multiscale image fusion using the undecimated wavelet transform with spectral factorization and nonorthogonal filter banks multiscale transforms are among the most popular techniques in the field of pixel level image fusion however the fusion performance of these methods often deteriorates for images derived from different sensor modalities in this paper we demonstrate that for such images results can be improved using a novel undecimated wavelet transform uwt based fusion scheme which splits the image decomposition process into two successive filtering operations using spectral factorization of the analysis filters the actual fusion takes place after convolution with the first filter pair its significantly smaller support size leads to the minimization of the unwanted spreading of coefficient values around overlapping image singularities this usually complicates the feature selection process and may lead to the introduction of reconstruction errors in the fused image moreover we will show that the nonsubsampled nature of the uwt allows the design of nonorthogonal filter banks which are more robust to artifacts introduced during fusion additionally improving the obtained results the combination of these techniques leads to a fusion framework which provides clear advantages over traditional multiscale fusion approaches independent of the underlying fusion rule and reduces unwanted side effects such as ringing artifacts in the fused reconstruction

the discrete wavelet transform for a symmetric anti symmetric multiwavelet family on the interval the chui lian multiwavelet family of approximation order three is extended to serve as a basis for an orthogonal discrete multiwavelet transform on the interval this requires in addition to the original symmetric and antisymmetric pairs of scaling functions and wavelets four new functions of each type at each edge of the interval i e one more than required just to preserve the approximation order the criteria are given that such edge functions must satisfy in order to allow multiresolution analysis of data sequences without special constraints on endpoint behavior or use of periodic boundary conditions and a specific choice is examined in applications to both noiseless and noisy data these are accomplished through the development of interval basis extensions for both an earlier multiwavelet projection based prefilter postfilter combination and a multiwavelet vector thresholding algorithm

a multicarrier architecture based upon the affine fourier transform recently innovative multicarrier schemes have been proposed that exploit the transmission of chirp shaped waves e sup j2 spl pi ct2 by optimally choosing the chirp parameter c on the basis of the channel characteristics and are more robust to time varying channels than ordinary ofdm schemes this concept was applied to continuous time and to discrete time systems in the present paper we aim at developing those ideas using the affine fourier transform aft which is a very general formulation of chirp transforms we present a multicarrier modulation based upon the discrete form of the aft that is therefore inherently discrete and strictly invertible moreover it allows to define a circular prefix concept that is coherent with the chirp nature of the transmission the system can be efficiently implemented by adding a simple phase correction block to standard ofdm modulators demodulators and can effectively combat interchannel interference when the propagation channel is made of few multipath components affected by independent frequency offsets our discrete time multicarrier scheme is an improved version of martone s approach as we also show by simulation results and exhibits analogous characteristics to barbarossa s continuous time system

the scale representation the authors considers scale a physical attribute of a signal and develop its properties he presents an operator which represents scale and study its characteristics and representation this allows one to define the scale transform and the energy scale density spectrum which is an indication of the intensity of scale values in a signal he obtains explicit expressions for the mean scale scale bandwidth instantaneous scale and scale group delay furthermore he derives expressions for mean time mean frequency duration frequency bandwidth in terms of the scale variable the short time transform is defined and used to obtain the conditional value of scale for a given time he shows that as the windows narrows one obtains instantaneous scale convolution and correlation theorems for scale are derived a formulation is devised for studying linear scale invariant systems he derives joint representations of time scale and frequency scale general classes for each are presented using the same methodology as for the time frequency case as special cases the joint distributions of marinovich altes 1978 1986 and bertrand bertrand 1984 are recovered also joint representations of the three quantities time frequency scale are devised a general expression for the local scale autocorrelation function is given uncertainty principles for scale and time and scale and frequency are derived

mimo biorthogonal partners and applications multiple input multiple output mimo biorthogonal partners arise in many different contexts one of them being multiwavelet theory they also play a central role in the theory of mimo channel equalization especially with fractionally spaced equalizers in this paper we first derive some theoretical properties of mimo biorthogonal partners we develop conditions for the existence of mimo biorthogonal partners and conditions under which fir solutions are possible in the process of constructing fir mimo biorthogonal partners we exploit the nonuniqueness of the solution this will lead to the design of flexible fractionally spaced mimo zero forcing equalizers the additional flexibility in design makes these equalizers more robust to channel noise finally other situations where mimo biorthogonal partners occur are also considered such as prefiltering in multiwavelet theory and deriving the vector version of the least squares signal projection problem

uniqueness of nash equilibria in a quantum cournot duopoly game a quantum cournot game whose classical form game has multiple nash equilibria is examined although the classical equilibria fail to be pareto optimal the quantum equilibrium exhibits the following two properties i if the measurement of entanglement between strategic variables chosen by the competing firms is sufficiently large the multiplicity of equilibria vanishes and ii the more strongly the strategic variables are entangled the more closely the unique equilibrium approaches to the optimal one

a class of scaled bessel sampling theorems sampling theorems for a class of scaled bessel unitary transforms are presented the derivations are based on the properties of the generalized laguerre functions this class of scaled bessel unitary transforms includes the classical sine and cosine transforms but also novel chirp sine and modified hankel transforms the results for the sine and cosine transform can also be utilized to yield a sampling theorem different from shannon s for the fourier transform

