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

sample adaptive product quantization asymptotic analysis and examples vector quantization vq is an efficient data compression technique for low bit rate applications however the major disadvantage of vq is that its encoding complexity increases dramatically with bit rate and vector dimension even though one can use a modified vq such as the tree structured vq to reduce the encoding complexity it is practically infeasible to implement such a vq at a high bit rate or for large vector dimensions because of the huge memory requirement for its codebook and for the very large training sequence requirement to overcome this difficulty a structurally constrained vq called the sample adaptive product quantizer sapq has recently been proposed we extensively study the sapq that is based on scalar quantizers in order to exploit the simplicity of scalar quantization through an asymptotic distortion result we discuss the achievable performance and the relationship between distortion and encoding complexity we illustrate that even when sapq is based on scalar quantizers it can provide vq level performance we also provide numerical results that show a 2 3 db improvement over the lloyd max 1982 1960 quantizers for data rates above 4 b point

a pyramid vector quantizer the geometric properties of a memoryless laplacian source are presented and used to establish a source coding theorem motivated by this geometric structure a pyramid vector quantizer pvq is developed for arbitrary vector dimension the pvq is based on the cubic lattice points that lie on the surface of an l dimensional pyramid and has simple encoding and decoding algorithms a product code version of the pvq is developed and generalized to apply to a variety of sources analytical expressions are derived for the pvq mean square error mse and simulation results are presented for pvq encoding of several memoryless sources for large rate and dimension pvq encoding of memoryless laplacian gamma and gaussian sources provides rose improvements of 5 64 8 40 and 2 39 db respectively over the corresponding optimum scalar quantizer although suboptimum in a rate distortion sense because the pvq can encode large dimensional vectors it offers significant reduction in rose distortion compared with the optimum lloyd max scalar quantizer and provides an attractive alternative to currently available vector quantizers

joint wyner ziv dirty paper coding by modulo lattice modulation the combination of source coding with decoder side information the wyner ziv problem and channel coding with encoder side information the gel fand pinsker problem can be optimally solved using the separation principle in this work we show an alternative scheme for the quadratic gaussian case which merges source and channel coding this scheme achieves the optimal performance by applying a modulo lattice modulation to the analog source thus it saves the complexity of quantization and channel decoding and remains with the task of ldquoshapingrdquo only furthermore for high signal to noise ratio snr the scheme approaches the optimal performance using an snr independent encoder thus it proves for this special case the feasibility of universal joint source channel coding

decay rate for the incompressible flows in half spaces we show that the time decay rate of l 2 norm of weak solution for the stokes equations and for the navier stokes equations on the half spaces are t frac n 2 frac 1 r frac 1 2 frac 1 2 if the initial data u 0 in l 2 cap l r and int mathbb r n y n u 0 y r dy infty for 1 r 2 we also show that the decay rate is determined by the linear part of the weak solution we use the heat kernel and ukai s solution formula for the stokes equations it has been known up to now that the decay rate on the half space was t frac n 2 frac 1 r frac 1 2 which was obtained by borchers and miyakawa 1 and ukai 9

dcrvq a new strategy for efficient entropy coding of vector quantized images this paper presents a novel predictive coding scheme for image data compression by vector quantization vq on the basis of a prediction further compression is achieved by using a dynamic codebook reordering strategy that allows a more efficient huffman encoding of vector addresses the proposed method is lossless for it increases the compression performances of a baseline vector quantization scheme without causing any further image degradation results are presented and a comparison with cache vq is made

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

extension of two stage vector quantization lattice vector quantization this paper is the extension of two stage vector quantization spherical lattice vector quantization vq s lvq recently introduced by pan and fischer see ieee trans inform theory vol 41 p 155 1995 first according to high resolution quantization theory generalized vector quantization lattice vector quantization g vq lvq is formulated in order to release the constraint of the spherical boundary for the second stage lattice vector quantization lvq which would provide possibilities of improving this kind of two stage unstructured structured quantizer by using more efficient lvq second among g vq lvq vector quantization pyramidal lattice vector quantization vq plvq is developed which is slightly superior or comparable to vq s lvq in performance but has a much lower complexity simulation results show that for memoryless sources vq plvq achieves a rate distortion performance that is among the best of the fixed rate quantization that we found in the literature therefore vq plvq is an attractive alternative to vq s lvq in practice third transform vq plvq tvq plvq is proposed for sources with memory for encoding 16 d vectors of the gauss markov source t vq plvq has an advantage of close to 1 0 db over vq plvq and is about 0 5 db better than vq s lvq

