efficient scalar quantization of exponential and laplacian random variables this paper presents solutions to the entropy constrained scalar quantizer ecsq design problem for two sources commonly encountered in image and speech compression applications sources having the exponential and laplacian probability density functions we use the memoryless property of the exponential distribution to develop a new noniterative algorithm for obtaining the optimal quantizer design we show how to obtain the optimal ecsq either with or without an additional constraint on the number of levels in the quantizer in contrast to prior methods which require a multidimensional iterative solution of a large number of nonlinear equations the new method needs only a single sequence of solutions to one dimensional nonlinear equations in some laplacian cases one additional two dimensional solution is needed as a result the new method is orders of magnitude faster than prior ones we show that as the constraint on the number of levels in the quantizer is relaxed the optimal ecsq becomes a uniform threshold quantizer utq for exponential but not for laplacian sources we then further examine the performance of the utq and optimal ecsq and also investigate some interesting alternatives to the utq including a uniform reconstruction quantizer urq and a constant dead zone ratio quantizer cdzrq

conformal invariance in two dimensional percolation the word percolation borrowed from the latin refers to the seeping or oozing of a liquid through a porous medium usually to be strained in this and related senses it has been in use since the seventeenth century it was introduced more recently into mathematics by s r broadbent and j m hammersley bh and is a branch of probability theory that is especially close to statistical mechanics broadbent and hammersley distinguish between two types of spreading of a fluid through a medium or between two aspects of the probabilistic models of such processes diffusion processes in which the random mechanism is ascribed to the fluid and percolation processes in which it is ascribed to the medium a percolation process typically depends on one or more probabilistic parameters for example if molecules of a gas are absorbed at the surface of a porous solid as in a gas mask then their ability to penetrate the solid depends on the sizes of the pores in it and their positions both conceived to be distributed in some random manner a simple mathematical model of such a process is often defined by taking the pores to be distributed in some regular manner that could be determined by a periodic graph and to be open thus very large or closed thus smaller than the molecules with probabilities p and 1 p as p increases the probability of deeper penetration of the gas into the interior of the solid grows

a note on the optimal stability of bases of univariate functions this note is concerned with the characterizations and uniqueness of bases of finite dimensional spaces of univariate continuous functions which are optimally stable for evaluation with respect to bases whose elements have no sign changes

a weyl group for the virasoro and n 1 super virasoro algebras we introduce a weyl group for the highest weight modules over the virasoro algebra and the neveu schwarz and ramond superalgebras using this group we rewrite the character formulae for the irreducible highest weight modules over these algebras in the form of the classical weyl character formula for the finite dimensional irreducible representations of semi simple lie algebras and also of the weyl kac character formula for the integrable highest weight modules over affine kac moody algebras this is the same group we introduced recently in order to rewrite in a similar manner the characters of the singular highest weight modules over the affine kac moody algebraa 1 1

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

running relative error for the evaluation of polynomials running relative error for evaluating polynomials is analyzed and applied to different evaluation algorithms horner vs de casteljau and clenshaw algorithms numerical experiments show that the derived running relative error bounds are sharp and that the de casteljau algorithm presents better stability properties than the remaining algorithms

on a class of weak tchebycheff systems in this paper we study the approximation power the existence of a normalized b basis and the structure of a degree raising process for spaces of the form requiring suitable assumptions on the functions u and v the results about degree raising are detailed for special spaces of this form which have been recently introduced in the area of cagd

on the optimal stability of the bernstein basis we show that the bernstein polynomial basis on a given interval is optimally stable in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or roots of arbitrary polynomials on that interval this result follows from a partial ordering of the set of all nonnegative bases that is induced by nonnegative basis transformations we further show by means of some low degree examples that the bernstein form is not uniquely optimal in this respect however it is the only optimally stable basis whose elements have no roots on the interior of the chosen interval these ideas are illustrated by comparing the stability properties of the power bernstein and generalized ball bases