multiwavelet moments and projection prefilters an efficient projection procedure is derived for use of orthogonal multiwavelets in the analysis of discrete data sequences a family of simple prefilters corresponding to numerical quadrature evaluation of the projection integrals provides exact results for locally polynomial data the full approximation order of the multiwavelet basis can thus always be enabled for nonpolynomial signals the prefilters provide approximations to the coefficients of the multiwavelet series whose convergence accelerates quickly with increase in sampling rate comparison is also made with previous time invariant multiwavelet prefilters

theory of regular m band wavelet bases orthonormal m band wavelet bases have been constructed and applied by several authors this paper makes three main contributions first it generalizes the minimal length k regular 2 band wavelets of daubechies 1988 to the m band case by deriving explicit formulas for k regular m band scaling filters several equivalent characterizations of k regularity are given and their significance explained second two approaches to the construction of the m 1 wavelet filters and associated wavelet bases are described one relies on a state space characterization with a novel technique to obtain the unitary wavelet filters the other uses a factorization approach third this paper gives a set of necessary and sufficient condition on the m band scaling filter for it to generate an orthonormal wavelet basis the conditions are very similar to those obtained by cohen 1990 and lawton 1990 for 2 band wavelets

interpolating between periodicity and discreteness through the fractional fourier transform periodicity and discreteness are fourier duals in the same sense as operators such as coordinate multiplication and differentiation and translation and phase shift the fractional fourier transform allows interpolation between such operators which gradually evolve from one member of the dual pair to the other as the fractional order goes from zero to one here we similarly discuss the interpolation between the dual properties of periodicity and discreteness showing how one evolves into the other as the order goes from zero to one we also discuss the concepts of partial discreteness and partial periodicity and relate them to fractional discreteness and periodicity

zur theorie der linearen und nichtlinearen integralgleichungen fredholm hat eine auflosungsformel der inhomogenen linearen integralgleichung f left s right varphi left s right lambda int limits a b k left s t right varphi left t right dt entdeckt welche das resultat enthielt das diese gleichung nach s stets aufgelost werden kann wenn nicht eine der nullstellen einer gewissen ganzen transzendenten ist fur diese werte von in hilbertscher bezeichnung die sogenannten eigenwerte des kernes k s t und nur fur diese last wie fredholm ferner zeigte die homogene gleichung 0 varphi left s right int limits a b k left s t right varphi left t right dt losungen zu die nach hilbert zum betreffenden eigenwert des kernes k s t gehorige eigenfunktionen genannt werden hilbert hat die in der theorie der partiellen und gewohnlichen differentialgleichungen so wichtigen fragen nach der existenz sogenannter normalfunktionen und der entwickelbarkeit willkurlicher funktionen nach ihnen durch einfuhrung der greenschen funktion auf das viel allgemeinere problem zuruckgefuhrt die existenz von eigenfunktionen eines in s und t symmetrischen kernes k s t zu beweisen und die gesetze der entwickelbarkeit willkurlicher funktionen nach ihnen aufzustellen die greensche funktion selbst wird durch eine integralgleichung mit unsymmetrischem kerne bestimmt wobei die fredholmschen formeln zur anwendung kommen

short wavelets and matrix dilation equations scaling functions and orthogonal wavelets are created from the coefficients of a lowpass and highpass filter in a two band orthogonal filter bank for multifilters those coefficients are matrices this gives a new block structure for the filter bank and leads to multiple scaling functions and wavelets geronimo hardin and massopust see j approx theory vol 78 p 373 401 1994 constructed two scaling functions that have extra properties not previously achieved the functions spl phi sub 1 and spl phi sub 2 are symmetric linear phase and they have short support two intervals or less while their translates form an orthogonal family for any single function spl phi apart from haar s piecewise constants those extra properties are known to be impossible the novelty is to introduce 2 spl times 2 matrix coefficients while retaining orthogonality of the multiwavelets this note derives the properties of spl phi sub 1 and spl phi sub 2 from the matrix dilation equation that they satisfy then our main step is to construct associated wavelets two wavelets for two scaling functions the properties were derived by geronimo hardin and massopust from the iterated interpolation that led to spl phi 1 and spl phi sub 2 one pair of wavelets was found earlier by direct solution of the orthogonality conditions using mathematica our construction is in parallel with recent progress by hardin and geronimo to develop the underlying algebra from the matrix coefficients in the dilation equation in another language to build the 4 spl times 4 paraunitary polyphase matrix in the filter bank the short support opens new possibilities for applications of filters and wavelets near boundaries

a unified framework for the fractional fourier transform the paper investigates the possibility for giving a general definition of the fractional fourier transform frt for all signal classes one dimensional 1 d and multidimensional continuous and discrete periodic and aperiodic since the definition is based on the eigenfunctions of the ordinary fourier transform ft the preliminary conditions is that the signal domain periodicity be the same as the ft domain periodicity within these classes a general frt definition is formulated and the frt properties are established in addition the multiplicity which is intrinsic in a fractional operator is clearly developed the general definition is checked in the case in which the frt is presently available and moreover to establish the frt in new classes of signals

