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

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

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

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

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

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

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

the asymptotic iteration method revisited the asymptotic iteration method is a technique for solving analytically and approximately the linear second order differential equation especially the eigenvalue problems that frequently appear in theoretical and mathematical physics the analysis and mathematical justifications of the success and failure of the asymptotic iteration method are detailed in this work a theorem explaining why the asymptotic iteration method works for the eigenvalue problem is presented as a byproduct a new procedure to generate unlimited classes of exactly solvable differential equations is also introduced

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

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

random eigenvalue problems in structural dynamics experimental investigations an experimental investigation of uncertainty in natural frequencies of linear structures estimated from measured frequency response function frf under dynamic loading is presented experiments were conducted on one hundred nominally identical realizations of a fixed fixed beam each realisation was obtained by placin g twelve identical masses at random spatial locations generated by a computer along the length of the beam the total random mass is about 2 of the total mass of the beam these experiments contain parametric uncertainty in the mass matrix and hence may be useful for the validation of many random eigenvalue analysis and prediction methods currently available to structural dynamicist predictions from monte carlo simulation of deterministic finite elements model are compared with experiments it is co ncluded that the method of estimation of natural frequencies from frfs and the spatial location of the measurements has significant influence upon the first two moments mean and standard deviation of the natural frequency ensemble furthermore whilst the monte carlo simulation estimates of the mean and standard deviation are in reasonable agreement with experiments at higher frequencies the probability density function differ appreciably within the limits of the sample size investigated in this study

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