wavelets and filter banks theory and design the wavelet transform is compared with the more classical short time fourier transform approach to signal analysis then the relations between wavelets filter banks and multiresolution signal processing are explored a brief review is given of perfect reconstruction filter banks which can be used both for computing the discrete wavelet transform and for deriving continuous wavelet bases provided that the filters meet a constraint known as regularity given a low pass filter necessary and sufficient conditions for the existence of a complementary high pass filter that will permit perfect reconstruction are derived the perfect reconstruction condition is posed as a bezout identity and it is shown how it is possible to find all higher degree complementary filters based on an analogy with the theory of diophantine equations an alternative approach based on the theory of continued fractions is also given these results are used to design highly regular filter banks which generate biorthogonal continuous wavelet bases with symmetries

design of regular wavelets using a three step lifting scheme we propose structural multidimensional multichannel filter banks with desirable numbers of vanishing moments for the analysis and synthesis banks for a two channel filter bank we use a three step lifting scheme as opposed to the conventional two step lifting method in order to provide more symmetry between the analysis and synthesis filters we show that the resulting filters have more regularity lower frame bounds ratio and better frequency selectivity we also extend our design to a general multichannel framework three steps of lifting scheme provides us a degree of freedom which we can benefit from toward a more flexible design

scalability of 2 d wavelet transform algorithms analytical and experimental results on mpps this paper studies the scalability of two dimensional 2 d discrete wavelet transform dwt algorithms on massively parallel processors mpps the principal operation in the 2 d dwt is the filtering operation used to implement the filter banks of the 2 d subband decomposition this filtering operation can be implemented as a convolution in the time domain or as a multiplication in the frequency domain we demonstrate that there exist combinations of machine size image size and wavelet kernel size for which the time domain algorithms outperform the frequency domain algorithms and vice versa we therefore demonstrate that a hybrid approach that combines time and frequency domain approaches can yield linear scalability for a broad range of problem and machine sizes furthermore we show the effect of processor speed versus communication overhead and the use of separable versus nonseparable wavelets on the crossover points between the algorithm approaches

solution of inverse problems in image processing by wavelet expansion we describe a wavelet based approach to linear inverse problems in image processing in this approach both the images and the linear operator to be inverted are represented by wavelet expansions leading to a multiresolution sparse matrix representation of the inverse problem the constraints for a regularized solution are enforced through wavelet expansion coefficients a unique feature of the wavelet approach is a general and consistent scheme for representing an operator in different resolutions an important problem in multigrid multiresolution processing this and the sparseness of the representation induce a multigrid algorithm the proposed approach was tested on image restoration problems and produced good results

adaptive 2 d wavelet transform based on the lifting scheme with preserved vanishing moments in this paper we propose novel adaptive wavelet filter bank structures based on the lifting scheme the filter banks are nonseparable based on quincunx sampling with their properties being pixel wise adapted according to the local image features despite being adaptive the filter banks retain a desirable number of primal and dual vanishing moments the adaptation is introduced in the predict stage of the filter bank with an adaptation region chosen independently for each pixel based on the intersection of confidence intervals ici rule the image denoising results are presented for both synthetic and real world images it is shown that the obtained wavelet decompositions perform well especially for synthetic images that contain periodic patterns for which the proposed method outperforms the state of the art in image denoising

coset sum an alternative to the tensor product in wavelet construction a multivariate biorthogonal wavelet system can be obtained from a pair of multivariate biorthogonal refinement masks in multiresolution analysis setup some multivariate refinement masks may be decomposed into lower dimensional refinement masks tensor product is a popular way to construct a decomposable multivariate refinement mask from lower dimensional refinement masks we present an alternative method which we call coset sum for constructing multivariate refinement masks from univariate refinement masks the coset sum shares many essential features of the tensor product that make it attractive in practice 1 it preserves the biorthogonality of univariate refinement masks 2 it preserves the accuracy number of the univariate refinement mask and 3 the wavelet system associated with it has fast algorithms for computing and inverting the wavelet coefficients the coset sum can even provide a wavelet system with faster algorithms in certain cases than the tensor product these features of the coset sum suggest that it is worthwhile to develop and practice alternative methods to the tensor product for constructing multivariate wavelet systems some experimental results using 2 d images are presented to illustrate our findings