downsampling of bounded bandlimited signals and the bandlimited interpolation analytic properties and computability downsampling and the computation of the bandlimited interpolation of discrete time signals are two important concepts in signal processing in this paper we analyze the downsampling operation regarding its impact on the existence and computability of the bounded bandlimited interpolation we assume that the discrete time signal is obtained by downsampling the samples of a bounded bandlimited signal that vanishes at infinity and we study two problems first we investigate the existence of the bounded bandlimited interpolation for such discrete time signals from a signal theoretic perspective and show that there exist signals for which the bounded bandlimited interpolation does not exist second we analyze the algorithmic generation of the bounded bandlimited interpolation using the concept of turing computability turing computability models what is theoretically implementable on a digital computer interestingly it turns out that even if the bounded bandlimited interpolation exists analytically it is not always computable which implies that there exists no algorithm on a digital computer that can always compute it computability is important in order that the approximation error be controlled if a signal is not computable we cannot ascertain whether the computed signal is sufficiently close to the true signal i e we cannot verify every approximation accuracy

a sampling theorem for wavelet subspaces the classical shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory under weak hypotheses these subspaces are first shown to have a riesz basis formed from the reproducing kernels these in turn are used to construct the sampling sequences examples are given

m band compactly supported orthogonal symmetric interpolating scaling functions in many applications wavelets are usually expected to have the following properties compact support orthogonality linear phase regularity and interpolation to construct such wavelets it is crucial designing scaling functions with the above properties in two and three band cases except for the haar functions there exists no scaling function with the above five properties in m band case m spl ges 4 more free degrees available in design enable us to construct such scaling functions a novel approach to designing such scaling functions is proposed first we extend the two band dubuc 1986 filters to the m band case next the m band fir regular symmetric interpolating scaling filters are parameterized and then m band fir regular orthogonal symmetric interpolating scaling filters osisfs are designed via optimal selection of parameters finally two family of four band and five band osisfs and scaling functions are developed and their smoothness are estimated

generalized random walks for fusion of multi exposure images a single captured image of a real world scene is usually insufficient to reveal all the details due to under or over exposed regions to solve this problem images of the same scene can be first captured under different exposure settings and then combined into a single image using image fusion techniques in this paper we propose a novel probabilistic model based fusion technique for multi exposure images unlike previous multi exposure fusion methods our method aims to achieve an optimal balance between two quality measures i e local contrast and color consistency while combining the scene details revealed under different exposures a generalized random walks framework is proposed to calculate a globally optimal solution subject to the two quality measures by formulating the fusion problem as probability estimation experiments demonstrate that our algorithm generates high quality images at low computational cost comparisons with a number of other techniques show that our method generates better results in most cases

balanced multiwavelets with interpolatory property balanced multiwavelets with interpolatory property are discussed in this paper this kind of multiwavelets can have a sampling property like shannon s sampling theorem it has been shown that the corresponding matrix valued refinable mask has special structure and an orthogonal multifilter bank h z g z can be reduced to a scalar valued conjugate quadrature filter cqf a z but it does not mean that any scalar cqf can form a good multifilter bank which can generate a vector valued refinable function with some degree of smoothness in the context of balanced multiwavelets we give the definition of transferring balance order which a scalar cqf a z satisfies to guarantee that the multiwavelet generated is balanced on the basis of the parametrization of a scalar cqf with any length and conditions of transferring balance order parametrization of multifilter banks which can generate interpolatory multiwavelet and interpolatory scaling function is gotten moreover some balanced interpolatory multiwavelets have been constructed interpolatory analysis ready multiwavelets armlets are also discussed in this paper it is known that conditions of armlets are easy to validate compared with balanced multiwavelets but it will be present that if the corresponding scaling function is interpolatory the multiwavelet is balanced of order n if and only if it is an armlet of order n finally the application of balanced multiwavelets with interpolatory property in image processing is also discussed

approximation by translates of refinable functions the functions f 1 x dots f r x are refinable if they are combinations of the rescaled and translated functions f i 2x k this is very common in scientific computing on a regular mesh the space v 0 of approximating functions with meshwidth h 1 is a subspace of v 1 with meshwidth h 1 2 these refinable spaces have refinable basis functions the accuracy of the computations depends on p the order of approximation which is determined by the degree of polynomials 1 x dots x p 1 that lie in v 0 most refinable functions such as scaling functions in the theory of wavelets have no simple formulas the functions f i x are known only through the coefficients c k in the refinement equation scalars in the traditional case r times r matrices for multiwavelets the scalar sum rules that determine p are well known we find the conditions on the matrices c k that yield approximation of order p from v 0 these are equivalent to the strang fix conditions on the fourier transforms hat f i omega but for refinable functions they can be explicitly verified from the c k

simultaneous image fusion and denoising by using fractional order gradient information abstract image fusion and denoising are significant in image processing because of the availability of multi sensor and the presence of the noise the first order and second order gradient information have been effectively applied to deal with fusing the noise free source images in this paper we utilize the fractional order derivatives to represent image features and propose two new convex variational models for fusing noisy source images furthermore we apply an alternating direction method of multiplier admm to solve the minimization problems in the proposed models numerical experiments show that the proposed methods outperform the conventional total variation methods for simultaneously fusing and denoising

