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

ill posedness in three dimensional plastic flow this paper examines partial differential equations for frictional materials flowing via plastic yield including the equations given by the critical state theory of soil mechanics in particular the material density is considered as a dependent variable in previous work we demonstrated that two dimensional plastic flow may be ill posed due to an instability along two rays in fourier transform space in this paper we show that in three dimensions the equations are linearly well posed provided all three strain rates are nonzero

three point boundary value problems for second order discrete equations we formulate existence results for solutions to discrete equations which approximate three point boundary value problems for second order ordinary differential equations c 2003 elsevier science ltd all rights reserved

a new class of orthonormal symmetric wavelet bases using a complex allpass filter this paper considers the design of the whole sample symmetric wss paraunitary filterbanks composed of a single complex allpass filter and gives a new class of real valued orthonormal symmetric wavelet bases first the conditions that the complex allpass filter has to satisfy are derived from the symmetry and orthonormality conditions of wavelets and its transfer function is given to satisfy these conditions second the paraunitary filter banks are designed by using the derived transfer function from the viewpoints of the regularity and frequency selectivity a new method for designing the proposed paraunitary filterbanks with a given degrees of flatness is presented the proposed method is based on the formulation of a generalized eigenvalue problem by using the remez exchange algorithm therefore the filter coefficients can be easily obtained by solving the eigenvalue problem and the optimal solution is attained through a few iterations furthermore both the maximally flat and minimax solutions are also included in the proposed method as two specific cases the maximally flat filters have a closed form solution without any iteration finally some design examples are presented to demonstrate the effectiveness of the proposed method

mathematical properties of the jpeg2000 wavelet filters the legall 5 3 and daubechies 9 7 filters have risen to special prominence because they were selected for inclusion in the jpeg2000 standard we determine their key mathematical features riesz bounds order of approximation and regularity holder and sobolev we give approximation theoretic quantities such as the asymptotic constant for the l sup 2 error and the angle between the analysis and synthesis spaces which characterizes the loss of performance with respect to an orthogonal projection we also derive new asymptotic error formulae that exhibit bound constants that are proportional to the magnitude of the first nonvanishing moment of the wavelet the daubechies 9 7 stands out because it is very close to orthonormal but this turns out to be slightly detrimental to its asymptotic performance when compared to other wavelets with four vanishing moments

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

embedded image coding using zerotrees of wavelet coefficients the embedded zerotree wavelet algorithm ezw is a simple yet remarkably effective image compression algorithm having the property that the bits in the bit stream are generated in order of importance yielding a fully embedded code the embedded code represents a sequence of binary decisions that distinguish an image from the null image using an embedded coding algorithm an encoder can terminate the encoding at any point thereby allowing a target rate or target distortion metric to be met exactly also given a bit stream the decoder can cease decoding at any point in the bit stream and still produce exactly the same image that would have been encoded at the bit rate corresponding to the truncated bit stream in addition to producing a fully embedded bit stream the ezw consistently produces compression results that are competitive with virtually all known compression algorithms on standard test images yet this performance is achieved with a technique that requires absolutely no training no pre stored tables or codebooks and requires no prior knowledge of the image source the ezw algorithm is based on four key concepts 1 a discrete wavelet transform or hierarchical subband decomposition 2 prediction of the absence of significant information across scales by exploiting the self similarity inherent in images 3 entropy coded successive approximation quantization and 4 universal lossless data compression which is achieved via adaptive arithmetic coding

on the representation of operators in bases of compactly supported wavelets this paper describes exact and explicit representations of the differential operators d n dx n n 1 2 cdots in orthonormal bases of compactly supported wavelets as well as the representations of the hilbert transform and fractional derivatives the method of computing these representations is directly applicable to multidimensional convolution operators also sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring o n log n operations for computing the wavelet coefficients of all n circulant shifts of a vector of the length n 2 n is constructed as an example of an application of this algorithm it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector see g beylkin r r coifman and v rokhlin comm pure appl math 44 1991 pp 141 183 may be reduced from o n to o log 2 n significant entries