efficient quadtree coding of images and video the quadtree data structure is commonly used in image coding to decompose an image into separate spatial regions to adaptively identify the type of quantizer used in various regions of an image the authors describe the theory needed to construct quadtree data structures that optimally allocate rate given a set of quantizers a lagrange multiplier method finds these optimal rate allocations with no monotonicity restrictions they use the theory to derive a new quadtree construction method that uses a stepwise search to find the overall optimal quadtree structure the search can be driven with either actual measured quantizer performance or ensemble average predicted performance they apply this theory to the design of a motion compensated interframe video coding system using a quadtree with vector quantization

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

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

new permutation codes using hadamard unscrambling corresp a new class of codes for data compression is described that combines permutations with the fast hadamard transform fht it was invented for digital speech compression based on linear predictive coding lpc but may be useful for other data compression applications one particular code with rate frac 1 2 is considered a 16 bit code for a block length of 32 samples all coding and decoding steps are fast so that real time applications with cheap hardware can be anticipated

on uniform quantization with various distortion measures corresp upper bounds are presented for the difference in entropy between that of a uniform scalar quantizer and that of any n dimensional quantizer the bounds are universal in the sense that they suit every input density and every value of distortion bounds were found for some common distortion criteria

fourier transform vector quantization for speech coding design algorithms and simulation results are presented for vector quantizers for fourier transformed data transforming the data prior to quantization has two potential advantages first each sample in the transform domain depends on many samples in the original domain thus even scalar quantization in the transform domain is a form of vector quantization or block source coding in the original waveform domain and the basic coding theorems of information theory show that such block codes can provide better performance than scalar codes even for memoryless sources second vector quantization of fourier transformed speech waveforms provides distinctly better subjective quality than ordinary vector quantization of the waveform using codes of comparable complexity while the system is of course more complicated due to the need to take fourier transforms its envisioned application is as a coder for the output of fft chips currently available or under development the proposed implementation of a fourier transform vector quantizer ftvq uses a product code structure providing different codes for different coefficient vectors corresponding to different frequency bands this is a form of subband coding and yields a simple means of optimizing bit allocations among the subcodes two coding structures with corresponding distortion measures are considered those that quantize vectors of pairs of real and imaginary coefficients and those that quantize separate vectors of magnitude and phase coefficients both structures yield good performance for the given complexity in comparison to waveform vector quantizers for speech coding a magnitude phase ftvq yields better subjective quality than a real imaginary ftvq when the rate allocation is properly chosen

upper bounds to the asymptotic performance of block quantizers upper bounds to the asymptotic performance of block quantizers with difference distortion measures are derived in many eases these upper bounds approach known lower bounds as the block length of the quantizer approaches infinity a condition for the optimal point density function of the output levels is derived it is shown to particularize to a known result of gersho the behavior of the bounds for large block lengths is investigated

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

vector quantizer design for memoryless gaussian gamma and laplacian sources locally optimum vector quantizer vq designs are presented for memoryless gaussian gamma and laplacian sources for gaussian sources low 2 6 dimensional vector quantization provides relatively little improvement in mean squared error mse compared to the minimum mean squared error mmse scalar quantizer for laplacian or gamma sources however significant improvement in mse is available with vector quantization the laplacian and gamma 6 bit sixdimensional vector quantizers achieve respectively improvements of 2 and 4 5 db over the corresponding scalar mmse quantizer distortions