multi d wavelet filter bank design using quillen suslin theorem for laurent polynomials in this paper we present a new approach for constructing the wavelet filter bank our approach enables constructing nonseparable multidimensional non redundant wavelet filter banks with fir filters using the quillen suslin theorem for laurent polynomials our construction method presents some advantages over the traditional methods of multidimensional wavelet filter bank design first it works for any spatial dimension and for any sampling matrix second it does not require the initial lowpass filters to satisfy any additional assumption such as interpolatory condition third it provides an algorithm for constructing a wavelet filter bank from a single lowpass filter so that its vanishing moments are at least as many as the accuracy number of the lowpass filter

theory of two dimensional multirate filter banks results are presented on 2 d fir two dimensional finite impulse response filter banks for multirate applications the theory is valid for all sampling lattices conditions for alias free and perfect signal reconstruction are derived synthesis structures for paraunitary and nonparaunitary polynomial matrices are derived which yield perfect reconstruction filter banks the degrees of freedom are given for these systems linear phase conditions are posed on the polyphase form of filter banks which is used to derive a design structure for the restricted but important case of linear phase filter banks

image compression and denoising via nonseparable wavelet approximation it was shown in acta math sci 22 1 2002 19 that two dimensional nonseparable scaling function interpolation has better convergence results than scalar wavelet systems in some cases based on several experimental results in this paper we present applications of two dimensional nonseparable wavelet approximation the algorithms are developed by using two dimensional nonseparable scaling function interpolation to perform image compression and image denoising comparing with the separable counterparts our results show that there are some improvements and advantages of two dimensional nonseparable scaling function interpolation

polynomial wavelet regression for images with irregular boundaries in this paper we focus on denoising images for which observations are equally spaced except around the boundaries which are irregular such images are very common in many fields for example in geophysics the advantages of adding a low order polynomial term when implementing a wavelet regression for such images are presented besides removing the classical restriction of having a dyadic of number of observations this strategy reduces the bias at the edges without significantly increasing the risk in addition this method is simple to implement fast and efficient its utility is illustrated with simulation studies and a real example

adaptive prediction trees for image compression this paper presents a complete general purpose method for still image compression called adaptive prediction trees efficient lossy and lossless compression of photographs graphics textual and mixed images is achieved by ordering the data in a multicomponent binary pyramid applying an empirically optimized nonlinear predictor exploiting structural redundancies between color components then coding with hex trees and adaptive runlength huffman coders color palettization and order statistics prefiltering are applied adaptively as appropriate over a diverse image test set the method outperforms standard lossless and lossy alternatives the competing lossy alternatives use block transforms and wavelets in well studied configurations a major result of this paper is that predictive coding is a viable and sometimes preferable alternative to these methods

perfect reconstruction fir filter banks some properties and factorizations perfect reconstruction finite impulse response fir filter banks are analyzed both in the z transform and time domains showing the alternatives between designs in the two domains various classes of perfect reconstruction schemes are indicated and relations between previously known systems are given windowed modulated filter banks with low computational complexity and perfect reconstruction are shown new factorizations of polyphase filter matrices leading in particular to linear phase filters are given the computational complexity and the architecture of the new structures are indicated

orthogonal and biorthogonal fir hexagonal filter banks with sixfold symmetry recently hexagonal image processing has attracted attention the hexagonal lattice has several advantages in comparison with the rectangular lattice the conventionally used lattice for image sampling and processing for example a hexagonal lattice needs less sampling points it has better consistent connectivity it has higher symmetry and its structure is plausible to human vision systems the multiresolution analysis method has been used for hexagonal image processing since the hexagonal lattice has high degree of symmetry it is desirable that the hexagonal filter banks designed for multiresolution hexagonal image processing also have high order of symmetry which is pertinent to the symmetry structure of the hexagonal lattice the orthogonal or prefect reconstruction pr hexagonal filter banks that are available in the literature have only threefold symmetry in this paper we investigate the construction of orthogonal and pr finite impulse response fir hexagonal filter banks with sixfold symmetry we obtain block structures of 7 size refinement seven channel two dimensional orthogonal and pr fir hexagonal filter banks with sixfold rotational symmetry radic7 refinement orthogonal and biorthogonal wavelets based on these block structures are constructed in this paper we also consider fir hexagonal filter banks with axial line symmetry and we present a block structure of fir hexagonal filter banks with pseudo sixfold axial symmetry

