theory and lattice structure of complex paraunitary filterbanks with filters of hermitian symmetry antisymmetry properties the theory of the real coefficient linear phase filterbank lpfb is extended to the complex case in two ways leading to two generalized classes of m channel filterbanks one is the symmetric antisymmetric filterbank safb where all filters are symmetric or antisymmetric the other is the complex linear phase filterbank clpfb where all filters are hermitian symmetric or hermitian antisymmetric and hence have the linear phase property necessary conditions on the filter symmetry polarity and lengths for the existence of permissible solutions are investigated complete and minimal lattice structures are developed for the paraunitary safb and paraunitary clpfb where the channel number m is arbitrary even or odd and the subband filters could have different lengths with the elementary unitary matrices in the structure of the paraunitary safb constrained to be real and orthogonal the structure covers the most general real coefficient paraunitary lpfbs compared with the existing results the number of parameters is reduced significantly

a generalized algorithm for linear phase paraunitary filter banks a new algorithm for a subclass of linear phase paraunitary filter banks with generalized filter length and symmetry polarity is reported new properties in the polyphase matrix are derived and the lattice factorizations for filter banks with an even and odd number of channels are examined in a generalized algorithm

lapped transforms for efficient transform subband coding two lapped transforms for subband transform coding of signals are introduced a version of the lapped orthogonal transform lot which can be efficiently computed for any transform length and the modulated lapped transform mlt which is based on a modulated quadrature mirror qmf bank the mlt can also be efficiently computed by means of a type iv discrete sine transform dst iv the lot and the mlt are both asymptotically optimal lapped transforms for coding an ar 1 signal with a high intersample correlation coefficient the coding gains of the lot and mlt of length m are higher than that of the discrete cosine transform dct of the same length they are actually close to the coding gains obtained with a dct of length 2m an mlt based adaptive transform coder act for speech signals is simulated the code is essentially free from frame rate noise and has a better spectral resolution that dct based atc systems

a new active noise control algorithm that requires no secondary path identification based on the spr property active noise control anc has been widely applied in industry to reduce environmental noise and equipment vibrations most available control algorithms require the identification of the secondary path which increases the control system complexity contributes to an increased residual noise power and can even cause the control system to fail if the identified secondary path is not sufficiently close to the actual path in this paper based on the geometric analysis and the strict positive real spr property of the filtered x lms algorithm we introduce a new anc algorithm suitable for single tone noises as well as some specific narrowband noises that does not require the identification of the secondary path though its convergence can be very slow in some special cases we are able to extend the developed anc algorithm to the case of active control of broadband noises through our use of a subband implementation of the anc algorithm compared to other available control algorithms that do not require secondary path identification our developed method is simple to implement yields good performance and converges quickly simulation results confirm the effectiveness of our proposed algorithm

factorization approach to unitary time varying filter bank trees and wavelets a complete factorization of all optimal in terms of quick transition time varying fir unitary filter bank tree topologies is obtained this has applications in adaptive subband coding tiling of the time frequency plane and the construction of orthonormal wavelet and wavelet packet bases for the half line and interval for an m channel filter bank the factorization allows one to construct entry exit filters that allow the filter bank to be used on finite signals without distortion at the boundaries one of the advantages of the approach is that an efficient implementation algorithm comes with the factorization the factorization can be used to generate filter bank tree structures where the tree topology changes over time explicit formulas for the transition filters are obtained for arbitrary tree transitions the results hold for tree structures where filter banks with any number of channels or filters of any length are used time varying wavelet and wavelet packet bases are also constructed using these filter bank structures the present construction of wavelets is unique in several ways 1 the number of entry exit functions is equal to the number of entry exit filters of the corresponding filter bank 2 these functions are defined as linear combinations of the scaling functions other methods involve infinite product constructions 3 the functions are trivially as regular as the wavelet bases from which they are constructed

wiener filter based error resilient time domain lapped transform in this paper the design of the error resilient time domain lapped transform is formulated as a linear minimal mean squared error problem the optimal wiener solution and several simplifications with different tradeoffs between complexity and performance are developed we also prove the persymmetric structure of these wiener filters the existing mean reconstruction method is proven to be a special case of the proposed framework our method also includes as a special case the linear interpolation method used in dct based systems when there is no pre postfiltering and when the quantization noise is ignored the design criteria in our previous results are scrutinized and improved solutions are obtained various design examples and multiple description image coding experiments are reported to demonstrate the performance of the proposed method

frame theory based analysis and design of oversampled filter banks direct computational method this paper studies the frames corresponding to oversampled filter banks fbs for this class of fb frames we present a state space parameterization of perfect reconstruction fb frames and explicit and numerically efficient formulas to compute the tightest frame bounds to obtain the dual fb frame and to construct a tight paraunitary fb frame from a given untight nonparaunitary fb frame the derivation uses well developed techniques from modern control theory which results in the unified formulas for generic infinite impulse response iir and finite impulse response fir fbs these formulas involve only algebraic manipulations of real matrices and can be computed efficiently reliably and exactly without the approximation required in the existing methods for generic fbs the results provide a unified framework for frame theory based analysis and systematic design of oversampled filter banks