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

quantization based on a novel sample adaptive product quantizer sapq in this paper we propose a novel feedforward adaptive quantization scheme called the sample adaptive product quantizer sapq this is a structurally constrained vector quantizer that uses unions of product codebooks sapq is based on a concept of adaptive quantization to the varying samples of the source and is very different from traditional adaptation techniques for nonstationary sources sapq quantizes each source sample using a sequence of quantizers even when using scalar quantization in sapq we can achieve performance comparable to vector quantization with the complexity still close to that of scalar quantization we also show that important lattice based vector quantizers can be constructed using scalar quantization in sapq we mathematically analyze sapq and propose a algorithm to implement it we numerically study sapq for independent and identically distributed gaussian and laplacian sources through our numerical study we find that sapq using scalar quantizers achieves typical gains of 13 db in distortion over the lloyd max quantizer we also show that sapq can he used in conjunction with vector quantizers to further improve the gains

asymptotic behavior for the navier stokes equations in 2d exterior domains we show that the lp spatial temporal decay rates of solutions of incompressible flow in an 2d exterior domain when a domain has a boundary pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary to overcome the difficulty we adopt the ideas in he xin c he z xin weighted estimates for nonstationary navier stokes equations in exterior domain methods appl anal 7 3 2000 443 458 and our previous results h o bae b j jin asymptotic behavior of stokes solutions in 2d exterior domains j math fluid mech in press h o bae b j jin temporal and spatial decay rates of navier stokes solutions in exterior domains submitted for publication for the spatial decay rate estimate we first extend temporal decay rate result of the navier stokes solutions for general lp space when the initial velocity is in l 2 lr 1 r q 1 r q

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

a vector quantizer for the laplace source the low complexity nearly optimal vector quantizer vq is a generalization of t r fischer s 1986 pyramid vq and is similar in structure to the unrestricted polar quantizers previously presented for the independent gaussian source an analysis of performance is presented with results for both the product code pyramid vq and the unrestricted version this analysis although asymptotic in nature helps to demonstrate the performance advantages of the vq implementation issues of the vq are discussed nonasymptotic results are considered in particular the author presents an approximate design algorithm for finite bit rate and demonstrates the usefulness of this vq through several example designs with monte carlo simulations of performance for the restricted form the pyramid vq the author provides further implementational information and low dimension analytical results

relaxation and overlap probability function in the spherical and mean spherical models the problem of the equivalence of the spherical and mean spherical models which has been thoroughly studied and is understood in equilibrium is considered anew from the dynamical point of view during the time evolution following a quench from above to below the critical temperature it is found that there exists a crossover time t ensuremath sim v 2 d such that for tl t the two models are equivalent while for tg t macroscopic discrepancies arise the relation between the off equilibrium response function and the structure of the equilibrium state that usually holds for phase ordering systems is found to hold for the spherical model but not for the mean spherical one the latter model offers an explicit example of a system which is not stochastically stable

successive refinement lattice vector quantization lattice vector quantization lvq solves the complexity problem of lbg based vector quantizers yielding very general codebooks however a single stage lvq when applied to high resolution quantization of a vector may result in very large and unwieldy indices making it unsuitable for applications requiring successive refinement the goal of this work is to develop a unified framework for progressive uniform quantization of vectors without having to sacrifice the mean squared error advantage of lattice quantization a successive refinement uniform vector quantization methodology is developed where the codebooks in successive stages are all lattice codebooks each in the shape of the voronoi regions of the lattice at the previous stage such voronoi shaped geometric lattice codebooks are named voronoi lattice vqs vlvq measures of efficiency of successive refinement are developed based on the entropy of the indices transmitted by the vlvqs additionally a constructive method for asymptotically optimal uniform quantization is developed using tree structured subset vlvqs in conjunction with entropy coding the methodology developed here essentially yields the optimal vector counterpart of scalar bitplane wise refinement unfortunately it is not as trivial to implement as in the scalar case furthermore the benefits of asymptotic optimality in tree structured subset vlvqs remain elusive in practical nonasymptotic situations nevertheless because scalar bitplane wise refinement is extensively used in modern wavelet image coders we have applied the vlvq techniques to successively refine vectors of wavelet coefficients in the vector set partitioning vspiht framework the results are compared against spiht and the previous successive approximation wavelet vector quantization sa w vq results of sampson da silva and ghanbari 1996

