new perspectives and improvements on the symmetric extension filter bank for subband wavelet image compression in subband wavelet image coding size limited subband decompositions are ordinarily used to avoid increasing the number of samples that need to be coded to reduce coding distortions that can occur at the borders the symmetric extension filter bank is typically employed this paper introduces some new perspectives and improvements to that decomposition the symmetric extension filter bank is couched in the cyclic frequency domain providing a framework that accommodates fir and iir filters in a natural way all with perfect reconstruction iir filters with both rational and irrational transfer functions can be implemented and in the context of symmetric extension can accommodate iirs that effectively have perfect stopband suppression enhancements to the filter bank at a tree structured system level are also presented and include the application of spectral reversal correction and a transition band normalization approach to designing the constituent filters of the symmetric extension wavelet packet transform

a filter bank for the directional decomposition of images theory and design the authors introduce a directionally oriented 2 d filter bank with the property that the individual channels may be critically sampled without loss of information the passband regions of the component filters are wedge shaped and thus provide directional information it is shown that these filter bank outputs may be maximally decimated to achieve a minimum sample representation in a way that permits the original signal to be exactly reconstructed the authors discuss the theory for directional decomposition and reconstruction in addition implementation issues are addressed where realizations based on both recursive and nonrecursive filters are considered

handling borders in systolic architectures for the 1 d discrete wavelet transform for perfect reconstruction we propose a modified systolic architecture that implements the 1 d discrete wavelet transform dwt on the basis of the recursive pyramid algorithm rpa while correctly managing the border problem and obtaining perfect reconstruction all the architectures so far described in the literature do not explicitly address the handling of borders and usually assume a zero padding extension the rpa makes heavy use of this assumption which produces very efficient systolic architectures it is however well known that the zero padding extension does not allow perfect recovery of the original signal if we compute exactly n coefficients more coefficients are required either in the forward or in the inverse transform and their production within the rpa scheme lowers the efficiency of the systolic architecture to a minimum we introduce a modified rpa working on an extended transform with a compressed schedule that restores good values of efficiency ec rpa we also propose a second version of rpa based on a periodic extension of the signal and of the transform pe rpa that achieves perfect reconstruction by computing exactly n dwt coefficients a reduced version of the second algorithm r pe rpa is tailored to the efficient computations of a fixed number of levels in the transform the vlsi complexity of the management of borders is analyzed through synthesis from a vhdl description of the three algorithms synthesis results show that the controller required for pe rpa can be as large as a ten tap array r pe rpa instead requires half this area in both cases the controller complexity scales sublinearly with the length of the input

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

joint space frequency segmentation using balanced wavelet packet trees for least cost image representation we examine the question of how to choose a space varying filterbank tree representation that minimizes some additive cost function for an image the idea is that for a particular cost function e g energy compaction or quantization distortion some tree structures perform better than others while the wavelet tree represents a good choice for many signals it is generally outperformed by the best tree from the library of wavelet packet frequency selective trees the double tree library of bases performs better still by allowing different wavelet packet trees over all binary spatial segments of the image we build on this foundation and present efficient new pruning algorithms for both one and two dimensional 1 d and 2 d trees that will find the best basis from a library that is many times larger than the library of the single tree or double tree algorithms the augmentation of the library of bases overcomes the constrained nature of the spatial variation in the double tree bases and is a significant enhancement in practice use of these algorithms to select the least cost expansion for images with a rate distortion cost function gives a very effective signal adaptive compression scheme this scheme is universal in the sense that without assuming a model for the signal or making use of training data it performs very well over a large class of signal types in experiments it achieves compression rates that are competitive with the best training based schemes

on the implementation of nonseparable two dimensional haar wavelet transforms we introduce a transformation to localize the equations defining the successive levels of the mallat 1989 pyramid for two dimensional 2 d haar wavelets we propose a methodology for implementing these wavelet transforms in parallel architectures like systolic arrays more specifically we show that there is a perfect match between the wavelet algorithms and the multiphase multirate array mpra architectures

symmetric extension compatible reversible integer to integer wavelet transforms symmetric extension is explored as a means for constructing nonexpansive reversible integer to integer iti wavelet transforms for finite length signals two families of reversible iti wavelet transforms are introduced and their constituent transforms are shown to be compatible with symmetric extension one of these families is then studied in detail and several interesting results concerning its member transforms are presented in addition some new reversible iti structures are derived that are useful in conjunction with techniques like symmetric extension last the relationship between symmetric extension and per lifting step extension is explored and some new results are obtained in this regard

generation of a section of conjugation classes and lyndon word tree of limited length resume etant donne un alphabet ordonne un mot de lyndon est un mot primitif qui est le plus petit dans sa classe de conjugaison nous donnons ici un algorithme qui construit dans l ordre la liste des mots de lyndon de longueur donnee cet algorithme est optimal en ce sens que le calcul du mot de lyndon suivant de la liste se fait en temps lineaire a partir du mot precedent et sans utilisation d une memoire auxiliaire cet algorithme trouve son application dans la construction d une section des classes de conjugaison d une longueur donnee c est une response a une question posee a l auter par m p schutzenberger