on factorization of m channel paraunitary filterbanks we systematically investigate the factorization of causal finite impulse response fir paraunitary filterbanks with given filter length based on the singular value decomposition of the coefficient matrices of the polyphase representation a fundamental order one factorization form is first proposed for general paraunitary systems then we develop a new structure for the design and implementation of paraunitary system based on the decomposition of hermitian unitary matrices within this framework the linear phase filterbank and pairwise mirror image symmetry filterbank are revisited their structures are special cases of the proposed general structures compared with the existing structures more efficient ones that only use approximately half the number of free parameters are derived the proposed structures are complete and minimal although the factorization theory with or without constraints is discussed in the framework of m channel filterbanks the results can be applied to wavelets and multiwavelet systems and could serve as a general theory for paraunitary systems

design of lapped orthogonal transforms we investigate the design of lapped orthogonal transforms for data compression of images we present some properties and new results of paraunitary filter banks we concentrate on the case where the filter length l 2k where k is the number of channels the aim is to design perceptually relevant filters i e linear phase filters that smoothly decay to zero at the boundaries

design of linear phase m channel perfect reconstruction fir filter banks we propose two approaches to design m channel nonparaunitary filter banks that satisfy perfect reconstruction pr and linear phase lp properties in the first approach the pr condition is imposed on only a high pass filter although this method does not require nonlinear optimization it has a demerit in that the order of a high pass filter becomes high in the second approach two filters are optimized simultaneously using a lagrange newton method we can design pr filter banks that have the same length the pr constraint is also formulated as a linear and nonlinear equation of the analysis filter coefficients finally some design examples are included

parameterization of m band affine frames with pre assigned scaling filter affine wavelet frames related with subdivision scheme provide tools for building curves and surfaces in cagd cam at first we show that scaling filters can be parameterized by decomposed elementary matrices with the form im p pz k where k is a nonnegative integer and p is a one rank constant valued idempotent matrix p2 im theorem 2 2 then we use the method to construct m band affine wavelet frames with the given scaling filter and the dual frames general solutions for affine wavelet frames and its dual affine wavelet frames are obtained with pre assigned scaling filter theorems 3 5 and 3 6 at last a design example is given

the generalized lapped pseudo biorthogonal transform oversampled linear phase perfect reconstruction filterbanks with lattice structures we investigate a lattice structure for a special class of n channel oversampled linear phase perfect reconstruction filterbanks with a decimation factor m smaller than n we deal with systems in which all analysis and synthesis filters have the same finite impulse response fir length and share the same center of symmetry we provide the minimal lattice factorization of a polyphase matrix of a particular class of these oversampled filterbanks fbs all filter coefficients are parameterized by rotation angles and positive values the resulting lattice structure is able to provide fast implementation and allows us to determine the filter coefficients by solving an unconstrained optimization problem we consider next the case where we are given the generalized lapped pseudo biorthogonal transform glpbt lattice structure with specific parameters and we a priori know the correlation matrix of noise that is added in the transform domain in this case we provide an alternative lattice structure that suppress the noise we show that the proposed systems with the lattice structure cover a wide range of linear phase perfect reconstruction fbs we also introduce a new cost function for oversampled fb design that can be obtained by generalizing the conventional coding gain finally we exhibit several design examples and their properties

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

lattice structure for generalized support multidimensional linear phase perfect reconstruction filter bank multidimensional linear phase perfect reconstruction filter bank mdlpprfb can be designed and implemented via lattice structure the lattice structure for the mdlpprfb with filter support n m has been published by muramatsu where m is the decimation matrix is a positive integer diagonal matrix and n n denotes the set of integer vectors in the fundamental parallelepiped of the matrix n obviously if is chosen to be other positive diagonal matrices instead of only positive integer ones the corresponding lattice structure would provide more choices of filter banks offering better trade off between filter support and filter performance we call such resulted filter bank as generalized support mdlpprfb gsmdlpprfb the lattice structure for gsmdlpprfb however cannot be designed by simply generalizing the process that muramatsu employed furthermore the related theories to assist the design also become different from those used by muramatsu such issues will be addressed in this paper to guide the design of gsmdlpprfb the necessary and sufficient conditions are established for a generalized support multidimensional filter bank to be linear phase to determine the cases we can find a gsmdlpprfb the necessary conditions about the existence of it are proposed to be related with filter support and symmetry polarity i e the number of symmetric filters ns and antisymmetric filters na based on a process different from the one muramatsu used that combines several polyphase matrices to construct the starting block one of the core building blocks of lattice structure the lattice structure for gsmdlpprfb is developed and shown to be minimal additionally the result in this paper includes muramatsu s as a special case

