improvement of dct based compression algorithms using poisson s equation we propose two new image compression decompression methods that reproduce images with better visual fidelity less blocking artifacts and better psnr particularly in low bit rates than those processed by the jpeg baseline method at the same bit rates the additional computational cost is small i e linearly proportional to the number of pixels in an input image the first method the full mode polyharmonic local cosine transform phlct modifies the encoder and decoder parts of the jpeg baseline method the goal of the full mode phlct is to reduce the code size in the encoding part and reduce the blocking artifacts in the decoder part the second one the partial mode phlct or pphlct for short modifies only the decoder part and consequently accepts the jpeg files yet decompresses them with higher quality with less blocking artifacts the key idea behind these algorithms is a decomposition of each image block into a polyharmonic component and a residual the polyharmonic component in this paper is an approximate solution to poisson s equation with the neumann boundary condition which means that it is a smooth predictor of the original image block only using the image gradient information across the block boundary thus the residual obtained by removing the polyharmonic component from the original image block has approximately zero gradient across the block boundary which gives rise to the fast decaying dct coefficients which in turn lead to more efficient compression decompression algorithms for the same bit rates we show that the polyharmonic component of each block can be estimated solely by the first column and row of the dct coefficient matrix of that block and those of its adjacent blocks and can predict an original image data better than some of the other ac prediction methods previously proposed our numerical experiments objectively and subjectively demonstrate the superiority of phlct over the jpeg baseline method and the improvement of the jpeg compressed images when decompressed by pphlct

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

malvar wavelets with asymmetrically overlapped windows malvar see ibid vol 38 p 969 78 june 1990 wavelets or tapped orthogonal transforms with symmetric overlaps have been studied extensively we consider malvar wavelets with asymmetric overlaps in both continuous time and discrete time cases we present conditions on window functions in continuous time case so that malvar wavelets can be constructed from different local bases however in discrete time cases we show that malvar wavelets are impossible unless the overlaps are symmetric numerical examples are presented to illustrate the theory we consider real valued signals and bases

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

irreducible affine space curves and the uniform artin rees property on the prime spectrum abstract we give a negative answer to the problem open for twenty years as to whether the full uniform artin rees property holds on the prime spectrum of an excellent ring it was known to hold locally on the prime spectrum of such a ring

the change of the brunovsky structure of a controllable pair under one column perturbation in a particular case abstract we study the variation of the controllability indices of a pair a b a b 1 b c n n c n m 1 1 where a b 1 is controllable when we make small additive perturbations on the last column of b in a particular case

realizations of perturbations of an observable pair with prescribed indices it is well known that when a full rank observable pair c a is slightly perturbed the new observability indices k are majorized by the initial ones k k k conversely any indices k majorized by k can be obtained by perturbing c a the aim of this paper is the explicit construction of perturbations of c a which have the desired indices k by means of a sequence of uniparametrical versal perturbations even more using versal deformations we refine this construction in such a way that the perturbation has the maximum possible number of zeros and no parameters in the square part

an orientation selective orthogonal lapped transform a novel critically sampled orientation selective orthogonal lapped transform called the lapped hartley transform lht is derived in a first step overlapping basis functions are generated by modulating basis functions of a 2 d block hartley transform by a cosine wave to achieve invertibility and orthogonality an iterative filter is applied as prefilter in the analysis and as postfilter in the synthesis operation respectively alternatively filtering can be restricted to analysis or synthesis ending up with a biorthogonal transform lht pr lht po a statistical analysis based on a 4000 image data base shows that the lht and lht po have better redundancy removal properties than other block or lapped transforms finally image compression and noise removal examples are given showing the advantages of the lht especially in images containing oriented textures

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

general synthesis procedures for fir lossless transfer matrices for perfect reconstruction multirate filter bank applications a recently reported procedure see ibid vol assp 35 p 476 492 apr 1987 and proc ieee int conf acoust speech sig processing apr 1987 p 2169 2172 for the design of m channel perfect reconstruction quadrature mirror filter qmf bands uses the concept of lossless alias component matrices the synthesis of such qmf banks centers around the generation of a lossless m m finite impulse response fir transfer matrix e z recent results for such generation have been somewhat ad hoc i e not sufficiently general a general procedure is outlined for the generation of such transfer matrices the procedure is based on a cascaded lattice structure derived from a state space viewpoint the structure is such that it generates only m m lossless fir transfer matrices regardless of parameter values and conversely any m m lossless fir transfer matrix can be obtained by a suitable choice of parameters these parameters turn out to be angles theta sub k and the structure is such that the number of angles is minimal a design example is presented to demonstrate the main results

structured perturbation of the brunovsky form a particular case we study the variation of the feedback invariants of a complex rectangular n x n m matrix when we make small additive perturbations on the last m columns in the particular case when the square n x n submatrix is similar to a block diagonal matrix with two blocks in the diagonal associated with the controllable part and the noncontrollable part of the pair a b respectively

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

orthonormal and biorthonormal filter banks as convolvers and convolutional coding gain convolution theorems for filter bank transformers are introduced both uniform and nonuniform decimation ratios are considered and orthonormal as well as biorthonormal cases are addressed all the theorems are such that the original convolution reduces to a sum of shorter decoupled convolutions in the subbands that is there is no need to have cross convolution between subbands for the orthonormal case expressions for optimal bit allocation and the optimized coding gain are derived the contribution to coding gain comes partly from the nonuniformity of the signal spectrum and partly from nonuniformity of the filter spectrum with one of the convolved sequences taken to be the unit pulse function e coding gain expressions reduce to those for traditional subband and transform coding the filter bank convolver has about the same computational complexity as a traditional convolver if the analysis bank has small complexity compared to the convolution itself

sequences of linear type let a be a commutative ring 91 an ideal of a s 2l the symmetric algebra of 8 and r vl the recs algebra of vi there is a canonical surjection x s k which is homogeneous of degree zro and therefore gives canonical surjcctions r f 91 for every integer t 3 0 many recent papers 1 2 4 5 7 cc have investigated the problem of determining which ideals 91 have the property that a is an isomorphism such ideals have been dubbed ideals of lineur type s for example huneke showed that every ideal generated by a d sequence is of linear type 71 if ii is generated by n elements a a then for t 3 i there is an exact scquencc

superposition frames for adaptive time frequency analysis and fast reconstruction in this paper we introduce a broad family of adaptive linear time frequency representations termed superposition frames and show that they admit desirable fast overlap add reconstruction properties akin to standard short time fourier techniques this approach stands in contrast to many adaptive time frequency representations in the existing literature which while more flexible than standard fixed resolution approaches typically fail to provide for efficient reconstruction and often lack the regular structure necessary for precise frame theoretic analysis our main technical contributions come through the development of properties which ensure that our superposition construction provides for a numerically stable invertible signal representation our primary algorithmic contributions come via the introduction and discussion of two signal adaptation schemes based on greedy selection and dynamic programming respectively we conclude with two short enhancement examples that serve to highlight potential applications of our approach

fast search for best representations in multitree dictionaries we address the best basis problem or more generally the best representation problem given a signal a dictionary of representations and an additive cost function the aim is to select the representation from the dictionary which minimizes the cost for the given signal we develop a new framework of multitree dictionaries which includes some previously proposed dictionaries as special cases we show how to efficiently find the best representation in a multitree dictionary using a recursive tree pruning algorithm we illustrate our framework through several examples including a novel block image coder which significantly outperforms both the standard jpeg and quadtree based methods and is comparable to embedded coders such as jpeg2000 and spiht

perturbation of linear control systems abstract we study the variation of the controllability indices and the jordan structure of a pair of matrices a b under small perturbations

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

on the depth of the fiber cone of filtrations preprint enviat per a la seva publicacio en una revista cientifica journal of algebra 15 december 1997 volume 198 issue 2 pages 428 445 https doi org 10 1006 jabr 1997 7147

the change of similarity invariants under row perturbations the problem of the possible similarity classes of all the matrices obtained by small perturbations on some rows of a given complex square matrix is under consideration some necessary conditions on these similarity classes are provided the near converse problem is also studied under what conditions can a given collection of polynomials or partitions be the invariant factors or the weyr characteristic of a matrix obtained by perturbing some rows of a given matrix a complete answer is provided in some particular cases the obtained conditions can all be derived from the previously mentioned necessary conditions

reduction numbers of ideals abstract we give general bounds for the reduction numbers of ideals in arbitrary noetherian rings and multiplicity dependent bounds for m primary ideals in a noetherian local ring r m in the case of polynomial rings over fields the bound is a non elementary function with four levels of exponentiation for primary ideals the bound is linear in the samuel multiplicity of the ideal finally we extend these techniques to generic complete intersections of dimension one

the polynomial fourier transform with minimum mean square error for noisy data in 2006 naoki saito proposed a polyharmonic local fourier transform phlft to decompose a signal f l 2 w into the sum of a polyharmonic componentu and a residualv where w is a bounded and open domain in r d the solution presented in phlft in general does not have an error with minimal energy in resolving this issue we propose the least squares approximant to a given signal in l 2 1 1 using the combination of a set of algebraic polynomials and a set of trigonometric polynomials the maximum degree of the algebraic polynomials is chosen to be small and fixed we show in this paper that the least squares approximant converges uniformly for a holder continuous function therefore gibbs phenomenon will not occur around the boundary for such a function we also show that the phlft converges uniformly and is a near least squares approximation in the sense that it is arbitrarily close to the least squares approximant in l 2 norm as the dimension of the approximation space increases our experiments show that the proposed method is robust in approximating a highly oscillating signal even when the signal is corrupted by noise the method is still robust the experiments also reveal that an optimum degree of the trigonometric polynomial is needed in order to attain the minimal l 2 error of the approximation when there is noise present in the data set this optimum degree is shown to be determined by the intrinsic frequency of the signal we also discuss the energy compaction of the solution vector and give an explanation to it

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

the change of the jordan structure of a mtrix under small perturbations abstract in this paper a complete description including multiplicity is given for the jordan structure of a matrix which is a small perturbation of a matrix of known jordan structure the problem solved here was solved independently as noted and the other solution has been published in english however the solutions are different the problem is important and it seems desirable to have this solution widely available

the change of feedback invariants under one row perturbation in this paper we completely characterize possible feedback invariants of a rectangular matrix under small additive perturbations on one of its rows

smooth local cosine based galerkin method for scattering problems the smooth local trigonometric slt functions are employed as the basis and testing functions in the galerkin based method of moments mom and sparse impedance matrices are obtained the basic idea of slt is to use smooth cutoff functions to split the function and to fold overlapping parts back into the intervals so that the orthogonality of the system is preserved moreover by choosing the correct trigonometric basis rapid convergence in the case of smooth functions is ensured the slt system is particularly suitable to handle electrically large scatterers where the integral kernel behaves in a highly oscillatory manner numerical examples demonstrate the scattering of electromagnetic waves from two dimensional objects with smooth contours as well as with sharp edges a comparison of the new approach versus the traditional mom and wavelet methods is provided

fast imdct and mdct algorithms a matrix approach this paper presents a systematic investigation of the modified discrete cosine transform inverse modified discrete cosine transform mdct imdct algorithm using a matrix representation this approach results in new understanding of the mdct imdct enables the development of new algorithms and makes clear the connection between the algorithms we represent in a matrix form the imdct as the product of the type iv dct with simple scaling sign changing and permutation operations such that fast algorithms for the type iv dct can be simply modified for the imdct and vice versa then the simple symmetry and inversion properties of the type iv dct are used to develop new algorithms and establish the connection between existing fast imdct algorithms this approach also enables us to show that mdct and imdct share common core operation and present an efficient architecture for implementing both the mdct and the imdct in one hardware

stratification of full rank polynomial matrices we show that perturbations of polynomial matrices of full normal rank can be analyzed viathe study of perturbations of companion form linearizations of such polynomial matrices it is proved that a

global analytic block similarity to a brunoovsky form abstract necessary and sufficient conditions for reducing a pair of analytic matrix functions a z b z to a brunovsky canonical form by means of analytic passage matrices are given

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

undersampled boundary pre postfilters for low bit rate dct based block coders it has been well established that critically sampled boundary pre postfiltering operators can improve the coding efficiency and mitigate blocking artifacts in traditional discrete cosine transform based block coders at low bit rates in these systems both the prefilter and the postfilter are square matrices this paper proposes to use undersampled boundary pre and postfiltering modules where the pre postfilters are rectangular matrices specifically the prefilter is a fat matrix while the postfilter is a tall one in this way the size of the prefiltered image is smaller than that of the original input image which leads to improved compression performance and reduced computational complexities at low bit rates the design and vlsi friendly implementation of the undersampled pre postfilters are derived their relations to lapped transforms and filter banks are also presented two design examples are also included to demonstrate the validity of the theory furthermore image coding results indicate that the proposed undersampled pre postfiltering systems yield excellent and stable performance in low bit rate image coding

canonical form associated with the problem of perturbation of one column of a controllable pair abstract let a b be a completely controllable matrix pair when we consider the problem of characterizing the controllability indices of all the matrix pairs obtained by small perturbations on one column of b a new equivalence relation arises in a natural way this equivalence relation is a kind of partial feedback equivalence as a consequence the controllability indices are invariant but they do not form a complete system of invariants we have found new invariants for this equivalence relation and we have obtained a canonical form in terms of these invariants

role of anticausal inverses in multirate filter banks i system theoretic fundamentals in a maximally decimated filter bank with identical decimation ratios for all channels the perfect reconstructibility property and the nature of reconstruction filters causality stability fir property and so on depend on the properties of the polyphase matrix various properties and capabilities of the filter bank depend on the properties of the polyphase matrix as well as the nature of its inverse in this paper we undertake a study of the types of inverses and characterize them according to their system theoretic properties i e properties of state space descriptions mcmillan degree degree of determinant and so forth we find in particular that causal polyphase matrices with anticausal inverses have an important role in filter bank theory we study their properties both for the fir and iir cases techniques for implementing anticausal iir inverses based on state space descriptions are outlined it is found that causal fir matrices with anticausal fir inverses cafacafi have a key role in the characterization of fir filter banks

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

extended lapped transform in image coding a modulated lapped transform with extended overlap elt is investigated in image coding with the objective of verifying its potential to replace the discrete cosine transform dct in specific applications some of the criteria utilized for the performance comparison are reconstructed image quality both objective and subjective reduction of blocking artifacts robustness against transmission errors and filtering for scalability also a fast implementation algorithm for finite length signals using symmetric extensions is developed specially for the elt with overlap factor 2 elt 2 this comparison shows that elt 2 is superior to both dct and the lapped orthogonal transform lot

extended lapped transforms properties applications and fast algorithms the family of lapped orthogonal transforms is extended to include basis functions of arbitrary length within this new family the extended lapped transform elt is introduced as a generalization of the previously reported modulated lapped transform mlt design techniques and fast algorithms for the elt are presented as well as examples that demonstrate the good performance of the elt in signal coding applications therefore the elt is a promising substitute for traditional block transforms in transform coding systems and also a good substitute for less efficient filter banks in subband coding systems

on variable overlapped windows and weighted orthonormal bases a generalized lapped orthonormal transform glot formulation is presented for the analysis and synthesis of signals this formulation is composed of a variable width window and a linear combination of weighted orthonormal functions tradeoffs in the specification of windows are examined a sinusoidal example is considered and a fast algorithm is provided for its evaluation

interlacing inequalities and control theory abstract given a p by p matrix a we solve the problem of the existence of a p by q matrix b such that a b has prescribed controllability indices and i p a b has prescribed invariant polynomials the solution of this problem together with an earlier theorem of the author s is used to provide a new proof of the sa thompson interlacing inequalities for invariant polynomials

time domain oversampled lapped transforms theory structure and application in image coding this paper generalizes time domain lapped transforms tdlts proposed by tran et al to oversampled systems thus leading to time domain oversampled lapped transforms tdolts these new transforms correspond to a subclass of oversampled linear phase perfect reconstruction filterbanks olpprfbs which can be implemented by adding a prefilter before the discrete cosine transform dct and a post filter after the inverse discrete cosine transform idct structures of the pre and post filters are developed and the frame theoretic properties of tdolts are analyzed a new parameterization of lattice matrices through the givens qr factorization is proposed for unconstrained optimization comparisons with other parameterization methods are also included several design examples along with some image coding results are presented to demonstrate the validity of the theory and the potential of tdolts in image coding especially in error resilient coding

role of anticausal inverses in multirate filter banks ii the fir case factorizations and biorthogonal lapped transforms for pt i see ibid vol 43 no 5 p 1090 1990 in part i we studied the system theoretic properties of discrete time transfer matrices in the context of inversion and classified them according to the types of inverses they had in particular we outlined the role of causal fir matrices with anticausal fir inverses abbreviated cafacafi in the characterization of fir perfect reconstruction pr filter banks essentially all fir pr filter banks can be characterized by causal fir polyphase matrices having anticausal fir inverses in this paper we introduce the most general degree one cafacafi building block and consider the problem of factorizing cafacafi systems into these building blocks factorizability conditions are developed a special class of cafacafi systems called the biorthogonal lapped transform bolt is developed and shown to be factorizable this is a generalization of the well known lapped orthogonal transform lot examples of unfactorizable cafacafi systems are also demonstrated finally it is shown that any causal fir matrix with fir inverse can be written as a product of a factorizable cafacafi system and a unimodular matrix

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

on denoising and best signal representation we propose a best basis algorithm for signal enhancement in white gaussian noise the best basis search is performed in families of orthonormal bases constructed with wavelet packets or local cosine bases we base our search for the best basis on a criterion of minimal reconstruction error of the underlying signal this approach is intuitively appealing because the enhanced or estimated signal has an associated measure of performance namely the resulting mean square error previous approaches in this framework have focused on obtaining the most compact signal representations which consequently contribute to effective denoising these approaches however do not possess the inherent measure of performance which our algorithm provides we first propose an estimator of the mean square error based on a heuristic argument and subsequently compare the reconstruction performance based upon it to that based on the stein 1981 unbiased risk estimator we compare the two proposed estimators by providing both qualitative and quantitative analyses of the bias term having two estimators of the mean square error we incorporate these cost functions into the search for the best basis and subsequently provide a substantiating example to demonstrate their performance

integral degree of a ring reduction numbers and uniform artin rees numbers abstract the supremum of reduction numbers of ideals having principal reductions is expressed in terms of the integral degree a new invariant of the ring which is finite provided the ring has finite integral closure as a consequence one obtains bounds for the castelnuovo mumford regularity of the rees algebra and for the artin rees numbers

best basis search in lapped dictionaries this paper proposes analyzes and illustrates several best basis search algorithms for dictionaries consisting of lapped orthogonal bases it improves upon the best local cosine basis selection based on a dyadic tree by considering larger dictionaries of bases it is shown that this can result in sparser representations and approximate shift invariance an algorithm that is strictly shift invariant is also provided the experiments in this paper suggest that the new dictionaries can be advantageous for time frequency analysis compression and noise removal accelerated versions of the basic algorithm are provided that explore various tradeoffs between computational efficiency and adaptability it is shown that the proposed algorithms are in fact applicable to any finite dictionary comprised of lapped orthogonal bases one such novel dictionary is proposed that constructs the best local cosine representation in the frequency domain and it is shown that the new dictionary is better suited for representing certain types of signals

on a homeomorphism between orbit spaces of linear systems and matrix polynomials abstract the orbit space of controllable systems under system similarity and the orbit space of matrix polynomials with determinant degree equal to the order of the state matrix under right equivalence are proved to be homeomorphic when the quotient compact open topology is considered in the latter as a consequence the variation of the finite and left wiener hopf structures under small perturbations of matrix polynomials with fixed degree for their determinants is described

on matrix functions which commute with their derivative abstract we improve several results published from 1950 up to 1982 on matrix functions commuting with their derivative and establish two results of general interest the first one gives a condition for a finite dimensional vector subspace e t of a normed space not to depend on t when t varies in a normed space the second one asserts that if a is a matrix function defined on a set of the form a t u diag b1 t bp t u 1 t and if each matrix function bk has the polynomial form b k t i 0 k f ki t c k i t k 1 p then a itself has the polynomial form a t i 0 d 1 f i t c i t where d k 1 p d k dk being the degree of the minimal polynomial of the matrix ck for every k 1 p

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

matrices with prescribed rows and invariant factors abstract in this paper we solve the problem of the existence of an n n matrix over an arbitrary field when its invariant polynomials and either some rows or columns are prescribed the solution is given in terms of invariant factor inequalities and of majorization inequalities involving controllability indices and the degrees of the invariant polynomials

invariants of the block tensor product abstract we are concerned with the following problem let xi i 1 2 be vector spaces over c yi subspaces of xi i 1 2 and fi yi xi i 1 2 linear m defined on a subspace we characterize the invariants of block similarity of f1 f2 y1 y2 x1 x2 in terms of the block similarity invariants of f1 and f2

p and s decomposition in anisotropic media with localized low rank approximations abstractwe have developed a p and s wave decomposition algorithm based on windowed fourier transforms and a localized low rank approximation with improved scalability and efficiency for anisotropic wavefields the model and wavefield are divided into rectangular blocks that do not have to be geologically constrained low rank approximations and p and s decomposition are performed separately in each block an overlap add method reduces artifacts at block boundaries caused by fourier transforms at wavefield truncations limited communication is required between blocks localization allows a lower rank to be used than global low rank approximations while maintaining the same quality of decomposition the algorithm is scalable making p and s decomposition possible in complicated 3d models tests with 2d and 3d synthetic data indicate good p and s decomposition results

on the eigenvectors of symmetric toeplitz matrices this paper presents a number of results concerning the eigenvectors of a symmetric toeplitz matrix and the location of the zeros of the filters eigenfilters whose coefficients are the elements of the eigenvectors one of the results is that the eigenfilters corresponding to the maximum and minimum eigenvalues if distinct have their zeros on the unit circle while the zeros of the other eigenfilters may or may not have their zeros on the unit circle even if the zeros of the eigenfilters of a matrix are all on the unit circle the matrix need not be toeplitz examples are given to illustrate the different properties

subspace approach for the design of cosine modulated filter banks with linear phase prototype filter a method for designing perfect reconstruction pr prototypes for paraunitary cosine modulated filter banks is presented the design procedure is based on a subspace approach that allows linear combinations of even length linear phase pr prototype filters in such a way that the resulting filter is also a linear phase pr prototype within a given subspace the weights of the optimal linear combination can easily be computed via an eigenanalysis the filter design is carried out iteratively while the pr property is guaranteed throughout the design process no nonlinear optimization routine is needed as a special case the proposed approach allows the design of discrete coefficient prototypes which are of great interest for efficient hardware implementations

bounds on the a invariant and reduction numbers of ideals abstract let r be a d dimensional standard graded ring over an artinian local ring let m be the unique maximal homogeneous ideal of r let hi r n denote the length of the nth graded component of the local cohomology module h i m r define the eisenbud goto invariant eg r of r to be the number q 0 d 1 d 1 q h q m r 1 q we prove that the a invariant a r of the top local cohomology module h m d r satisfies the inequality a r e r l r1 d 1 l r0 1 eg r this bound is used to get upper bounds for the reduction number of an m primary ideal i of a cohen macaulay local ring r m when the associated graded ring of i has depth at least d 1

cosine modulated fir filter banks satisfying perfect reconstruction the authors obtain a necessary and sufficient condition on the 2m m number of channels polyphase components of a linear phase prototype filter of length n 2 mm where m an arbitrary positive integer such that the polyphase component matrix of the modulated filter is lossless the losslessness of the polyphase component matrix in turn is sufficient to ensure that the analysis synthesis system satisfies perfect reconstruction pr using this result a novel design procedure is presented based on the two channel lossless lattice this enables the design of a large class of fir finite impulse response pr filter banks and includes the n 2m case it is shown that this approach requires fewer parameters to be optimized than in the pseudo qmf quadrature mirror filter designs and in the lossless lattice based pr qmf designs for equal length filters in the three designs this advantage becomes significant when designing long filters for large m the design procedure and its other advantages are described in detail design examples and comparisons are included

multirate filter banks with block sampling multirate filter banks with block sampling were recently studied by khansari and leon garcia 1993 in this paper we want to systematically study multirate filter banks with block sampling by studying general vector filter banks where the input signals and transfer functions in conventional multirate filter banks are replaced by vector signals and transfer matrices respectively we show that multirate filter banks with block sampling studied by khansari and leon garcia are special vector filter banks where the transfer matrices are pseudocirculant we present some fundamental properties for the basic building blocks such as noble identities interchangeability of down up sampling polyphase representations of m channel vector filter banks and multirate filter banks with block sampling we then present necessary and sufficient conditions for the alias free property finite impulse response fir systems with fir inverses paraunitariness and lattice structures for paraunitary vector filter banks we also present a necessary and sufficient condition for paraunitary multirate filter banks with block sampling as an application of this theory we present all possible perfect reconstruction delay chain systems with block sampling we also show some examples that are not paraunitary for conventional multirate filter banks but are paraunitary for multirate filter banks with proper block sampling in this paper we also present a connection between vector filter banks and vector transforms studied by li vector filter banks also play important roles in multiwavelet transforms and vector subband coding

the algebraic structure of pencils and block toeplitz matrices abstract we restate several results on the kronecker structure obtainable after a given matrix pencil is affected by an arbitrarily small perturbation and state some new results on the kronecker structure after a new row is appended or a column deleted the results are simple consequences of a theory of majorization for semi infinite integer sequences the bounds after a new row is appended correspond to bounding the controllability indices after adding a single input to a linear time invariant dynamical system we illustrate the known relations between the kronecker indices the segre characteristics and the nullities of certain block toeplitz matrices constructed from the original pencils and discuss how those nullities change when the pencil is affected by a perturbation or a new row

direct sum cancellation for modules over one dimensional rings let r be a one dimensional noetherian domain with finite normalization r in the eighties the second named author and s wiegand developed a mechanism for studying the cancellation problem for finitely generated torsion free r modules the key idea described in wie84 and ww87 is to represent a given torsion free module m as a pullback m rm m fm rm fm this research was initiated while the author was a visitor at the university of nebraska supported by the fonds zur forderung der wissenschaftlichen forschung partially supported by grants from the national science foundation the national security agency and the mathematical sciences research institute where much of this research was done

encoding of images based on a lapped orthogonal transform traditional block transform image coding systems generate artifacts near block boundaries which degrade low bit rate coded images to reduce these artifacts a class of unitary transformations called lapped orthogonal transforms lot is investigated the basis function on which the signal is projected are overlapped for adjacent blocks an example of an lot optimized in terms of energy compaction is numerically derived using an augmented lagrangian optimization algorithm using this lot intraframe coding experiments for 256 240 pixel images were performed at bit rates between 0 1 and 0 35 bits pixel the lot improved the coded image subjective quality over other transforms the lot was also used in interframe full motion video coding experiments for head and shoulder sequences at 28 and 56 kb s significant improvement resulted at low data rates and if no motion compensation were used however the improvement was no longer significant at 56 kb s with full motion compensation

on computing the discrete fourier and cosine transforms an efficient algorithm based on real matrix decomposition is developed for computing a class of sinusoidal transforms that include the discrete fourier and cosine transform

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

reversible symmetric nonexpansive convolution an effective image boundary processing for mbi m channel lifting based linear phase filter banks we present an effective image boundary processing for m channel m in bbn m geq 2 lifting based linear phase filter banks that are applied to unified lossy and lossless image compression coding i e lossy to lossless image coding the reversible symmetric extension we propose is achieved by manipulating building blocks on the image boundary and reawakening the symmetry of each building block that has been lost due to rounding error on each lifting step in addition complexity is reduced by extending nonexpansive convolution called reversible symmetric nonexpansive convolution because the number of input signals does not even temporarily increase our method not only achieves reversible boundary processing but also is comparable with irreversible symmetric extension in lossy image coding and outperformed periodic extension in lossy to lossless image coding

fixed and adaptive predictors for hybrid predictive transform coding hybrid predictive transform coding is studied the usual formulation is to first apply a unitary transform and then code the transform coefficients with independent dpcm coders i e the prediction is performed in the transform domain this structure is compared to spatial domain prediction where a difference signal is formed in the spatial domain and then coded by a transform coder a linear spatial domain predictor which minimizes the mean square prediction error also minimizes the mean square of each transform coefficient the two structures are equivalent if the transform domain prediction scheme is extended to a more general predictor hence the structure that gives the easiest implementation can be chosen the spatial domain structure is preferred for motion compensation and for line interlaced video signals interframe hybrid coding experiments are performed on interlaced videophone scenes using an adaptive transform coder motion compensation gives a rate reduction of 25 35 percent compared to frame difference prediction with the same mean square error the subjective advantage is even greater since the dirty window effect is not present with motion compensation it is important to perform the motion estimation with fractional pel accuracy field coding with a switched predictor using previous field in moving areas is an interesting alternative to frame coding with frame difference prediction

quadrature mirror filter design for an arbitrary number of equal bandwidth channels the two channel quadrature mirror filter structure of croisier and esteban may be extended to an arbitrary number of equal bandwidth channels given certain restrictions on the bandpass filters the most serious restriction is that the stopband attenuation of eacli band pass filter must be high for all frequencies outside twice the nominal 3 db bandwidth of the filter this restriction is not really a limiting factor for speech subband waveform coding since high adjacent channel attenuation is a necessity for the confinement of quantization noise a key feature of our filter structure is that the number of multiplies adds and stored coefficients required for implementation is significantly less than those needed for the conventional qmf structure given the same number of channels fortran code for a 16 channel filter structure is listed as an example of efficient implementation

similarity and block similarity we present some equivalent characterizations of the block similarity of matrix pairs with an application to a characterization of the similarity of two square matrices also a survey of some generalizations of properties of the invariants of similarity to the invariants of block similarity is presented