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

general analysis of two band qmf banks the two channel perfect reconstruction quadrature mirror filter banks pr qmf banks are analyzed in detail by assuming arbitrary analysis and synthesis filters solutions where the filters are fir or iir correspond to the fact that a certain function is monomial or nonmonomial respectively for the monomial case the design problem is formulated as a nonlinear constrained optimization problem the formulation is quite robust and is able to design various two channel filter banks such as orthogonal and biorthogonal arbitrary delay linear phase filter banks to name a few same formulation is used for causal and stable pr iir filter bank solutions

an entropy coded lattice vector quantizer for transform and subband image coding a lattice based vector quantizer vq and noiseless code are proposed for transform and subband image coding the quantization is simple to implement and no vector codebooks need to be stored the noiseless code enumerates lattice codevectors based on their weighted l sub 1 norm a software implementation is able to handle lattice codebooks of size 2 sup 256 the image coding performance is shown to be comparable or superior to the best encoding methods reported in the literature

a new approach to the design of critically sampled m channel uniform band perfect reconstruction linear phase fir filter banks a new approach is presented for the design of uniform band m channel perfect reconstruction pr fir filter banks employing linear phase analysis and synthesis filters the technique designs on the impulse responses of the analysis filters directly the design problem is formulated as an optimization program the filter bank s pr feature can either be implicitly enforced through a set of mathematical relationships among the analysis filters coefficients or through a set of constraints in the optimization program the former approach results in a filter bank whose pr feature s dependency on hardware and software is eliminated or at least minimized the synthesis filters are then obtained by a set of relationships that describe each synthesis filter as a function of the analysis filters the criterion for optimality is least squares where the square of the difference between the ideal and actual frequency responses is integrated over the appropriate frequency bands for all m analysis filters and minimized

average cost of duval s algorithm for generating lyndon words the average cost of duval s algorithm for generating all lyndon words up to a given length in lexicographic order is proved to be asymptotically equal to q 1 q 1 where q is the size of the underlying alphabet in particular the average cost is independent of the length of the words generated a precise evaluation of the constants is also given

stochastic versus possibilistic programming abstract we consider a linear program with uncertain parameters which are modeled as either random variables or as fuzzy variables we first construct the distribution of the optimal valve of the objective function to show a major difference between stochastic and possibilistic programming we then introduce a new solution concept where both programming problems are modeled as multiobjective games against nature

filters for distortion free two band multirate filter banks in this paper conditions are given for a two band multi rate filter bank to be alias free and to have a unity frequency response it is shown that the class of quadrature mirror filters qmf s that satisfies these conditions is quite limited a class of filters which does satisfy these conditions is given and a simple procedure for designing filters from this class is presented with an example

matched sampling systems relation to wavelets and implementation using prcc filter banks this paper deals with two types of sampling systems namely the interpolation and approximation sampling systems closed form expressions are derived for the frequency responses of the filters used in these systems that are matched to the input process in the mean squared sense closed form expressions are also derived for the mean squared error between the input and the reconstructed processes for these matched sampling systems using these expressions it is shown that the meyer scaling function and wavelet or functions derived from these arise naturally in the context of subsampled bandlimited processes to implement these systems the perfect reconstruction circular convolution prcc filter bank is proposed as a framework for the frequency sampled implementation of these systems examples of matched interpolation and approximation sampling systems are provided and their performance is compared with some standard interpolators to demonstrate their efficacy

on using a cuts to evaluate fuzzy equations abstract consider an equation y f x1 xn where the extension principle is used to find y f x 1 x n when we subtitute fuzzy numbers x i for the xi 1 i n if x i denotes an cut of x i then we argue that for only certain simple equations will one obtain y equal to f x 1 x n where f is evaluated using interval arithmetic