generalized sampling stability and performance analysis generalized sampling provides a general mechanism for recovering an unknown input function f x spl epsiv spl hscr from the samples of the responses of m linear shift invariant systems sampled at 1 mth the reconstruction rate the system can be designed to perform a projection of f x onto the reconstruction subspace v spl phi span spl phi x k sub k spl epsiv z for example the family of bandlimited signals with spl phi x sinc x this implies that the reconstruction will be perfect when the input signal is included in v spl phi the traditional framework of papoulis 1977 generalized sampling theory otherwise one recovers a signal approximation f x spl epsiv v spl phi that is consistent with f x in the sense that it produces the same measurements to characterize the stability of the algorithm we prove that the dual synthesis functions that appear in the generalized sampling reconstruction formula constitute a riesz basis of v spl phi and we use the corresponding riesz bounds to define the condition number of the system we then use these results to analyze the stability of various instances of interlaced and derivative sampling next we consider the issue of performance which becomes pertinent once we have extended the applicability of the method to arbitrary input functions that is when spl hscr is considerably larger than v spl phi and the reconstruction is no longer exact we show that the generalized sampling solution is essentially equivalent to the optimal minimum error approximation we then perform a detailed analysis for the case in which the analysis filters are in l sub 2 and determine all relevant bound constants explicitly finally we use an interlaced sampling example to illustrate these various calculations

cellular automata models for the flow of granular materials abstract the importance of particle irregularity in the flow of granular material can be demonstrated using simple experiments we introduce cellular automata which model nonspherical particles using rules derived from experiments these automata can reproduce many features of the physical flows including the alignment of the particles the formation of a complex channel region the shape of the upper free surface and the formation of defects in the body of the material

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

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

analysis of low bit rate image transform coding calculations based on high resolution quantizations prove that the distortion rate d r of an image transform coding is proportional to 2 sup 2r when r is large enough in wavelet and block cosine bases we show that if r 1 bit pixel then d r varies like r sup 1 2 spl gamma where spl gamma remains of the order of 1 for most natural images the improved performance of embedded codings in wavelet bases is analyzed at low bit rates we show that the compression performance of an orthonormal basis depends mostly on its ability to approximate images with a few nonzero vectors

regular wavelets a discrete time approach regularity is a new filter property brought by wavelet theory for perfect reconstruction octave band filter banks tools for investigating its role in coding applications are provided first discrete time interpretations and optimal estimates of regularity are reviewed then a simple design procedure for paraunitary fir filter banks with optimal trade off between frequency selectivity and regularity is given finally the obtained filters are used to measure the effect of regularity versus frequency selectivity in a still image compression scheme with optimized rate distortion in this case regularity is shown to be more relevant than frequency selectivity especially for short filters

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

compactly supported orthogonal and biorthogonal sqrt 5 refinement wavelets with 4 fold symmetry recently radic5 refinement hierarchical sampling has been studied and radic5 refinement has been used for surface subdivision compared with other refinements such as the dyadic or quincunx refinement radic5 refinement has a special property that the nodes in a refined lattice form groups of five nodes with these five nodes having different x and y coordinates this special property has been shown to be very useful to represent adaptively and render complex and procedural geometry when radic5 refinement is used for multiresolution data processing radic5 refinement filter banks and wavelets are required while the construction of 2 d nonseparable bi orthogonal wavelets with the dyadic or quincunx refinement has been studied by many researchers the construction of bi orthogonal wavelets with radic5 refinement has not been investigated the main goal of this paper is to construct compactly supported orthogonal and biorthogonal wavelets with radic5 refinement in this paper we obtain block structures of orthogonal and biorthogonal radic5 refinement fir filter banks with 4 fold rotational symmetry we construct compactly supported orthogonal and biorthogonal wavelets based on these block structures

on the existence of two solutions of the four point problem we have proved the existence of at least two different solutions to the equation u t u u s where s r the solutions fulfill four point conditions u a u c u d u b where a c d b a b c d r

design of iir digital allpass filters based on eigenvalue problem a new method is proposed for designing iir digital allpass filters with an equiripple phase response that can be proven to be optimal in the chebyshev sense the proposed procedure is based on the formulation of an eigenvalue problem by using the remez exchange algorithm since there exists more than one eigenvalue in the general eigenvalue problem we introduce a new and very simple selection rule for the eigenvalue to be searched for where the rational interpolation is performed if and only if the real maximum eigenvalue is chosen therefore the solution of the rational interpolation problem can be gotten by computing only one eigenvector corresponding to the real maximum eigenvalue and the optimal filter coefficients are easily obtained through a few iterations without any initial guess of the solution the design algorithm proposed not only retains the speed inherent in the remez exchange algorithm but also simplifies the interpolation step because it has been reduced to the computation of the real maximum eigenvalue two examples are designed to demonstrate the effectiveness of this method