maximally decimated paraunitary linear phase fir filter bank design via iterative svd approach a necessary and sufficient condition for m channel maximally decimated paraunitary fir filter banks is derived when the lengths of all the filters are the same and they are integer multiples of m based on the derived condition a family of this kind of filter banks is defined these results are also applied to the corresponding linear phase filter banks then the design of this type of filter banks with the linear phase property is formulated as two orthogonal procrustes optimization problems the analytical solution of each optimization problem is derived based on a singular value decomposition approach since no numerical optimization computer aided design tool is required for finding the analytical solution of each optimization problem the computational power required for our proposed method is much lower than that for existing methods computer numerical simulation results show that our proposed method can obtain a design instantaneously

a class of m channel linear phase biorthogonal filter banks and their applications to subband coding this correspondence presents a new factorization for linear phase biorthogonal perfect reconstruction pr fir filter hanks using this factorization we propose a new family of lapped transform called the generalized lapped transform glt since the analysis and synthesis filters of the glt are not restricted to be time reverses of each other they can offer more freedom to avoid blocking artifacts and improve coding gain in subband coding applications the glt is found to have higher coding gain and smoother synthesis basis functions than the lapped orthogonal transform lot simulation results also demonstrated that the glt has significantly less blocking artifacts higher peak signal to noise ratio psnr and better visual quality than the lot in image coding simplified glt with different complexity performance tradeoff is also studied

boundary conditions for translation invariant gibbs measures of the potts model on cayley trees we consider translation invariant splitting gibbs measures tisgms for the q state potts model on a cayley tree of order two recently a full description of the tisgms was obtained and it was shown in particular that at sufficiently low temperatures their number is 2 q 1 in this paper for each tisgm mu we explicitly give the set of boundary conditions such that limiting gibbs measures with respect to these boundary conditions coincide with mu

on m channel linear phase fir filter banks and application in image compression this paper investigates the theory and structure of a large subclass of m channel linear phase perfect reconstruction fir filter banks systems with analysis and synthesis filters of length l sub i k sub i m spl beta where spl beta is an arbitrary integer 0 spl les spl beta m and k sub i is any positive integer for this subclass of systems we first investigate the necessary conditions for the existence of linear phase perfect reconstruction filter banks lpprfbs next we develop a complete and minimal factorization for all even channel linear phase paraunitary systems the most general lapped orthogonal transforms to date finally several design examples as well as comparisons with previous generalized lapped orthogonal transforms genlots in image compression are presented to confirm the validity of the theory

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

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

linear phase perfect reconstruction filter bank lattice structure design and application in image coding a lattice structure for an m channel linear phase perfect reconstruction filter bank lpprfb based on the singular value decomposition svd is introduced the lattice can be proven to use a minimal number of delay elements and to completely span a large class of lpprfbs all analysis and synthesis filters have the same fir length sharing the same center of symmetry the lattice also structurally enforces both linear phase and perfect reconstruction properties is capable of providing fast and efficient implementation and avoids the costly matrix inversion problem in the optimization process from a block transform perspective the new lattice can be viewed as representing a family of generalized lapped biorthogonal transform glbt with an arbitrary number of channels m and arbitrarily large overlap the relaxation of the orthogonal constraint allows the glbt to have significantly different analysis and synthesis basis functions which can then be tailored appropriately to fit a particular application several design examples are presented along with a high performance glbt based progressive image coder to demonstrate the potential of the new transforms

lapped unimodular transform and its factorization two types of lapped transforms have been studied in detail in the literature namely the lapped orthogonal transform lot and its extension the biorthogonal lapped transform bolt in this paper we will study the lapped unimodular transform lut all three transforms are first order matrices with finite impulse response fir inverses we will show that like lot and bolt all luts can be factorized into degree one unimodular matrices the factorization is both minimal and complete we will also show that all first order systems with fir inverses can be minimally factorized as a cascade of degree one lot bolt and lut building blocks two examples will be given to demonstrate that despite having a very small system delay the luts have a satisfactory performance in comparison with lot and bolt

the lot transform coding without blocking effects an exact derivation of an optimal lapped orthogonal transform lot is presented the optimal lot is related to the discrete cosine transform dct in such a way that a fast algorithm for a nearly optimal lot is derived compared to the dct the fast lot requires about 20 30 more computations mostly additions an image coding example demonstrates the effectiveness of the lot in reducing blocking effects the lot actually leads to slightly smaller signal reconstruction errors than does the dct

generalized unequal length lapped orthogonal transform for subband image coding generalized linear phase lapped orthogonal transforms with unequal length basis functions gullots are considered the length of each basis of the proposed gullot can be different from each other whereas all the bases of the conventional genlot are of equal length in general for image coding application the long basis for a low frequency band and the short basis for a high frequency one are desirable to reduce the blocking and the ringing artifact simultaneously therefore the gullot is suitable especially for a subband image coding in order to apply the gullot to a subband image coding we also investigate the size limited structure to process the finite length signal which is important in practice finally some design and image coding examples are shown to confirm the validity of the proposed gullot

