage of an n dimensional sphere instead of an n dimensional optimal polytope the analysis presented here explains this behavior readily while central quantizer cells on a uniform lattice are asymptotically optimal in high dimensions the present authors have shown that by using nonuniform rather than uniform central quantizer cells the central side distortion product in an mdsq can be reduced by 0 4 db at asymptotically high rate the asymptotic analysis derived here partially unifies these previous results in the same framework though a complete characterization is still beyond reach

bit error resilient index assignment for multiple description scalar quantizers this paper addresses the problem of increasing the robustness to bit errors for two description scalar quantizers our approach is to start with an m diagonal index assignment and further apply a permutation to the indexes of each description to increase the minimum hamming distance d rm min of the set of valid index pairs in particular we show how to construct linear permutation pairs achieving d rm min geq 3 and establish a lower bound in terms of the description rate r for the highest value of m for which such permutations exist

alphabet constrained data compression the optimal data compression problem is posed in terms of an alphabet constraint rather than an entropy constraint solving the optimal alphabet constrained data compression problem yields explicit source encoder decoder designs which is in sharp contrast to other approaches the alphabet constrained approach is shown to have the additional advantages that 1 classical waveform encoding schemes such as pulse code modulation pcm differential pulse code modulation dpcm and delta modulation dm as well as rate distortion theory motivated tree trellis coders fit within this theory 2 the concept of preposterior analysis in data compression is introduced yielding a rich new class of coders and 3 it provides a conceptual framework for the design of joint source channel coders for noisy channel applications examples are presented of single path differential encoding delayed or tree encoding preposterior analysis and source coding over noisy channels

asymmetric multiple description lattice vector quantizers we consider the design of asymmetric multiple description lattice quantizers that cover the entire spectrum of the distortion profile ranging from symmetric or balanced to successively refinable we present a solution to a labeling problem which is an important part of the construction along with a general design procedure the high rate asymptotic performance of the quantizer is also studied we evaluate the rate distortion performance of the quantizer and compare it to known information theoretic bounds the high rate asymptotic analysis is compared to the performance of the quantizer

minimum breakdown degradation in binary source encoding a memoryless binary equiprobable source produces one letter per second two people each are provided separately with private information about the source data at a rate of 1 2 bit per second suppose that by pooling their information they can produce a long mn reconstruction of the source output that has arbitrarily small error frequency we prove that then the least common asymptotic error frequency d that each can achieve without the other s help is sqrt 2 1 2 0 207 since it had been shown previously that 0 200 leq d leq 0 207 our result closes the so called 007 gap new analytical techniques introduced to effect the proof are of broader significance in multiuser information theory

optimum quantizer performance for a class of non gaussian memoryless sources the performance of optimum quantizers subject to an entropy constraint is studied for a wide class of memoryless sources for a general distortion criterion necessary conditions are developed for optimality and a recursive algorithm is described for obtaining the optimum quantizer under a mean square error criterion the performance of entropy encoded uniform quantization of memoryless gaussian sources is well known to be within 0 255 bits sample of the rate distortion bound at relatively high rates despite claims to the contrary it is demonstrated that similar performance can be expected for a wide range of memoryless sources indeed for the cases considered the worst case performance is observed to be less than 0 3 bits sample from the rate distortion bound and in most cases this disparity is less at iow rates

applications of littlewood richardson tableaux to computing generic extension of semisimple invariant subspaces of nilpotent linear operators abstract the main aim of the paper is to present a combinatorial algorithm that applying littlewood richardson tableaux with entries equal to 1 computes generic extensions of semisimple invariant subspaces of nilpotent linear operators moreover we discuss geometric properties of generic extensions and their connections with combinatorics

on codecell convexity of optimal multiresolution scalar quantizers for continuous sources it has been shown by earlier results that for fixed rate multiresolution scalar quantizers and for mean squared error distortion measure codecell convexity precludes optimality for certain discrete sources however it was unknown whether the same phenomenon can occur for any continuous source in this paper examples of continuous sources even with bounded continuous densities are presented for which optimal fixed rate multiresolution scalar quantizers cannot have only convex codecells proving that codecell convexity precludes optimality also for such regular sources