on the competitive ratio of evaluating priced functions let f be a function on a set of variables v for each x i v let c x be the cost of reading the value of x an algorithm for evaluating f is a strategy for adaptively identifying and reading a set of variables u s v whose values uniquely determine the value of f we are interested in finding algorithms which minimize the cost incurred to evaluate f in the above sense competitive analysis is employed to measure the performance of the algorithms we address two variants of the above problem we consider the basic model in which the evaluation algorithm knows the cost c x for each x i v we also study a novel model where the costs of the variables are not known in advance and some preemption is allowed in the reading operations this model has applications for example when reading a variable coincides with obtaining the output of a job on a cpu and the cost is the cpu time for the model where the costs of the variables are known we present a polynomial time algorithm with the best possible competitive ratio cf for each function f that is representable by a threshold tree and for each fixed cost function c remarkably the best known result for the same class of functions is a pseudo polynomial algorithm with competitiveness 2 cf still in the same model we introduce the linear programming approach lpa a framework that allows the design of efficient algorithms for evaluating functions we show that different implementations of this approach lead in general to the best algorithms known so far and in many cases to optimal algorithms for different classes of functions considered before in the literature via the lpa we are able to determine exactly the optimal extremal competitiveness of monotone boolean functions remarkably the upper bound which leads to this result holds for a much broader class of functions which also includes the whole set of boolean functions we also show how to extend the lpa together with these results to the model where the costs of the variables are not known beforehand in particular we show how to employ the extended lpa to design a polynomial time optimal with respect to competitiveness algorithm for the class of monotone boolean functions representable by threshold trees

nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r sup n new results on multidimensional filter banks and their connection to multidimensional nonseparable wavelets are presented among the topics discussed are sampling in multiple dimensions multidimensional perfect reconstruction filter banks the two channel case in multiple dimensions the synthesis of multidimensional filter banks and the design of compactly supported wavelets

a new family of nonredundant transforms using hybrid wavelets and directional filter banks we propose a new family of nonredundant geometrical image transforms that are based on wavelets and directional filter banks we convert the wavelet basis functions in the finest scales to a flexible and rich set of directional basis elements by employing directional filter banks where we form a nonredundant transform family which exhibits both directional and nondirectional basis functions we demonstrate the potential of the proposed transforms using nonlinear approximation in addition we employ the proposed family in two key image processing applications image coding and denoising and show its efficiency for these applications

design of regular nonseparable bidimensional wavelets using grobner basis techniques the design of two dimensional 2 d filter banks yielding orthogonality and linear phase filters and generating regular wavelet bases is a difficult task involving the algebraic properties of multivariate polynomials using cascade forms implies dealing with nonlinear optimization we turn the issue of optimizing the orthogonal linear phase cascade from kovacevic and vetterli 1992 into a polynomial problem and solve it using grobner basis techniques and computer algebra this leads to a complete description of maximally flat wavelets among the orthogonal linear phase family proposed by kovacevic and vetterli we obtain up to five degrees of flatness for a 16 spl times 16 filter bank whose sobolev exponent is 2 11 making this wavelet the most regular orthogonal linear phase nonseparable wavelet to the authors knowledge

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

pyramidal lattice vector quantization for multiscale image coding introduces a new image coding scheme using lattice vector quantization the proposed method involves two steps biorthogonal wavelet transform of the image and lattice vector quantization of wavelet coefficients in order to obtain a compromise between minimum distortion and bit rate we must truncate and scale the lattice suitably to meet this goal we need to know how many lattice points lie within the truncated area we investigate the case of laplacian sources where surfaces of equal probability are spheres for the l sup 1 metric pyramids for arbitrary lattices we give explicit generating functions for the codebook sizes for the most useful lattices like z sup n d sub n e sub s spl and sub 16

biorthogonal multivariate filter banks from centrally symmetric matrices we provide a practical characterization of block centrally symmetric and anti symmetric matrices which arise in the construction of multivariate filter banks and use these matrices for the construction of biorthogonal filter banks with linear phase

isotropic polyharmonic b splines scaling functions and wavelets in this paper we use polyharmonic b splines to build multidimensional wavelet bases these functions are nonseparable multidimensional basis functions that are localized versions of radial basis functions we show that rabut s elementary polyharmonic b splines do not converge to a gaussian as the order parameter increases as opposed to their separable b spline counterparts therefore we introduce a more isotropic localization operator that guarantees this convergence resulting into the isotropic polyharmonic b splines next we focus on the two dimensional quincunx subsampling scheme this configuration is of particular interest for image processing because it yields a finer scale progression than the standard dyadic approach however up until now the design of appropriate filters for the quincunx scheme has mainly been done using the mcclellan transform in our approach we start from the scaling functions which are the polyharmonic b splines and as such explicitly known and we derive a family of polyharmonic spline wavelets corresponding to different flavors of the semi orthogonal wavelet transform e g orthonormal b spline and dual the filters are automatically specified by the scaling relations satisfied by these functions we prove that the isotropic polyharmonic b spline wavelet converges to a combination of four gabor atoms which are well separated in the frequency domain we also show that these wavelets are nearly isotropic and that they behave as an iterated laplacian operator at low frequencies we describe an efficient fast fourier transform based implementation of the discrete wavelet transform based on polyharmonic b splines