interpolating multiwavelet bases and the sampling theorem this paper considers the classical sampling theorem in multiresolution spaces with scaling functions as interpolants as discussed by xia and zhang 1993 for an orthogonal scaling function to support such a sampling theorem the scaling function must be cardinal interpolating they also showed that the only orthogonal scaling function that is both cardinal and of compact support is the haar function which is not continuous this paper addresses the same question but in the multiwavelet context where the situation is different this paper presents the construction of compactly supported orthogonal multiscaling functions that are continuously differentiable and cardinal the scaling functions thereby support a shannon like sampling theorem such wavelet bases are appealing because the initialization of the discrete wavelet transform prefiltering is the identity operator

a new class of explicit interpolatory splines and related measurement estimation this paper proposes a new and rich class of cardinal splines called generalized mk splines gmk splines which manifest several beneficial properties especially the interpolation property and the ability to obtain a construction with a desired frequency response the gmk spline basis functions of given degree can be constructed by a linear combination of a finite number of shifted b splines of that degree in a unified framework the gmk spline bases form riesz bases of their generating spaces and the gmk spline frequency responses meet strang fix conditions furthermore we provide search methods for minimal support gmk splines and local optimal gmk spline frequency responses under different given passband frequencies several related estimates are theoretically assessed including energy magnitude maximum and approximation error of gmk spline interpolated signals in signal processing applications the gmk splines behave as interpolating fir filters with simple implementation and some of them attain lower error than other traditional interpolation system responses such as cubic cardinal b spline and linear spline when approximating to an ideal lowpass filter experimental results show that the gmk spline filters can be implemented efficiently and have significant characteristics in the process of bandlimited signal reconstruction

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

generalized sampling in shift invariant spaces with multiple stable generators abstract the aim of this paper is to derive stable generalized sampling in a shift invariant space with l stable generators this is done in the light of the theory of frames in the product hilbert space l l 2 0 1 l 2 0 1 l 2 0 1 l times the generalized samples are expressed as the frame coefficients of an appropriate function in l l 2 0 1 with respect to some particular frame in l l 2 0 1 since any multiply stable generated shift invariant space is the image of l l 2 0 1 by means of a bounded invertible operator the generalized sampling is obtained from some dual frame expansions in l l 2 0 1 an example in the setting of the hermite cubic splines is exhibited

a sampling theorem associated with quasi fourier transform we propose in this correspondence a unified generalized bandlimited sampling theorem associated with quasi fourier transform of the form f sub spl phi f spl xi spl int sub d f x exp i spl epsiv spl phi x dx where the phase function satisfies spl phi x 0 in its domain our result contains classical shannon s sampling theorem and the sundaram et al 1997 scale limited sampling theorem as a special case new results such as the chirp limited sampling theorem can be established

sampling theorem and irregular sampling theorem for multiwavelet subspaces we generalize the multiwavelet sampling theorem by reproducing a kernel that is easy to use multiscaling functions that have orthogonality regularity short compact support symmetry and high approximation order satisfy the conditions in the sampling theorem which is not possible in the case of the scalar wavelet we consider the general cases of the uniform noninteger and irregular sampling a reconstruction from more general sets of points is necessary since the measurements may not be made at uniform points finally we establish a general irregular sampling theorem for multiwavelet subspaces and derive an estimate for the perturbations of uniform noninteger sampling in shift invariant spaces

social optimality in quantum bayesian games a b s t r a c t a significant aspect of the study of quantum strategies is the exploration of the gametheoretic solution concept of the nash equilibrium in relation to the quantization of a game pareto optimality is a refinement on the set of nash equilibria a refinement on the set of pareto optimal outcomes is known as social optimality in which the sum of players payoffs is maximized this paper analyzes social optimality in a bayesian game that uses the setting of generalized einstein podolsky rosen experiments for its physical implementation we show that for the quantum bayesian game a direct connection appears between the violation of bell s inequality and the social optimal outcome of the game and that it attains a superior socially optimal outcome

multiwavelet bases with extra approximation properties this paper highlights the differences between traditional wavelet and multiwavelet bases with equal approximation order because multiwavelet bases normally lack important properties that traditional wavelet bases of equal approximation order possess the associated discrete multiwavelet transform is less useful for signal processing unless it is preceded by a preprocessing step prefiltering this paper examines the properties and design of orthogonal multiwavelet bases with approximation order 1 that possess those properties that are normally absent for these balanced bases so named by lebrun and vetterli see proc ieee int conf acoust speech signal process icassp munich germany vol 3 p 2473 76 1997 and ibid vol 46 p 1119 24 1998 prefiltering can be avoided by reorganizing the multiwavelet filter bank the development in this paper draws from results regarding the approximation order of m band wavelet bases the main result thereby obtained is a characterization of balanced multiwavelet bases in terms of the divisibility of certain transfer functions by powers of z sup 2r 1 z sup 1 1 for traditional wavelets r 1 this specializes to the usual factor z 1 sup k

