on optimal analysis synthesis filters for coding gain maximization we consider the use of pre and postfilters in conjunction with m channel uniform band paraunitary orthonormal filter banks we show that given any orthonormal filter bank the pre and postfilters that maximize the coding gain are determined entirely by the power spectrum of the input process regardless of the details of the orthonormal filter blank which could be fir iir or even the ideal brickwall filter bank the optimized coding gain however depends on the prefilter as well as the sandwiched orthonormal filter bank the coding gain improvement due to pre and postfiltering is often significant as we demonstrate with numerical examples and comparisons the validity of our results depends strongly on the orthonormality property of the filter bank in between the pre and postfilters in the nonorthonormal case most of these results are not true as is demonstrated

optimal biorthogonal wavelet bases for signal decomposition the selection of scaling functions for optimal signal representation by general multidimensional biorthogonal wavelet bases is investigated criterion for optimality is the minimization of the mean square approximation error at each level of the decomposition conditions are given under which the approximation error of the decomposition approaches zero as the level increases given arbitrary synthesis filters the optimal corresponding analysis filters are determined globally optimal families of filters are also found and suboptimal linear and nonlinear phase filters for the realization of the optimal scaling functions are explicitly determined

analysis synthesis techniques for subband image coding analysis synthesis systems designed for low bit rate image coding their impact on overall system quality and their computational complexity are discussed the investigation focuses on the design of analysis synthesis systems for image coding and the perceptual impact of these systems at low bit rates two objectives are emphasized in developing these systems confining the total size of the subband images to be equal to the original image size and designing the filters so that perceptual distortion is not introduced by the analysis synthesis system methods based on circular convolution and symmetric extensions are developed and discussed in detail the theory design and implementation of both recursive and nonrecursive filtering systems are discussed methods are introduced which display advantages over conventional quadrature mirror filter based approaches

scalar quantization error analysis for image subband coding using qmfs the authors analyze the coding errors due to quantization by explicitly incorporating a mathematical model for a lloyd max quantizer into a quadrature mirror filter qmf splitting and reconstruction scheme p h westerink et al 1988 this approach explicitly incorporates quantization errors into a qmf system by means of a quantizer model this makes it possible to discriminate between different types of coding errors such as the aliasing error other errors that can be distinguished are a qmf design error a signal error and a random error which is uncorrelated with the original image both a mean squared error calculation and a subjective judgment of the coding errors show that the aliasing errors can be neglected for filter lengths of 12 taps or more the signal error determines the sharpness of the reconstructed image while the random error is most visible in the flat areas

stability and evolution of solitary waves in perturbed generalized nonlinear schrodinger equations in this paper we study the stability and evolution of solitary waves in perturbed generalized nonlinear schrodinger nls equations our method is based on the completeness of the bounded eigenstates of the associated linear operator in l2 space and a standard multiple scale perturbation technique unlike the adiabatic perturbation method our method details all instability mechanisms caused by perturbations of such equations and explicitly specifies when such instabilities will occur in particular our method uncovers the instability caused by bifurcation of nonzero discrete eigenvalues of the linearization operator as an example we consider the perturbed cubic quintic nls equation in detail and determine the stability regions of its solitary waves in the instability region we also specify where the solitary waves decay collapse develop movingfronts or evolve into a stable spatially localized and temporally periodic state the generalization of this method to other perturbed nonlinear wave systems is also discussed

subband coding systems incorporating quantizer models a new method for dealing with the effects of quantization in a subband system is proposed it uses the gain plus additive noise linear model for the lloyd max quantizer based on this it is demonstrated how by an appropriate choice of synthesis filters one can cancel all signal dependent errors at the output of the system the only remaining error is random in nature and not correlated with the input signal we therefore have a tradeoff between the error being only random or having signal dependent components as well since the error variances in both cases are comparable as a result of having only a random error it is possible to reduce this error using for example a noise removal technique the result is then extended to the case where the input is a multidimensional signal and arbitrary sampling lattices are used as well as to the qmf alias cancellation case to demonstrate the validity of the proposed approach two types of experiments on images are carried out in a toy example it is shown that using noise removal could be beneficial for a more realistic coding scheme however it is demonstrated that even in the case when the model is no longer valid when some of the subbands are discarded the output error is still much less correlated with the input signal as opposed to the commonly used subband system while visually the reconstructed images look very similar