on degenerations and extensions of finite dimensional modules abstract we derive a cancellation theorem for degenerations of modules that says in particular that projective or injective common direct summands can always be neglected combining the cancellation result with the existence of almost split sequences we characterize the orbit closure of a module living on preprojective components by the fact that the dimension of the homomorphism space to any other module does not decrease for representation directed algebras whence in particular for path algebras of dynkin quivers we provide an alternative proof which shows in addition that any minimal degenerationnofmcomes from an exact sequence with middle termmwhose end terms add up ton by a careful examination the same is true for degenerations of matrix pencils having used so far the existence of certain extensions to obtain degenerations we then turn the tables and use degenerations to produce a lot of interesting short exact sequences in particular we show that any non simple indecomposable over a tame quiver is an extension of an indecomposable and a simple

bounded natural frequencies of completely free rectangular plates this paper presents dimensionless natural frequency parameters for completely free rectangular plates calculated using the superposition method sm and the finite difference method fdm the convergence tests show that while the sm gives upper bound results the fdm gives lower bound results the results obtained using the fdm appears to be the best lower bound results available to date together with the upper bound results from the sm the maximum possible error in the first 12 non zero natural frequency parameters for completely free rectangular plates of various aspects ratios ranging from 1 1 to 3 1 can be determined from the results presented here

n channel entropy constrained multiple description lattice vector quantization in this paper we derive analytical expressions for the central and side quantizers which under high resolution assumptions minimize the expected distortion of a symmetric multiple description lattice vector quantization md lvq system subject to entropy constraints on the side descriptions for given packet loss probabilities we consider a special case of the general n channel symmetric multiple description problem where only a single parameter controls the redundancy tradeoffs between the central and the side distortions previous work on two channel md lvq showed that the distortions of the side quantizers can be expressed through the normalized second moment of a sphere we show here that this is also the case for three channel md lvq furthermore we conjecture that this is true for the general n channel md lvq for given source target rate and packet loss probabilities we find the optimal number of descriptions and construct the md lvq system that minimizes the expected distortion we verify theoretical expressions by numerical simulations and show in a practical setup that significant performance improvements can be achieved over state of the art two channel md lvq by using three channel md lvq

representation types of the category of subprojective representations of a finite poset over k t t m and a solution of a birkhoff type problem we determine the representation type wild tame polynomial growth of the category fspr i fm of filtered subprojective fm representations of a finite poset i in terms of m and i w herefm k t t m m 1 and k is an algebraically closed field criteria for tameness wildness and tameness of nonpolynomial growth of fspr i fm are given in theorems 1 1 and 1 2 as an application a solution of birkhoff s type problem g birkhoff subgroups of abelian groups proc london math soc 38 1934 385 401 for the category rep ft i fm of filtered i chains of fm modules is given in section 5 by determining the representation type repft i fm 2007 elsevier inc all rights reserved

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

an improvement to multiple description transform coding a multiple description transform coding mdtc method has been reported previously the redundancy rate distortion rrd performance of this coding scheme for the independent and identically distributed i i d two dimensional 2 d gaussian source has been analyzed using mean squared error mse as the distortion measure at the small redundancy region the mdtc scheme can achieve excellent rrd performance because a small increase in redundancy can reduce the single description distortion at a rate faster than exponential but the performance of mdtc becomes increasingly poor at larger redundancies this paper describes a generalization of the mdtc gmdtc scheme which introduces redundancy both by transform and through correcting the error resulting from a single description its rrd performance is closer to the theoretical bound in the entire range of redundancy analysis both for a single pair of variables and for multiple variables is presented

design of entropy constrained multiple description scalar quantizers the problem of entropy constrained multiple description scalar quantizer design is posed as an optimization problem necessary conditions for optimality are derived and an iterative design algorithm is presented performance results are presented for a gaussian source along with comparisons to the multiple description rate distortion bound and a reference system

optimal index assignment for multiple description scalar quantization with translated lattice codebooks we design a k description scalar quantizer whose construction is based on a structure of translated scalar lattices and a lattice in k 1 dimensional space the use of translated lattices provides a performance advantage by exploiting a so called staggering gain the use of the k 1 dimensional lattice facilitates analytic insight into the performance and significantly speeds up the computation of the index assignment compared to state of the art methods using a common decoding method the proposed index assignment is proven to be optimal for the k description case it is shown that the optimal index assignment is not unique this is illustrated for the two description case where a periodic index assignment is selected from possible optimal assignments and described in detail the performance of the proposed quantizer accurately matches theoretic analysis over the full range of operational redundancies moreover the quantizer outperforms the state of the art md scheme as the redundancy among the description increases