scale periodicity and its sampling theorem the scalar transform is a new representation for signals offering a perspective that is different from the fourier transform we introduce the notion of a scalar periodic function these functions are then represented through the discrete scale series we also define the notion of a strictly scale limited signal analogous to the shannon interpolation formula we show that such signals can be exactly reconstructed from exponentially spaced samples of the signal in the time domain as an interesting practical application we show how properties unique to the scale transform make it very useful in computing depth maps of a scene

gradient based multiresolution image fusion a novel approach to multiresolution signal level image fusion is presented for accurately transferring visual information from any number of input image signals into a single fused image without loss of information or the introduction of distortion the proposed system uses a fuse then decompose technique realized through a novel fusion decomposition system architecture in particular information fusion is performed on a multiresolution gradient map representation domain of image signal information at each resolution input images are represented as gradient maps and combined to produce new fused gradient maps fused gradient map signals are processed using gradient filters derived from high pass quadrature mirror filters to yield a fused multiresolution pyramid representation the fused output image is obtained by applying on the fused pyramid a reconstruction process that is analogous to that of conventional discrete wavelet transform this new gradient fusion significantly reduces the amount of distortion artefacts and the loss of contrast information usually observed in fused images obtained from conventional multiresolution fusion schemes this is because fusion in the gradient map domain significantly improves the reliability of the feature selection and information fusion processes fusion performance is evaluated through informal visual inspection and subjective psychometric preference tests as well as objective fusion performance measurements results clearly demonstrate the superiority of this new approach when compared to conventional fusion systems

nonideal sampling and regularization theory shannon s sampling theory and its variants provide effective solutions to the problem of reconstructing a signal from its samples in some shift invariant space which may or may not be bandlimited in this paper we present some further justification for this type of representation while addressing the issue of the specification of the best reconstruction space we consider a realistic setting where a multidimensional signal is prefiltered prior to sampling and the samples are corrupted by additive noise we adopt a variational approach to the reconstruction problem and minimize a data fidelity term subject to a tikhonov like continuous domain l2 regularization to obtain the continuous space solution we present theoretical justification for the minimization of this cost functional and show that the globally minimal continuous space solution belongs to a shift invariant space generated by a function generalized b spline that is generally not bandlimited when the sampling is ideal we recover some of the classical smoothing spline estimators the optimal reconstruction space is characterized by a condition that links the generating function to the regularization operator and implies the existence of a b spline like basis to make the scheme practical we specify the generating functions corresponding to the most popular families of regularization operators derivatives iterated laplacian as well as a new generalized one that leads to a new brand of matern splines we conclude the paper by proposing a stochastic interpretation of the reconstruction algorithm and establishing an equivalence with the minimax and minimum mean square error mmse wiener solutions of the generalized sampling problem

geometric measures of entanglement and the schmidt decomposition in the standard geometric approach the entanglement of a pure state is sin 2 where is the angle between the entangled state and the closest separable state of products of normalized qubit states we consider here a generalization of this notion by considering separable states that consist of products of unnormalized states of different dimensions the distance between the target entangled state and the closest unnormalized product state can be interpreted as a measure of the entanglement of the target state the components of the closest product state and its norm have an interpretation in terms of respectively the eigenvectors and eigenvalues of the reduced density matrices arising in the schmidt decomposition of the state vector for several cases where the target state has a large degree of symmetry we solve the system of equations analytically and look specifically at the limit where the number of qubits is large

vector valued wavelets and vector filter banks in this paper we introduce vector valued multiresolution analysis and vector valued wavelets for vector valued signal spaces we construct vector valued wavelets by using paraunitary vector filter bank theory in particular we construct vector valued meyer wavelets that are band limited we classify and construct vector valued wavelets with sampling property as an application of vector valued wavelets multiwavelets can be constructed from vector valued wavelets we show that certain linear combinations of known scalar valued wavelets may yield multiwavelets we then present discrete vector wavelet transforms for discrete time vector valued or blocked signals which can be thought of as a family of unitary vector transforms

interpolatory orthogonal multiwavelets and refinable functions multiwavelet bases of l sub 2 consist of families of functions 2 sup j 2 spl psi sub 2 2 sup j x k by allowing more than one function spl psi sub 1 spl psi sub 2 multiwavelets provide some useful applications in signal processing and nice features such as symmetry and orthogonality the elementary structure for multiwavelets is the multiresolution analysis of multiplicity two v sub j generated by dilating the basic subspace v sub 0 this subspace v sub 0 is generated by a multiple refinable function spl phi spl phi sub 1 spl phi sub 2 sup t refinable vector of functions satisfying a vector refinement equation spl phi x spl sigma a k spl phi 2x k here each a k is a 2 spl times 2 matrix in this paper we investigate interpolatory orthogonal multiple refinable functions and multiwavelets the interpolatory property here means that spl phi sub 1 and spl phi sub 2 vanish at all integers and half integers except that spl phi sub 1 0 1 and spl phi sub 2 1 2 1 when spl phi is both interpolatory and orthogonal which is impossible for scalar refinable functions the coefficients in the multiresolution representation can be realized by sampling instead of inner products if f x spl sigma c sub 1 k spl phi sub 1 2 sup n x k c sub 2 k spl phi sub 2 2 sup n x k then c sub 1 k f k 2 sup n and c sub 2 k f k 2 sup n 1 2 sup n 1 for k spl isin z what is more the orthogonal multiwavelets we construct here are also interpolatory we show that the refinement mask for an interpolatory orthogonal multiple refinable function and multiwavelets filterbank is reduced to a scalar cqf the approximation order of interpolatory multiple refinable functions is described a complete characterization of interpolatory orthogonal multiple refinable functions is given in this paper however interpolatory orthogonal multiple refinable functions cannot be symmetric examples are presented to illustrate the general theory