the recursive pyramid algorithm for the discrete wavelet transform the recursive pyramid algorithm rpa is a reformulation of the classical pyramid algorithm pa for computing the discrete wavelet transform dwt the rpa computes the n point dwt in real time running dwt using just l log n spl minus 1 words of storage as compared with o n words required by the pa l is the length of the wavelet filter the rpa is combined with the short length fir filter algorithms to reduce the number of multiplications and additions

using fuzzy bases to resolve nonlinear programming problems in this paper we introduce the concept of fuzzy bases and its usefulness in solving optimization problems with a nonlinear objective function and linear constraints we investigate the properties of fuzzy bases and operationalize them in fuzzy interpolation the nlp can be relaxed into a bilinear program with a simple structure using fuzzy interpolation irrespective of whether the objective function is convex or not if the objective function is convex we prove that the optimization problem can be transformed into an ordinary lp using fuzzy linear bases

constant time generation of free trees an algorithm of beyer and hedetniemi siam j comput 9 1980 pp 706 712 for generating rooted unlabeled trees is extended to generate unlabeled free trees all the nonisomorphic trees of a given size are generated without repetition in time proportional to the number of trees

neural net solutions to fuzzy problems the quadratic equation abstract this paper continues our research on using neural nets to solve fuzzy equations after outlining our general method of employing neural nets to solve fuzzy problems we concentrate on solving the fuzzy quadratic equation we show how neural nets can produce both real and complex fuzzy number solutions

orthogonal time varying filter banks and wavelet packets considers the construction of orthogonal time varying filter banks by examining the time domain description of the two channel orthogonal filter bank the authors find it possible to construct a set of orthogonal boundary filters which allows to apply the filter bank to one sided or finite length signals without redundancy or distortion the method is constructive and complete there is a whole space of orthogonal boundary solutions and there is considerable freedom for optimization this may be used to generate subband tree structures where the tree varies over time and to change between different filter sets the authors also show that the iteration of discrete time time varying filter banks gives continuous time bases just as in the stationary case this gives rise to wavelet or wavelet packet bases for half line and interval regions

fast algorithms for discrete and continuous wavelet transforms several algorithms are reviewed for computing various types of wavelet transforms the mallat algorithm 1989 the a trous algorithm and their generalizations by shensa the goal of this work is to develop guidelines for implementing discrete and continuous wavelet transforms efficiently and to compare the various algorithms obtained and give an idea of possible gains by providing operation counts most wavelet transform algorithms compute sampled coefficients of the continuous wavelet transform using the filter bank structure of the discrete wavelet transform although this general method is already efficient it is shown that noticeable computational savings can be obtained by applying known fast convolution techniques such as the fft fast fourier transform in a suitable manner the modified algorithms are termed fast because of their ability to reduce the computational complexity per computed coefficient from l to log l within a small constant factor for large filter lengths l for short filters smaller gains are obtained fast running fir finite impulse response filtering techniques allow one to achieve typically 30 savings in computations

the genlot generalized linear phase lapped orthogonal transform the general factorization of a linear phase paraunitary filter bank lppufb is revisited from this new perspective a class of lapped orthogonal transforms with extended overlap generalized linear phase lapped orthogonal transforms genlots is developed as a subclass of the general class of lppufb in this formulation the discrete cosine transform dct is the order 1 genlot the lapped orthogonal transform is the order 2 genlot and so on for any filter length that is an integer multiple of the block size the genlots are based on the dct and have fast implementation algorithms the implementation of genlots is explained including the method to process finite length signals the degrees of freedom in the design of genlots are described and design examples are presented along with image compression tests

fuzzy hierarchical analysis revisited abstract we present a new method of finding the fuzzy weights in fuzzy hierarchical analysis which is the direct fuzzification of the original method used by saaty in the analytic hierarchy process we test our new procedure in two cases where there are formulas for the crisp weights an example is presented where there are five criteria and three alternatives

lagrange multiplier approaches to the design of two channel perfect reconstruction linear phase fir filter banks the authors present two approaches to the design of two channel perfect reconstruction linear phase finite impulse response fir filter banks both approaches analyze and design the impulse responses of the analysis filter bank directly the synthesis filter bank is then obtained by simply changing the signs of odd order coefficients in the analysis filter bank the approach deals with unequal length filter banks by designing the lower length filters first one can take advantage of the fact that the number of variables for designing the higher length filters is more than the number of perfect reconstruction constraint equations the second approach generalizes the first and covers the design for all parts of linear phase perfect reconstruction constraint equations