linear phase paraunitary filter banks theory factorizations and designs m channel maximally decimated filter banks have been used in the past to decompose signals into subbands the theory of perfect reconstruction filter banks has also been studied extensively nonparaunitary systems with linear phase filters have also been designed the authors study paraunitary systems in which each individual filter in the analysis synthesis banks has linear phase specific instances of this problem have been addressed by other authors and linear phase paraunitary systems have been shown to exist this property is often desirable for several applications particularly in image processing they begin by answering several theoretical questions pertaining to linear phase paraunitary systems next they develop a minimal factorization for a large class of such systems this factorization will be proved to be complete for even m further they structurally impose the additional condition that the filters satisfy pairwise mirror image symmetry in the frequency domain this significantly reduces the number of parameters to be optimized in the design process they then demonstrate the use of these filter banks in the generation of m band orthonormal wavelets several design examples are also given to validate the theory

directional lapped orthogonal transform theory and design this paper proposes a directional design method of 2 d nonseparable linear phase paraunitary filter banks the proposed method is based on a lattice structure consisting of the 2 d separable dct block and nonseparable support extension processes because of the nonseparability the bases are allowed to be directional with the critically fixed subsampling overlapping orthogonal symmetric real valued and compact support properties first a novel vanishing moment vm condition is introduced as a suitable directional constraint where the moment is referred to as the trend vm the condition forces wavelet filters i e high pass and bandpass filters to annihilate trend surface components second some theoretical properties of tvms are discussed for general 2 d paraunitary systems and then the properties are applied to the lattice parameters in order to verify the significance several design examples are shown the trend surface annihilation properties are numerically confirmed and the denoising capability is evaluated for images through shrinkage it is shown that our proposed transforms yield perceptually preferable results

the polyphase with advance representation and linear phase lifting factorizations a matrix theory is developed for the noncausal polyphase representation that underlies the theory of lifted filter banks and wavelet transforms the theory presented here develops an extensive matrix algebra framework for analyzing and implementing linear phase two channel filter banks via lifting cascade schemes whole sample symmetric and half sample symmetric linear phase filter banks are characterized completely in terms of the polyphase with advance representation and new proofs are given of linear phase lifting factorization theorems for these two principal classes of linear phase filter banks the theory benefits significantly from a number of group theoretic structures arising in the polyphase with advance representation and in the lifting factorization of linear phase filter banks these results form the foundations of the lifting methodology employed in part 2 of the iso iec jpeg 2000 still image coding standard

image coding using wavelet transform a scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed this method involves two steps first a wavelet transform used in order to obtain a set of biorthogonal subclasses of images the original image is decomposed at different scales using a pyramidal algorithm architecture the decomposition is along the vertical and horizontal directions and maintains constant the number of pixels required to describe the image second according to shannon s rate distortion theory the wavelet coefficients are vector quantized using a multiresolution codebook to encode the wavelet coefficients a noise shaping bit allocation procedure which assumes that details at high resolution are less visible to the human eye is proposed in order to allow the receiver to recognize a picture as quickly as possible at minimum cost a progressive transmission scheme is presented it is shown that the wavelet transform is particularly well adapted to progressive transmission

a new class of biorthogonal wavelet systems for image transform coding we construct general biorthogonal coifman wavelet systems a new class of compactly supported biorthogonal wavelet systems with vanishing moments equally distributed for a scaling function and wavelet pair a time domain design method is employed and closed form expressions for the impulse responses and the frequency responses of the corresponding dual filters are derived the resulting filter coefficients are all dyadic fractions which is an attractive feature in the realization of multiplication free discrete wavelet transform even ordered systems in this family are symmetric which correspond to linear phase dual filters in particular three filterbanks fbs in this family are systematically verified to have competitive compression potential to the 9 7 tap biorthogonal wavelet fb by cohen et al 1992 which is currently the most widely used one in the field of wavelet transform coding in addition the proposed fb s have much smaller computational complexity in terms of floating point operations required in transformation and therefore indicate a better tradeoff between compression performance and computational complexity

wavelets and recursive filter banks it is shown that infinite impulse response iir filters lead to more general wavelets of infinite support than finite impulse response fir filters a complete constructive method that yields all orthogonal two channel filter banks where the filters have rational transfer functions is given and it is shown how these can be used to generate orthonormal wavelet bases a family of orthonormal wavelets that have a maximum number of disappearing moments is shown to be generated by the halfband butterworth filters when there is an odd number of zeros at pi it is shown that closed forms for the filters are available without need for factorization a still larger class of orthonormal wavelet bases having the same moment properties and containing the daubechies and butterworth filters as the limiting cases is presented it is shown that it is possible to have both linear phase and orthogonality in the infinite impulse response case and a constructive method is given it is also shown how compactly supported bases may be orthogonalized and bases for the spline function spaces are constructed