regularity constrained pre and post filtering for block dct based systems it is well known that the traditional block transform can only have at most one degree of regularity in other words by retaining only one subband these transforms including the popular discrete cosine transform dct can only capture the constant signal the ability to capture polynomials of higher orders is critical in smooth signal approximation minimizing blocking effects this paper presents the theory design and fast implementation of regularity constrained pre post filters for block based decomposition systems we demonstrate that simple pre post filtering modules added to the current block based infrastructure can help the block transform capture not only the constant signal but the ramp signal as well moreover our proposed framework can be used to generate various fast symmetric m band wavelets with up to two degrees of regularity

efficient large scale filter filterbank design via lmi characterization of trigonometric curves many filter and filterbank design problems can be posed as the optimization of linear or convex quadratic objectives over trigonometric semi infinite constraints recent advances in design methodology are based on various linear matrix inequality lmi characterizations of the semi infinite constraints and semidefinite programming sdp solutions despite these advances the design of filters of several hundredth order which typically arise in multicarrier communication and signal compression cannot be accommodated this hurdle is due mainly to the large number of additional variables incurred in the lmi characterizations this paper proposes a novel lmi characterization of the semi infinite constraints that involves additional variables of miminal dimensions consequently the design of high order filters required in practical applications can be achieved examples of designs of up to 1200 tap filters are presented to verify the viability of the proposed approach

periodic p adic gibbs measures of q state potts model on cayley trees i the chaos implies the vastness of the set of p adic gibbs measures we study the set of p adic gibbs measures of the q state potts model on the cayley tree of order three we prove the vastness of the set of the periodic p adic gibbs measures for such model by showing the chaotic behavior of the corresponding potts bethe mapping over mathbb q p for the prime numbers p equiv 1 mathrm mod 3 in fact for 0 theta 1 p q p 2 1 where theta exp p j and j is a coupling constant there exists a subsystem that is isometrically conjugate to the full shift on three symbols meanwhile for 0 q p 2 le theta 1 p q p 1 there exists a subsystem that is isometrically conjugate to a subshift of finite type on r symbols where r ge 4 however these subshifts on r symbols are all topologically conjugate to the full shift on three symbols the p adic gibbs measures of the same model for the prime numbers p 2 3 and the corresponding potts bethe mapping are also discussed on the other hand for 0 theta 1 p q p 1 we remark that the potts bethe mapping is not chaotic when p 3 and p equiv 2 mathrm mod 3 and we could not conclude the vastness of the set of the periodic p adic gibbs measures in a forthcoming paper with the same title we will treat the case 0 q p le theta 1 p 1 for all prime numbers p

design of linear phase recombination nonuniform filter banks this correspondence proposes a novel method for designing a class of recombination nonuniform filter banks rnfbs with the linear phase lp property in the structure of the proposed nonuniform filter bank fb certain channels of an m channel uniform fb are merged by synthesis filters of transmultiplexers tmuxs yielding nonuniform subbands by a detailed analysis of the frequency characteristics of the rnfbs we derive the existence and matching conditions that are necessary for constructing the proposed rnfbs having good frequency responses with those conditions being satisfied the design problem becomes that of a set of lp uniform fbs by using the parks mcclellan algorithm we are able to design the near perfect reconstruction lp rnfbs in a very simple and efficient way as demonstrated by two examples

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

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

computational formulation for periodic vibration of geometrically nonlinear structures part 1 theoretical background abstract a general computational formulation for geometrically nonlinear structures excited by harmonic forces and executing periodic motion in a steady state is presented the equations of both continuous and discretized models are reformulated to obtain the motion equation in a more suitable form to a further analysis the multi harmonic solution of motion equation is written in a form of truncated fourier series next the galerkin ritz and the harmonic balance method are discussed in a context of their equivalency in derivation of the matrix amplitude equation the matrix amplitude equation as well as the associated tangent matrix are given in an explicit form the stability of steady state solution is discussed by using the floquet theory the numerical algorithm and an example application are described in a companion paper by lewandowski lewandowski r computational formulation for periodic vibration of geometrically nonlinear structures part 2 numerical strategy and numerical examples international journal of solids and structures in preparation

lattice structure for regular paraunitary linear phase filterbanks and m band orthogonal symmetric wavelets parauninary linear phase pulp m channel uniform filterbanks which are also known as the generalized lapped orthogonal transforms genlots can be designed and implemented using lattice structures this paper discusses how to impose regularity constraints onto the lattice structure of pulp filterbanks these conditions are expressed in term of the rotation angles of the lattice components by which the resulting filterbanks are guaranteed to have one or two degrees of regularity iterating these new regular filterbanks on the lowpass subband generates a large family of symmetric m band orthonormal wavelets design procedures with many design examples are presented smooth interpolation using regular pulp filterbanks is illustrated through image coding experiments where the novel m band wavelets consistently yield smoother reconstructed images and better perceptual quality

non linear free vibrations of beams by the finite element and continuation methods abstract a computational method is presented for determining backbone curves of arbitrary geometry as an example beam vibration is considered the beam response is expanded into a truncated fourier series with respect to time after starting from the integral of action the variational approach and the finite element method are used to formulate the non linear eigenvalue problem the continuation method is adopted to solve the resulting non linear eigenvalue problem and to obtain the non linear frequencies and modes of vibration numerical results for various beams are presented and compared with available results to demonstrate the accuracy and applicability of the method moreover the bifurcation points on some beams backbone curves are found and reported for the first time