an optimal quadtree based motion estimation and motion compensated interpolation scheme for video compression we propose an optimal quadtree qt based motion estimator for video compression it is optimal in the sense that for a given bit budget for encoding the displacement vector field dvf and the qt segmentation the scheme finds a dvf and a qt segmentation which minimizes the energy of the resulting displaced frame difference dfd we find the optimal qt decomposition and the optimal dvf jointly using the lagrangian multiplier method and a multilevel dynamic program we introduce a new very fast convex search for the optimal lagrangian multiplier spl lambda which results in a very fast convergence of the lagrangian multiplier method the resulting dvf is spatially inhomogeneous since large blocks are used in areas with simple motion and small blocks in areas with complex motion we also propose a novel motion compensated interpolation scheme which uses the same mathematical tools developed for the qt based motion estimator one of the advantages of this scheme is the globally optimal control of the tradeoff between the interpolation error energy and the dvf smoothness another advantage is that no interpolation of the dvf is required since we directly estimate the dvf and the qt segmentation for the frame which needs to be interpolated we present results with the proposed qt based motion estimator which show that for the same dfd energy the proposed estimator uses about 25 fewer bits than the commonly used block matching algorithm we also experimentally compare the interpolated frames using the proposed motion compensated interpolation scheme with the reconstructed original frames

borel subalgebras and categories of highest weight modules for toroidal lie algebras abstract in this article we begin an investigation of the conjugacy classes of borel subalgebras together with verma modules induced from standard borel subalgebras of a toroidal lie algebra t in two variables we define for each highest weight a category o of representations of t that contain these verma modules and show that when a certain central element acts nontrivially this category is equivalent to the category o m for an extension of a suitable affine kac moody algebra m t from this equivalence we obtain a bgg type resolution and bgg duality theorem in the setting of o

progressive coding of a gaussian source using matching pursuit in this paper the application of matching pursuit mp in progressive coding of memoryless gaussian sources is studied in addition a detailed analysis of the rate distortion performance of our proposed mp encoder is presented and the distortion of the mp encoder is derived in terms of dictionary size and number of quantization levels and the optimum parameters are calculated our analysis is based on a probabilistic model for matching pursuit residual vectors our simulation results verify the accuracy of our analysis and show that matching pursuit can produce an embedded bitstream with comparable quality to existing quantizers the mp encoder outperforms quantizers that are capable of producing embedded bitstreams

adaptive rate distortion optimal in loop quantization for matching pursuit in this paper an adaptive in loop quantization technique is proposed for quantizing inner product coefficients in matching pursuit for each matching pursuit mp stage a different quantizer is used based on the probability distribution of mp coefficients the quantizers are optimized for a given rate budget constraint additionally our proposed adaptive quantization scheme finds the optimal quantizers for each stage based on the already encoded inner product coefficients experimental results show that our proposed adaptive quantization scheme outperforms existing quantization methods used in matching pursuit image coding

sparse identification of nonlinear functions and parametric set membership optimality analysis sparse identification can be relevant in the automatic control field to solve several problems for nonlinear systems such as identification control filtering fault detection however identifying a maximally sparse approximation of a nonlinear function is in general an np hard problem the common approach is to use relaxed or greedy algorithms that under certain conditions can provide sparsest solutions in this technical note a combined l1 relaxed greedy algorithm is proposed and conditions are given under which the approximation derived by the algorithm is a sparsest one differently from other conditions available in the literature the ones provided here can be actually verified for any choice of the basis functions defining the sparse approximation a set membership analysis is also carried out assuming that the function to approximate is a linear combination of unknown basis functions belonging to a known set of functions it is shown that the algorithm is able to exactly select the basis functions which define the unknown function and to provide an optimal estimate of their coefficients

on the optimal stability of bases of univariate functions this paper is concerned with bases of finite dimensional spaces of univariate continuous functions which are optimally stable for evaluation the only bases considered are those whose elements have no sign changes among these an optimally stable basis is characterized under the assumption that the set of points where each basis function is nonzero is an interval a uniqueness result and many examples of such optimally stable bases are also provided

geometry based distributed scene representation with omnidirectional vision sensors this paper addresses the problem of efficient representation of scenes captured by distributed omnidirectional vision sensors we propose a novel geometric model to describe the correlation between different views of a 3 d scene we first approximate the camera images by sparse expansions over a dictionary of geometric atoms since the most important visual features are likely to be equivalently dominant in images from multiple cameras we model the correlation between corresponding features in different views by local geometric transforms for the particular case of omnidirectional images we define the multiview transforms between corresponding features based on shape and epipolar geometry constraints we apply this geometric framework in the design of a distributed coding scheme with side information which builds an efficient representation of the scene without communication between cameras the wyner ziv encoder partitions the dictionary into cosets of dissimilar atoms with respect to shape and position in the image the joint decoder then determines pairwise correspondences between atoms in the reference image and atoms in the cosets of the wyner ziv image in order to identify the most likely atoms to decode under epipolar geometry constraints experiments demonstrate that the proposed method leads to reliable estimation of the geometric transforms between views in particular the distributed coding scheme offers similar rate distortion performance as joint encoding at low bit rate and outperforms methods based on independent decoding of the different images

