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

advances in residual vector quantization a review advances in residual vector quantization rvq are surveyed definitions of joint encoder optimality and joint decoder optimality are discussed design techniques for rvqs with large numbers of stages and generally different encoder and decoder codebooks are elaborated and extended fixed rate rvqs and variable rate rvqs that employ entropy coding are examined predictive and finite state rvqs designed and integrated into neural network based source coding structures are revisited successive approximation rvqs that achieve embedded and refinable coding are reviewed a new type of successive approximation rvq that varies the instantaneous block rate by using different numbers of stages on different blocks is introduced and applied to image waveforms and a scalar version of the new residual quantizer is applied to image subbands in an embedded wavelet transform coding system

a fast encoding method for lattice codes and quantizers in an earlier paper the authors described a very fast method which for the root lattices a n d n e n their duals and certain other lattices finds the closest lattice point to an arbitrary point of the underlying space if the lattices are used as codes for a gaussian channel the algorithm provides a fast decoding procedure or if they are used as vector quantizers the algorithm performs the analog to digital conversion efficiently the present paper offers a solution to the inverse problem for the same lattices the encoding problem for channel codes or the digital to analog part of quantizing namely given an integer k to find the kth code vector and to the closely related problem of finding the index k of a given code vector

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

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

gorenstein biliaison and acm sheaves abstract let x be a normal arithmetically gorenstein scheme in p n we give a criterion for all codimension two acm subschemes of x to be in the same gorenstein biliaison class on x in terms of the category of acm sheaves on x these are sheaves that correspond to the graded maximal cohen macaulay modules on the homogeneous coordinate ring of x using known results on mcm modules we are able to determine the gorenstein biliaison classes of codimension two subschemes of certain varieties including the nonsingular quadric surface in p 3 and the cone over it in p 4 as an application we obtain a new proof of some theorems of lesperance about curves in p 4 and answer some questions he raised

bound and resonant states in coulomb like potentials the potential separable expansion method has been generalised to calculating bound and resonant states in coulomb like potentials the complete set of coulomb sturmian functions was taken as the basis to expand the short range potential on this basis the matrix elements of the coulomb green functions have been given in closed form as functions of the complex energy the feasibility of the method is demonstrated by a numerical example

robust memoryless quantization for minimum signal distortion robust quantizers are designed for situations where there is only an incomplete statistical description of the quantizer input the goal of the design is to closely approximate quantizer inputs by quantizer outputs without using more than a specified number of quantization levels the exact probability distribution of the input is unknown but this distribution is known to belong to some set c the primary set c considered is the set of all unimodal probability distributions which satisfy generalized moment constraint e g mean square value less than or equal to a constant a quantizer is derived which minimizes over all quantizers the maximum distortion over all distributions in c this robust quantizer guarantees a significanfiy lower worst case distortion than the classical gaussian optimal quantizer while performing nearly as well as the gaussian optimal quantizer when the input is in fact gaussian

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

design of quantizers from histograms the design of signal quantizers when the statistical description of the source is a histogram on a finite domain is considered minimum mean and minimax error criteria are discussed both leading to the design of piecewise linear compressor functions the problem of allocating the histogram intervals in the symmetric source density case is addressed through the use of chebyshev probability inequalities finally an adaptive implementation of the proposed system is described

unrestricted multistage vector quantizers a low complexity high performance vector quantization technique for medium to high bit rate encoding based on multistage residual vq is introduced the design a variable rate version of multistage vq is the vector extension of the piecewise uniform scalar quantizers of p f swaszek and j b thomas 1984 and is related to the piecewise uniform vqs of f kuhlmann and j a buckview 1988 examples for the independent gaussian and laplacian sources are included

multivariate block polar quantization we introduce multivariate block polar quantization mbpq mbpq minimizes the weighted squared error distortion for a set of complex variables representing one block of a signal under a resolution constraint for the entire block mbpq performs below the lower bound for classical bivariate quantization both for gaussian complex variables and for complex variables found from sinusoidal analysis of audio data still it is of similar complexity as traditional polar quantizers in the case of audio data we found a performance gain of about 2 5 db over the best performing conventional resolution constrained polar quantization an extension of unrestricted polar quantization

liaison equivalence classes we will work in ip n 2 over an algebraically closed field the notion of liaison among subvarieties of ip was introduced in 3 we will consider here liaison in codimension two generalizing results in 4 recall therefore that two subschemes v 1 v 2 of ip that are locally cohenmacaulay and equidimensional of codimension two are said to be linked algebraically via the complete intersection x of two hypersurfaces denoted v l v z via x if d e v j x v r o v x t j they are said to be linked geometrically if i 1 and v2 in the above definition have no components in common according to 4 theorem 1 7 the two notions of linkage generate the same equivalence relation among those subschemes v which are also generically locally complete intersections we will hence consider only algebraic linkage

on the cohen macaulay type of the general hypersurface section of a curve a theorem of strano 24 states that if c is a reduced irreducible curve in lp 3 lying on no quadric surface and if a general hyperplane section c n h of c is a complete intersection then c itself must be a complete intersection this was extended by re 22 to the case where c is a locally cohen macaulay curve in lw it was then shown by huneke and ulrich 14 that if c is reduced and connected not lying on a quadric hypersurface and if c a h is merely gorenstein then the same is true of c finally the first author 18 extended the techniques of 14 to show that if h is replaced by a general hypersurface of degree d then similar results hold recall that one definition of a gorenstein scheme is arithmetically cohenmacaulay with cohen macaulay type one in other words in a minimal free resolution

multidimensional rotations for robust quantization of image data laplacian and generalized gaussian data arise in the transform and subband coding of images this paper describes a method of rotating independent identically distributed i i d laplacian like data in multiple dimensions to significantly improve the overload characteristics for quantization the rotation is motivated by the geometry of the laplacian probability distribution and can be achieved with only additions and subtractions using a walsh hadamard transform its theoretical and simulated results for scalar lattice and polar quantization are presented in this paper followed by a direct application to image compression we show that rotating the image data before quantization not only improves compression performance but also increases robustness to the channel noise and deep fades often encountered in wireless communication

variable length coding for performance improvement of asymptotically optimal unrestricted polar quantization of bivariate gaussian source this paper considers the application of variable length coding using two unrestricted polar quantizers upqs for performance improvement of unrestricted polar quantization for bivariate gaussian source we propose the use of two upqs both designed for the bivariate gaussian source of unit variance having different sizes of codebooks and different optimal compressor functions we show that the fixed rate upq is a subset of our model and we perform rigorous optimization procedure in order to obtain optimal parameter values maximizing the signal to quantization noise ratio sqnr for the given average bit rate in addition we study how gain in sqnr over the fixed rate upq depends on the average bit rate and we show that the gain ranges from 0 619db to 0 869db depending on the average bit rate discussion is also provided about the proposed quantizer complexity and its performance in comparison to shannon limit the proposed upq provides a sophisticated choice of average bit rate compared to the fixed rate upq features of the proposed quantizer indicate that the obtained model should be of high theoretical and practical significance

orthogonality condition model for bound and resonant states with a separable expansion of the potential the author shows a very efficient solution of the equation of saito s orthogonality condition model ocm for bound and resonant states by means of a separable expansion of the potential pse method he derives some simplifications of the published formulae of the pse method which facilitate its application to the ocm and may be useful in solving the schrodinger equation as well

piecewise uniform vector quantizers the companding model for quantizer design and analysis has been widely applied in the scalar quantization case however if the signal to be quantized is a vector then the optimum companding system can be designed for only a limited number of distributions on the other hand multidimensional piecewise linear companders can be designed for any signal density generating quantizers that are uniform on each region of the compander these systems while not optimal can have asymptotic performance arbitrarily close to the optimum their analysis and implementation can be simpler than those of optimal systems piecewise linear companders for asymptotic multidimensional quantization are analyzed and a method for their design is suggested

green s matrix from jacobi matrix hamiltonian we propose two ways for determining the green s matrix for problems admitting hamiltonians that have infinite symmetric tridiagonal i e jacobi matrix form on some basis representation in addition to the recurrence relation coming from the jacobi matrix the first approach also requires the matrix elements of the green s operator between the first elements of the basis in the second approach the recurrence relation is solved directly by continued fractions and the solution is continued analytically to the whole complex plane both approaches are illustrated with the non trivial but calculable example of the d dimensional coulomb green s matrix we give the corresponding formulas for the d dimensional harmonic oscillator as well

multidimensional spherical coordinates quantization several investigators have considered polar coordinates quantization of a circularly symmetric source in particular the independent bivariate gaussian source their schemes quantize the polar coordinates independently in an attempt to reduce the mean square error below that of an analogous rectangular coordinates quantizer yet retain an implementation simpler than that of the optimal bivariate quantizer the design of a spherical coordinates quantizer in k dimensions with k 2 k 2 matches published results is considered examples are presented along with comparisons to the rectangular one dimensional and optimal schemes

liaison addition and the structure of a gorenstein liaison class we study the concept of liaison addition for codimension two subschemes of an arithmetically gorenstein projective scheme we show how it relates to liaison and biliaison classes of subschemes and use it to investigate the structure of gorenstein liaison equivalence classes extending the known theory for complete intersection liaison of codimension two subschemes in particular we show that on the non singular quadric threefold in projective 4 space every non licci acm curve can be obtained from a single line by successive liaison additions with lines and ci biliaisons

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

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

uniform and piecewise uniform lattice vector quantization for memoryless gaussian and laplacian sources lattice vector quantizer design procedures for nonuniform sources are presented the procedures yield lattice vector quantizers with excellent performance and retaining the structure required for fast quantization analytical methods for truncating and scaling lattices to be used in vector quantizations are given and their utility is demonstrated for independent and identically distributed i i d gaussian and laplacian sources an analytical technique for piecewise linear multidimensional compandor designs is evaluated for i i d gaussian and laplacian sources by comparing its performance to that of the other vector quantizers

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

uniform spherical coordinate quantization of spherically symmetric sources for signal quantization with the minimum mean square error criterion optimum spherical coordinate quantizers have been shown to outperform optimum rectangular coordinate qnantizers for many spherically symmetric sources in this paper spherical coordinate quantizers are designed and analyzed under the added constraint that each scalar quantizer is a uniform equal step size quantizer it is shown that for small block length uniform spherical coordinate quantizers outperform the uniform rectangular coordinate quantizer as well as the uniform qnantizer consisting of a tesselation of the minimum inertia space filling polytope

geometric piecewise uniform lattice vector quantization of the memoryless gaussian source the aim of this paper is to find a quantization technique that has low implementation complexity and asymptotic performance arbitrarily close to the optimum more specifically it is of interest to develop a new vector quantizer design procedure for a memoryless gaussian source that yields vector quantizers with excellent performance and the structure required for fast quantization to achieve this we combined a fast lattice encoding algorithm with a geometric approach to generate a model of a geometric piecewise uniform lattice vector quantizer expressions for granular distortion and the optimal number of outputs points in each region were derived both exact and approximative asymptotic analyses were carried out during this process the constant probability density function of the input signal vector was kept inside the whole region the analysis demonstrated the existence of piecewise constant approximations to the input vector probability density function which is optimal for the proposed geometric piecewise uniform vector quantizer the considered quantization technique is near optimal for a memoryless gaussian source in other words this paper proposes a method for a near optimum low complex vector quantizer design based on probability density function discretization the presented methodology gives a signal to quantization noise ratio that in some cases differs from the optimum by 0 1db or less improvements of the considered model in performance and complexity over some of the existing techniques are also demonstrated

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

asymptotic performance of unrestricted polar quantizers corresp unrestricted polar quantizers upq s were previously introduced to overcome the weak mean squared error performance of polar coordinate quantizers for the independent bivariate gaussian source when the number of levels was small bits dimension an asymptotic ntean squared error analysis of the upq s with a general circularly symmetric source is presented the performance of the upq s is shown to be asymptotically within 0 17 db of that of the optimum bivariate quantizer the results are extended to dimensions greater than two examples for the independent gaussian source are included