the characterization of a class of multivariate mra and semi orthogonal parseval frame wavelets abstract let a be a d d expansive matrix with deta 2 this paper addresses parseval frame wavelets pfws in the setting of reducing subspaces of l 2 r d we prove that all semi orthogonal pfws semi orthogonal mra pfws are precisely the ones with their dimension functions being non negative integer valued 0 or 1 we also characterize all mra pfws some examples are provided

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

subband decomposition of signals with generalized sampling the authors generalize down and up sampling operations by proposing block sampling perfect reconstruction conditions for two band subband coding with block sampling are derived by generalizing the sampling operation new degrees of freedom are introduced and as a result filter banks which were not previously possible become possible generalized down sampler introduces different aliasing components than that of the traditional down sampler this can be used to ease some the requirements of the filter bank design problem a constructive sampling method is proposed so that coprimeness of the transfer functions of the analysis filter banks is not only a necessary but also sufficient condition for perfect reconstruction the results are extended to the case where the filter banks are linear periodically time varying the multichannel case is analyzed and the relation between unimodular matrices and perfect reconstruction filter banks is discussed

a geometric construction of multivariate sinc functions we present a geometric framework for explicit derivation of multivariate sampling functions sinc on multidimensional lattices the approach leads to a generalization of the link between sinc functions and the lagrange interpolation in the multivariate setting our geometric approach also provides a frequency partition of the spectrum that leads to a nonseparable extension of the 1 d shannon sinc wavelets to the multivariate setting moreover we propose a generalization of the lanczos window function that provides a practical and unbiased approach for signal reconstruction on sampling lattices while this framework is general for lattices of any dimension we specifically characterize all 2 d and 3 d lattices and show the detailed derivations for 2 d hexagonal body centered cubic bcc and face centered cubic fcc lattices both visual and numerical comparisons validate the theoretical expectations about superiority of the bcc and fcc lattices over the commonly used cartesian lattice

modeling analysis and optimum design of quantized m band filter banks this paper provides a rigorous modeling and analysis of quantization effects in m band subband codecs followed by optimal filter bank design and compensation the codec is represented by a polyphase decomposition of the analysis synthesis filter banks and an embedded nonlinear gain plus additive noise model for the pdf optimized scalar quantizers we construct an equivalent time invariant but nonlinear structure operating at the slow clock rate that allows us to compute the exact expression for the mean square quantization error in the reconstructed output this error is shown to consist of two components a distortion component and a dominant random noise component uncorrelated with the input signal we determine the optimal paraunitary and biorthogonal fir filter coefficients compensators and integer bit allocation to minimize this mse subject to the constraints of filter length average bit rate and perfect reconstruction pr in the absence of quantizers the biorthogonal filter bank results in a smaller mse but the filter coefficients are very sensitive to signal statistics and to average bit constraints by comparison the paraunitary structure is much more robust we also show that the null compensated design that eliminates the distortion component is more robust than the optimally compensated case that minimizes the total mse but only at nominal conditions both modeling and optimal design are validated by simulation in the two channel case

finding optimal satisficing strategies for and or trees many tasks require evaluating a specified boolean expression f over a set of probabilistic tests whose costs and success probabilities are each known a strategy specifies when to perform which test towards determining the overall outcome of f we are interested in finding the strategy with the minimum expected cost as this task is typically np hard for example when tests can occur many times within f or when there are probabilistic correlations between the test outcomes we consider those cases in which the tests are probabilistically independent and each appears only once in f in such cases f can be written as an and or tree where each internal node corresponds to either the and or or of its children and each leaf node is a probabilistic test in this paper we investigate probabilistic and or tree resolution paotr namely the problem of finding optimal strategies for and or trees we first consider a depth first approach evaluate each penultimate rooted subtree in isolation replace each such subtree with a single mega test and recurse on the resulting reduced tree we show that the strategies produced by this approach are optimal for and or trees with depth at most two but can be arbitrarily sub optimal for deeper trees each depth first strategy can be described by giving the linear relative order in which tests are to be executed with the understanding that any test whose outcome becomes irrelevant is skipped the class of linear strategies is strictly larger than depth first strategies we show that even the best linear strategy can also be arbitrarily sub optimal we next prove that an optimal strategy honors a natural partial order among tests with a common parent node leaf sibling tests and use this to produce a dynamic programming algorithm that finds the optimal strategy in time o d 2 r 1 d where r is the maximum number of leaf siblings and d is the number of leaf parents hence for trees with a bounded number of internal nodes this run time is polynomial in the tree size we also present another special class of and or trees for which this task takes polynomial time we close by presenting a number of other plausible approaches to paotr together with counterexamples to show their limitations