tracking the best quantizer an algorithm is presented for online prediction that allows to track the best expert efficiently even when the number of experts is exponentially large provided that the set of experts has a certain additive structure as an example we work out the case where each expert is represented by a path in a directed graph and the loss of each expert is the sum of the weights over the edges in the path these results are then used to construct universal limited delay schemes for lossy coding of individual sequences in particular we consider the problem of tracking the best scalar quantizer that is adaptively matched to the source sequence with piecewise different behavior a randomized algorithm is presented which can perform on any source sequence asymptotically as well as the best scalar quantization algorithm that is matched to the sequence and is allowed to change the employed quantizer for a given number of times the complexity of the algorithm is quadratic in the sequence length but at the price of some deterioration in performance the complexity can be made linear analogous results are obtained for sequential multiresolution and multiple description scalar quantization of individual sequences

generalized multiple description coding with correlating transforms multiple description md coding is source coding in which several descriptions of the source are produced such that various reconstruction qualities are obtained from different subsets of the descriptions unlike multiresolution or layered source coding there is no hierarchy of descriptions thus md coding is suitable for packet erasure channels or networks without priority provisions generalizing work by orchard wang vaishampayan and reibman see proc ieee int conf image processing vol i santa barbara ca p 608 11 1997 a transform based approach is developed for producing m descriptions of an n tuple source m spl les n the descriptions are sets of transform coefficients and the transform coefficients of different descriptions are correlated so that missing coefficients can be estimated several transform optimization results are presented for memoryless gaussian sources including a complete solution of the n 2 m 2 case with arbitrary weighting of the descriptions the technique is effective only when independent components of the source have differing variances numerical studies show that this method performs well at low redundancies as compared to uniform md scalar quantization

box moves on littlewood richardson tableaux and an application to invariant subspace varieties abstract in his 1951 book infinite abelian groups kaplansky gives a combinatorial characterization of the isomorphism types of embeddings of a cyclic subgroup in a finite abelian group in this paper we first use partial maps on littlewood richardson tableaux to generalize this result to finite direct sums of such embeddings our main interest is an application to invariant subspaces of nilpotent linear operators we develop a criterion to decide if two irreducible components in the representation space are in the boundary partial order

quantization as histogram segmentation optimal scalar quantizer design in network systems an algorithm for scalar quantizer design on discrete alphabet sources is proposed the proposed algorithm can be used to design fixed rate and entropy constrained conventional scalar quantizers multiresolution scalar quantizers multiple description scalar quantizers and wyner ziv scalar quantizers the algorithm guarantees globally optimal solutions for conventional fixed rate scalar quantizers and entropy constrained scalar quantizers for the other coding scenarios the algorithm yields the best code among all codes that meet a given convexity constraint in all cases the algorithm run time is polynomial in the size of the source alphabet the algorithm derivation arises from a demonstration of the connection between scalar quantization histogram segmentation and the shortest path problem in a certain directed acyclic graph

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

multiple description coding by dithered delta sigma quantization we address the connection between the multiple description md problem and delta sigma quantization the inherent redundancy due to oversampling in delta sigma quantization and the simple linear additive noise model resulting from dithered lattice quantization allow us to construct a symmetric and time invariant md coding scheme we show that the use of a noise shaping filter makes it possible to trade off central distortion for side distortion asymptotically as the dimension of the lattice vector quantizer and order of the noise shaping filter approach infinity the entropy rate of the dithered delta sigma quantization scheme approaches the symmetric two channel md rate distortion function for a memoryless gaussian source and mean square error mse fidelity criterion at any side to central distortion ratio and any resolution in the optimal scheme the infinite order noise shaping filter must be minimum phase and have a piecewise flat power spectrum with a single jump discontinuity an important advantage of the proposed design is that it is symmetric in rate and distortion by construction so the coding rates of the descriptions are identical and there is therefore no need for source splitting

