design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

parameter selection for wyner ziv coding of laplacian sources with additive laplacian or gaussian innovation a large number of practical coding scenarios deal with sources such as transform coefficients that can be well modeled as laplacians for regular coding of such sources samples are often quantized by a family of uniform quantizers possibly with a deadzone and then entropy coded for the wyner ziv coding problem when correlated side information is available at the decoder the side information can be modeled as obtained by independent additive laplacian or gaussian innovation on the source this paper deals with optimal choice of parameters for practical wyner ziv coding in such scenarios using the same quantizer family as in the regular codec to cover a range of rate distortion tradeoffs given the variances of the source and additive noise we propose and analyze a general encoding model based on multilevel coset codes that combines source coding and channel coding and show that at practical block lengths and code complexities not pure channel coding but a hybrid combination of source coding and channel coding with right parameters provide optimal rate distortion performance we also provide a framework for on the fly parameter choice based on nonparametric representation of a set of seed functions for use in scenarios where variances are estimated during encoding a good insight in the optimal parameter selection mechanism is essential for building practical distributed codecs

cohomological characterizations of projective spaces and hyperquadrics projective spaces and hyperquadrics are the simplest projective algebraic varieties and they can be characterized in many ways the aim of this paper is to provide a new characterization of them in terms of positivity properties of the tangent bundle theorem 1 1 the first result in this direction was mori s proof of the hartshorne conjecture in mor79 see also siu and yau sy80 that characterizes projective spaces as the only manifolds having ample tangent bundle then in wah83 wahl characterized projective spaces as the only manifolds whose tangent bundles contain ample invertible subsheaves interpolating mori s and wahl s results andreatta and wi niewski gave the following characterization

on essential components of the solution set of generalized games in this paper we prove that for any generalized game there exists at least one essential connected component of the set of it s equilibrium points

the stability of the set of kkm points abstract let m be the collection of all kkm mappings g and denote by f g the set of all kkm points of g in this paper we prove that there exists a dense residual subset q of m such that for each g q g is stable and there exists at least one essential component of f g for each g m

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

distributed source coding using syndromes discus design and construction we address the problem of compressing correlated distributed sources i e correlated sources which are not co located or which cannot cooperate to directly exploit their correlation we consider the related problem of compressing a source which is correlated with another source that is available only at the decoder this problem has been studied in the information theory literature under the name of the slepian wolf 1973 source coding problem for the lossless coding case and as rate distortion with side information for the lossy coding case we provide a constructive practical framework based on algebraic trellis codes dubbed as distributed source coding using syndromes discus that can be applicable in a variety of settings simulation results are presented for source coding of independent and identically distributed i i d gaussian sources with side information available at the decoder in the form of a noisy version of the source to be coded our results reveal the promise of this approach using trellis based quantization and coset construction the performance of the proposed approach is 2 5 db from the wyner ziv 1976 bound

a decomposition theorem for singular spaces with trivial canonical class of dimension at most five in this paper we partly extend the beauville bogomolov decomposition theorem to the singular setting we show that any complex projective variety of dimension at most five with canonical singularities and numerically trivial canonical class admits a finite cover etale in codimension one that decomposes as a product of an abelian variety and singular analogues of irreducible calabi yau and irreducible holomorphic symplectic varieties

adaptive binary slepian wolf decoding using particle based belief propagation a major difficulty that plagues the practical use of slepian wolf sw coding and distributed source coding in general is that the precise correlation among sources needs to be known a priori to resolve this problem we propose an adaptive asymmetric sw decoding scheme using particle based belief propagation pbp we explain the adaptive scheme for asymmetric setup in detail and then further extend it to the non asymmetric setup based on the code partitioning approach moreover we introduce a metropolis hastings mh algorithm in the resampling step which efficiently decreases the number of simulation iterations we show through experiments that the proposed algorithm can simultaneously reconstruct the compressed sources and estimate the joint correlation among sources further comparing to the conventional sw decoder based on standard belief propagation the proposed approach can achieve higher compression under varying correlation statistics

duality between source coding and channel coding and its extension to the side information case we explore the information theoretic duality between source coding with side information at the decoder and channel coding with side information at the encoder we begin with a mathematical characterization of the functional duality between classical source and channel coding formulating the precise conditions under which the optimal encoder for one problem is functionally identical to the optimal decoder for the other problem we then extend this functional duality to the case of coding with side information by invoking this duality we are able to generalize the result of wyner and ziv 1976 relating to no rate loss for source coding with side information from gaussian to more arbitrary distributions we consider several examples corresponding to both discrete and continuous valued cases to illustrate our formulation for the gaussian cases of coding with side information we invoke geometric arguments to provide further insights into their duality our geometric treatment inspires the construction and dual use of practical coset codes for a large class of emerging applications for coding with side information such as distributed sensor networks watermarking and information hiding communication systems