a simple method to build oversampled filter banks and tight frames this paper presents conditions under which the sampling lattice for a filter bank can be replaced without loss of perfect reconstruction this is the generalization of common knowledge that removing up downsampling will not lose perfect reconstruction the results provide a simple way of building over sampled filter banks if the original filter banks are orthogonal these oversampled banks construct tight frames of l2 z n when iterated as an example a quincunx lattice is used to replace the rectangular one of the standard wavelet transform this replacement leads to a tight frame that has a higher sampling in both time and frequency the frame transform is nearly shift invariant and has intermediate scales an application of the transform to image fusion is also presented

an orthogonal family of quincunx wavelets with continuously adjustable order we present a new family of two dimensional and three dimensional orthogonal wavelets which uses quincunx sampling the orthogonal refinement filters have a simple analytical expression in the fourier domain as a function of the order spl lambda which may be noninteger we can also prove that they yield wavelet bases of l sub 2 r sup 2 for any spl lambda 0 the wavelets are fractional in the sense that the approximation error at a given scale a decays like o a sup spl lambda they also essentially behave like fractional derivative operators to make our construction practical we propose a fast fourier transform based implementation that turns out to be surprisingly fast in fact our method is almost as efficient as the standard mallat algorithm for separable wavelets

representation and compression of multidimensional piecewise functions using surflets we study the representation approximation and compression of functions in m dimensions that consist of constant or smooth regions separated by smooth m 1 dimensional discontinuities examples include images containing edges video sequences of moving objects and seismic data containing geological horizons for both function classes we derive the optimal asymptotic approximation and compression rates based on kolmogorov metric entropy for piecewise constant functions we develop a multiresolution predictive coder that achieves the optimal rate distortion performance for piecewise smooth functions our coder has near optimal rate distortion performance our coder for piecewise constant functions employs surflets a new multiscale geometric tiling consisting of m dimensional piecewise constant atoms containing polynomial discontinuities our coder for piecewise smooth functions uses surfprints which wed surflets to wavelets for piecewise smooth approximation both of these schemes achieve the optimal asymptotic approximation performance key features of our algorithms are that they carefully control the potential growth in surflet parameters at higher smoothness and do not require explicit estimation of the discontinuity we also extend our results to the corresponding discrete function spaces for sampled data we provide asymptotic performance results for both discrete function spaces and relate this asymptotic performance to the sampling rate and smoothness orders of the underlying functions and discontinuities for approximation of discrete data we propose a new scale adaptive dictionary that contains few elements at coarse and fine scales but many elements at medium scales simulation results on synthetic signals provide a comparison between surflet based coders and previously studied approximation schemes based on wedgelets and wavelets

on the design of multi dimensional compactly supported parseval framelets with directional characteristics abstract in this paper we propose a new method for the construction of multi dimensional wavelet like families of affine frames commonly referred to as framelets with specific directional characteristics small and compact support in space directional vanishing moments dvm and axial symmetries or anti symmetries the framelets we construct arise from readily available refinable functions the filters defining these framelets have few non zero coefficients custom selected orientations and can act as finite difference operators

on two channel filter banks with directional vanishing moments the contourlet transform was proposed to address the limited directional resolution of the separable wavelet transform one way to guarantee good approximation behavior is to let the directional filters in the contourlet filter bank have sharp frequency response this requires filters with large support size we seek to isolate the key filter property that ensures good approximation in this direction we propose filters with directional vanishing moments dvm these filters we show annihilate information along a given direction we study two channel filter banks with dvm filters we provide conditions under which the design of dvm filter banks is possible a complete characterization of the product filter is thus obtained we propose a design framework that avoids 2 d factorization using the mapping technique the filters designed when used in the contourlet transform exhibit nonlinear approximation comparable to the conventional filters while being shorter and therefore providing better visual quality with less ringing artifacts furthermore experiments show that the proposed filters outperform the conventional ones in image approximation and denoising