on properties of locally optimal multiple description scalar quantizers with convex cells it is known that the generalized lloyd method is applicable to locally optimal multiple description scalar quantizer mdsq design however it remains unsettled when the resulting mdsq is also globally optimal we partially answer the above question by proving that for a fixed index assignment there is a unique locally optimal fixed rate mdsq of convex cells under trushkin s sufficient conditions for the uniqueness of locally optimal fixed rate single description scalar quantizer this result holds for fixed rate multiresolution scalar quantizer mrsq of convex cells as well thus the well known log concave probability density function pdf condition can be extended to the multiple description and multiresolution cases

multiple description vector quantization with a coarse lattice a multiple description md lattice vector quantization technique for two descriptions was previously introduced in which fine and coarse codebooks are both lattices the encoding begins with quantization to the nearest point in the fine lattice this encoding is an inherent optimization for the decoder that receives both descriptions performance can be improved with little increase in complexity by considering all decoders in the initial encoding step the altered encoding relies only on the symmetries of the coarse lattice this allows us to further improve performance without a significant increase in complexity by replacing the fine lattice codebook with a nonlattice codebook that respects many of the symmetries of the coarse lattice examples constructed with the two dimensional 2 d hexagonal lattice demonstrate large improvement over time sharing between previously known quantizers

a rayleigh ritz approach to transverse vibration of isotropic and anisotropic trapezoidal plates using orthogonal plate functions abstract to overcome the lack of a more general method for free vibration analysis of trapezoidal plates a computationally efficient and highly accurate rayleigh ritz approach with the newly developed orthogonal plate functions is proposed to solve these problems with any combination of clamped simply supported and free edge support conditions the deflection of the plate is approximated by a set of two dimensional orthogonal plate functions generated using the gram schmidt procedure which expresses the entire plate domain into two implicitly related variables in the present paper the effects of the fibre orientation on the vibrational behaviour of the plates are considered the numerical results for isotropic and anisotropic trapezoidal plates are presented where possible the numerical results are verified with other existing values in the literature

successive refinement of information the successive refinement of information consists of first approximating data using a few bits of information then iteratively improving the approximation as more and more information is supplied the goal is to achieve an optimal description at each stage in general an ongoing description which is rate distortion optimal whenever it is interrupted is sought it is shown that in order to achieve optimal successive refinement the necessary and sufficient conditions are that the solutions of the rate distortion problem can be written as a markov chain in particular all finite alphabet signals with hamming distortion satisfy these requirements it is also shown that the same is true for gaussian signals with squared error distortion and for laplacian signals with absolute error distortion a simple counterexample with absolute error distortion and a symmetric source distribution which shows that successive refinement is not always achievable is presented

multiple description quantization via gram schmidt orthogonalization the multiple description md problem has received considerable attention as a model of information transmission over unreliable channels a general framework for designing efficient md quantization schemes is proposed in this paper we provide a systematic treatment of the el gamal cover egc achievable md rate distortion region and show it can be decomposed into a simplified egc segc region and a superimposed refinement operation furthermore any point in the segc region can be achieved via a successive quantization scheme along with quantization splitting for the quadratic gaussian case the proposed scheme has an intrinsic connection with the gram schmidt orthogonalization which implies that the whole gaussian md rate distortion region is achievable with a sequential dithered lattice based quantization scheme as the dimension of the optimal lattice quantizers becomes large moreover this scheme is shown to be universal for all independent and identically distributed i i d smooth sources with performance no worse than that for an i i d gaussian source with the same variance and asymptotically optimal at high resolution a class of md scalar quantizers in the proposed general framework is also constructed and is illustrated geometrically the performance is analyzed in the high resolution regime which exhibits a noticeable improvement over the existing md scalar quantization schemes

achievable rates for multiple descriptions consider a sequence of independent identically distributed i i d random variables x l x 2 cdots x n and a distortion measure d x i x i on the estimates x i of x i two descriptions i x in 1 2 cdots 2 nr 1 and j x in 1 2 cdots 2 nr 2 are given of the sequence x x 1 x 2 cdots x n from these two descriptions three estimates i x x2 j x and hat x o i x j x are formed with resulting expected distortions e frac 1 n sum n k 1 d x k hat x mk d m m 0 1 2 we find that the distortion constraints d 0 d 1 d 2 are achievable if there exists a probability mass distribution p x p hat x 1 hat x 2 hat x 0 x with ed x hat x m leq d m such that r 1 i x hat x 1 r 2 i x hat x 2 where i cdot denotes shannon mutual information these rates are shown to be optimal for deterministic distortion measures