video signal coding with dct and vector quantization an image coding scheme using the discrete cosine transform is analyzed when the transform coefficients are vector quantized the coding method is based on the known scheme proposed by chen and smith 1977 which sorts the picture blocks into classes according to the level of image activity the coding scheme is modified to allow for vector quantization of the ac coefficients in particular a pyramid vector quantizer pvq is used this is based on the statistical and geometric properties of a laplacian source which in fact is the best model for the ac coefficients of the two dimensional discrete cosine transform 2d dct of an image a method for forming almost statistically independent vectors is also suggested and improves quantization performance images are encoded with both the pvq and standard scalar quantizer transform coders demonstrating that the pvq coder reduces the mean square encoding error and improves image quality in particular emphasis is given to how the use of fractional bit rates affects the objective and subjective gains obtained the results presented i e mean square error values and printed images have been obtained experimentally working with a statistical criterion in a group of images whose size was in accordance with the 50 hz ccir recommendation 601 standard

near ellipsoidal voronoi coding we consider a special case of voronoi coding where a lattice spl lambda in spl ropf sup n is shaped or truncated using a lattice spl lambda m sub 1 x sub 1 m sub n x sub n x sub 1 x sub n spl isin spl lambda for a fixed m m sub 1 m sub n spl isin spl nopf spl bsol 0 1 sup n using this technique the shaping boundary is near ellipsoidal it is shown that the resulting codes can be indexed by standard voronoi indexing algorithms plus a conditional modification step as far as spl lambda is a sublattice of spl lambda we derive the underlying conditions on m and present generic near ellipsoidal voronoi indexing algorithms examples of constraints on m and conditional modification are provided for the lattices a sub 2 d sub n n spl ges 2 and 2d sub n sup n even spl ges 4

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

decay results of higher order norms for the navier stokes flows in 3d exterior domains using the nonstationary stokes solution formula and the regularity theory of the steady stokes equations we establish the decay results including the weighted cases of higher order spatial derivatives for the navier stokes flows in 3d exterior domains which is a long standing problem the higher order norm weighted decay results in this article are inspired by the work in bae and jin proc r soc edinburgh sect a 135 461 477 2005 schonbek and wiegner proc r soc edinburgh sect a 126 677 685 1996 respectively which are considered in the whole space

the fluctuation dissipation relation in phase ordering kinetics abstract the properties of the response function and the fluctuation dissipation relation in phase ordering systems are discussed these issues are shown to be highly nontrivial even in the relatively simple case of coarsening dynamics considered here it is discussed how the connection between statics and dynamics fails in particular cases and the origin of this phenomenon is considered

a vector quantization approach to universal noiseless coding and quantization a two stage code is a block code in which each block of data is coded in two stages the first stage codes the identity of a block code among a collection of codes and the second stage codes the data using the identified code the collection of codes may be noiseless codes fixed rate quantizers or variable rate quantizers we take a vector quantization approach to two stage coding in which the first stage code can be regarded as a vector quantizer that quantizes the input data of length n to one of a fixed collection of block codes we apply the generalized lloyd algorithm to the first stage quantizer using induced measures of rate and distortion to design locally optimal two stage codes on a source of medical images two stage variable rate vector quantizers designed in this way outperform standard one stage fixed rate vector quantizers by over 9 db the tail of the operational distortion rate function of the first stage quantizer determines the optimal rate of convergence of the redundancy of a universal sequence of two stage codes we show that there exist two stage universal noiseless codes fixed rate quantizers and variable rate quantizers whose per letter rate and distortion redundancies converge to zero as k 2 n sup 1 log n when the universe of sources has finite dimension k this extends the achievability part of rissanen s theorem from universal noiseless codes to universal quantizers further we show that the redundancies converge as o n sup 1 when the universe of sources is countable and as o n sup 1 spl epsiv when the universe of sources is infinite dimensional under appropriate conditions

the art of signaling fifty years of coding theory in 1948 shannon developed fundamental limits on the efficiency of communication over noisy channels the coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero fifty years later codes for the gaussian channel have been discovered that come close to these fundamental limits there is now a substantial algebraic theory of error correcting codes with as many connections to mathematics as to engineering practice and the last 20 years have seen the construction of algebraic geometry codes that can be encoded and decoded in polynomial time and that beat the gilbert varshamov bound given the size of coding theory as a subject this review is of necessity a personal perspective and the focus is reliable communication and not source coding or cryptography the emphasis is on connecting coding theories for hamming and euclidean space and on future challenges specifically in data networking wireless communication and quantum information theory