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

wavelet neural networks for function learning a wavelet based neural network is described the structure of this network is similar to that of the radial basis function rbf network except that in the present paper the radial basis functions are replaced by orthonormal scaling functions that are not necessarily radial symmetric the efficacy of this type of network in function learning and estimation is demonstrated through theoretical analysis and experimental results in particular it has been shown that the wavelet network has universal and l sup 2 approximation properties and is a consistent function estimator convergence rates associated with these properties are obtained for certain function classes where the rates avoid the curse of dimensionality in the experiments the wavelet network performed well and compared favorably to the mlp and rbf networks

finding optimal derivation in redundant knowledge bases abstract a backward chaining process uses a collection of rules to reduce a given goal to a sequence of database retrievals a derivation strategy is an ordering on these steps specifying when to use each rule and when to perform each retrieval given the costs of reductions and retrievals and the a priori likelihood that each particular retrieval will succeed one can compute the expected cost of any strategy for answering a specific query from a given knowledge base smith 19 presents an algorithm that finds the minimal cost strategy in time essentially linear in the number of rules for any disjunctive irredundant knowledge base this paper proves that the addition of redundancies renders this task np hard many explanation based learning systems work by adding in redundancies this shows the complexities inherent in their task

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

multidimensional orthogonal filter bank characterization and design using the cayley transform we present a complete characterization and design of orthogonal infinite impulse response iir and finite impulse response fir filter banks in any dimension using the cayley transform ct traditional design methods for one dimensional orthogonal filter banks cannot be extended to higher dimensions directly due to the lack of a multidimensional md spectral factorization theorem in the polyphase domain orthogonal filter banks are equivalent to paraunitary matrices and lead to solving a set of nonlinear equations the ct establishes a one to one mapping between paraunitary matrices and para skew hermitian matrices in contrast to the paraunitary condition the para skew hermitian condition amounts to linear constraints on the matrix entries which are much easier to solve based on this characterization we propose efficient methods to design md orthogonal filter banks and present new design results for both iir and fir cases

multiresolution analysis haar bases and self similar tilings of r sup n orthonormal bases for l sup 2 r sup n are constructed that have properties that are similar to those enjoyed by the classical haar basis for l sup 2 r for example each basis consists of appropriate dilates and translates of a finite collection of piecewise constant functions the construction is based on the notion of multiresolution analysis and reveals an interesting connection between the theory of compactly supported wavelet bases and the theory of self similar tilings

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

theory and design of m channel maximally decimated quadrature mirror filters with arbitrary m having the perfect reconstruction property based on the concept of losslessness in digital filter structures this paper derives a general class of maximally decimated m channel quadrature mirror filter banks that lead to perfect reconstruction the perfect reconstruction property guarantees that the reconstructed signal hat x n is a delayed version of the input signal x n i e hat x n x n n 0 it is shown that such a property can be satisfied if the alias component matrix ac matrix for short is unitary on the unit circle of the z plane the number of channels m is arbitrary and when m is two the results reduce to certain recently reported 2 channel perfect reconstruction qmf structures a procedure based on recently reported fir cascaded lattice structures is presented for optimal design of such fir m channel filter banks design examples are included

multiresolution detection of spiculated lesions in digital mammograms we present a novel multiresolution scheme for the detection of spiculated lesions in digital mammograms first a multiresolution representation of the original mammogram is obtained using a linear phase nonseparable two dimensional 2 d wavelet transform a set of features is then extracted at each resolution in the wavelet pyramid for every pixel this approach addresses the difficulty of predetermining the neighborhood size for feature extraction to characterize objects that may appear in different sizes detection is performed from the coarsest resolution to the finest resolution using a binary tree classifier this top down approach requires less computation by starting with the least amount of data and propagating detection results to finer resolutions experimental results using the mias image database have shown that this algorithm is capable of detecting spiculated lesions of very different sizes at low false positive rates

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

on a class of bidimensional nonseparable wavelet multipliers abstract let e 1 1 1 1 or 0 2 1 0 a measurable function v is called an e wavelet multiplier if v is an e wavelet whenever is an e wavelet in this paper some characterizations of e wavelet multiplier are obtained as an application of these techniques we prove that the set of e wavelets is arcwise connected

fco sampling of digital video using perfect reconstruction filter banks three dimensional nonseparable perfect reconstruction filter banks using three dimensional nonseparable sampling by two fco are proposed filter structures are derived and applied to digital video separation into two bands is obtained and it is shown to perform better from the perceptual point of view than interlaced sequences resulting from the quincunx sampling of a progressively scanned signal in time vertical dimensions