a fuzzy linear basis algorithm for nonlinear separable programming problems in the paper we develop a theory of fuzzy linear bases the theory is useful for transforming nonlinear separable programming problems nlsp into a finite sequence of fuzzy linear programming relaxations at a given level of accuracy e the key concepts of the theory are fuzzy linear interpolation and the maximal profile of the polyhedron generated from a set of break points for each variable dimension the maximal profile is divided into adjacent convex sub intervals in which the nonlinear problem is transformed into a sequence of fuzzy linear sub problems all discontinuities are equipped with a break point whereby the fuzzy linear basis flb algorithm is applicable to separable nlps with a finite number of discontinuities we prove that the solution to the original nonlinear problem is included in the sequence of fuzzy linear sub problems at the prespecified accuracy e the principles of the fuzzy linear basis algorithm are illustrated in an example

a class of fuzzy random optimization expected value models fuzzy random variable is a measurable mapping from a probability space to a collection of fuzzy variables this concept may be regard as an extension of both random variable and fuzzy variable in this paper the linearity of a scalar value expected value operator of fuzzy random variable is discussed and a fuzzy random simulation approach is suggested to evaluate the expected value of a fuzzy random variable in addition three types of fuzzy random expected value models are presented to model fuzzy random decision systems moreover a hybrid intelligent algorithm which incorporates simulation neural network and genetic algorithm is designed in order to solve general fuzzy random expected value models at the end of this paper the effectiveness of this algorithm is showed via three numerical examples

theory and design of dyadic subband tree structures with embedded pdf optimized quantizers we develop a methodology for the analysis of signal quantization effects in critically sampled dyadic subband tree structures using a nonlinear gain plus additive noise model for the probability density function pdf optimized quantizer we constrain the two band nonquantized and uncompensated structure at each level to be perfect reconstruction pr we develop an equivalent uniform filter bank followed by its polyphase structure described by primitive submatrices and compute a rigorously correct mean squared error mse in the frequency domain using cyclostationary concepts in terms of 1 the allocated quantizer bits 2 the filter coefficients 3 an embedded compensation parameter vector this mse is then minimized over all three items above our optimization method is applied to the specific case of a four channel dyadic tree with average bit rate constraint this tree is represented by an eight channel polyphase equivalent whose interchannel signals are correlated we show how to represent rigorously the correlation of random noise between channels due to the embedded quantizers our design of paraunitary and biorthogonal structures with identical and nonidentical stages is performed compared and validated by computer simulation under the assumption of uncorrelated cross band noise the nonidentical stage biorthogonal filter bank turned out to have the best performance in mse sense but the most robust structure is the nonidentical stage paraunitary filter bank

possibilities as cumulative subjective probabilities and a norm on the space of congruence classes of fuzzy numbers motivated by an expected utility functional abstract in our prior paper jamison and lodwick 1996 we defined a fuzzy function and discussed the theory for optimizing unconstrained fuzzy functions in that paper we described the need for a measure for comparing different possibility distributions in order to determine the decision makers preference for one distribution over another in this paper we develop one such measure we begin by developing possibility distributions from a set of axioms to show that possibility levels can be viewed as cumulative subjective probabilities we use this development to motivate a norm on the vector space of congruence classes of fuzzy numbers this norm is based on an expected utility functional we show that our normed space is complete if the supports of our fuzzy numbers are contained within a bounded subset of the real line

fuzzy initial value problem for n th order linear differential equations we present two methods of solving nth order linear differential equation that have fuzzy initial conditions for the second order constant coefficient case we show that 1 the first method always gives a solution and 2 the second solution quite often does not exist and when it does exist it may be equal to the solution determined using the first method we apply the first method to three example problems including an electrical circuit and a vibrating mass

optimal pyramidal and subband decompositions for hierarchical coding of noisy and quantized images optimal hierarchical coding is sought for progressive or scalable image transmission by minimizing the variance of the error difference between the original image and its lower resolution renditions the optimal according to the above criterion pyramidal and subband image coders are determined for images subject to corruption by quantization or transmission noise given arbitrary analysis filters and assuming adequate knowledge of the noise statistics optimal synthesis filters are found the optimal analysis filters are subsequently determined leading to formulas for globally optimal structures for pyramidal and subband image decompositions experimental results illustrate the implementation and performance of the optimal coders

subband coding of images subband coding has become quite popular for the source encoding of speech this paper presents a simple yet efficient extension of this concept to the source coding of images we specify the constraints for a set of two dimensional quadrature mirror filters qmf s for a particular frequency domain partition and show that these constraints are satisfied by a separable combination of one dimensional qmf s bits are then optimally allocated among the subbands to minimize the mean squared error for dpcm coding of the subbands also an adaptive technique is developed to allocate the bits within each subband by means of a local variance mask optimum quantization is employed with quantizers matched to the laplacian distribution subband coded images are presented along with their signal to noise ratios snr s the snr performance of the subband coder is compared to that of the adaptive discrete cosine transform dct vector quantization and differential vector quantization for bit rates of 0 67 1 0 and 2 0 bits per pixel for 256 256 monochrome images the adaptive subband coder has the best snr performance