best wavelet packet bases in a rate distortion sense a fast rate distortion r d optimal scheme for coding adaptive trees whose individual nodes spawn descendents forming a disjoint and complete basis cover for the space spanned by their parent nodes is presented the scheme guarantees operation on the convex hull of the operational r d curve and uses a fast dynamic programing pruning algorithm to markedly reduce computational complexity applications for this coding technique include r coefman et al s yale univ 1990 generalized multiresolution wavelet packet decomposition iterative subband coders and quadtree structures applications to image processing involving wavelet packets as well as discrete cosine transform dct quadtrees are presented

a posteriori quantization of progressive matching pursuit streams this paper proposes a rate distortion optimal a posteriori quantization scheme for matching pursuit mp coefficients the a posteriori quantization applies to an mp expansion that has been generated offline and cannot benefit of any feedback loop to the encoder in order to compensate for the quantization noise the redundancy of the mp dictionary provides an indicator of the relative importance of coefficients and atom indices and subsequently on the quantization error it is used to define a universal upper bound on the decay of the coefficients sorted in decreasing order of magnitude a new quantization scheme is then derived where this bound is used as an oracle for the design of an optimal a posteriori quantizer the latter turns the exponentially distributed coefficient entropy constrained quantization problem into a simple uniform quantization problem using simulations with random dictionaries we show that the proposed exponentially upper bounded quantization euq clearly outperforms classical schemes stepping on the ideal oracle based approach a suboptimal adaptive scheme is then designed that approximates the euq but still outperforms competing quantization methods in terms of rate distortion characteristics finally the proposed quantization method is studied in the context of image coding it performs similarly to state of the art coding methods and even better at low rates while interestingly providing a progressive stream that is very easy to transcode and adapt to changing rate constraints

the adapted ordering method for lie algebras and superalgebras and their generalizations in 1998 the adapted ordering method was developed for the representation theory of the superconformal algebras in two dimensions it allows us to determine maximal dimensions for a given type of space of singular vectors to identify all singular vectors by only a few coefficients to spot subsingular vectors and to set the basis for constructing embedding diagrams in this paper we present the adapted ordering method for general lie algebras and superalgebras and their generalizations provided they can be triangulated we also review briefly the results obtained for the virasoro algebra and for the n 2 and ramond n 1 superconformal algebras

b splines and optimal stability it is proved that among all nonnegative bases of its space the b spline basis is optimally stable for evaluating spline functions

in loop atom modulus quantization for matching pursuit and its application to video coding this paper provides a precise analytical study of the selection and modulus quantization of matching pursuit mp coefficients we demonstrate that an optimal rate distortion trade off is achieved by selecting the atoms up to a quality dependent threshold and by defining the modulus quantizer in terms of that threshold in doing so we take into account quantization error re injection resulting from inserting the modulus quantizer inside the mp atom computation loop in loop quantization not only improves coding performance but also affects the optimal quantizer design for both uniform and nonuniform quantization we measure the impact of our work in the context of video coding for both uniform and nonuniform quantization the precise understanding of the relation between atom selection and quantization results in significant improvements in terms of coding efficiency at high bitrates the proposed nonuniform quantization scheme results in 0 5 to 2 db improvement over the previous method

filter design with low complexity coefficients we introduce a heuristic for designing filters that have low complexity coefficients as measured by the total number of nonzeros digits in the binary or canonic signed digit csd representations of the filter coefficients while still meeting a set of design specifications such as limits on frequency response magnitude phase and group delay numerical examples show that the method is able to attain very low complexity designs with only modest relaxation of the specifications

characters of irreducible representations of the virasoro algebra on obtient les caracteres des modules irreductibles non integrables de plus haut poids l h 0 on determine les caracteres des familles l t 25 et l t 26 duales de l h 1 et de l h 0

on a conjecture of huber concerning the convergence of projection pursuit regression we generalize the projection pursuit procedure of friedman and stuetzle abstract version and prove strong convergence this answers a question of huber