the free flexural vibration of right triangular isotropic and orthotropic plates abstract a simple straighforward rayleigh ritz analysis is given for the determination of an eigenproblem for the free flexural vibration of thin right triangular plates having any combination of the classical free simply supported or clamped boundary conditions simple polynomials are used as the admissible functions and a recurrence relationship facilitates the evaluation of the necessary integrals the analysis is developed for specially orthotropic plates but numerical results are given primarily for the isotropic case

multiple description trellis coded quantization of sinusoidal parameters a new scheme for sinusoidal audio coding named multiple description spherical trellis coded quantization is proposed and analytic expressions for the point densities and expected distortion of the quantizers are derived based on a high resolution assumption the proposed quantizers are of variable dimensions meaning that any number of sinusoids can be quantized jointly for each audio segment whereby a lower distortion is achieved compared to previously published scalar spherical quantizers the quantizers are designed to minimize a perceptual distortion measure subject to an entropy constraint for a given packet loss probability in experiments the performance of the quantizers is assessed and compared to the corresponding single description spherical quantizer and associated bounds under various conditions and is found to increase robustness towards packet loss

multiple description distribution preserving quantization the notion of a multiple description quantizer mdq includes providing multiple distortion levels if a human observer is involved the design of mdq requires a suitable distortion measure to achieve a graceful quality degradation in the case of description losses while the mean squared error is a ubiquitous distortion measure for many classic mdq schemes it is known to be perceptually relevant only at low distortions we propose a new mdq designed according to an unconventional distortion criterion that combines the mean squared error with a constraint on the probability distribution of the reconstructed signal the performance of the new mdq is shown to approach that of the classic mdq asymptotically as rate increases however once applied in the context of transform audio coding the new mdq significantly outperforms a classic mdq in perceptual tests the new scheme is suitable for a wide range of distortions and renders a seamless transition between coding that preserves signal features and coding of a waveform

an efficient differential quadrature methodology for free vibration analysis of arbitrary straight sided quadrilateral thin plates in this paper a new differential quadrature dq methodology is employed to study free vibration of irregular quadrilateral straight sided thin plates a four nodded super element is used to map the irregular physical domain into a square domain in the computational domain second order transformation schemes with relative ease and less computation are employed to transform the fourth order governing equation of thin plates between the two domains the only degree of freedom within the domain is the displacement whereas along the boundaries the displacement as well as the second order derivative of the displacement with respect to associated normal co ordinate variable in computational domain are the two degrees of freedom implementing the method the formulation for the dq method for the free vibration analysis of plates of straight sided shapes was presented together with the implementation procedure for the different boundary conditions to demonstrate the accuracy convergency and stability of the new methodology detail studies are made on isotropic plates at acute angles with different geometries boundary and loading conditions including dq free edge boundary condition implementations accurate results even with fewer degrees of freedom than for those of comparable numerical algorithms were achieved

m description lattice vector quantization index assignment and analysis in this paper we investigate the design of symmetric entropy constrained multiple description lattice vector quantization mdlvq more specifically mdlvq index assignment we consider a fine lattice containing clean similar sublattices with s similarity due to the s similarity of the sublattices an m fraction lattice can be used to regularly partition the fine lattice with smaller voronoi cells than a sublattice does with the partition the mdlvq index assignment design can be translated into a transportation problem in operations research both greedy and general algorithms are developed to pursue optimality of the index assignment under high resolution assumption we compare the proposed schemes with other relevant techniques in terms of optimality and complexity following our index assignment design we also obtain an asymptotical close form expression of k description side distortion simulation results on coding different sources of gaussian speech and image are presented to validate the effectiveness of the proposed schemes