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

maximally decimated perfect reconstruction fir filter banks with pairwise mirror image analysis and synthesis frequency responses structures are presented for the perfect reconstruction quadrature mirror filter bank that are based on lossless building blocks these structures ensure that the frequency responses of the analysis and synthesis filters have pairwise symmetry with respect to pi 2 and require fewer parameters than recently reported structures also based on lossless building blocks the design time for the proposed structures is correspondingly much less than for the earlier methods which did not incorporate such symmetry

optimal design of oversampled synthesis fbs with lattice structure constraints oversampled filter banks fbs with lattice structure allow fast implementation and guarantee the linear phase and perfect reconstruction property for such fbs the synthesis fb is not unique this nonuniqueness provides extra design freedom for optimal reduction of subband noise required in many applications however the existing design methods of synthesis fb for optimal reduction of subband noise cannot guarantee the linear phase and lattice structure property this paper studies the design of oversampled synthesis fbs with guaranteed lattice structure for optimal reduction of subband noise based on a state space parameterization of all synthesis fbs with lattice structure the explicit formulae are derived to find the optimal solution for noise reduction numerical examples are presented to demonstrate the effectiveness of the new design methods obtained

structures factorizations and design criteria for oversampled paraunitary filterbanks yielding linear phase filters we present here a special class of oversampled filterbanks fbs namely paraunitary fbs with linear phase filters we propose some necessary conditions for the existence of such banks based on the repartition between type i ii and type ii iv linear phase filters in the bank for a subset of these fbs we develop a factorization that leads to a minimal implementation as well as a direct parameterization of the fbs in terms of elementary rotation angles this factorization is applied to some design examples with two different optimization criteria coding gain and reconstructibility of lost coefficients

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

design of orthogonal and biorthogonal lapped transforms satisfying perception related constraints we propose a new efficient method for the design of orthogonal and biorthogonal lapped transforms for image coding applications it is shown how perception related constraints such as decay and smoothness of the filters impulse responses can be incorporated in the optimization procedure a decomposition of lapped transforms orthogonal and biorthogonal with 50 overlap leads to an efficient recursive optimization procedure which is robust with respect to initial solutions the importance of this decomposition lies in the fact that it allows to decouple the design of the even symmetric and the odd symmetric filters and hence drastically reduces the number of variables to be optimized it furthermore reveals all the variables predetermined by perception related and coding efficiency related constraints imposed on the filters we present design and coding examples demonstrating the perceptual performance and the rate distortion performance of the resulting transforms

on the perfect reconstruction problem in n band multirate maximally decimated fir filter banks the problem of finding n k filters of an n band maximally decimated fir analysis filter bank given k filters so that fir perfect reconstruction can be achieved is considered the perfect reconstruction condition is expressed as a requirement of unimodularity of the polyphase analysis filter matrix based on the theory of euclidean division for matrix polynomials the conditions the given transfer functions must satisfy are given and a complete parameterization of the solution is obtained this approach provides an interesting alternative to the method of the complementary filter in the case of n 2 k n 1 where the latter leads to a system of nonlinear equations moreover it yields the polyphase synthesis filter matrix as a byproduct as an application a complete characterization of all paraunitary matrices with fixed first row is derived the problem of appropriately choosing the parameters characterizing the complementary filters to lead to filters of practical useful frequency responses is studied and analytical solutions for k n 1 are given it is demonstrated through an example that the complementary filters found via euclid s algorithm are not necessarily linear phase even if the given filters are the problem of obtaining linear phase solutions of given orders is investigated for the general case n spl ges 2 k spl les n 1 and systematic ways are developed to compute such solutions in the k n 1 and k 1 cases it is also shown that the resulting synthesis filters are linear phase design examples illustrating the theory are presented

description of the translation invariant splitting gibbs measures for the potts model on a cayley tree for the q state potts model on a cayley tree of order k ge 2 it is well known that at sufficiently low temperatures there are at least q 1 translation invariant gibbs measures which are also tree indexed markov chains such measures are called translation invariant splitting gibbs measures tisgms in this paper we find all tisgms and show in particular that at sufficiently low temperatures their number is 2 q 1 we prove that there are q 2 where a is the integer part of a critical temperatures at which the number of tisgms changes and give the exact number of tisgms for each intermediate temperature for the binary tree we give explicit formulae for the critical temperatures and the possible tisgms while we show that these measures are never convex combinations of each other the question which of these measures are extremals in the set of all gibbs measures will be treated in future work