generalized interpolating refinable function vectors interpolating scalar refinable functions with compact support are of interest in several applications such as sampling theory signal processing computer graphics and numerical algorithms in this paper we shall generalize the notion of interpolating scalar refinable functions to compactly supported interpolating d refinable function vectors with any multiplicity r and dilation factor d more precisely we are interested in a d refinable function vector f f 1 f r t such that f is an rx1 column vector of compactly supported continuous functions with the following interpolation property f mr k email protected email protected 1 m email protected z m 0 r 1 1 r where d 0 1 and d k 0 for k 0 now for any function f email protected c the function f can be interpolated and approximated by email protected email protected email protected email protected email protected zf 1r k f k email protected email protected z f k f 1r k f r 1r k f k since f is interpolating email protected k r f k r for all email protected z that is email protected agrees with f on r 1z moreover for r 2 or d 2 such interpolating refinable function vectors can have the additional orthogonality property r email protected d k k for all k k z and 1

method for defining a class of fractional operations the fractional fourier transform frt permits a variety of associated fractional operations this correspondence proposes a systematic method based on the structure of the frt which not only provides unambiguous extensions of ordinary operations but permits writing the applicable expressions simply by inspection the approach also exposes the possible paths for implementing such operations

a focus fusion framework with anisotropic depth map smoothing focus fusion is the task of combining a set of images focused at different depths into a single image that is entirely in focus the crucial point of all focus fusion methods is the decision about the in focus areas to this end we present a general framework for focus fusion that introduces a modern regularisation strategy on these per pixel decisions we assume that neighbouring pixels in the fused image belong to similar depth layers following this assumption we smooth the depth map with a sophisticated anisotropic diffusion process combined with a robust data fidelity term the experiments with synthetic and real world data demonstrate that our new model yields a better quality than several existing focus fusion methods moreover our methodology is general and can be applied to improve many fusion approaches highlightswe show that depth map regularisation is an important tool for focus fusion we incorporate modern concepts such as a coupled anisotropic diffusion term we substantially improve the runtime with a fast gpu implementation we evaluate different in focus measures we compare the overall performance to several methods from the literature

generalized quantization scheme for two person non zero sum games we proposed a generalized quantization scheme for non zero sum games which can be reduced to the two existing quantization schemes under an appropriate set of parameters some other important situations are identified which are not apparent in the two existing quantization schemes

quantum stackelberg duopoly in the presence of correlated noise we study the influence of entanglement and correlated noise using correlated amplitude damping depolarizing and phase damping channels on the quantum stackelberg duopoly our investigations show that under the influence of an amplitude damping channel a critical point exists for an unentangled initial state at which firms get equal payoffs the game becomes a follower advantage game when the channel is highly decohered two critical points corresponding to two values of the entanglement angle are found in the presence of correlated noise within the range of these limits of the entanglement angle the game is a follower advantage game in the case of a depolarizing channel the payoffs of the two firms are strongly influenced by the memory parameter the presence of quantum memory ensures the existence of the nash equilibrium for the entire range of decoherence and entanglement parameters for both the channels a local maximum in the payoffs is observed which vanishes as the channel correlation increases moreover under the influence of the depolarizing channel the game is always a leader advantage game furthermore it is seen that the phase damping channel does not affect the outcome of the game

digital computation of the fractional fourier transform an algorithm for efficient and accurate computation of the fractional fourier transform is given for signals with time bandwidth product n the presented algorithm computes the fractional transform in o nlogn time a definition for the discrete fractional fourier transform that emerges from our analysis is also discussed

multisource image fusion method using support value transform with the development of numerous imaging sensors many images can be simultaneously pictured by various sensors however there are many scenarios where no one sensor can give the complete picture image fusion is an important approach to solve this problem and produces a single image which preserves all relevant information from a set of different sensors in this paper we proposed a new image fusion method using the support value transform which uses the support value to represent the salient features of image this is based on the fact that in support vector machines svms the data with larger support values have a physical meaning in the sense that they reveal relative more importance of the data points for contributing to the svm model the mapped least squares svm mapped ls svm is used to efficiently compute the support values of image the support value analysis is developed by using a series of multiscale support value filters which are obtained by filling zeros in the basic support value filter deduced from the mapped ls svm to match the resolution of the desired level compared with the widely used image fusion methods such as the laplacian pyramid discrete wavelet transform methods the proposed method is an undecimated transform based approach the fusion experiments are undertaken on multisource images the results demonstrate that the proposed approach is effective and is superior to the conventional image fusion methods in terms of the pertained quantitative fusion evaluation indexes such as quality of visual information qab f the mutual information etc