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

complete to overcomplete discrete wavelet transforms theory and applications a new transform is proposed that derives the overcomplete discrete wavelet transform odwt subbands from the critically sampled dwt subbands complete representation this complete to overcomplete dwt codwt has certain advantages in comparison to the conventional approach that performs the inverse dwt to reconstruct the input signal followed by the a spl grave trous or the lowband shift algorithm specifically the computation of the input signal is not required as a result the minimum number of downsampling operations is performed and the use of upsampling is avoided the proposed codwt computes the odwt subbands by using a set of prediction filter matrices and filtering and downsampling operators applied to the dwt this formulation demonstrates a clear separation between the single rate and multirate components of the transform this can be especially significant when the codwt is used in resource constrained environments such as resolution scalable image and video codecs to illustrate the applicability of the proposed transform in these emerging applications a new scheme for the transform calculation is proposed and existing coding techniques that benefit from its usage are surveyed the analysis of the proposed codwt in terms of arithmetic complexity and delay reveals significant gains as compared with the conventional approach

two fast architectures for the direct 2 d discrete wavelet transform we propose two architectures for the direct two dimensional 2 d discrete wavelet transform dwt the first one is based on a modified recursive pyramid algorithm mrpa and performs a nonstandard decomposition i e mallat s 1989 tree of an n spl times n image in approximately 2n sup 2 3 clock cycles ccs this result consistently speeds up other known architectures that commonly need approximately n sup 2 ccs furthermore the proposed architecture is simpler than others in terms of hardware complexity subsequently we show how symmetric anti symmetric properties of linear phase wavelet filter bases can be exploited in order to further reduce the vlsi area this is used to design a second architecture that provides one processing unit for each level of decomposition pipelined approach and performs a decomposition in approximately n sup 2 2 ccs in many practical cases even this architecture is simpler than general mrpa based devices having only one processing unit

solving systems of linear fuzzy equations abstract in this paper we wish to construct solutions for x to the fuzzy matrix equation ax b when the elements in a and b are triangular fuzzy numbers a is square and always non singular we argue that the previous method of solving for x based on the extension principle and regular fuzzy arithmetic should be abandoned since it too often fails to produce a solution we present six new solutions of which we show that five are identical we then adopt the common value x of these five new solutions as our solution to ax b we show that x is a fuzzy vector which is the generalization to rn of real fuzzy numbers we also show how these results pertain to and extend previous research on this problem within the area of interval analysis

the structure of single track gray codes single track gray codes are cyclic gray codes with codewords of length n such that all the n tracks which correspond to the n distinct coordinates of the codewords are cyclic shifts of the first track we investigate the structure of such binary codes and show that there is no such code with 2 sup n codewords when n is a power of 2 this implies that the known codes with 2 sup n 2n codewords when n is a power of 2 are optimal this result is then generalized to codes over gf p where p is a prime a subclass of single track gray codes called single track gray codes with k spaced heads is also defined all known systematic constructions for single track gray codes result in codes from this subclass we investigate this class and show it has a strong connection with two classes of sequences the full order words and the full order self dual words we present an iterative construction for binary single track gray codes which are asymptotically optimal if an infinite family of asymptotically optimal seed codes exists this construction is based on an effective way to generate a large set of distinct necklaces and a merging method for cyclic gray codes based on necklaces representatives

design of iir orthogonal wavelet filter banks using lifting scheme the lifting scheme is well known to be an efficient tool for constructing second generation wavelets and is often used to design a class of biorthogonal wavelet filter banks for its efficiency the lifting implementation has been adopted in the international standard jpeg2000 it is known that the orthogonality of wavelets is an important property for many applications this paper presents how to implement a class of infinite impulse response iir orthogonal wavelet filter banks by using the lifting scheme with two lifting steps it is shown that a class of iir orthogonal wavelet filter banks can be realized by using allpass filters in the lifting steps then the design of the proposed iir orthogonal wavelet filter banks is discussed the designed iir orthogonal wavelet filter banks have approximately linear phase responses finally the proposed iir orthogonal wavelet filter banks are applied to the image compression and then the coding performance of the proposed iir filter banks is evaluated and compared with the conventional wavelet transforms