optimization of integer wavelet transforms based on difference correlation structures in this paper a novel lifting integer wavelet transform based on difference correlation structure dccs liwt is proposed first we establish a relationship between the performance of a linear predictor and the difference correlations of an image the obtained results provide a theoretical foundation for the following construction of the optimal lifting filters then the optimal prediction lifting coefficients in the sense of least square prediction error are derived dccs liwt puts heavy emphasis on image inherent dependence a distinct feature of this method is the use of the variance normalized autocorrelation function of the difference image to construct a linear predictor and adapt the predictor to varying image sources the proposed scheme also allows respective calculations of the lifting filters for the horizontal and vertical orientations experimental evaluation shows that the proposed method produces better results than the other well known integer transforms for the lossless image compression

design of iir digital filters based on eigenvalue problem this paper presents a new method for designing iir digital filters with optimum magnitude response in the chebyshev sense and different order numerator and denominator the proposed procedure is based on the formulation of a generalized eigenvalue problem by using remez exchange algorithm since there exist more than one eigenvalue in the general eigenvalue problem we introduce a very simple selection rule for the eigenvalue to be sought for where the rational interpolation is performed if and only if the positive minimum eigenvalue is chosen therefore the solution of the rational interpolation problem can be obtained by computing only one eigenvector corresponding to the positive minimum eigenvalue and the optimal filter coefficients are easily obtained through a few iterations the design algorithm proposed in this paper not only retains the speed inherent in the remez exchange algorithm but also simplifies the interpolation step because it has been reduced to the computation of the positive minimum eigenvalue some properties of the filters such as lowpass filters bandpass filters and so on are discussed and several design examples are presented to demonstrate the effectiveness of this method

a unified approach to some two point three point and four point boundary value problems with caratheodory functions this paper deals with boundary value problems formula and formula wherea b 0 0 aandbare given real numbers gis a continuous strictly increasing function withg r r andfis a caratheodory function satisfying a nagumo condition the problems have solutions if and only if there exist a lower solutionxand an upper solutionyandx t y t for alla t b

generalized daubechies wavelet families we present a generalization of the orthonormal daubechies wavelets and of their related biorthogonal flavors cohen daubechies feauveau 9 7 our fundamental constraint is that the scaling functions should reproduce a predefined set of exponential polynomials this allows one to tune the corresponding wavelet transform to a specific class of signals thereby ensuring good approximation and sparsity properties the main difference with the classical construction of daubechies is that the multiresolution spaces are derived from scale dependent generating functions however from an algorithmic standpoint mallat s fast wavelet transform algorithm can still be applied the only adaptation consists in using scale dependent filter banks finite support ensures the same computational efficiency as in the classical case we characterize the scaling and wavelet filters construct them and show several examples of the associated functions we prove that these functions are square integrable and that they converge to their classical counterparts of the corresponding order

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

multidimensional nonseparable oversampled lapped transforms theory and design this paper extends the theory of the one dimensional oversampled linear phase perfect reconstruction filter banks olpprfbs developed by gan et al to multidimensional md cases and proposes md nonseparable oversampled lapped transforms nsolts nsolts allow us to achieve an overcomplete analysis synthesis system with nonseparable symmetric real valued overlapping and compact supported filters the proposed systems are based on lattice structures and the redundancy is flexibly controlled by the number of channels and downsampling ratio the proposed structure is shown to be capable of constructing parseval tight frames in any number of dimensions the number of design parameters are examined under the parseval tight frame constraint in order to design nsolts specified for sparse approximation of image or volume data an example based design procedure is introduced the effectiveness of this method is verified by examining design samples and evaluating their sparse approximation performance using the iterative hard thresholding algorithm for a natural image and mri volume data patch

a design method of multidimensional linear phase paraunitary filter banks with a lattice structure a lattice structure of multidimensional md linear phase paraunitary filter banks lppufbs is proposed which makes it possible to design such systems in a systematic manner our proposed structure can produce md lppufbs whose filters all have the region of support spl nscr m spl xi where m and spl xi are the decimation and positive integer diagonal matrices respectively and spl nscr n denotes the set of integer vectors in the fundamental parallelepiped of a matrix n it is shown that if spl nscr m is reflection invariant with respect to some center then the reflection invariance of spl nscr m spl xi is guaranteed this fact is important in constructing md linear phase filter banks because the reflection invariance is necessary for any linear phase filter since our proposed system structurally restricts both the paraunitary and linear phase properties an unconstrained optimization process can be used to design md lppufbs our proposed structure is developed for both an even and an odd number of channels and includes the conventional 1 d system as a special case it is also shown to be minimal and the no dc leakage condition is presented some design examples show the significance of our proposed structure for both the rectangular and nonrectangular decimation cases

oriented wavelet transform for image compression and denoising in this paper we introduce a new transform for image processing based on wavelets and the lifting paradigm the lifting steps of a unidimensional wavelet are applied along a local orientation defined on a quincunx sampling grid to maximize energy compaction the orientation minimizing the prediction error is chosen adaptively a fine grained multiscale analysis is provided by iterating the decomposition on the low frequency band in the context of image compression the multiresolution orientation map is coded using a quad tree the rate allocation between the orientation map and wavelet coefficients is jointly optimized in a rate distortion sense for image denoising a markov model is used to extract the orientations from the noisy image as long as the map is sufficiently homogeneous interesting properties of the original wavelet are preserved such as regularity and orthogonality perfect reconstruction is ensured by the reversibility of the lifting scheme the mutual information between the wavelet coefficients is studied and compared to the one observed with a separable wavelet transform the rate distortion performance of this new transform is evaluated for image coding using state of the art subband coders its performance in a denoising application is also assessed against the performance obtained with other transforms or denoising methods

a new algorithm for the embedding of a prediction mechanism into the jpeg2000 coding chain in this paper we propose the embedding of a prediction mechanism into a part of the coding structure of jpeg2000 image compression standard in order to reduce the amount of bits sent to the arithmetic coder without any significant changes into the standard architecture and without loosing performance the prediction is based upon an innovative processing of the data structures used by the standard jpeg2000 in progressive coding and the addition of a prediction matrix whose computation does not add any overhead at the decoder side experiments are performed to test the efficacy of the prediction mechanism and results are compared to the standard jpeg2000 and other similar approaches tests are documented over a set of well known images from literature also against different kinds of added noise performance in terms of saved bits are reported and a new figure of merit is defined to test the efficiency of prediction the results prove that the new proposal overcomes the standard and other related approaches for the entire set of referenced images with significant gain in synthetic images also in presence of noise

two dimensional 2 times oversampled dft modulated filter banks and critically sampled modified dft modulated filter banks this paper investigates two dimensional 2d 2 oversampled dft modulated filter banks and 2d critically sampled modified dft mdft modulated filter banks as well as their design the structure and perfect reconstruction pr condition of 2d 2 oversampled dft modulated filter banks are presented in terms of the polyphase decompositions of prototype filters pfs in the double prototype case the part solutions of the pr condition are parameterized by imposing the 2d two channel lifting structure on each pair of the polyphase components of analysis and synthesis pfs based on the parametric structure the analysis and synthesis pfs are separately designed by constrained quadratic programs the obtained filter banks are of structurally pr moreover 2d critically sampled mdft modulated filter banks are proposed it is proved that 2d critically sampled pr mdft modulated filter banks can be rebuilt from 2d 2 oversampled pr dft modulated filter banks when the decimation matrices satisfy a permissible condition and the analysis and synthesis pfs are identical and symmetric with respect to the origin a numerical algorithm is given to design 2d critically sampled pr mdft modulated filter banks and the obtained filter banks are of numerically pr

design of almost symmetric orthogonal wavelet filter bank via direct optimization it is a well known fact that compact support dyadic wavelets based on the two channel filter banks fbs cannot be simultaneously orthogonal and symmetric although orthogonal wavelets have the energy preservation property biorthogonal wavelets are preferred in image processing applications because of their symmetric property in this paper a novel method is presented for the design of almost symmetric orthogonal wavelet fb orthogonality is structurally imposed by using the unnormalized lattice structure and this leads to an objective function which is relatively simple to optimize the designed filters have good frequency response flat group delay almost symmetric filter coefficients and symmetric wavelet function