perceptual image fusion using wavelets a perceptual image fusion method is proposed that employs explicit luminance and contrast masking models these models are combined to give the perceptual importance of each coefficient produced by the dual tree complex wavelet transform of each input image this combined model of perceptual importance is used to select which coefficients are retained and furthermore to determine how to present the retained information in the most effective way this paper is the first to give a principled approach to image fusion from a perceptual perspective furthermore the proposed method is shown to give improved quantitative and qualitative results compared with previously developed methods

unified fractional fourier transform and sampling theorem the fractional fourier transform frt is an extension of the ordinary fourier transform ft applying the language of the unified ft we develop frt expressions for discrete and continuous signals introducing a particular form of periodicity chirp periodicity the frt sampling theorem is derived as an extension of its ordinary counterpart

quantum game theory based on the schmidt decomposition we present a novel formulation of quantum game theory based on the schmidt decomposition which has the merit that the entanglement of quantum strategies is manifestly quantified we apply this formulation to 2 player 2 strategy symmetric games and obtain a complete set of quantum nash equilibria apart from those available with the maximal entanglement these quantum nash equilibria are extensions of the nash equilibria in classical game theory the phase structure of the equilibria is determined for all values of entanglement and thereby the possibility of resolving the dilemmas by entanglement in the game of chicken the battle of the sexes the prisoners dilemma and the stag hunt is examined we find that entanglement transforms these dilemmas with each other but cannot resolve them except in the stag hunt game where the dilemma can be alleviated to a certain degree

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

a maximum likelihood approach to joint image registration and fusion both image registration and fusion can be formulated as estimation problems instead of estimating the registration parameters and the true scene separately as in the conventional way we propose a maximum likelihood approach for joint image registration and fusion in this paper more precisely the fusion performance is used as the criteria to evaluate the registration accuracy hence the registration parameters can be automatically tuned so that both fusion and registration can be optimized simultaneously the expectation maximization algorithm is employed to solve this joint optimization problem the cramer rao bound crb is then derived our experiments use several types of sensory images for performance evaluation such as visual images ir thermal images and hyperspectral images it is shown that the mean square error of estimating the registration parameters using the proposed method is close to the crbs at the mean time an improved fusion performance can be achieved in terms of the edge preservation measure qab f compared to the laplacian pyramid fusion approach

formulas for orthogonal iir wavelet filters explicit solutions are given for the rational function p z for two classes of iir orthogonal two band wavelet bases for which the scaling filter is maximally flat p z denotes the rational transfer function h z h 1 z where h z is the lowpass scaling filter the first is the class of solutions that are intermediate between the daubechies 1992 and the butterworth wavelets it is found that the daubechies the butterworth and the intermediate solutions are unified by a single formula the second is the class of scaling filters realizable as a parallel sum of two allpass filters a particular case of which yields the class of symmetric iir orthogonal wavelet bases for this class a closed form solution is provided by the solution to an older problem in group delay approximation by digital allpole filters

the m band cardinal orthogonal scaling function the m band symmetric cardinal orthogonal scaling function with compact support is of interest in several applications such as sampling theory signal processing computer graphics and numerical algorithms in this paper we provide a complete mathematical analysis for the m band symmetric cardinal orthogonal scaling function firstly we generalize some results of the cardinal orthogonal scaling function from the special case m 2 to the most general case m 2 also we find some new results secondly we obtain the characterizations of the m band symmetric cardinal orthogonal scaling function and revisit some known examples to prove our theory

interpolating refinable function vectors with reflection properties in this paper we investigate interpolating refinable function vectors whose component functions constitute reflection pairs a necessary condition for an interpolating refinable function vector to satisfy the reflection property is provided for any dilation factor and multiplicity although the converse is not true in general for some special cases it becomes a sufficient condition as well we concentrate on the relation between a dilation factor and multiplicity and provide the necessary and sufficient condition on the coefficients of its refinement mask in order for such refinable function vector to form a reflection pair finally trivial solutions consisting of translated versions of one component function which is symmetric by itself are discussed to illustrate the results various numerical examples are provided

on sampling theorem wavelets and wavelet transforms the classical shannon sampling theorem has resulted in many applications and generalizations from a multiresolution point of view it provides the sine scaling function in this case for a band limited signal its wavelet series transform wst coefficients below a certain resolution level can be exactly obtained from the samples with a sampling rate higher than the nyquist rate the authors study the properties of cardinal orthogonal scaling functions cosf which provide the standard sampling theorem in multiresolution spaces with scaling functions as interpolants they show that cosf with compact support have and only have one possibility which is the haar pulse they present a family of cosf with exponential decay which are generalizations of the haar function with these cosf an application is the computation of wst coefficients of a signal by the mallat 1989 algorithm they present some numerical comparisons for different scaling functions to illustrate the advantage of cosf for signals which are not in multiresolution spaces they estimate the aliasing error in the sampling theorem by using uniform samples