compression of sar raw data through range focusing and variable rate trellis coded quantization there is an ever growing interest in the compression of sar data because of the huge resources required for storage and transmission this is especially true for spaceborne sensors given the limited capacity of the downlink channel unfortunately sar data lack the useful properties on which compression algorithms rely indeed these are present in the focused images but focusing is too complex for on board implementation at this time poggi et al 2000 proposed to perform on the satellite only the low complexity range focusing which increases the data correlation and better concentrates their energy these properties were then exploited by adopting a variable rate vector quantizer with a clear performance improvement with respect to reference techniques however vector quantization vq is too complex for actual on board implementation and therefore here we replace vq with trellis coded vq to limit complexity only small vectors are used which reduces vq s ability to exploit data dependencies on the other hand trellis coding allows one to encode large blocks of data at once and to obtain a better partition of the input space experiments on real sar data show that the overall performance is comparable to that of poggi et al but the complexity is much lower making on board implementation possible

an algorithm for uniform vector quantizer design a vector quantizer maps a k dimensional vector into one of a finite set of output vectors or points although certain lattices have been shown to have desirable properties for vector quantization applications there are as yet no algorithms available in the quantization literature for building quantizers based on these lattices an algorithm for designing vector quantizers based on the root lattices a n d n and e n and their duals is presented also a coding scheme that has general applicability to all vector quantizers is presented a four dimensional uniform vector quantizer is used to encode laplacian and gamma distributed sources at entropy rates of one and two bits sample and is demonstrated to achieve performance that compares favorably with the rate distortion bound and other scalar and vector quantizers finally an application using uniform four and eight dimensional vector quantizers for encoding the discrete cosine transform coefficients of an image at 0 5 bit pel is presented which visibly illustrates the performance advantage of vector quantization over scalar quantization

optimal weighted estimates of the flows in exterior domains abstract in this paper we prove some decay properties of global solutions for the navier stokes equations in an exterior domain r n n 2 3 when a domain has a boundary the pressure term is troublesome since we do not have enough information on the pressure near the boundary to overcome this difficulty by multiplying a special form of test functions we obtain an integral equation he xin 2000 12 first introduced this method and then bae jin 2006 2007 1 13 modified their method to obtain better decay rates also bae roh 2009 11 improved bae jin s results unfortunately their results were not optimal because there exists an unpleasant positive small in their rates in this paper we obtain the following optimal rate without x u t l p c t n 2 1 r 1 p 2 for the initial condition u 0 with x u 0 l r and for any n

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

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

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

complexity constrained trellis quantizers a reasonable measure of quantizer complexity is the expected number of quanta per input sample for which distortion is computed given this measure a rate distortion complexity theory is obtained by extending earlier work in alphabet constrained rate distortion theory numerical results show that operation on the alphabet constrained rate distortion bound can be obtained with a complexity of two furthermore lloyd max conditions are shown to describe the minimum of a slightly constrained version of the rate distortion complexity problem complexity constrained design methods are applied first to trellis coded quantizers where they are shown to reduce arithmetic operations by at least 25 they are then used to develop model based trellis quantizers the trellises of which are derived from a markov model of the source simulation results confirm that excellent performance can be obtained with modest complexity

analyticity and asymptotics for the stokes solutions in a weighted space abstract we first show the analyticity of stokes operator in a weighted space l p r 3 we also show that the hodge decomposition holds on lp r 3 then we estimate the asymptotic behavior of the stokes solutions in space and time directions for 1 r 3 u x t q x q dx 1 q c t 3 2 1 r 1 q 1 2 2 where is a constant with 0