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

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

existence and multiplicity of solutions for four point boundary value problems at resonance abstract this paper deals with the second order four point boundary value problem x t f t x t 0 t 1 x 0 ax x 1 bx where f 0 1 r r is continuous 0 1 0 a 1 1 0 b 1 a 1 b 1 a s 1 b 0 by using the coincidence degree theory the method of lower and upper solutions is established as applications some existence and multiplicity results are obtained

separable and nonseparable multiwavelets in multiple dimensions we report on a method of constructing multidimensional biorthogonal interpolating multiwavelets the approach is based upon polynomial interpolation in rd c de boor and a ron math comput 58 198 1997 and an extension of the lifting scheme j kovacevic and w sweldens ieee trans image process 9 no 3 480 2000 the constructed wavelets have compact support are nearly isotropic and retain partial scale invariance leading to a fast and efficient multidimensional wavelet transform we demonstrate an implementation for these wavelets of variable polynomial order up to four dimensions finally we show that these wavelets have a much sparser representation of discontinuous functions as compared to tensor product wavelets which allows for a more compact and efficient representation

from differential equations to the construction of new wavelet like bases in this paper an approach is introduced based on differential operators to construct wavelet like basis functions given a differential operator l with rational transfer function elementary building blocks are obtained that are shifted replicates of the green s function of l it is shown that these can be used to specify a sequence of embedded spline spaces that admit a hierarchical exponential b spline representation the corresponding b splines are entirely specified by their poles and zeros they are compactly supported have an explicit analytical form and generate multiresolution riesz bases moreover they satisfy generalized refinement equations with a scale dependent filter and lead to a representation that is dense in l sub 2 this allows us to specify a corresponding family of semi orthogonal exponential spline wavelets which provides a major extension of earlier polynomial spline constructions these wavelets are completely characterized and it is proven that they satisfy the following remarkable properties 1 they are orthogonal across scales and generate riesz bases at each resolution level 2 they yield unconditional bases of l sub 2 either compactly supported b spline type or with exponential decay orthogonal or dual type 3 they have n vanishing exponential moments where n is the order of the differential operator 4 they behave like multiresolution versions of the operator l from which they are derived and 5 their order of approximation is n m where n and m give the number of poles and zeros respectively last but not least the new wavelet like decompositions are as computationally efficient as the classical ones they are computed using an adapted version of mallat s filter bank algorithm where the filters depend on the decomposition level

construction of biorthogonal wavelets starting from any two multiresolutions starting from any two given multiresolution analyses of l sub 2 v sub j sup 1 sub j spl isin z and v sub j sup 2 sub j spl isin z we construct biorthogonal wavelet bases that are associated with this chosen pair of multiresolutions thus our construction method takes a point of view opposite to the one of cohen daubechies feauveau 1992 which starts from a well choosen pair of biorthogonal discrete filters in our construction the necessary and sufficient condition is the nonperpendicularity of the multiresolutions

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

two dimensional orthogonal filter banks and wavelets with linear phase two dimensional 2 d compactly supported orthogonal wavelets and filter banks having linear phase are presented two cases are discussed wavelets with two fold symmetry centrosymmetric and wavelets with four fold symmetry that are symmetric or anti symmetric about the vertical and horizontal axes we show that imposing the requirement of linear phase in the case of order factorable wavelets imposes a simple constraint on each of its polynomial order 1 factors we thus obtain a simple and complete method of constructing orthogonal order factorable wavelets with linear phase this method is exemplified by design in the case of four band separable sampling an interesting result that is similar to the one well known in the one dimensional 1 d case is obtained orthogonal order factorable wavelets cannot be both continuous and have four fold symmetry

approximation power of biorthogonal wavelet expansions this paper looks at the effect of the number of vanishing moments on the approximation power of wavelet expansions the strang fix conditions imply that the error for an orthogonal wavelet approximation at scale a 2 sup i globally decays as a sup n where n is the order of the transform this is why for a given number of scales higher order wavelet transforms usually result in better signal approximations we prove that this result carries over for the general biorthogonal case and that the rate of decay of the error is determined by the order properties of the synthesis scaling function alone we also derive asymptotic error formulas and show that biorthogonal wavelet transforms are equivalent to their corresponding orthogonal projector as the scale goes to zero these results strengthen sweldens earlier analysis and confirm that the approximation power of biorthogonal and semi orthogonal wavelet expansions is essentially the same finally we compare the asymptotic performance of various wavelet transforms and briefly discuss the advantages of splines we also indicate how the smoothness of the basis functions is beneficial in reducing the approximation error