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

multicriterion optimized qmf bank design the quadrature mirror filter qmf bank with multicriterion constraints such as minimal aliasing and or minimal error coding is among the most important problems in filterbank design for solving which linear algebra based methods are still heuristic and do not always work especially for large filter length it is shown in this paper that this problem can be reduced either to convex linear matrix inequality lmi optimization when filters are of nonlinear phase or to semi infinite linear sip programming when filters are of linear phase which can be very efficiently solved either by the standard lmi solvers or our previously developed sip solver the proposed computationally tractable optimization formulations are confirmed by several simulations

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

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

linear phase paraunitary filter bank with filters of different lengths and its application in image compression in this paper the theory structure design and implementation of a new class of linear phase paraunitary filter banks lppufbs are investigated the novel filter banks with filters of different lengths can be viewed as the generalized lapped orthogonal transforms genlots with variable length basis functions our main motivation is the application in block transform based image coding besides having all of the attractive properties of other lapped orthogonal transforms the new transform takes advantage of its long overlapping basis functions to represent smooth signals in order to reduce blocking artifacts whereas it reserves short basis functions for high frequency signal components like edges and texture thereby limiting ringing artifacts two design methods are presented each with its own set of advantages the first is based on a direct lattice factorization and the second enforces certain relationships between the lattice coefficients to obtain variable length filters various necessary conditions for the existence of meaningful solutions are derived and discussed in both cases finally several design and image coding examples are presented to confirm the validity of the theory

parameterizations of masks for 3 band tight wavelet frames by symmetric extension of polyphase matrix this paper deals with the construction of 3 band tight wavelet frames with symmetric properties using symmetric extension and parameterizations of the paraunitary matrix firstly we construct a paraunitary matrix which is a symmetric extension of polyphase matrix corresponding to compactly supported wavelet frames with the least number generators further we present symmetric characters and uniformity of wavelet frames with different filter lengths in terms of paraunitary matrix then we focus on investigating the parameterizations of masks for 3 band tight wavelet frames with 3n filter length finally examples of 3 band wavelet frames with good smoothness are constructed based on the parameterizations of masks

coding gain in paraunitary analysis synthesis systems a formal proof that bit allocation results hold for the entire class of paraunitary subband coders is presented the problem of finding an optimal paraunitary subband coder so as to maximize the coding gain of the system is discussed the bit allocation problem is analyzed for the case of the paraunitary tree structured filter banks such as those used for generating orthonormal wavelets the even more general case of nonuniform filter banks is also considered in all cases it is shown that under optimal bit allocation the variances of the errors introduced by each of the quantizers have to be equal expressions for coding gains for these systems are derived

hierarchical finite element analyses of geometrically non linear vibration of beams and plane frames geometrically non linear vibrations of beams and plane frameworks are analyzed by the hierarchical finite element method hfem two main points are of interest the first is to compare polynomials trigonometric functions and beam eigenfunctions as displacement shape functions for beam hierarchical finite elements the second is to examine the suitability of the hfem for time domain non linear analyses

robust image watermarking based on multiband wavelets and empirical mode decomposition in this paper we propose a blind image watermarking algorithm based on the multiband wavelet transformation and the empirical mode decomposition unlike the watermark algorithms based on the traditional two band wavelet transform where the watermark bits are embedded directly on the wavelet coefficients in the proposed scheme we embed the watermark bits in the mean trend of some middle frequency subimages in the wavelet domain we further select appropriate dilation factor and filters in the multiband wavelet transform to achieve better performance in terms of perceptually invisibility and the robustness of the watermark the experimental results show that the proposed blind watermarking scheme is robust against jpeg compression gaussian noise salt and pepper noise median filtering and con vfilter attacks the comparison analysis demonstrate that our scheme has better performance than the watermarking schemes reported recently

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

m band scaling function with filter having vanishing moments two and minimal length abstract in this paper we consider the holder continuity local linearity linear independence and interpolation problem of the m band scaling function with a filter having vanishing moments two and minimal length in particular we find some new properties that are not true when m 2 such as local linearity local linear dependence differentiability at adjoint m adic points and interpolation problem at integer knots