measuring the satisfaction of constraints in fuzzy linear programming the paper proposes a new kind of method for solving fuzzy linear programming problems based on the satisfaction or fulfillment degree of the constraints using a new ranking method of fuzzy numbers the fulfillment of the constraints can be measured then the properties of the ranking index are discussed with this ranking index the decision maker can make the constraints tight or loose based on his optimistic or pessimistic attitude and get the optimal solution from the fuzzy constraint space the corresponding value of objective distribution function can be obtained a numerical example illustrates the merits of the approach

solving fuzzy equations in economics and finance abstract in this paper we apply our new solution procedure for solving fuzzy equations linear non linear differential to three problems in economics and finance the three applications are 1 leontief s input output model 2 the internal rate of return and 3 a dynamic supply demand model where price and supply are governed by a system of differential equations

solving linear and quadratic fuzzy equations abstract we present necessary and sufficient conditions for some linear and quadratic equations to have a solution when the parameters are either real or complex fuzzy numbers the solution will be a real or complex fuzzy number applications in chemistry economics finance and physics are presented for these types of equations

perfect reconstruction iir digital filter banks supporting nonexpansive linear signal extensions in this paper perfect reconstruction polyphase infinite impulse response iir filter banks involving causal and anticausal inverses are examined for finite length signals first a novel and efficient nonexpansive perfect reconstruction algorithm based on the state space implementation is presented then the proposed method is extended to support linear signal extensions at the boundaries in a nonexpansive manner the powerfulness of the proposed algorithm is demonstrated with the image compression results

possibilistic linear programming with triangular fuzzy numbers abstract we consider a mathematical programming problem where all the parameters may be fuzzy variables specified by their possibility distribution and we define the possibility distribution of the objective function the ideas are illustrated on possibilistic linear programming

m channel nonlinear phase filter banks in image compression structure design and signal extension this paper describes a transform coding technique based on m channel perfect reconstruction filter banks with a nonlinear phase in contrast with linear phase filter banks the nonlinearity provides an extra degree of freedom that can be used to design a more efficient transform we present new lattice structures of paraunitary and perfect reconstruction biorthogonal filter banks which can be implemented with a lower computational complexity and or represented by a few free parameters through the decomposition of the lattice blocks and the displacement across the delay block we further discuss a smooth extension method for nonlinear phase filter banks to obtain a nonexpansive transform the promise of our proposed approaches is demonstrated through several design examples extended signals and compression results

construction of optimal subband coders using optimized and optimal quantizers a method is presented for the optimization of arbitrary quantizers by use of a compensating postfilter it is shown that the resulting optimized quantizers fit the model of a linear time invariant filter followed by additive noise uncorrelated with the input which also characterizes the optimal lloyd max quantizers on the basis of this model an expression for the variance of the error of a subband coder using optimized quantizers is explicitly determined given analysis filters which statistically separate the subbands it is shown that this variance is minimized if these synthesis filters are chosen which would achieve perfect reconstruction in lossless coding the globally optimum filter bank minimizing the coder error variance is further obtained by proper choice of its analysis filters a novel method for the determination of optimal bit allocation to subbands of the filter banks with optimized quantizers is also developed the results are evaluated experimentally by comparison of the optimum uniformly split subband image coding scheme to classical logarithmically split filter bank wavelet coding methods

minimizing unconstrained fuzzy functions in this paper we introduce a definition for a fuzzy function as a possibility distribution over the set of all functions with certain properties we then examine some of the implications of this definition in particular we show that the image of a fuzzy functions is a fuzzy vector as defined herein we also show that the set of values at which a real valued fuzzy function achieves its minimum is a fuzzy number and if the fuzzy function is convex in a particular sense then the set of vectors for which the function achieves the minimum values is connected lastly we discuss minimization of an unconstrained fuzzy function and how one might proceed to defuzzify the solution

possibility and necessity in optimization abstract we employ possibility and necessity in fuzzy programming and in other optimization problems that traditionally use random variables the possibility approach to multiobjective programming produces the same results as fuzzy programming a detailed example is presented to illustrate the use of possibility and necessity in optimization problems that usually employ random variables to describe uncertainty

on the calculation of a membership function for the solution of a fuzzy linear optimization problem in the present paper the fuzzy linear optimization problem with fuzzy coefficients in the objective function is considered recent concepts of fuzzy solution to the fuzzy optimization problem based on the level cut and the set of pareto optimal solutions of a multiobjective optimization problem are applied chanas and kuchta suggested one approach to determine the membership function values of fuzzy optimal solutions of the fuzzy optimization problem which is based on calculating the sum of lengths of certain intervals the purpose of this paper is to determine a method for realizing this idea we derive explicit formulas for the bounds of these intervals in the case of triangular fuzzy numbers and show that only one interval needs to be considered