eigenvalues of spl darr 2 h and convergence of the cascade algorithm this paper is about the eigenvalues and eigenvectors of spl darr 2 h the ordinary fir filter h is a convolution with a vector h h o h n which is the impulse response the operator spl darr 2 downsamples the output y h x keeping the even numbered components y 2n where h is represented by a constant diagonal matrix this is a toeplitz matrix with h k on its kth diagonal the odd numbered rows are removed in spl darr 2 h the result is a double shift between rows yielding a block toeplitz matrix with 1 spl times 2 blocks iteration of the filter is governed by the eigenvalues if the transfer function h z spl sigma h k z sup k has a zero of order p at z 1 corresponding to spl omega spl pi then spl darr 2 h has p special eigenvalues 1 2 1 4 1 2 sup p we show how each additional zero at spl pi divides all eigenvalues by 2 and creates a new eigenvector for spl lambda 1 2 this eigenvector solves the dilation equation spl phi t 2 spl sigma h k spl phi 2t k at the integers t n the left eigenvectors show how 1 t t sup p 1 can be produced as combinations of spl phi t k the dilation equation is solved by the cascade algorithm which is an infinite iteration of m spl darr 2 2h convergence in l sup 2 is governed by the eigenvalues of t spl darr 2 2hh sup t corresponding to the response 2h z h z sup 1 we find a simple proof of the necessary and sufficient condition for convergence

coiflet systems and zero moments the coifman wavelets created by daubechies 1992 have more zero moments than imposed by specifications this results in systems with approximately equal numbers of zero scaling function and wavelet moments and gives a partitioning of the systems into three well defined classes the nonunique solutions are more complex than for daubechies wavelets

wavelet filter evaluation for image compression choice of filter bank in wavelet compression is a critical issue that affects image quality as well as system design although regularity is sometimes used in filter evaluation its success at predicting compression performance is only partial a more reliable evaluation can be obtained by considering an l level synthesis analysis system as a single input single output linear shift variant system with a response that varies according to the input location module 2 sup l 2 sup l by characterizing a filter bank according to its impulse response and step response in addition to regularity we obtain reliable and relevant for image coding filter evaluation metrics using this approach we have evaluated all possible reasonably short less than 36 taps in the synthesis analysis pair minimum order biorthogonal wavelet filter banks of this group of over 4300 candidate filter banks we have selected and present here the filters best suited to image compression while some of these filters have been published previously others are new and have properties that make them attractive in system design

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

data compression and harmonic analysis in this paper we review some recent interactions between harmonic analysis and data compression the story goes back of course to shannon s r d theory in the case of gaussian stationary processes which says that transforming into a fourier basis followed by block coding gives an optimal lossy compression technique practical developments like transform based image compression have been inspired by this result in this paper we also discuss connections perhaps less familiar to the information theory community growing out of the field of harmonic analysis recent harmonic analysis constructions such as wavelet transforms and gabor transforms are essentially optimal transforms for transform coding in certain settings some of these transforms are under consideration for future compression standards we discuss some of the lessons of harmonic analysis in this century typically the problems and achievements of this field have involved goals that were not obviously related to practical data compression and have used a language not immediately accessible to outsiders nevertheless through an extensive generalization of what shannon called the sampling theorem harmonic analysis has succeeded in developing new forms of functional representation which turn out to have significant data compression interpretations we explain why harmonic analysis has interacted with data compression and we describe some interesting recent ideas in the field that may affect data compression in the future

quantitative fourier analysis of approximation techniques i interpolators and projectors we present a general fourier based method that provides an accurate prediction of the approximation error as a function of the sampling step t our formalism applies to an extended class of convolution based signal approximation techniques which includes interpolation generalized sampling with prefiltering and the projectors encountered in wavelet theory we claim that we can predict the l sup 2 approximation error by integrating the spectrum of the function to approximate not necessarily bandlimited against a frequency kernel e spl omega that characterizes the approximation operator this prediction is easier yet more precise than was previously available our approach has the remarkable property of providing a global error estimate that is the average of the true approximation error over all possible shifts of the input function our error prediction is exact for stationary processes as well as for bandlimited signals we apply this method to the comparison of standard interpolation and approximation techniques our method has interesting implications for approximation theory in particular we use our results to obtain some new asymptotic expansions of the error as t spl rarr 0 as well as to derive improved upper bounds of the kind found in the strang fix 1971 theory we finally show how we can design quasi interpolators that are near optimal in the least squares sense

a viscoplastic approach to model the flow of granular solids the flow of granular solids within rigid walls is modeled using continuum mechanics the problem is represented as a viscoplastic flow in which the discontinuity function is taken as in previous works by gray and stiles while the flow rule is modeled by the von mises criterion the resulting model is incompressible and based on non associated viscoplasticity the apparent viscosity results in a non linear function of the second invariant of the symmetric rate of deformation tensor and of the pressure friction cohesion and fluidity of the granular model are taken into account the constitutive model has been implemented assuming steady state in which the granular material flows under a critical state incompressible behavior discretization of the problem has been carried out by finite elements with direct iteration techniques to solve the non linear system of equations the model has been applied to the massive flow of granular material stored in vertical silos and hoppers with axisymmetric or planar shape comparisons with experimental tests performed by other authors are presented together with parametric investigations to identify the main variables affecting the response

instability in critical state theories of granular flow conditions characterizing the occurrence of instability in the partial differential equations arising in critical state theories for granular flow are derived this continues earlier work studying ill posedness in these equations implications of the stability results for the formation of shear bands are discussed