on the design of optimal noisy channel scalar quantizer with random index assignment the general approach in a noisy channel scalar quantizer design is an iterative descent algorithm which guarantees only a locally optimal solution while sufficient conditions under which the local optimum becomes a global optimum are known in the noiseless channel case such sufficient conditions were not derived for the noisy counterpart moreover efficient globally optimal design techniques for general discrete distributions in the noiseless case exist however they seem not to extend to the noisy scenario when a fixed index assignment is assumed recently the design of noisy channel scalar quantizer with random index assignment ria was proposed using a locally optimal iterative algorithm in this paper we derive sufficient conditions for the uniqueness of a local optimum which thus guarantee the global optimality of the solution these sufficient conditions are satisfied for a log concave probability density function which is additionally symmetric around its mean furthermore we show that assuming an ria the globally optimal design for general discrete sources can also be carried out efficiently

quantizer monotonicities and globally optimal scalar quantizer design new monotonicity properties of optimal scalar quantizers are discussed these monotonicities reveal a globally optimal scalar quantizer structure depending on the probability mass functions and on the number of quantizer levels by incorporating the monotone quantizer structure into a dynamic programming process the time complexities of previous algorithms for designing globally optimal scalar quantizers can be significantly reduced for very general classes of distortion measures

filter bank frame expansions with erasures we study frames for robust transmission over the internet in our previous work we used quantized finite dimensional frames to achieve resilience to packet losses here we allow the input to be a sequence in l sub 2 z and focus on a filter bank implementation of the system we present results in parallel r sup n or c sup n versus l sub 2 z and show that uniform tight frames as well as newly introduced strongly uniform tight frames provide the best performance

multiple description vector quantization with lattice codebooks design and analysis the problem of designing a multiple description vector quantizer with lattice codebook spl lambda is considered a general solution is given to a labeling problem which plays a crucial role in the design of such quantizers numerical performance results are obtained for quantizers based on the lattices a sub 2 and z sup i i 1 2 4 8 that make use of this labeling algorithm the high rate squared error distortions for this family of l dimensional vector quantizers are then analyzed for a memoryless source with probability density function pdf p and differential entropy h p spl infin for any a spl epsiv 0 1 and rate pair r r it is shown that the two channel distortion d sub o and the channel 1 or channel 2 distortion d sub s satisfy lim sub r spl rarr spl infin d sub o 2 sup 2r 1 a 1 4 g spl lambda 2 sup 2h p and lim sub r spl rarr spl infin d sub s 2 sup 2r 1 a g s sub l 2 sup 2h p where g spl lambda is the normalized second moment of a voronoi cell of the lattice spl lambda and g s sub l is the normalized second moment of a sphere in l dimensions

asymptotic analysis of multiple description quantizers a high rate analysis of multiple description quantizers is presented for rth power distortions and general source densities both fixed length and variable length encoding of the quantizer indices are considered optimal companding functions are shown to be the same as for single channel quantizers as compared to the bound of ozarow 1980 a gap of 8 69 db and 3 07 db exists between the entropy constrained and level constrained cases respectively for a memoryless gaussian source and r 2

vector gaussian multiple description with individual and central receivers t multiple descriptions of a vector gaussian source for individual and central receivers are investigated the sum rate of the descriptions with covariance distortion measure constraints in a positive semidefinite ordering is exactly characterized for two descriptions the entire rate region is characterized the key component of the solution is a novel information theoretic inequality that is used to lower bound the achievable multiple description rates jointly gaussian descriptions are optimal in achieving the limiting rates we also show the robustness of this description scheme the distortions achieved are no larger when used to describe any non gaussian source with the same covariance matrix

the free flexural vibration of isotropic and orthotropic general triangular shaped plates abstract the rayleigh ritz method is used with simple polynomials as admissible functions to obtain an eigenvalue equation for the free flexural vibration of thin isotropic and specially orthotropic general triangular plates the plates may have any combination of the classical free simply supported or clamped boundary conditions the convergence and accuracy of the solution is illustrated and numerical results are presented for a variety of isotropic plates and for orthotropic cantilever plates