random matrices and complexity of spin glasses cern y abstract we give an asymptotic evaluation of the complexity of spherical p spin spin glass models via random matrix theory this study enables us to obtain detailed infor mation about the bottom of the energy landscape including the absolute minimum the ground state the other local minima and describe an interesting layered structure of the low critical values for the hamiltonians of these models we also show that our ap proach allows us to compute the related tap complexity and extend the results known in the physics literature as an independent tool we prove a ldp for the k th largest eigenvalue of the goe extending the results of bdg01 how many critical values of given index and below a given level does a typical random morse function have on a high dimensional manifold our work addresses this question in a very special case we look at certain natural random gaussian functions on the n dimensional sphere known as p spin spherical spin glass models we cannot yet answer the question above about the typical number but we can study thoroughly the mean number which we show is exponentially large in n we introduce a new identity based on the classical kac rice formula relating random matrix theory and the problem of counting these critical values using this identity and tools from random matrix theory we give an asymptotic evaluation of the complexity of these spherical spin glass models the complexity mentioned here is defined as the mean number of critical points of given index whose value is below or above a given level this includes the important question of counting the mean number of local minima below a given level and in particular the question of finding the ground state energy the minimal value of the hamiltonian we show that this question is directly related to the study of the edge of the spectrum of the gaussian orthogonal ensemble goe the question of computing the complexity of mean field spin glass models has recently been thoroughly studied in the physics literature see for example clr03 and the refer ences therein mainly for a different measure of the complexity i e the mean number of solutions to the thouless anderson palmer equations or tap complexity our approach to the complexity enables us to recover known results in the physics literature about tap complexity to compute the ground state energy when p is even and to describe an interesting layered structure of the low energy levels of the hamiltonians of these models which might prove useful for the study of the metastability of langevin dynamics for these models in longer time scales than those studied in bdg01 the paper is organised as follows in section 2 we give our main results in section 3 we prove two main formulas theorem 2 1 and 2 2 relating random matrix theory specif ically the goe and spherical spin glasses these formulas are direct consequences of the kac rice formula we learned the version needed here in the book at07 for another modern account see aw09 the main new ingredient is the fact that for spherical spin glass models the hessian of the hamiltonian at a critical point conditioned on the value of the hamiltonian is a symmetric gaussian random matrix with independent entries up to symmetry plus a diagonal matrix this implies in particular that it is possible to

free energy and complexity of spherical bipartite models we investigate both free energy and complexity of the spherical bipartite spin glass model we first prove a variational formula in high temperature for the limiting free energy based on the well known crisanti sommers representation of the mixed p spin spherical model next we show that the mean number of local minima at low levels of energy is exponentially large in the size of the system and we derive a bound on the location of the ground state energy

algebraic solitary wave solutions of a nonlinear schrodinger equation we present algebraic solitary wave solutions of a nonlinear schroedinger equation that includes two terms with power law nonlinearity both bright and dark types are found numerical stability analyses show that the solitary wave exhibits a solitonic feature

neural networks retrieving boolean patterns in a sea of gaussian ones restricted boltzmann machines are key tools in machine learning and are described by the energy function of bipartite spin glasses from a statistical mechanical perspective they share the same gibbs measure of hopfield networks for associative memory in this equivalence weights in the former play as patterns in the latter as boltzmann machines usually require real weights to be trained with gradient descent like methods while hopfield networks typically store binary patterns to be able to retrieve the investigation of a mixed hebbian network equipped with both real e g gaussian and discrete e g boolean patterns naturally arises we prove that in the challenging regime of a high storage of real patterns where retrieval is forbidden an additional load of boolean patterns can still be retrieved as long as the ratio between the overall load and the network size does not exceed a critical threshold that turns out to be the same of the standard amit gutfreund sompolinsky theory assuming replica symmetry we study the case of a low load of boolean patterns combining the stochastic stability and hamilton jacobi interpolating techniques the result can be extended to the high load by a non rigorous but standard replica computation argument

optimal construction of subband coders using lloyd max quantizers a new method is presented for the analysis of the effects of lloyd max quantization in subband filterbanks and for the optimal design of such filterbanks a rigorous statistical model of a vector lloyd max quantizer is established first consisting of a linear time invariant filter followed by additive noise uncorrelated with the input on the basis of this model an expression for this variance of the error of a subband coder using lloyd max quantizers is explicitly determined given analysis filters that statistically separate the subbands it is shown that this variance is minimized if the synthesis filters are chosen which mould achieve perfect reconstruction in lossless coding the globally optimum of such a filterbank minimizing the coder error variance is further obtained by proper choice of its analysis filters an alternative design method is also evaluated and optimized in this the errors correlated with the signal are set to zero leaving a random error residue uncorrelated with the signal this design method is optimized by choosing the analysis filters so as to minimize the random error variance the results are evaluated experimentally in the realistic setting of a logarithmically split subband image coding scheme

theory of non kerr law generalized vector solitons the soliton perturbation theory is used to obtain the adiabatic parameter dynamics of the perturbed vector solitons that are governed by the generalized non linear schrodinger s equation the perturbation terms considered in this paper are of dispersive dissipative as well of non local type

complexity of random smooth functions on the high dimensional sphere we analyze the landscape of general smooth gaussian functions on the sphere in dimension n when n is large we give an explicit formula for the asymptotic complexity of the mean number of critical points of finite and diverging index at any level of energy and for the mean euler characteristic of level sets we then find two possible scenarios for the bottom landscape one that has a layered structure of critical values and a strong correlation between indexes and critical values and another where even at levels below the limiting ground state energy the mean number of local minima is exponentially large we end the paper by discussing how these results can be interpreted in the language of spin glasses models