improved direction estimation for di zenzo s multichannel image gradient operator gradient estimation is one of the most important tasks in image video processing for multichannel images a classical and widely used gradient method is di zenzo s gradient operator which is based on the measure of squared local contrast variation of multichannel images however up to now the indetermination of di zenzo s gradient direction has not been well solved which results in errors occurring in most of the subsequent studies in which di zenzo s vector gradient is used in this paper this problem is solved thoroughly furthermore the ranges of the values that the gradient angle should take in various cases are also analyzed as an application in color image processing a color version of canny edge detector is implemented by introducing the new gradient estimator to the traditional grayscale image canny operator the experimental results indicate that the improved di zenzo s gradient operator is currently one of the best color gradient estimators and outperforms other state of the art color image gradient methods the improved multichannel gradient operator not only provides accurate gradient estimation but also is efficient and easy to implement

high order balanced multiwavelets theory factorization and design this paper deals with multiwavelets and the different properties of approximation and smoothness associated with them in particular we focus on the important issue of the preservation of discrete time polynomial signals by multifilterbanks we introduce and detail the property of balancing for higher degree discrete time polynomial signals and link it to a very natural factorization of the refinement mask of the lowpass synthesis multifilter this factorization turns out to be the counterpart for multiwavelets of the well known zeros at spl pi condition in the usual scalar wavelet framework the property of balancing also proves to be central to the different issues of the preservation of smooth signals by multifilterbanks the approximation power of finitely generated multiresolution analyses and the smoothness of the multiscaling functions and multiwavelets using these new results we describe the construction of a family of orthogonal multiwavelets with symmetries and compact support that is indexed by increasing order of balancing in addition we also detail for any given balancing order the orthogonal multiwavelets with minimum length multifilters

multiplicity of fractional fourier transforms and their relationships the multiplicity of the fractional fourier transform frt which is intrinsic in any fractional operator has been claimed by several authors but never systematically developed the paper starts with a general frt definition based on eigenfunctions and eigenvalues of the ordinary fourier transform which allows us to generate all possible definitions the multiplicity is due to different choices of both the eigenfunction and the eigenvalue classes a main result obtained by a generalized form of the sampling theorem gives explicit relationships between the different frts

orthonormal wavelets with simple closed form expressions two classes of orthonormal wavelets that have simple closed form expressions are derived from pulses with the raised cosine spectrum these wavelets which are bandlimited and polynomial decaying in time are found to be particular examples of the lemarie meyer 1992 1993 wavelets the derivation reveals interesting connections between wavelet construction and intersymbol interference isi free signaling for digital communications

m band biorthogonal interpolating wavelets via lifting scheme recently the lifting scheme was generalized to the multidimensional and multiband cases and was used to design m band interpolating scaling filters and their duals based on this idea we develop a new lifting pattern namely the progressive lifting pattern this pattern allows us to pairwise generate m band interpolating filterbanks and wavelets by the order from lowpass to highpass filters a complete lifting procedure is divided into m 1 simple steps in each step a pair of filters the l th filter and its dual are generated in this way an m band biorthogonal interpolating filterbank wavelet is determined by m m 1 lifting filters the first 2 m 1 lifting filters completely characterize the two scaling filters as well as the vanishing moments of bandpass and highpass filters the residual m 1 m 2 lifting filters are used to pairwise optimize the bandpass and highpass filters in terms of the criterion of stopband energy minimization the obtained m band biorthogonal interpolating filterbanks and wavelets provide excellent frequency characteristics in particular low stopband sidelobes furthermore the pattern is also utilized to design signal adapted interpolating filterbanks and their rational coefficient counterparts in terms of subband coding gain the obtained filterbanks achieve large subband coding gains the rational coefficient filterbanks preserve the biorthogonality and allow wavelet transforms from integers to integers and a unifying lossy lossless coding framework at the cost of a slight degradation in subband coding gain

constructing quantum games from nonfactorizable joint probabilities a probabilistic framework is developed that gives a unifying perspective on both the classical and quantum versions of two player games we suggest exploiting peculiar joint probabilities involved in einstein podolsky rosen epr experiments to construct a quantum game when the corresponding classical game is obtained from factorizable joint probabilities we analyze how nonfactorizability changes nash equilibria in three well known games of prisoner s dilemma stag hunt and chicken in this framework we find that for the game of prisoner s dilemma even nonfactorizable epr joint probabilities cannot be helpful to escape from the classical outcome of the game for a particular version of the chicken game however we find that the two nonfactorizable sets of joint probabilities which maximally violate the clauser holt shimony horne sum of correlations indeed result in new nash equilibria

balanced multiwavelets theory and design this article deals with multiwavelets which are a generalization of wavelets in the context of time varying filter banks and with their applications to signal processing and especially compression by their inherent structure multiwavelets are fit for processing multichannel signals this is the main issue in which we are interested first we review material on multiwavelets and their links with multifilter banks and especially time varying filter banks then we have a close look at the problems encountered when using multiwavelets in applications and we propose new solutions for the design of multiwavelets filter banks by introducing the so called balanced multiwavelets