design of optimal entropy constrained unrestricted polar quantizer for bivariate circularly symmetric sources this paper proposes an algorithm for the design of entropy constrained unrestricted polar quantizer ecupq for bivariate circularly symmetric sources the algorithm is globally optimal for the class of ecupqs with magnitude quantizer thresholds confined to a finite set the optimization problem is formulated as the minimization of a weighted sum of distortion and entropy and the proposed solution is based on modeling the problem as a minimum weight path problem in a certain weighted directed acyclic graph each graph edge corresponds to a possible magnitude quantizer bin and computing its weight involves solving another optimization problem we develop a fast strategy for evaluating all edge weights leading to a o k 2 textit kp mathrm max time solution algorithm where k is the size of the set of possible magnitude thresholds and p mathrm max is the maximum number of phase levels the practical performance of the proposed algorithm is assessed for a bivariate circularly symmetric gaussian source at rates ranging from 0 5 to 6 bits sample our results demonstrate that the proposed approach achieves performance very close to the asymptotically optimal ecupq at all rates while at low rates it significantly outperforms all previous upq schemes notably peak improvement of 0 755 db can be achieved for rates below 2 5

combinatorial message sharing and a new achievable region for multiple descriptions this paper presents a new achievable rate distortion region for the general l channel multiple descriptions mds problem a well known general region for this problem is due to venkataramani kramer and goyal vkg their encoding scheme is an extension of the el gamal cover ec and zhang berger zb coding schemes to the l channel case and includes a combinatorial number of refinement codebooks one for each subset of the descriptions as in zb the scheme also allows for a single common codeword to be shared by all descriptions this paper proposes a novel encoding technique involving combinatorial message sharing cms where every subset of the descriptions may share a distinct common message this introduces a combinatorial number of shared codebooks along with the refinement codebooks of these shared codebooks provide a more flexible framework to tradeoff redundancy across the messages for resilience to descriptions loss we derive an achievable rate distortion region for the proposed technique and show that it subsumes the vkg region for general sources and distortion measures we further show that cms provides a strict improvement of the achievable region for any source and distortion measures for which some two description subset is such that zb achieves points outside the ec region we then show a more surprising result cms outperforms vkg for a general class of sources and distortion measures including scenarios where the zb and ec regions coincide for all two description subsets in particular we show that cms strictly improves on vkg for the l channel quadratic gaussian md problem for all l geq 3 despite the fact that the ec region is complete for the corresponding two descriptions problem consequently the correlated quantization scheme an extreme special case of vkg that has been proven to be optimal for several cross sections of the l channel quadratic gaussian md problem is strictly suboptimal in general using the encoding principles derived we show that the cms scheme achieves the complete rate distortion region for several asymmetric cross sections of the l channel quadratic gaussian md problem

optimization of the index assignments for multiple description vector quantizers the optimization criterion and a practically feasible new algorithm is stated for the optimization of the index assignments of a multiple description unconstrained vector quantizer with an arbitrary number of descriptions in the simulations the index optimized multiple description vector quantizer achieves significant gains in source signal to noise ratio over scalar multiple description schemes

upper and lower bounds for frequencies of trapezoidal and triangular plates upper and lower bounds are given for the two lowest frequencies of vibration of clamped trapezoidal and triangular plates it is believed that this is the first time that rigorous bounds have been calculated for the vibrational frequencies of plates of these shapes

n channel symmetric multiple descriptions part i n k source channel erasure codes in this two part paper we present a new achievable rate region for the general n channel symmetric multiple descriptions problem in part i inspired by the concept of maximum distance separable mds erasure channel codes we consider a special case of this rate region where the source is encoded into n descriptions each with rate r these descriptions are transmitted over n bandwidth constrained and errorless channels during transmission a subset of these channels can break down thus erasing the corresponding descriptions the decoder is interested in recovering the source with the reception of at least k descriptions thus the encoder is allowed to sample only one realization of this breakdown process during the entire transmission for gaussian sources we have the following interesting result when any k descriptions arrive the achievable distortion exactly matches the optimal distortion rate performance corresponding to a source rate of kr bits with the reception of any m k descriptions the source reconstruction quality is strictly better the improvement being nearly linear in the number of descriptions received

optimal filter banks for multiple description coding analysis and synthesis multiple description md coding is a source coding technique for information transmission over unreliable networks in md coding the coder generates several different descriptions of the same signal and the decoder can produce a useful reconstruction of the source with any received subset of these descriptions in this paper we study the problem of md coding of stationary gaussian sources with memory first we compute an approximate md rate distortion region for these sources which we prove to be asymptotically tight at high rates this region generalizes the md rate distortion region of el gamal and cover 1982 and ozarow 1980 for memoryless gaussian sources then we develop an algorithm for the design of optimal two channel biorthogonal filter banks for md coding of gaussian sources we show that optimal filters are obtained by allocating the redundancy over frequency with a reverse water filling strategy finally we present experimental results which show the effectiveness of our filter banks in the low complexity low rate regime

vibrational frequencies of clamped plates of variable thickness abstract a method is presented for computing lower and upper bounds for vibrational frequencies of clamped plates with general thickness variations the method is illustrated for plates with linear tapers