design of filter banks using transformation of variables new results the design problem of multirate filter banks can be divided into two parts the first part involves the issue of reconstruction errors the second part involves the issue of designing good quality subband filters there is a class of techniques that we refer to as transformation of variables which satisfactorily addresses the two parts of the design problem this class of filter banks is specified by its prototype filters and transformation kernel it is the flexibility and relative ease in designing the kernel that enables the technique to satisfactorily address the second part of the design problem we present two new methods of designing the kernel that will enhance the design techniques flexibility and effectiveness the first is the combination of the mcclellan transformation and rotation operators the second introduces and uses the concept of directional singular value decomposition svd

design of signal adapted multidimensional lifting scheme for lossy coding this paper proposes a new method for the design of lifting filters to compute a multidimensional nonseparable wavelet transform our approach is stated in the general case and is illustrated for the 2 d separable and for the quincunx images results are shown for the jpeg2000 database and for satellite images acquired on a quincunx sampling grid the design of efficient quincunx filters is a difficult challenge which has already been addressed for specific cases our approach enables the design of less expensive filters adapted to the signal statistics to enhance the compression efficiency in a more general case it is based on a two step lifting scheme and joins the lifting theory with wiener s optimization the prediction step is designed in order to minimize the variance of the signal and the update step is designed in order to minimize a reconstruction error application for lossy compression shows the performances of the method

a family of nonseparable scaling functions and compactly supported tight framelets abstract given integers b and d with d 1 and b 1 we construct even nonseparable compactly supported refinable functions with dilation factor b that generate multiresolution analyses on l 2 r d these refinable functions are nonseparable in the sense that they cannot be expressed as the product of two functions defined on lower dimensions we use these scaling functions and a slight generalization of a theorem of lai and stockler to construct smooth compactly supported tight framelets both the refinable functions and the framelets they generate can be made as smooth as desired estimates for the supports of these refinable functions and framelets are given

the nonsubsampled contourlet transform theory design and applications in this paper we develop the nonsubsampled contourlet transform nsct and study its applications the construction proposed in this paper is based on a nonsubsampled pyramid structure and nonsubsampled directional filter banks the result is a flexible multiscale multidirection and shift invariant image decomposition that can be efficiently implemented via the a trous algorithm at the core of the proposed scheme is the nonseparable two channel nonsubsampled filter bank nsfb we exploit the less stringent design condition of the nsfb to design filters that lead to a nsct with better frequency selectivity and regularity when compared to the contourlet transform we propose a design framework based on the mapping approach that allows for a fast implementation based on a lifting or ladder structure and only uses one dimensional filtering in some cases in addition our design ensures that the corresponding frame elements are regular symmetric and the frame is close to a tight one we assess the performance of the nsct in image denoising and enhancement applications in both applications the nsct compares favorably to other existing methods in the literature

directional filtering transform for image intra frame compression while directional adaption is introduced into traditional transforms different orders of two 1 d transforms will result in different results of one 2 d transform based upon an anisotropic image model this paper analyzes the effect of transform orders in terms of theoretical coding gain our results reveal that the transform orders have little effect on the coding gain with full decomposition good directional modes and good interpolation however in practical compression schemes since high pass bands are not decomposed fully because of the consideration on complexity different transform orders have different coding performances which can be solved by an adaptive transform order motivated by our analyzed results a directional filtering transform dft in order to distinguish from the common usage on dft is proposed in this paper to better exploit correlations among samples in h 264 intraframe coding it provides an evenly distributed set of prediction modes with an adaptive transform order both interblock and intrablock correlations are exploited in this scheme experimental results in h 264 intraframe coding demonstrate its superiority both objectively and subjectively

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

tomographic reconstruction using nonseparable wavelets in this paper the use of nonseparable wavelets for tomographic reconstruction is investigated local tomography is also presented the algorithm computes both the quincunx approximation and detail coefficients of a function from its projections simulation results showed that nonseparable wavelets provide a reconstruction improvement versus separable wavelets

results on the factorization of multidimensional matrices for paraunitary filterbanks over the complex field this paper undertakes the study of multidimensional finite impulse response fir filterbanks one way to design a filterbank is to factorize its polyphase matrices in terms of elementary building blocks that are fully parameterized factorization of one dimensional 1 d paraunitary pu filterbanks has been successfully accomplished but its generalization to the multidimensional case has been an open problem in this paper a complete factorization for multichannel two dimensional 2 d fir pu filterbanks is presented this factorization is based on considering a two variable fir pu matrix as a polynomial in one variable whose coefficients are matrices with entries from the ring of polynomials in the other variable this representation allows the polyphase matrix to be treated as a one variable matrix polynomial to perform the factorization the definition of paraunitariness is generalized to the ring of polynomials in addition a new degree one building block in the ring setting is defined this results in a building block that generates all two variable fir pu matrices a similar approach is taken for pu matrices with higher dimensions however only a first level factorization is always possible in such cases further factorization depends on the structure of the factors obtained in the first level