oversampled linear phase perfect reconstruction filterbanks theory lattice structure and parameterization the paper presents the theory lattice structure and parameterization for a general class of p channel oversampled linear phase perfect reconstruction filterbanks olpprfbs systems with sampling factor m p spl ges m and filter length of l km k spl ges 1 each for these olpprfbs the necessary existence conditions on the number of symmetric filters n sub spl beta and antisymmetric filters n sub spl alpha i e symmetry polarity are first investigated vlsi friendly lattice structures are then developed for two types of olpprfbs type i system n sub spl beta n sub spl alpha and type ii system n sub spl beta spl ne n sub spl alpha the completeness and minimality of each type of lattice are also analyzed compared with existing work the proposed lattices are the most general and efficient ones for olpprfbs besides through the lattice structures the sufficiency of the existence conditions is also verified next lifting based structures are proposed to parameterize a left invertible matrix and all of its left inverses which leads to unconstrained optimization as well as robust implementation of olpprfbs finally several design examples are presented to confirm the validity of the theory and demonstrate the versatility of synthesis filterbanks in the oversampled system

design and factorization of fir paraunitary filter banks given several analysis filters this correspondence discusses the design and factorization of fir paraunitary filter banks where several analysis filters are given a lattice factorization based algorithm is presented that provides complete and minimal characterization for nonlinear phase and balanced linear phase systems as a result the design problem can be formulated as an unconstrained optimization of the lattice coefficients whereas paraunitariness and linear phase properties are structurally guaranteed the second presented algorithm formulates the design problem in quadratic form that provides better control on the frequency responses of the embedding solution an unitary scaling filter design algorithm is also presented that can be paired with the paraunitary embedding solution to design an orthonormal wavelet basis with high regularity a design example is presented

symmetric extension methods for m channel linear phase perfect reconstruction filter banks the symmetric extension method has been shown to he an efficient way for subband processing of finite length sequences this paper presents an extension of this method to general linear phase perfect reconstruction filter banks we derive constraints on the length and symmetry polarity of the permissible filter banks and propose a new design algorithm in the algorithm different symmetric sequences are formulated in a unified form based on the circular symmetry framework the length constraints in symmetrically extending the input sequence and windowing the subband sequences are investigated the effect of shifting the input sequence is included when the algorithm is applied to equal length filter banks we explicitly show that symmetric extension methods can always be constructed to replace the circular convolution approach

dct based general structure for linear phase paraunitary filterbanks in this paper we present a general structure for linear phase paraunitary filterbanks lppufb via both time domain pre processing and frequency domain post processing of the discrete cosine transform dct the proposed structure not only covers existing frameworks with either pre processing or post processing of the dct but enables the design of dct based lppufb with both partial block overlapping and variable length filters as well some high performance and low complexity transforms can be obtained by fine tuning the structure parameters furthermore a dct oriented initialization method is developed to improve the convergence and to simplify the parameterization in the filterbank optimization process the application in image compression is demonstrated

a progressive transmission image coder using linear phase uniform filterbanks as block transforms this paper presents a novel image coding scheme using m channel linear phase perfect reconstruction filterbanks lpprfbs in the embedded zerotree wavelet ezw framework introduced by shapiro 1993 the innovation here is to replace the ezws dyadic wavelet transform by m channel uniform band maximally decimated lpprfbs which offer finer frequency spectrum partitioning and higher energy compaction the transform stage can now be implemented as a block transform which supports parallel processing and facilitates region of interest coding decoding for hardware implementation the transform boasts efficient lattice structures which employ a minimal number of delay elements and are robust under the quantization of lattice coefficients the resulting compression algorithm also retains all the attractive properties of the ezw coder and its variations such as progressive image transmission embedded quantization exact bit rate control and idempotency despite its simplicity our new coder outperforms some of the best image coders published previously in the literature for almost all test images especially natural hard to code ones at almost all bit rates

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

a group theoretic approach to multidimensional filter banks theory and applications in this paper we provide a new method for analyzing multidimensional filter banks this method enables us to solve various open problems in multidimensional filter bank characterization and design the essential element in this new approach is the redefinition of polyphase components it will be shown that a rich set of mathematical tools in particular algebraic group theory will become available for use in the analysis of filter banks we demonstrate the elegance and power of the tool set by employing it for the characterization of multidimensional filter banks and applying it to two open problems the first problem is concerned with the development of a general method to design multichannel spl ges 2 multidimensional filter banks using transformations while the second problem is concerned with the derivation of general restrictions on group delays in linear phase filter banks the treatment of these problems is only an illustration of the power of the tool set of algebraic group theory employed for the first time in the context of multidimensional filter banks

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

on minimal lattice factorizations of symmetric antisymmetric multifilterbanks this paper introduces two minimal lattice structures for symmetric antisymmetric multiwavelets samwts and symmetric antisymmetric multifilterbanks samfbs first by exploring the relation of the symmetric antisymmetric property in multifilterbanks and the linear phase property in traditional scalar filterbanks we show that the implementation and design of an samfb can be converted into that of a four channel scalar linear phase perfect reconstruction filterbank lpprfb then based on the lattice factorization for lpprfbs we propose two fast modular minimal structures for samfbs to demonstrate the effectiveness of the proposed lattice structures several rational or dyadic coefficient samwt design examples are presented along with their application in image coding