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

multiplicity of sign changing solutions for some four point boundary value problem abstract in this paper we obtain using leray schauder degree theory some multiplicity results for sign changing solutions of a four point boundary value problem we assume the existence of a pair of strict lower and upper solutions and some additional conditions on the nonlinear term f

quantitative fourier analysis of approximation techniques ii wavelets for pt i see ibid vol 47 no 10 p 2783 95 1999 in a previous paper we proposed a general fourier method that provides an accurate prediction of the approximation error irrespective of the scaling properties of the approximating functions here we apply our results when these functions satisfy the usual two scale relation encountered in dyadic multiresolution analysis as a consequence of this additional constraint the quantities introduced in our previous paper can be computed explicitly as a function of the refinement filter this is in particular true for the asymptotic expansion of the approximation error for biorthonormal wavelets as the scale tends to zero one of the contributions of this paper is the computation of sharp asymptotically optimal upper bounds for the least squares approximation error another contribution is the application of these results to b splines and daubechies 1988 1992 scaling functions which yields explicit asymptotic developments and upper bounds thanks to these explicit expressions we can quantify the improvement that can be obtained by using b splines instead of daubechies wavelets in other words we can use a coarser spline sampling and achieve the same reconstruction accuracy as daubechies specifically we show that this sampling gain converges to spl pi as the order tends to infinity

a general sampling theory for nonideal acquisition devices the authors first describe the general class of approximation spaces generated by translation of a function spl psi x and provide a full characterization of their basis functions they then present a general sampling theorem for computing the approximation of signals in these subspaces based on a simple consistency principle the theory puts no restrictions on the system input which can be an arbitrary finite energy signal bandlimitedness is not required in contrast to previous approaches this formulation allows for an independent specification of the sampling analysis and approximation synthesis spaces in particular when both spaces are identical the theorem provides a simple procedure for obtaining the least squares approximation of a signal they discuss the properties of this new sampling procedure and present some examples of applications involving bandlimited and polynomial spline signal representations they also define a spectral coherence function that measures the similarity between the sampling and approximation spaces and derive a relative performance bound for the comparison with the least squares solution

two dimensional orthogonal wavelets with vanishing moments we investigate a very general subset of 2 d orthogonal compactly supported wavelets this subset includes all the wavelets with a corresponding wavelet polyphase matrix that can be factored as a product of factors of degree 1 in one variable in this paper we consider in particular wavelets with vanishing moments the number of vanishing moments that can be achieved increases with the increase in the mcmillan degrees of the wavelet matrix we design wavelets with the maximal number of vanishing moments for given mcmillan degrees by solving a set of nonlinear constraints on the free parameters defining the wavelet matrix and discuss their relation to regular smooth wavelets design examples are given for two fundamental sampling schemes the quincunx and the four band separable sampling the relation of the wavelets to the well known 1 d daubechies wavelets with vanishing moments is discussed

a multiscale relaxation algorithm for snr maximization in nonorthogonal subband coding develops a technique for improving the applicability of complete nonorthogonal multiresolution transforms to image coding as is well known the l sup 2 norm of the quantization errors is not preserved by nonorthogonal transforms so the l sup 2 reconstruction error may be unacceptably large however given the quantizers and synthesis filters the authors show that this artifact can be eliminated by formulating the coding problem as that of minimizing the l sup 2 reconstruction error over the set of possible encoded images with this new formulation the coding problem becomes a high dimensional discrete optimization problem and features a coupling between the redundancy removing and quantization operations a practical solution to the optimization problem is presented in the form of a multiscale relaxation algorithm using inter and intrascale quantization noise feedback filters bounds on the coding gain over the standard coding technique are derived a simple extension of the algorithm allows for the use of a weighted l sup 2 error criterion and deadband non mmse quantizers experiments using biorthogonal spline filter banks demonstrate appreciable snr gains over the standard coding technique and comparable visual improvements