hierarchical partition priority wavelet image compression image compression methods for progressive transmission using optimal hierarchical decomposition partition priority coding ppc and multiple distribution entropy coding mdec are presented in the proposed coder a hierarchical subband wavelet decomposition transforms the original image the analysis filter banks are selected to maximize the reproduction fidelity in each stage of progressive image transmission an efficient triple state differential pulse code modulation dpcm method is applied to the smoothed subband coefficients and the corresponding prediction error is lloyd max quantized such a quantizer is also designed to fit the characteristics of the detail transform coefficients in each subband which are then coded using novel hierarchical ppc hppc and predictive hppc phppc algorithms more specifically given a suitable partitioning of their absolute range the quantized detail coefficients are ordered based on both their decomposition level and partition and then are coded along with the corresponding address map space filling scanning further reduces the coding cost by providing a highly spatially correlated address map of the coefficients in each ppc partition finally adaptive mdec is applied to both the dpcm and hppc phppc outputs by considering a division of the source quantized coefficients into multiple subsources and adaptive arithmetic coding based on their corresponding histograms experimental results demonstrate the great performance of the proposed compression methods

free energy of the spherical mean field model we compute at any temperature the free energy of the multi p spin spherical model when only terms for p even are considered

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

aging of spherical spin glasses sompolinski and zippelius 1981 propose the study of dynamical systems whose invariant measures are the gibbs measures for hard to analyze statistical physics models of interest in the course of doing so physicists often report of an aging phenomenon for example aging is expected to happen for the sherrington kirkpatrick model a disordered mean field model with a very complex phase transition in equilibrium at low temperature we shall study the langevin dynamics for a simplified spherical version of this model the induced rotational symmetry of the spherical model reduces the dynamics in question to an n dimensional coupled system of ornstein uhlenbeck processes whose random drift parameters are the eigenvalues of certain random matrices we obtain the limiting dynamics for n approaching infinity and by analyzing its long time behavior explain what is aging mathematically speaking what causes this phenomenon and what is its relationship with the phase transition of the corresponding equilibrium invariant measures

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

phase diagram of restricted boltzmann machines and generalized hopfield networks with arbitrary priors restricted boltzmann machines are described by the gibbs measure of a bipartite spin glass which in turn can be seen as a generalized hopfield network this equivalence allows us to characterize the state of these systems in terms of their retrieval capabilities both at low and high load of pure states we study the paramagnetic spin glass and the spin glass retrieval phase transitions as the pattern i e weight distribution and spin i e unit priors vary smoothly from gaussian real variables to boolean discrete variables our analysis shows that the presence of a retrieval phase is robust and not peculiar to the standard hopfield model with boolean patterns the retrieval region becomes larger when the pattern entries and retrieval units get more peaked and conversely when the hidden units acquire a broader prior and therefore have a stronger response to high fields moreover at low load retrieval always exists below some critical temperature for every pattern distribution ranging from the boolean to the gaussian case

hamiltonian versus energy diagrams in soliton theory parametric curves featuring hamiltonian versus energy are useful in the theory of solitons in conservative nonintegrable systems with local nonlinearities these curves can be constructed in various ways we show here that it is possible to find the hamiltonian h and energy q for solitons of non kerr law media with local nonlinearities without specific knowledge of the functional form of the soliton itself more importantly we show that the stability criterion for solitons can be formulated in terms of h and q only this allows us to derive all the essential properties of solitons based only on the concavity of the curve h vs q we give examples of these curves for various nonlinearity laws and show that they confirm the general principle we also show that solitons of an unstable branch can transform into solitons of a stable branch by emitting small amplitude waves as a result we show that simple dynamics like the transformation of a soliton of an unstable branch into a soliton of a stable branch can also be predicted from the h q diagram s1063 651x 99 09805 0

stability of solitons described by nonlinear schrodinger type equations with higher order dispersion abstract steady solitons described by the fourth order nonlinear schrodinger type equations in one two and three dimensions with power law nonlinearities are studied conditions of existence and stability of such solitons are found numerical results demonstrating soliton existence stability and instability are described they agree with the developed theory

equilibrium statistical mechanics of bipartite spin systems the aim of this paper is to give an extensive treatment of bipartite mean field spin systems pure and disordered at first bipartite ferromagnets are investigated and an explicit expression for the free energy is achieved through a new minimax variational principle then via the hamilton jacobi technique the same structure of the free energy is obtained together with the existence of its thermodynamic limit and the minimax principle is connected to a standard max one the same is investigated for bipartite spin glasses by the borel cantelli lemma we obtain the control of the high temperature regime while via the double stochastic stability technique we also obtain the explicit expression of the free energy in the replica symmetric approximation uniquely defined by a minimax variational principle again we also obtain a general result that states that the free energies of these systems are convex linear combinations of their independent one party model counterparts for the sake of completeness we show further that at zero temperature the replica symmetric entropy becomes negative and consequently such a symmetry must be broken the treatment of the fully broken replica symmetry case is deferred to a forthcoming paper as a first step in this direction we start deriving the linear and quadratic constraints to overlap fluctuations