distributed coding using punctured quasi arithmetic codes for memory and memoryless sources this correspondence considers the use of punctured quasi arithmetic qa codes for the slepian wolf problem these entropy codes are defined by finite state machines for memoryless and first order memory sources puncturing an entropy coded bit stream leads to an ambiguity at the decoder side the decoder makes use of a correlated version of the original message in order to remove this ambiguity a complete distributed source coding dsc scheme based on qa encoding with side information at the decoder is presented together with iterative structures based on qa codes the proposed schemes are adapted to memoryless and first order memory sources simulation results reveal that the proposed schemes are efficient in terms of decoding performance for short sequences compared to well known dsc solutions using channel codes

hypersurfaces quasi invariant by codimension one foliations we present a variant of the classical darboux jouanolou theorem our main result provides a characterization of foliations which are pull backs of foliations on surfaces by rational maps as an application we provide a structure theorem for foliations on 3 folds admitting an infinite number of extremal rays

practical codes for collaborative estimation in collaborative estimation a network of nodes each indirectly observing an underlying source communicate with each other to form improved estimates of the underlying source this paper derives an efficient collaborative estimation algorithm achieving high performance estimates with low communication and complexity utilizing tools from multiterminal source coding theory and modern practical coding theory relevant theoretical limits based on achievability from multiterminal source coding are first presented in order to characterize efficient tradeoffs between communication and estimation performance and inspire the architecture of the collaborative estimation code next low complexity tools from modern practical coding theory are utilized to get a practical collaborative estimation algorithm approaching these performance limits the developed algorithm utilizes successively refined trellis coded quantization sr tcq to provide necessary diverse descriptions of the source and low density parity check ldpc codes to provide an efficient means of compressing these descriptions for low complexity belief propagation decoding with side information comparing the communication versus estimate quality tradeoff performance attained by the developed low complexity scheme with that obtainable with an inner bound an average distortion gap of only 1 0 and 1 6 db at average rates of 3 32 and 2 33 b s respectively is observed

nested linear lattice codes for structured multiterminal binning network information theory promises high gains over simple point to point communication techniques at the cost of higher complexity however lack of structured coding schemes limited the practical application of these concepts so far one of the basic elements of a network code is the binning scheme wyner 1974 1978 and other researchers proposed various forms of coset codes for efficient binning yet these schemes were applicable only for lossless source or noiseless channel network coding to extend the algebraic binning approach to lossy source or noisy channel network coding previous work proposed the idea of nested codes or more specifically nested parity check codes for the binary case and nested lattices in the continuous case these ideas connect network information theory with the rich areas of linear codes and lattice codes and have strong potential for practical applications we review these developments and explore their tight relation to concepts such as combined shaping and precoding coding for memories with defects and digital watermarking we also propose a few novel applications adhering to a unified approach

on the relation among some definitions of strategic stability in this paper we examine a number of different definitions of strategic stability and the relations among them in particular we show that the stability requirement given by hillas 1990 is weaker than the requirements involved in the various definitions of stability in mertens reformulation of stability mertens 1989 1991 to this end we introduce a new definition of stability and show that it is equivalent to a variant of the definition given by hillas 1990 we also use the equivalence of our new definition with the definition of hillas to provide correct proofs of some of the results that were originally claimed and incorrectly proved in hillas 1990

hamming distance spectrum of dac codes for equiprobable binary sources distributed arithmetic coding dac is an effective technique for implementing slepian wolf coding swc it has been shown that a dac code partitions source space into unequal size codebooks so that the overall performance of dac codes depends on the cardinality and structure of these codebooks the problem of dac codebook cardinality has been solved by the so called codebook cardinality spectrum ccs this paper extends the previous work on ccs by studying the problem of dac codebook structure we define hamming distance spectrum hds to describe dac codebook structure and propose a mathematical method to calculate the hds of dac codes the theoretical analyses are verified by experimental results

on fano foliations abstract in this paper we address fano foliations on complex projective varieties these are foliations whose anti canonical class k is ample we focus our attention on a special class of fano foliations namely del pezzo foliations on complex projective manifolds we show that these foliations are algebraically integrable with one exceptional case when the ambient space is p n we also provide a classification of del pezzo foliations with mild singularities

lossless and near lossless source coding for multiple access networks a multiple access source code masc is a source code designed for the following network configuration a pair of correlated information sequences x sub i sub i 1 sup spl infin and y sub i sub i 1 sup spl infin is drawn independent and identically distributed i i d according to joint probability mass function p m f p x y the encoder for each source operates without knowledge of the other source the decoder jointly decodes the encoded bit streams from both sources the work of slepian and wolf describes all rates achievable by mascs of infinite coding dimension n spl rarr spl infin and asymptotically negligible error probabilities p sub e sup n spl rarr 0 in this paper we consider the properties of optimal instantaneous mascs with finite coding dimension n spl infin and both lossless p sub e sup n 0 and nearlossless p sub e sup n spl rarr 0 performance the interest in near lossless codes is inspired by the discontinuity in the limiting rate region at p sub e sup n 0 and the resulting performance benefits achievable by using near lossless mascs as entropy codes within lossy mascs our central results include generalizations of huffman and arithmetic codes to the masc framework for arbitrary p x y n and p sub e sup n and polynomial time design algorithms that approximate these optimal solutions

adaptive rate reconstruction of time varying signals with application in compressive foreground extraction we propose and analyze an online algorithm for reconstructing a sequence of signals from a limited number of linear measurements the signals are assumed sparse with unknown support and evolve over time according to a generic nonlinear dynamical model our algorithm based on recent theoretical results for ell 1 ell 1 minimization is recursive and computes the number of measurements to be taken at each time on the fly as an example we apply the algorithm to online compressive video foreground extraction a problem stated as follows given a set of measurements of a sequence of images with a static background simultaneously reconstruct each image while separating its foreground from the background the performance of our method is illustrated on sequences of real images we observe that it allows a dramatic reduction in the number of measurements or reconstruction error with respect to state of the art compressive background subtraction schemes

analysis of sum product decoding of low density parity check codes using a gaussian approximation density evolution is an algorithm for computing the capacity of low density parity check ldpc codes under message passing decoding for memoryless binary input continuous output additive white gaussian noise awgn channels and sum product decoders we use a gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm we convert the infinite dimensional problem of iteratively calculating message densities which is needed to find the exact threshold to a one dimensional problem of updating the means of the gaussian densities this simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better but also makes it easier to design good irregular ldpc codes for awgn channels for various regular ldpc codes we have examined thresholds can be estimated within 0 1 db of the exact value for rates between 0 5 and 0 9 codes designed using the gaussian approximation perform within 0 02 db of the best performing codes found so far by using density evolution when the maximum variable degree is 10 we show that by using the gaussian approximation we can visualize the sum product decoding algorithm we also show that the optimization of degree distributions can be understood and done graphically using the visualization

on the rate distortion function for binary source coding with side information we present an in depth analysis of the problem of lossy compression of binary sources in the presence of correlated side information where the correlation is given by a generic binary asymmetric channel and the hamming distance is the distortion metric our analysis is motivated by systematic rate distortion gains observed when applying asymmetric correlation models in wyner ziv video coding first we derive for the first time the rate distortion function for conventional predictive coding in the binary asymmetric correlation channel scenario second we propose a new bound for the case where the side information is only available at the decoder wyner ziv coding we conjecture this bound to be tight we show that the maximum rate needed to encode as well as the maximum rate loss of wyner ziv coding relative to predictive coding corresponds to uniform sources and symmetric correlations importantly we show that the upper bound on the rate loss established by zamir is not tight and that the maximum value is actually significantly lower moreover we prove that the only binary correlation channel that incurs no rate loss for wyner ziv coding compared with predictive coding is the z channel finally we complement our analysis with new compression performance results obtained with our state of the art wyner ziv video coding system

the equivalence between slepian wolf coding and channel coding under density evolution we consider slepian wolf code design based on low density parity check ldpc coset codes the density evolution formula for slepian wolf coding is derived an intimate connection between slepian wolf coding and channel coding is then established specifically we show that under density evolution each slepian wolf coding problem is equivalent to a channel coding problem for a binary input output symmetric channel

generalized compute compress and forward compute and forward cf harnesses interference in wireless communications by exploiting structured coding the key idea of cf is to compute integer combinations of code words from multiple source nodes rather than to decode individual code words by treating others as noise compute compress and forward ccf can further enhance the network performance by introducing compression operations at receivers in this paper we develop a more general compression framework termed generalized ccf gccf where the compression function involves the selection of message segments over finite fields we show that gccf achieves a broader compression rate region than ccf we also compare our compression rate region with the fundamental slepian wolf sw region we show that gccf is optimal in the sense of achieving the minimum total compression rate we also establish the criteria under which gccf achieves the sw region in addition we consider a two hop relay network employing the gccf scheme we formulate a sum rate maximization problem and develop an approximate algorithm to solve the problem numerical results are presented to demonstrate the performance superiority of gccf over ccf and other schemes

comparison of constructions of irregular gallager codes the low density parity check codes whose performance is closest to the shannon limit are gallager codes based on irregular graphs we compare alternative methods for constructing these graphs and present two results first we find a super poisson construction which gives a small improvement in empirical performance over a random construction second whereas gallager codes normally take n sup 2 time to encode we investigate constructions of regular and irregular gallager codes that allow more rapid encoding and have smaller memory requirements in the encoder we find that these fast encoding gallager codes have equally good performance

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

distributed arithmetic coding for the slepian wolf problem distributed source coding schemes are typically based on the use of channels codes as source codes in this paper we propose a new paradigm named ldquodistributed arithmetic coding rdquo which extends arithmetic codes to the distributed case employing sequential decoding aided by the side information in particular we introduce a distributed binary arithmetic coder for the slepian wolf coding problem along with a joint decoder the proposed scheme can be applied to two sources in both the asymmetric mode wherein one source acts as side information and the symmetric mode wherein both sources are coded with ambiguity at any combination of achievable rates distributed arithmetic coding provides several advantages over existing slepian wolf coders especially good performance at small block lengths and the ability to incorporate arbitrary source models in the encoding process e g context based statistical models in much the same way as a classical arithmetic coder we have compared the performance of distributed arithmetic coding with turbo codes and low density parity check codes and found that the proposed approach is very competitive

rational morita equivalence for holomorphic poisson modules abstract we introduce a weak concept of morita equivalence in the birational context for poisson modules on complex normal poisson projective varieties we show that poisson modules on projective varieties with mild singularities are either rationally morita equivalent to a flat partial holomorphic sheaf or a sheaf with a meromorphic flat connection or a co higgs sheaf as an application we study the geometry of rank two meromorphic rank two sl 2 poisson modules which can be interpreted as a poisson analogous to transversally projective structures for codimension one holomorphic foliations moreover we describe the geometry of the symplectic foliation induced by the poisson connection on the projectivization of the poisson module

compress forward coding with bpsk modulation for the half duplex gaussian relay channel this paper studies compress forward cf coding with bpsk modulation for the half duplex gaussian relay channel in cf relaying wyner ziv coding is applied at the relay to exploit the joint statistics between signals at the relay and the destination we propose slepian wolf coded nested scalar quantization swcnsq for practical wyner ziv coding at the relay we first provide the achievable rate of swcnsq based cf relaying as a performance benchmark and then present a practical code design using low density parity check ldpc codes for error protection at the source and nested scalar quantization plus irregular repeat accumulation ira codes for cf coding at the relay the degree distributions of the ldpc and ira codes are optimized using extrinsic information transfer charts and gaussian approximation under discretized density evolution for asymptotically large block lengths our optimized code design operates 0 11 0 21 db away from the swcnsq limit for cf relaying simulations with ldpc ira codes of length 2 times 105 bits show a performance gap of 0 27 0 38 db from the achievable rate

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

on successive refinement for the wyner ziv problem achievable rates are characterized for successive refinement sr in the wyner ziv scenario namely in the presence of correlated side information si at the receivers in this setting the encoder is assumed to operate in two stages where the first corresponds to relatively low rate and high distortion and the second comprising a refinement code on top of the first code is aimed at reproduction at reduced distortion both decoders for low rate high distortion and for high rate low distortion are equipped with si streams correlated to the source but unavailable to the encoder furthermore it is assumed that the decoder that receives the higher rate bitstream i e the additional refinement bits accesses also si of better quality than that of the lower resolution decoder by better quality we mean that the source and the si are modeled together as a stochastically degraded joint source where the encoded symbols the si at the refinement stage and the si at the initial coarse stage form a markov chain in this order for a memoryless joint process that includes the source to be encoded and its instantaneously correlated si streams necessary and sufficient conditions are furnished in terms of single letter formulas for the achievability of a pair of rates corresponding to two given distortion levels special attention is devoted to the degenerate but important case where the two si streams at the two decoders are identical for this case conditions are provided for successive refinability in the sense of the existence of codes that asymptotically achieve the wyner ziv rate distortion function simultaneously at both distortion levels in this context the doubly symmetric binary source with the hamming distortion measure and the jointly gaussian source with the squared error distortion measure are successively refinable in the wyner ziv setting it is demonstrated that a source that is not successively refinable in the ordinary sense i e without si may become successively refinable in the presence of si at the decoders

wyner ziv coding based on tcq and ldpc codes this paper considers trellis coded quantization tcq and low density parity check ldpc codes for the quadratic gaussian wyner ziv coding problem after tcq of the source x ldpc codes are used to implement slepian wolf coding of the quantized source q x with side information y at the decoder assuming 256 state tcq and ideal slepian wolf coding in the sense of achieving the theoretical limit h q x y we experimentally show that slepian wolf coded tcq performs 0 2 db away from the wyner ziv distortion rate function dwz r at high rate this result mirrors that of entropy constrained tcq in classic source coding of gaussian sources furthermore using 8 192 state tcq and assuming ideal slepian wolf coding our simulations show that slepian wolf coded tcq performs only 0 1 db away from dwz r at high rate these results establish the practical performance limit of slepian wolf coded tcq for quadratic gaussian wyner ziv coding practical designs give performance very close to the theoretical limit for example with 8 192 state tcq irregular ldpc codes for slepian wolf coding and optimal non linear estimation at the decoder our performance gap to dwz r is 0 20 db 0 22 db 0 30 db and 0 93 db at 3 83 bit per sample b s 1 83 b s 1 53 b s and 1 05 b s respectively when 256 state 4 d trellis coded vector quantization instead of tcq is employed the performance gap to dwz r is 0 51 db 0 51 db 0 54 db and 0 80 db at 2 04 b s 1 38 b s 1 0 b s and 0 5 b s respectively

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

dac spectrum of binary sources with equally likely symbols though distributed arithmetic coding dac is an effective implementation of slepian wolf coding its performance which is closely linked with its decoding complexity has not received a thorough analysis with binary sources with equally likely symbols as the research object this paper develops the dac spectrum and makes use of it as a tool to answer the complexity problem of the ideal dac decoder based on an in depth analysis on dac decoding process we define the dac spectrum and propose to find it by a recursive formulation firstly the initial dac spectrum is constrained by a functional equation and the fourier transform is utilized to obtain its general explicit form secondly an equation is given through which stage i 1 dac spectrum can be deduced from stage i dac spectrum a numerical method is also proposed for calculating dac spectrum whose convergency is proved to measure the complexity of the ideal dac decoder we define the expansion factor and relate it to dac spectrum it is proved that if binary symbols 0 and 1 are mapped onto partially overlapped intervals 0 q and 1 q 1 respectively the expansion factor will converge to 2q i e the complexity of the ideal rate dac decoder is approximately o 2 n 1

coset codes ii binary lattices and related codes for pt i see ibid vol 34 no 5 p 1123 51 1988 the family of barnes wall lattices including d sub 4 and e sub 8 of lengths n 2 sup n and their principal sublattices which are useful in constructing coset codes are generated by iteration of a simple construction called the squaring construction the closely related reed muller codes are generated by the same construction the principal properties of these codes and lattices are consequences of the general properties of iterated squaring constructions which also exhibit the interrelationships between codes and lattices of different lengths an extension called the cubing construction generates good codes and lattices of lengths n 3 2 sup n including the golay code and leech lattice with the use of special bases for 8 space another related construction generates the nordstrom robinson code and an analogous 16 dimensional nonlattice packing these constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum likelihood decoding algorithms

universal bound on the performance of lattice codes we present a lower bound on the probability of symbol error for maximum likelihood decoding of lattices and lattice codes on a gaussian channel the bound is tight for error probabilities and signal to noise ratios of practical interest as opposed to most existing bounds that become tight asymptotically for high signal to noise ratios the bound is also universal it provides a limit on the highest possible coding gain that may be achieved at specific symbol error probabilities using any lattice or lattice code in n dimensions in particular it is shown that the effective coding gains of the densest known lattices are much lower than their nominal coding gains the asymptotic as n spl rarr spl infin behavior of the new bound is shown to coincide with the shannon 1948 limit for gaussian channels

wyner ziv coding based on multidimensional nested lattices distributed source coding addresses the compression of correlated sources without communication links among them this paper is concerned with the wyner ziv problem coding of an information source with side information available only at the decoder in the form of a noisy version of the source both the problems of theoretical analysis and code design are addressed in the framework of multi dimensional nested lattice coding for theoretical analysis accurate computation of the rate distortion function is given under the high resolution assumption and a new upper bound using the derivative of the theta series is derived for practical code design several low complexity techniques are proposed compared to the existing slepian wolf coded nested quantization for wyner ziv coding based on one or two dimensional lattices our proposed multi dimensional lattice coding can offer better performance at arguably lower complexity since it does not require the second stage of slepian wolf coding

two terminal video coding following recent works on the rate region of the quadratic gaussian two terminal source coding problem and limit approaching code designs this paper examines multiterminal source coding of two correlated i e stereo video sequences to save the sum rate over independent coding of both sequences two multiterminal video coding schemes are proposed in the first scheme the left sequence of the stereo pair is coded by h 264 avc and used at the joint decoder to facilitate wyner ziv coding of the right video sequence the first i frame of the right sequence is successively coded by h 264 avc intracoding and wyner ziv coding an efficient stereo matching algorithm based on loopy belief propagation is then adopted at the decoder to produce pixel level disparity maps between the corresponding frames of the two decoded video sequences on the fly based on the disparity maps side information for both motion vectors and motion compensated residual frames of the right sequence are generated at the decoder before wyner ziv encoding in the second scheme source splitting is employed on top of classic and wyner ziv coding for compression of both i frames to allow flexible rate allocation between the two sequences experiments with both schemes on stereo video sequences using h 264 avc ldpc codes for slepian wolf coding of the motion vectors and scalar quantization in conjunction with ldpc codes for wyner ziv coding of the residual coefficients give a slightly lower sum rate than separate h 264 avc coding of both sequences at the same video quality

slepian wolf coded nested lattice quantization for wyner ziv coding high rate performance analysis and code design nested lattice quantization provides a practical scheme for wyner ziv coding this paper examines the high rate performance of nested lattice quantizers and gives the theoretical performance for general continuous sources in the quadratic gaussian case as the rate increases we observe an increasing gap between the performance of finite dimensional nested lattice quantizers and the wyner ziv distortion rate function we argue that this is because the boundary gain decreases as the rate of the nested lattice quantizers increases to increase the boundary gain and ultimately boost the overall performance a new practical wyner ziv coding scheme called slepian wolf coded nested lattice quantization swc nq is proposed where slepian wolf coding is applied to the quantization indices of the source for the purpose of compression with side information at the decoder theoretical analysis shows that for the quadratic gaussian case and at high rate swc nq performs the same as conventional entropy coded lattice quantization with the side information available at both the encoder and the decoder furthermore a nonlinear minimum mean square error mse estimator is introduced at the decoder which is theoretically proven to degenerate to the linear minimum mse estimator at high rate and experimentally shown to outperform the linear estimator at low rate practical designs of one and two dimensional nested lattice quantizers together with multilevel low density parity check ldpc codes for slepian wolf coding give performance close to the theoretical limits of swc nq

writing on dirty paper corresp a channel with output y x s z is examined the state s sim n 0 qi and the noise z sim n 0 ni are multivariate gaussian random variables i is the identity matrix the input x in r n satisfies the power constraint l n sum i 1 n x i 2 leq p if s is unknown to both transmitter and receiver then the capacity is frac 1 2 ln 1 p n q nats per channel use however if the state s is known to the encoder the capacity is shown to be c ast frac 1 2 ln 1 p n independent of q this is also the capacity of a standard gaussian channel with signal to noise power ratio p n therefore the state s does not affect the capacity of the channel even though s is unknown to the receiver it is shown that the optimal transmitter adapts its signal to the state s rather than attempting to cancel it

maximum likelihood laplacian correlation channel estimation in layered wyner ziv coding the distributed source coding theory teaches that efficient compression can be achieved by exploiting the correlation between encoded sources at the decoder only a critical issue in this coding paradigm is how to accurately estimate the correlation statistics at the decoder this paper proposes a novel correlation channel estimation method designed for generic layered wyner ziv wz coding driven by real life applications i e video coding the correlation noise is assumed to be memoryless zero mean laplacian with a block stationary behavior unlike existing methods the proposed technique derives a novel maximum likelihood estimate of the noise scaling parameter per stationarity block adhering to layered coding the derived estimate is successively refined as more stages are decoded the cramer rao lower bound crlb for our estimator is derived and the mean squared error performance of the proposed successive refinement algorithm is evaluated through simulations apart from achieving an accuracy close to the derived crlb the proposed algorithm is shown to yield a wz rate distortion performance approaching the ideal but unrealistic scenario where the decoder knows a priori the correlation statistics moreover the proposed technique has been integrated into our latest wz video codec for wireless capsule endoscopy experimentation reveals that the proposed method delivers improved compression performance compared to state of the art techniques while simultaneously reducing the overall decoding complexity

successive refinement for the wyner ziv problem and layered code design we examine successive refinement for the wyner ziv problem described in a recent paper by steinberg and merhav where the authors showed that if the side information for all stages is identical then the jointly gaussian source with squared error distortion measure is successively refinable we first extend this result to the case where the difference between the source and the side information is gaussian and independent of the side information as a byproduct we give an alternative proof that the wyner ziv problem for these sources has no rate loss a result that was recently shown by pradhan et al through invoking the duality between the gaussian wyner ziv problem and the costa problem we then perform layered wyner ziv code design for this general type of source based on nested scalar quantization bit plane coding and low density parity check ldpc code based slepian wolf coding source coding with side information we show that density evolution can be used to analyze the slepian wolf code performance provided that certain symmetry conditions which have been dubbed dual symmetry are satisfied by the hypothetical channel between the source and the side information we also show that the dual symmetry condition is indeed satisfied by the hypothetical channel in our slepian wolf coding setup this justifies the use of density evolution in our ldpc code based slepian wolf code design for wyner ziv coding for the jointly gaussian source our layered coder performs 1 29 to 3 45 db from the wyner ziv bound for rates ranging from 0 47 to 4 68 bits per sample when the side information is laplacian and the source equals the side information plus an independent gaussian noise term our layered coder performs 1 33 to 3 90 db from the wyner ziv bound for rates ranging from 0 48 to 4 64 bits per sample

the rate loss in the wyner ziv problem the rate distortion function for source coding with side information at the decoder the wyner ziv problem is given in terms of an auxiliary random variable which forms a markov chain with the source and the side information this markov chain structure typical to the solution of multiterminal source coding problems corresponds to a loss in coding rate with respect to the conditional rate distortion function i e to the case where the encoder is fully informed we show that for difference or balanced distortion measures this loss is bounded by a universal constant which is the minimax capacity of a suitable additive noise channel furthermore in the worst case this loss is equal to the maximin redundancy over the rate distortion function of the additive noise test channel for example the loss in the wyner ziv problem is less than 0 5 bit sample in the squared error distortion case and it is less than 0 22 bit for a binary source with hamming distance these results have implications also in universal quantization with side information and in more general multiterminal source coding problems

residual quantization based code design for compressing noisy sources with arbitrary decoder side information this paper considers noisy wyner ziv coding wzc in which a remote noisy source is compressed with side information at the decoder only the decoder side information is not limited to being gaussian thus this noisy wzc problem cannot be transformed into the conventional one tailored for noiseless sources a new coding structure named the residual quantization rq based noisy wzc is proposed to solve the problem this scheme explicitly constructs the theoretical auxiliary random variable to facilitate optimal reconstruction of the noisy source in this two stage encoder the noisy source is quantized twice and the quantization error residue of the first stage is the input of the second stage by sending only the quantization index of the second stage to the decoder the corresponding code rate can theoretically approach the noisy wzc bound moreover the rq based noisy wzc is implemented using graph based codes the main challenge is that it is necessary to design a codebook that is simultaneously good for source and channel coding for the first stage quantization since this quantization code also acts as a channel code at the decoder this problem is solved by constructing a low density parity check ldpc code with edge degrees optimized for channel coding and enhancing its performance for source coding by using a modified reinforced belief propagation quantization algorithm simulation results show that the noisy wyner ziv bounds can be practically approached by our implementation in addition the proposed implementation offers more flexibility in the code rates compared with the existing practical designs making it more suitable for emerging applications such as fronthaul compression

rigorous estimates of the tails of the probability distribution function for the random linear shear model in previous work majda and mclaughlin and majda computed explicit expressions for the 2nth moments of a passive scalar advected by a linear shear flow in the form of an integral over r n in this paper we first compute the asymptotics of these moments for large moment number we are able to use this information about the large n behavior of the moments along with some basic facts about entire functions of finite order to compute the asymptotics of the tails of the probability distribution function we find that the probability distribution has gaussian tails when the energy is concentrated in the largest scales as the initial energy is moved to smaller and smaller scales we find that the tails of the distribution grow longer and the distribution moves smoothly from gaussian through exponential and stretched exponential we also show that the derivatives of the scalar are increasingly intermittent in agreement with experimental observations and relate the exponents of the scalar derivative to the exponents of the scalar

rate compatible punctured low density parity check codes with short block lengths we consider the problem of rate compatible puncturing of low density parity check ldpc codes over additive white gaussian noise awgn channels in previous work it was shown that good puncturing distributions exist for ldpc codes but the code length was large in this correspondence we give a procedure for determining the puncturing distributions for ldpc codes with short block lengths a few thousand bits and show that careful puncturing can produce good performance we compare the performance of the rate compatible punctured ldpc codes with dedicated ldpc codes across a range of rates and see that the rate compatible codes have a favorable complexity performance tradeoff

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

on multiterminal source code design multiterminal mt source coding refers to separate lossy encoding and joint decoding of multiple correlated sources recently the rate region of both direct and indirect mt source coding in the quadratic gaussian setup with two encoders was determined we are thus motivated to design practical mt source codes that can potentially achieve the entire rate region in this paper we present two practical mt coding schemes under the framework of slepian wolf coded quantization swcq for both direct and indirect mt problems the first asymmetric swcq scheme relies on quantization and wyner ziv coding and it is implemented via source splitting to achieve any point on the sum rate bound in the second conceptually simpler scheme symmetric swcq the two quantized sources are compressed using symmetric slepian wolf coding via a channel code partitioning technique that is capable of achieving any point on the slepian wolf sum rate bound our practical designs employ trellis coded quantization and turbo low density parity check ldpc codes for both asymmetric and symmetric slepian wolf coding simulation results show a gap of only 0 139 0 194 bit per sample away from the sum rate bound for both direct and indirect mt coding problems

on codimension 1 del pezzo foliations on varieties with mild singularities in this paper we extend to the singular setting the theory of fano foliations developed in our previous paper araujo and druel adv math 238 70 118 2013 a mathbb q fano foliation on a complex projective variety x is a foliation fancyscript f subsetneq t x whose anti canonical class k fancyscript f is an ample mathbb q cartier divisor in the spirit of kobayashi ochiai theorem we prove that under some conditions the index i fancyscript f of a mathbb q fano foliation is bounded by the rank r of fancyscript f and classify the cases in which i fancyscript f r next we consider mathbb q fano foliations fancyscript f for which i fancyscript f r 1 these are called del pezzo foliations we classify codimension 1 del pezzo foliations on mildly singular varieties

the essential components of the set of equilibrium points for set valued maps in this article we discuss the stability of equilibrium points for set valued maps we introduce the notion of essential components of the set of equilibrium points for set valued maps and prove an existence theorem of essential components as applications we deduce the existence of essential components of the set of coincidence points and show that every marchaud dynamical economy cf j p aubin dynamic economic theory springer verlag 1997 possesses at least one essential component of the set of its viable equilibrium points

low delay prediction and transform based wyner ziv coding this paper studies low delay wyner ziv coding i e lossy source coding with side information at the decoder with emphasis on the extreme of zero delay to achieve zero delay a scalar quantizer is followed by scalar coding of quantization indices in the fixed length coding scenario under high resolution assumptions and appropriately defined decodability constraints the optimal quantization level density is conjectured to be periodic this conjecture which is provable when the correlation is high allows for a precise analysis of the rate distortion tradeoff the performance of variable length coding with periodic quantization is also characterized the results are then incorporated in predictive wyner ziv coding for gaussian sources with memory and optimal prediction filters are numerically designed so as to strike a balance between maximally exploiting both temporal and spatial correlation and limiting the propagation of distortion due to occasional decoding errors finally the zero delay schemes are also employed in transform coding with small block lengths where the gaussian source and side information are transformed separately with the premise that corresponding transform coefficient pairs exhibit good spatial correlation and minimal temporal correlation for the specific source side information pairs studied it is shown that transform coding even with a small block length outperforms predictive coding performances of both predictive and transform coding are also compared with the asymptotic rate distortion bounds

rate distortion bounds for wyner ziv coding with gaussian scale mixture correlation noise the objective of this work is the characterization of the wyner ziv rate distortion function for memoryless continuous sources when the correlation between the sources is modeled via an additive noise channel modeling the distribution of the correlation noise via a gaussian mixture with discrete or continuous mixing variable provides a unified signal model able to describe a wide class of distributions useful in the context of practical applications the wyner ziv rate distortion function associated with this signal model cannot in general be obtained in analytical form this work contributes a method for its analysis by providing computable upper and lower bounds