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

voronoi regions of lattices second moments of polytopes and quantization if a point is picked at random inside a regular simplex octahedron 600 cell or other polytope what is its average squared distance from the centroid in n dimensional space what is the average squared distance of a random point from the closest point of the lattice a n or d n e n a n ast or d n ast the answers are given here together with a description of the voronoi or nearest neighbor regions of these lattices the results have applications to quantization and to the design of signals for the gaussian channel for example a quantizer based on the eight dimensional lattice e8 has a mean squared error per symbol of 0 0717 cdots when applied to uniformly distributed data compared with 0 08333 cdots for the best one dimensional quantizer

lower bounds for an integral involving fractional laplacians and the generalized navier stokes equations in besov spaces when estimating solutions of dissipative partial differential equations in l p related spaces we often need lower bounds for an integral involving the dissipative term if the dissipative term is given by the usual laplacian lower bounds can be derived through integration by parts and embedding inequalities however when the laplacian is replaced by the fractional laplacian the approach of integration by parts no longer applies in this paper we obtain lower bounds for the integral involving by combining pointwise inequalities for with bernstein s inequalities for fractional derivatives as an application of these lower bounds we establish the existence and uniqueness of solutions to the generalized navier stokes equations in besov spaces the generalized navier stokes equations are the equations resulting from replacing in the navier stokes equations by

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

sequential scalar quantization of vectors an analysis proposes an efficient vector quantization vq technique called sequential scalar quantization ssq the scalar components of the vector are individually quantized in a sequence with the quantization of each component utilizing conditional information from the quantization of previous components unlike conventional independent scalar quantization isq ssq has the ability to exploit intercomponent correlation at the same time since quantization is performed on scalar rather than vector variables ssq offers a significant computational advantage over conventional vq techniques and is easily amenable to a hardware implementation in order to analyze the performance of ssq the authors appeal to asymptotic quantization theory where the codebook size is assumed to be large closed form expressions are derived for the quantizer mean squared error mse these expressions are used to compare the asymptotic performance of ssq with other vq techniques the authors also demonstrate the use of asymptotic theory in designing ssq for a practical application color image quantization where the codebook size is typically small theoretical and experimental results show that ssq far outperforms isq with respect to mse while offering a considerable reduction in computation over conventional vq at the expense of a moderate increase in mse

distortion rate models for entropy coded lattice vector quantization the increasing demand for real time applications requires the use of variable rate quantizers having good performance in the low bit rate domain in order to minimize the complexity of quantization as well as maintaining a reasonably high psnr ratio we propose to use an entropy coded lattice vector quantizer eclvq these quantizers have proven to outperform the well known ezw algorithm s performance in terms of rate distortion tradeoff we focus our attention on the modeling of the mean squared error mse distortion and the prefix code rate for eclvq first we generalize the distortion model of jeong and gibson 1993 on fixed rate cubic quantizers to lattices under a high rate assumption second we derive new rate models for eclvq efficient at low bit rates without any high rate assumptions simulation results prove the precision of our models

lattice vector quantization of generalized gaussian sources many important sources of data including sub band image and speech coefficients are highly peaked and can be well modeled by the family of generalized gaussian gg pdfs parameterized by a shape parameter spl nu we present here algorithms for defining and quantizing to a z lattice in which the boundary is optimized to the characteristics of gg sources and show that these techniques lead to high performance and low complexity for bit rates and dimensions that are of interest in a number of practical coding applications we derive an analytical description of the granular and overload distortion valid for low and moderate bit rates and also provide a description of quantizer performance in the limit of high rate and dimension

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

on finite block length quantization distortion we investigate the upper and lower bounds on the quantization distortions for independent and identically distributed continuous sources in the finite block length regime we derive a lower bound on the quantization distortion as well as an upper bound on the quantization distortion based on random quantization codebooks moreover we apply the obtained bounds to the continuous gaussian source for the gaussian source we propose a computationally tractable method to numerically compute the upper and lower bounds for both bounded and unbounded quantization codebooks numerical results show that the gap between the upper and lower bounds is small for the block length of several hundreds and the upper and lower bounds are tighter than those proposed in the existing works

trellis based scalar vector quantizer for memoryless sources the paper describes a structured vector quantization approach for stationary memoryless sources that combines the scalar vector quantizer svq ideas laroia and farvardin 1993 with trellis coded quantization marcellin and fischer 1990 the resulting quantizer is called the trellis based scalar vector quantizer tb svq the svq structure allows the tb svq to realize a large boundary gain while the underlying trellis code enables it to achieve a significant portion of the total granular gain for large block lengths and powerful possibly complex trellis codes the tb svq can in principle achieve the rate distortion bound as indicated by the results obtained even for reasonable block lengths and relatively simple trellis codes the tb svq outperforms all other fixed rate quantizers at reasonable complexity

a lower bound on the average error of vector quantizers corresp a lower bound is proposed for the mean squared error of an n dimensional vector quantizer with a large number of output points

robust vector quantization by a linear mapping of a block code in this paper we propose a novel technique for vector quantizer design where the reconstruction vectors are given by a linear mapping of a binary block code lmbc the lmbc framework provides a relation between the index bits and the reconstruction vectors through mapping properties we define a framework show its flexibility and give optimality conditions we consider source optimized vector quantization vq where the objective is to directly obtain a vq with inherent good channel robustness properties several instructive theoretical results and properties of the distortion experienced due to channel noise are demonstrated these results are used to guide the design process both optimization algorithms and a block code selection procedure are devised experimental results for gauss markov sources show that quantization performance close to an unconstrained vq is obtained with a short block code which implies a constrained vq the resulting vqs have better channel noise robustness than conventional vqs designed with the generalized lloyd algorithm gla and splitting initialization even when a post processing index assignment algorithm is applied to the gla based vq we have thus demonstrated a unique method for direct design resulting in an inherent good index assignment combined with small losses in quantization performance

on universal quantization the quantization of n dimensional vectors in r n with an arbitrary probability measure under a mean square error constraint is discussed it is demonstrated that a uniform one dimensional quantizer followed by a noiseless digital variable rate encoder entropy encoding can yield a rate that is for any n no more than 0 754 bit per sample higher than the rate associated with the optimal n dimensionai quantizer regardless of the probabilistic characterization of the input n vector for the allowable mean square error

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

kinetics of the long range spherical model the kinetic spherical model with long range interactions is studied after a quench to t tc or to t tc for the two time response and correlation functions of the order parameter as well as for composite fields such as the energy density the ageing exponents and the corresponding scaling functions are derived the results are compared to the predictions which follow from local scale invariance

optimization of lattices for quantization a training algorithm for the design of lattices for vector quantization is presented the algorithm uses a steepest descent method to adjust a generator matrix in the search for a lattice whose voronoi regions have minimal normalized second moment the numerical elements of the found generator matrices are interpreted and translated into exact values experiments show that the algorithm is stable in the sense that several independent runs reach equivalent lattices the obtained lattices reach as low second moments as the best previously reported lattices or even lower specifically we report lattices in nine and ten dimensions with normalized second moments of 0 0716 and 0 0708 respectively and nonlattice tessellations in seven and nine dimensions with 0 0727 and 0 0711 which improves on previously known values the new nine and ten dimensional lattices suggest that conway and sloane s 1993 conjecture on the duality between the optimal lattices for packing and quantization might be false a discussion of the application of lattices in vector quantizer design for various sources uniform and nonuniform is included

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

the progressive transmission disadvantage the disadvantage of progressive transmission compared to transmission in a single step is analyzed for the high rate case using a polytope that is self similar under binary subdivision this disadvantage is calculated to be 0 3424 db or 0 057 bit compared to scalar quantization obtained with a step size of 1 bit per ensemble

trellis based scalar vector quantization of sources with memory the trellis based scalar vector quantizer tb svq can achieve the rate distortion performance bound for memoryless sources this paper extends the scope of this quantizer to coding of sources with memory first considered is a simple extension called the predictive tb svq which applies a closed loop predictive coding operation in each survivor path of the viterbi codebook search algorithm although the predictive tb svq outperforms all other known structured fixed rate vector quantizers due to practical reasons it may not approach the rate distortion limit a new quantization scheme motivated by the precoding idea of laroia et al 1993 called the precoded tb svq is also considered the granular gain is realized by the underlying trellis code while the combination of the precoder and the svq structure provides the boundary gain this new quantization scheme is asymptotically optimal and can in principle approach the rate distortion bound for markov sources

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

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

vector quantizers and predictive quantizers for gauss markov sources low rate vector quantizers are designed and simulated for highly correlated gauss markov sources and the resulting performance is compared with arnstein s optimized predictive quantizer and with huang and schultheiss optimized transform coder two implementations of vector quantizers are considered full search vector quantizers which are optimal but require large codebook searches and tree searched vector quantizers which are suboptimal but require far less searching the various systems are compared on the basis of performance complexity and generality of design techniques