multiobjective possibilistic linear programming abstract this paper presents two solutions to multiobjective linear programs when all the parameters may be fuzzy variables whose values are restricted by a possibility distribution in the first solution we first find possibility distribution of the objective function and then a solution consists of those values of the decision variables which produce the largest undominated values of the objective function where these values also have maximum possibility in the second solution we first find the possibility distribution of the undominated sets and then a solution consists of those values of the decision variables which are undominated with maximum possibility but also produce the largest undominated values of the objective function for certain multiobjective possibilistic linear programs we show that the two solution concepts are identical

fuzzy programming and the pareto optimal set it is shown how fuzzy programming using either the min or product operator may be used to generate the whole pareto optimal set for nonlinear concave or convex multiobjective programming problems also a new solution procedure for a fuzzy program when the min operator is employed is presented

neural net solutions to fuzzy linear programming we show how a neural net with sign restrictions on its weights can be trained to produce approximate solutions to fuzzy linear programming problems the neural net approximates the joint solution buckley 1995 to the fuzzy linear program

solving fuzzy equations a new solution concept abstract we have previously shown 2 that many fuzzy equations do not have solutions when the solution concept is based on the extension principle we therefore introduce two new solution procedures one based on the unified extension 6 and the other based on possibility theory after we solve the non fuzzy equation for the unknown variable we show for many types of equations that 1 the two new solutions are identical 2 the solution is either a real or generalized complex fuzzy number all uncertain parameters are modeled as real fuzzy numbers and 3 the previous solution based on the extension principle when it exists is a subset of the new solution in particular we show that the fuzzy quadratic equation with real fuzzy number coefficients always has a new solution

extending the scope of wavelet regression methods by coefficient dependent thresholding abstract various aspects of the wavelet approach to nonparametric regression are considered with the overall aim of extending the scope of wavelet techniques to irregularly spaced data to regularly spaced datasets of arbitrary size to heteroscedastic and correlated data and to data that contain outliers the core of the methodology is an algorithm for finding ail of the variances and within level covariances in the wavelet table of a sequence with given covariance structure if the original covariance matrix is band limited then the algorithm is linear in the length of the sequence the variance calculation algorithm allows data on any set of independent variable values to be treated by first interpolating to a fine regular grid of suitable length and then constructing a wavelet expansion of the gridded data various thresholding methods are discussed and investigated exact risk formulas for the mean square error of the methodology for given design are derived good performance is obtained by nois

fuzzy linear programming using a penalty method in this paper we begin with a standard form of the linear programming problem we replace each constant in the problem with a fuzzy number we then reformat the objective and constraints into an unconstrained fuzzy function by penalizing the objective for possible constraint violations the range of this fuzzy function lies in the space of fuzzy numbers the objective is then redefined as optimizing the expected midpoint of the image of this fuzzy function we show that this objective defines a concave function which therefore can be maximized globally we present an algorithm for finding the optimum

solving fuzzy equations using neural nets abstract in this paper we show how a neural net can be used to solve ax c for x even though for some values of a and c there is no fuzzy arithmetic solution for x the neural net solution is identified with our new solution 6 to fuzzy equations

second order statistical characterization of the filter bank and its elements this correspondence proposes an alternative scheme for characterizing the wide sense cyclostationary wscs processes and applies it for the analysis of the filter banks whose output is wscs for wide sense stationary and wscs inputs the proposed representation clearly illustrates the kind of transformation taking place in multiscale stochastic systems these relationships are extended to the filter bank to make clear the effect of the various stages of the filter bank on wscs processes

discrete wavelet transform based on cyclic convolutions a filterbank in which cyclic convolutions are used in place of linear convolutions will be referred to as a cyclic convolution filterbank ccfb a dyadic tree structured ccfb can be used to perform a discrete wavelet transform suitable for coding based on symmetric extension methods this paper derives two types of efficient implementation techniques for the tree structured ccfb one using complex arithmetic and one using only real arithmetic in addition the present paper analyzes in detail the perfect reconstruction pr condition for the two channel ccfb and shows that this condition is much less restrictive than that of usual two channel filterbanks when each of the two channel ccfbs constituting a tree structured ccfb is designed to be pr the whole system is pr a quadrature mirror filter qmf ccfb having the pr property is demonstrated to be easily designed using a standard filter design subroutine in contrast designing a linear phase pr fir qmf bank that has good frequency response is not possible when filters are realized by linear convolutions