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

high rate analysis of symmetric l channel multiple description coding this paper studies the tight rate distortion bound for l channel symmetric multiple description coding of a scalar gaussian source with two levels of receivers each of the first level receivers obtains of the l descriptions l the second level receiver obtains all l descriptions we find that if the central distortion corresponding to the second level receiver is much smaller than the side distortion corresponding to the first level receivers the product of a function of the side distortions and the central distortion is asymptotically independent of the redundancy between the descriptions using this property we analyze the asymptotic behavior of a practical multiple description lattice vector quantizer mdlvq our analysis includes the treatment of the mdlvq system from a new geometric viewpoint which results in an expression for the side distortions using the normalized second moment of a sphere of higher dimensionality than the quantization space the expression of the distortion product derived from the lower bound is then applied as a criterion to assess the performance loss of the considered mdlvq system in principle the efficiency of other practical md systems can also be evaluated using the derived distortion product

index assignment for two channel quantization this paper concerns the design of a multiple description scalar quantization mdsq system for two identical channels for an unbounded discrete information source this translates to the combinatorial problem of finding an arrangement of the integers into the infinite plane square grid so that each row and each column contains exactly n numbers such that the difference between any two numbers in the same row or column is at most d with d to be minimized for a given n the best previous lower and upper bounds on the lowest d were n sup 2 3 o n and n sup 2 2 o n we give new lower and upper bounds both of the form 3n sup 2 8 o n we also consider minimizing the maximal variance in any row or column and show that it must be at least n sup 4 60 o n sup 3 and that it does not have to be more than 3n sup 4 160 o n sup 3

the analysis and design of windowed fourier frame based multiple description source coding schemes in this paper the windowed fourier encoding decoding scheme applied to the multiple description compression problem is analyzed in the general case four window functions are needed to define the encoder and decoder although this number can be reduced to three or two by using time shift or frequency shift division schemes the encoding coefficients are next divided into two groups according to the eveness of either the modulation or translation index the distortion on each channel is analyzed using the zak transform for the optimal windows explicit representation formulas are obtained and nonlocalization results are proved asymptotic formulas of the total distortion and transmission rate are established and the redundancy is shown to trade off between these two

entropy constrained index assignments for multiple description quantizers multiple description coding has revealed itself highly useful for the transmission of digital signals over packet switched networks or diversity systems the index assignment problem is central in the theory and practice of multiple description quantizers we propose a novel local optimization algorithm that produces index assignments for such quantizers it locally minimizes the distortion due to description losses with constraints on the entropy of each description it works in any vector dimension and is easily generalizable to various cases such as asymmetric descriptions or m description systems we present experimental comparisons with previously known index assignment methods in both scalar and vector cases and show that the proposed algorithm performs significantly better

design of optimal scalar quantizer for sequential coding of correlated sources this paper addresses the design of a sequential scalar quantizer ssq for finite alphabet correlated sources in the fixed rate fr and entropy constrained ec cases the optimization problem is formulated as the minimization of a weighted sum of distortions and rates the proposed solution is globally optimal for the class of ssqs with convex cells and is based on solving the minimum weight path mwp problem in the ec case respectively a length constrained mwp problem in the fr case in a series of weighted directed acyclic graphs the asymptotic time complexity is o k 1 2 k 2 2 where k 1 and k 2 are the respective sizes of the alphabets of the two sources additionally it is proved that by applying the proposed algorithms to discretizations of correlated sources with continuous joint probability density function the performance approaches that of the optimal ec ssq respectively fr ssq with convex cells for the original sources as the accuracy of the discretization increases extensive experiments performed with correlated gaussian sources validate the effectiveness in practice of the proposed approach in approximating the optimal ssq for the case of continuous alphabet sources