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

special paraunitary matrices cayley transform and multidimensional orthogonal filter banks we characterize and design multidimensional md orthogonal filter banks using special paraunitary matrices and the cayley transform orthogonal filter banks are represented by paraunitary matrices in the polyphase domain we define special paraunitary matrices as paraunitary matrices with unit determinant we show that every paraunitary matrix can be characterized by a special paraunitary matrix and a phase factor therefore the design of paraunitary matrices and thus of orthogonal filter banks becomes the design of special paraunitary matrices which requires a smaller set of nonlinear equations moreover we provide a complete characterization of special paraunitary matrices in the cayley domain which converts nonlinear constraints into linear constraints our method greatly simplifies the design of md orthogonal filter banks and leads to complete characterizations of such filter banks

lattice structures for optimal design and robust implementation of two channel perfect reconstruction qmf banks a lattice structure and an algorithm are presented for the design of two channel qmf quadrature mirror filter banks satisfying a sufficient condition for perfect reconstruction the structure inherently has the perfect reconstruction property while the algorithm ensures a good stopband attenuation for each of the analysis filters implementations of such lattice structures are robust in the sense that the perfect reconstruction property is preserved in spite of coefficient quantization the lattice structure has the hierarchical property that a higher order perfect reconstruction qmf bank can be obtained from a lower order perfect reconstruction qmf bank simply by adding more lattice sections several numerical examples are provided in the form of design tables

nonseparable two and three dimensional wavelets we present two and three dimensional nonseparable wavelets they are obtained from discrete time bases by iterating filter banks we consider three sampling lattices quincunx separable by two in two dimensions and fco the design methods are based either on cascade structures or on the mcclellan transformation in the quincunx case we give a few design examples in particular the first example of an orthonormal 2 d wavelet basis with symmetries is constructed

perfect reconstruction filtering with unitary operators and projections abstract perfect reconstruction filter banks wavelets and multiresolution analysis have generated many new results in harmonic analysis and function theory traditionally these techniques are based on families of convolutions and decimation interpolation operations when viewed as linear algebraic operations those building blocks can be generalized to unitary operators and related projections we have developed a general approach to perfect reconstruction and multiresolution analysis that uses polynomial functions of unitary operators and closely related projection operations this general framework is powerful in that it easily captures much of the known theory as special cases and gives a simple mechanism for constructing arbitrary perfect reconstruction families of operators using the general framework we derive necessary and sufficient conditions for the perfect reconstruction property even for higher dimensional non separable sampling lattices and operators the necessary conditions can be interpreted in terms of the spectral mapping theorem and although explicity stated are not directly computable

optimal two sided diagonal scaling for filter bank frames two sided diagonal scaling for transfer matrices is formulated an efficient algorithm is proposed to compute globally optimal solutions using the iterated bi section and linear matrix inequality lmi solver it is shown that the two sided scaling of filter bank fb frames can be implemented by the adjustment of channel gains and the periodic precoding of source signal and that the frame bound ratio of fb frames can be effectively improved by such scaling explicit formulas are established for both uniform and nonuniform fb frames including detail formulas for discrete weyl heisenberg frames and tree structured fbs discrete wavelets different examples show the effectiveness of the obtained results

efficient architectures for 1 d and 2 d lifting based wavelet transforms the lifting scheme reduces the computational complexity of the discrete wavelet transform dwt by factoring the wavelet filters into cascades of simple lifting steps that process the input samples in pairs we propose four compact and efficient hardware architectures for implementing lifting based dwts namely one dimensional 1 d and two dimensional 2 d versions of what we call recursive and dual scan architectures the 1 d recursive architecture exploits interdependencies among the wavelet coefficients by interleaving on alternate clock cycles using the same datapath hardware the calculation of higher order coefficients along with that of the first stage coefficients the resulting hardware utilization exceeds 90 in the typical case of a five stage 1 d dwt operating on 1024 samples the 1 d dual scan architecture achieves 100 datapath hardware utilization by processing two independent data streams together using shared functional blocks the recursive and dual scan architectures can be readily extended to the 2 d case the 2 d recursive architecture is roughly 25 faster than conventional implementations and it requires a buffer that stores only a few rows of the data array instead of a fixed fraction typically 25 or more of the entire array the 2 d dual scan architecture processes the column and row transforms simultaneously and the memory buffer size is comparable to existing architectures