non linear vibration of beams with internal resonance by the hierarchical finite element method the hierarchical nite element hfem and the harmonic balance methods hbm are used to investigate the geometrically non linear free and steady state forced vibrations of uniform slender beams the beam analogue of von kah rmah n s non linear strain displacement relationships are employed and the middle plane in plane displacements are included in the model the equations of motion are developed by applying the principle of virtual work and are solved by a continuation method 1 3 and 1 5 internal resonances are discovered and their consequences are discussed the convergence properties of the hfem are analyzed and it is demonstrated that the hfem model requires far fewer degrees of freedom than the h version of the fem models presented in the literature 1999 academic press

tight frames of compactly supported affine wavelets this paper extends the class of orthonormal bases of compactly supported wavelets recently constructed by daubechies commun pure appl math 41 909 1988 for each integer n 1 a family of wavelet functions having support 0 2n 1 is constructed such that jk x 2j 2 2jx k kj k z is a tight frame of l2 r i e for every f l2 r f c jk jk f jk for some c 0 this family is parametrized by an algebraic subset vn of r4n furthermore for n 2 a proper algebraic subset wn of vn is specified such that all points in vn outside of wn yield orthonormal bases the relationship between these tight frames and the theory of group representations and coherent states is 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

digital filter bank design quadratic constrained formulation formulate the filter bank design problem as an quadratic constrained least squares minimization problem the solution of the minimization problem converges very quickly since the cost function as well as the constraints are quadratic functions with respect to the unknown parameters the formulations of the perfect reconstruction cosine modulated filter bank of the near perfect reconstruction pseudo qmf bank and of the two channel biorthogonal linear phase filter bank are derived using the proposed approach compared with other design methods the proposed technique yields pr filter banks with much higher stopband attenuation the proposed technique can also be extended to design multidimensional filter banks

complete factorization for poly phase matrix with linear phase based on semi rank orthogonal projection matrix abstract this paper puts forward a new factorization named the semi rank factorization for para unitary poly phase matrix with linear phase and reveals the equivalent relationship between the orthogonal projection matrix and the linear phase and characterizes the orthogonal projection matrix the semi rank factorization has the completeness property no matter for even or odd number of channels in addition a decomposition technique for the starting block matrix is developed using the reflection matrix the semi rank factorization provides an efficient and compact representation for multi band filter banks with linear phase in particular the complete representation for multi band symmetry filter banks with odd number of channels is brand new as a result the programmable algorithms are obtained for designs of optimal filter banks based on the complete representation the optimal examples are presented to illustrate the excellent performance on image compression

symmetric delay factorization generalized framework for paraunitary filter banks the symmetric delay factorization sdf was introduced to synthesize linear phase paraunitary filter banks lppufbs with uniform order i e filter length equal to nm for arbitrary n and real valued coefficients the sdf presents the advantage of decomposing the polyphase transfer matrix ptm into only orthogonal matrices even at the boundary of finite duration signals simplifying significantly the design of time bounded filter banks tbfbs or of time varying filter banks tvfbs however the symmetric delay factorization applies only to lppufbs on the other hand lattice structures as well as finite size lattice structures are proposed for classes of nonlinear phase paraunitary filter banks as the modulated lapped transform mlt and the extended tapped transform elt this paper describes a new minimal and complete symmetric delay factorization valid for a larger class of paraunitary filter banks encompassing paraunitary cosine modulated filter banks with nonlinear phase basis functions as well as for a set of lppufbs including the linear phase lapped orthogonal transforms lots and the generalized tapped orthogonal transforms genlots the derivations for filter banks with even and odd numbers of channels are formulated in a unified form this approach opens new perspectives in the design of time varying filter banks used for image and video compression especially in the framework of region or object based coding

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

symmetric nearly shift invariant tight frame wavelets k regular two band orthogonal filterbanks have been applied to image processing such filters can be extended into a case of downsampling by two and more than two filters provided that they satisfy a set of conditions such a setup allows for more degrees of freedom but also at the cost of higher redundancy the latter depends directly on the number of the wavelet filters involved tight frame filters allow the design of smooth scaling functions and wavelets with a limited number of coefficients moreover such filters are nearly shift invariant a desirable feature in many applications we explore a family of symmetric tight frame finite impulse response fir filters characterized by the relations h sub 3 z h sub 0 z and h sub 2 z h sub 1 z they are simple to design and exhibit a degree of near orthogonality in addition to near shift invariance both properties are desirable for noise removal purposes

a study of two channel complex valued filterbanks and wavelets with orthogonality and symmetry properties we investigate two channel complex valued filterbanks and wavelets that simultaneously have orthogonality and symmetry properties first the conditions for the filterbank to be orthogonal symmetric and regular for generating smooth wavelets are presented then a complete and minimal lattice structure is developed which enables a general design approach for filterbanks and wavelets with arbitrary length and arbitrary order of regularity finally two integer implementation methods that preserve the perfect reconstruction property of the filterbank are proposed their performances are evaluated via experimental results