a jointly optimized subband coder the mainstream approach to subband coding has been to partition the input signal into subband signals and to code those signals separately with optimal or near optimal quantizers and entropy coders a more effective approach however is one where the subband coders are optimized jointly so that the average distortion introduced by the subband quantizers is minimized subject to a constraint on the output rate of the subband encoder a subband coder with jointly optimized multistage residual quantizers and entropy coders is introduced and applied to image coding the high performance of the coder is attributed to its ability to exploit statistical dependencies within and across the subbands the efficiency of the multistage residual quantization structure and the effectiveness of the statistical modeling algorithm result in an attractive balance among the reproduction quality rate and complexity

solving fuzzy equations we start in the next section 10 2 with looking at possible solutions to the simple fuzzy linear equation overline a cdot overline x overline b overline c we discuss three different types of solution which we have studied before in solving fuzzy equations then we present a fourth type of solution based on our fuzzy monte carlo method in section 10 2 2 this new solution is based on random fuzzy numbers in section 10 3 we look at only classical solutions to the fuzzy quadratic equation and apply our fuzzy monte carlo method to obtain new solutions then in section 10 4 we consider the fuzzy matrix equation overline a cdot overline x overline b and a number of solution types for overline x and then another solution based on fuzzy monte carlo techniques the last section contains a brief summary and our conclusions

the advanced oca for 2 d discrete periodized wavelet transformation we develop a novel computation method for the 2 d discrete periodized wavelet transformation dpwt the new algorithm is based on the operator correlation algorithm oca compared with the classical 2 d pyramid algorithm the new advanced operator correlation algorithm aoca has two major advantages it requires half the number of multiplications and can yield the same output snr with half the precision of the dpwt filter coefficients furthermore the modular structure of the adca makes it particularly suitable for a vlsi implementation

an efficient algorithm for generating necklaces with fixed density a k ary necklace is an equivalence class of k ary strings under rotation a necklace of fixed density is a necklace where the number of zeros is fixed we present a fast simple recursive algorithm for generating i e listing fixed density k ary necklaces or aperiodic necklaces the algorithm is optimal in the sense that it runs in time proportional to the number of necklaces produced

a new framework for complex wavelet transforms although the discrete wavelet transform dwt is a powerful tool for signal and image processing it has three serious disadvantages shift sensitivity poor directionality and lack of phase information to overcome these disadvantages we introduce two stage mapping based complex wavelet transforms that consist of a mapping onto a complex function space followed by a dwt of the complex mapping unlike other popular transforms that also mitigate dwt shortcomings the decoupled implementation of our transforms has two important advantages first the controllable redundancy of the mapping stage offers a balance between degree of shift sensitivity and transform redundancy this allows us to create a directional non redundant complex wavelet transform with potential benefits for image coding systems to the best of our knowledge no other complex wavelet transform is simultaneously directional and non redundant the second advantage of our approach is the flexibility to use any dwt in the transform implementation as an example we can exploit this flexibility to create the complex double density dwt cddwt a shift insensitive directional complex wavelet transform with a low redundancy of 3 sup m 1 2 sup m 1 in m dimensions to the best of our knowledge no other transform achieves all these properties at a lower redundancy

subband coder design incorporating recursive quadrature filters and optimum adpcm coders this paper presents the results of an analytical and experimental study of the use of iir quadrature mirror filters and forward adapting adpcm with optimum quantizers for octave band subband coders for speech it is shown that the underlying analysis synthesis systems can be designed such that there is no interband aliasing distortion in the coded speech and such that the analysis synthesis transfer function has no frequency distortion or no phase distortion but not both the experimental study showed that subband coders based on a mixture of infinite impulse response iir and finite duration impulse response fir filters resulted in higher quality coding systems using fewer multiplies than for systems based on fir filters alone it was also found that a signal to noise ratio snr gain of about 4 db could be obtained using optimum quantizers for forward adapting adpcm coders over linear quantizers in backward adapting apcm codes but no perceptual quality gain was observed

joint solution to fuzzy programming problems we propose a new solution concept for fuzzy programming problems it is based on our new method of solving fuzzy equations 101 for simplicity we discuss in detail only fuzzy linear programming in this paper we define and obtain the basic properties of the joint solution a fuzzy vector in rn and the optimal value of the objective function a fuzzy number three examples are presented illustrating these concepts

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed