using polygons to recognize and locate partially occluded objects we present computer vision algorithms that recognize and locate partially occluded objects the scene may contain unknown objects that may touch or overlap giving rise to partial occlusion the algorithms revolve around a generate test paradigm the paradigm iteratively generates and tests hypotheses for compatibility with the scene until it identifies all the scene objects polygon representations of the object s boundary guide the hypothesis generation scheme choosing the polygon representation turns out to have powerful consequences in all phases of hypothesis generation and verification special vertices of the polygon called corners help detect and locate the model in the scene polygon moment calculations lead to estimates of the dissimilarity between scene and model corners and determine the model corner location in the scene an association graph represents the matches and compatibility constraints extraction of the largest set of mutually compatible matches from the association graph forms a model hypothesis using a coordinate transform that maps the model onto the scene the hypothesis gives the proposed model s location and orientation hypothesis verification requires checking for region consistency the union of two polygons and other polygon operations combine to measure the consistency of the hypothesis with the scene experimental results give examples of all phases of recognizing and locating the objects

multiple description coding using pairwise correlating transforms the objective of multiple description coding mdc is to encode a source into multiple bitstreams supporting multiple quality levels of decoding in this paper we only consider the two description case where the requirement is that a high quality reconstruction should be decodable from the two bitstreams together while lower but still acceptable quality reconstructions should be decodable from either of the two individual bitstreams this paper describes techniques for meeting mdc objectives in the framework of standard transform based image coding through the design of pairwise correlating transforms the correlation introduced by the transform helps to reduce the distortion when only a single description is received but it also increases the bit rate beyond that prescribed by the rate distortion function of the source we analyze the relation between the redundancy i e the extra bit rate and the single description distortion using this transform based framework we also describe an image coder that incorporates the pairwise transform and show its redundancy rate distortion performance for real images

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

how to prove that a preconditioner cannot be superlinear in the general case of multilevel toeplitz matrices we recently proved that any multilevel circulant preconditioner is not superlinear a cluster it may provide cannot be proper the proof was based on the concept of quasi equimodular matrices although this concept does not apply for example to the sine transform matrices in this paper with a new concept of partially equimodular matrices we cover all trigonometric matrix algebras widely used in the literature we propose a technique for proving the nonsuperlinearity of certain frequently used preconditioners for some representative sample multilevel matrices at the same time we show that these preconditioners are in a certain sense the best among the sublinear preconditioners with only a general cluster for multilevel toeplitz matrices

fast parallel solvers for elliptic problems in an earlier work 1 we have considered the use of t 2 preconditioners for the solution of dirichlet boundary value problems by means of the preconditioned conjugate gradients methods in this paper we propose the use of toeplitz matrices for the fast parallel solution of differential problems we propose some algorithms to solve preconditioned systems which are especially suitable in parallel environments we also present several numerical results to prove the effectiveness of our technique

decision region approximation by polynomials or neural networks we give degree of approximation results for decision regions which are defined by polynomial and neural network parametrizations the volume of the misclassified region is used to measure the approximation error and results for the degree of l sub 1 approximation of functions are used for polynomial parametrizations we show that the degree of approximation is at least 1 whereas for neural network parametrizations we prove the slightly weaker result that the degree of approximation is at least r where r can be any number in the open interval 0 1

the solution of ill conditioned symmetric toeplitz systems via two grid and wavelet methods abstract in this paper we present a generalized multigrid method combined with wavelet filters for solving ill conditioned symmetric toeplitz systems t n x b where t n r n n is generated by nonnegative functions with zeros first we propose the construction of general cohen daubechies and feauveau cdf 9 7 biorthogonal wavelet systems so that a new class of compactly supported biorthogonal wavelet systems gcdf are achieved with specified vanishing moments for scaling functions in order to solve ill conditioned toeplitz systems by using the two grid method tgm we use the constructed gcdf wavelets to get prolongation and restriction operators as a result the proposed tgm is proved by numerical experiment to be more efficient than the classic tgm especially when t n is seriously ill conditioned for the vanishing moments being n experimental tests illustrate that the tgm with damped jacobi smoother converges when the generating function has zeros of order less than or equal to 2n n 8 besides we prove in theory that the proposed method converges for toeplitz systems that are generated by functions with zeros of order less than or equal to four

capacity regions and sum rate capacities of vector gaussian interference channels the capacity regions of vector or multiple input multiple output gaussian interference channels are established for very strong interference and aligned strong interference furthermore the sum rate capacities are established for z interference noisy interference and mixed aligned weak intermediate and aligned strong interference these results generalize known results for scalar gaussian interference channels

spectral behavior of matrix sequences and discretized boundary value problems abstract in this paper we provide theoretical tools for dealing with the spectral properties of general sequences of matrices of increasing dimension more specifically we give a unified treatment of notions such as distribution equal distribution localization equal localization clustering and sub clustering as a case study we consider the matrix sequences arising from the finite difference fd discretization of elliptic and semielliptic boundary value problems bvps the spectral analysis is then extended to toeplitz based preconditioned matrix sequences with special attention to the case where the coefficients of the differential operator are not regular belong to l 1 and to the case of multidimensional problems the related clustering properties allow the establishment of some ergodic formulas for the eigenvalues of the preconditioned matrices

multigrid methods for multilevel circulant matrices we introduce a multigrid technique for the solution of multilevel circulant linear systems whose coefficient matrix has eigenvalues of the form f x j n where f is continuous and independent of n n 1 ldots n d and x j n equiv 2 pi j n 2 pi j 1 n 1 ldots 2 pi j d n d 0 le j r le n r 1 the interest of the proposed technique pertains to the multilevel banded case where the total cost is optimal i e o n arithmetic operations ops n prod r 1 d n r instead of o n log n ops arising from the use of ffts in fact multilevel banded circulants are used as preconditioners for elliptic and parabolic pdes with dirichlet or periodic boundary conditions and for some two dimensional image restoration problems where the point spread function psf is numerically banded so that the overall cost is reduced from o k varepsilon n n log n to o k varepsilon n n where k varepsilon n is the number of pcg iterations to reach the solution within an accuracy of varepsilon several numerical experiments concerning one rank regularized circulant discretization of elliptic 2q differential operators over one dimensional and two dimensional square domains with mixed boundary conditions are performed and discussed

optimal subband filter banks for multiple description coding multiple description coding refers to the encoding of an information source into multiple bit streams in a manner that each bit stream independently represents a coarse description of the source while multiple descriptions jointly convey a higher fidelity source representation thus multiple description coding allows enhanced resilience to loss of individual descriptions and can be viewed as a form of source based diversity coding this paper addresses the design of subband filter banks to optimize the rate distortion performance of multiple description subband coding the problem is formulated in a lagrangian optimization framework and is solved directly in the frequency domain to produce the optimal filter spectral responses by varying the lagrangian parameter our design obtains all points on the redundancy distortion curve for gaussian wide sense stationary input processes we consider the cases of both orthonormal and more general biorthogonal filter banks we also analyze the connections between these two solutions and quantify the gap in their optimal performance application of the proposed methods to the popular first order autoregressive ar 1 model yields very interesting and insightful results in the two extreme cases of maximum and minimum redundancy our solutions degenerate to previously known results finally comparisons with popularly deployed filter banks reveal large gaps from the theoretical performance bound

compact fourier analysis for designing multigrid methods the convergence of multigrid methods can be analyzed based on a fourier analysis of the method or by proving certain inequalities that have to be fulfilled by the smoother and by the coarse grid correction separately here we analyze the multigrid method for the constant coefficient poisson equation with a compact fourier analysis using the formalism of multilevel toeplitz matrices and their generating functions or symbols the fourier analysis is applied for determining the smoothing factor and the overall error of the combined smoothing and coarse grid correction error reduction of a twogrid step tgs by representing the twogrid step explicitly by a symbol if the effects of the smoothing correction and the coarse grid correction are orthogonal to each other then in a twogrid step the error is removed in one step and the twogrid method can be considered as a direct solver if the coarse linear system is identical to the original matrix then the same projection and smoother again make the twogrid step a direct solver hence the multigrid cycle has to be applied only once with one smoothing step on each level and therefore the whole multigrid method can be considered as a direct solver in this paper we want to identify multigrid as a direct solver in one dimension with a coarse system derived by linear and constant interpolation and in two dimensions for a modified projection related to u trottenberg c w osterlee and a schuller multigrid academic press san diego 2001 paragraph a 2 3 by studying not only smoothing and coarse grid correction separately but the symbol of the full twogrid step as well we are furthermore able to derive better convergence estimates and information on how to find efficient combinations of projection and smoother as smoothers we consider also approximate inverse smoothers colored smoothers that take into account the different character of grid points and rank reducing smoothers that coincide with the given matrix on a large number of entries numerical examples show the effectiveness of the new approach

recursive based pcg methods for toeplitz systems with nonnegative generating functions in this paper we consider the solutions of symmetric positive definite but ill conditioned toeplitz systems an x b here we propose to solve the system by the recursive based preconditioned conjugate gradient method the idea is to use the inverse of am the principal submatrix of an with the gohberg semencul formula as a preconditioner for an the inverse of am can be generated recursively by using the formula until m is small enough the construction of the preconditioners requires only the entries of an and does not require the explicit knowledge of the generating function f of an we show that if f is a nonnegative bounded and piecewise continuous even function with a finite number of zeros of even order the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers hence the conjugate gradient method when applied to solving the preconditioned system converges very quickly numerical results are included to illustrate the effectiveness of our approach

on the matrix equation x a tx 12ma i in this paper we study iterative methods for finding the extremal positive definite solutions of the matrix equation x a tx 1ma i first a condition on the existence of a positive definite solution of this matrix equation is given then the existence as well as the rate of convergence of some proposed algorithms to obtain the extremal positive definite solutions of this equation is presented moreover a generalization of computationally simple and efficient known algorithm is applied for obtaining the extremal positive definite solutions in addition both the necessary and sufficient conditions for this matrix equation to have positive definite solution are presented numerical examples are given to illustrate the performance and the effectiveness of the algorithms

preconditioning strategies for hermitian indefinite toeplitz linear systems in this paper we propose and analyze preconditioning strategies for hermitian indefinite linear systems by using indefinite preconditioners under very elementary assumptions we show that the eigenvalues are real moreover in the case of multilevel toeplitz structures we prove distributional and localization results these techniques used in connection with the cg gmres bicgstab and qmr algorithms allow us to solve in an optimal way the corresponding linear systems a wide numerical experimentation confirms the efficiency of the proposed procedures

multi iterative methods abstract a general class of iterative methods is introduced for solving positive definite linear systems ax b these methods use two or more different iterative techniques and each of them reduces the error by a constant factor in a different subspace of r n the multigrid as an example of a multi iterative method combines these partial results and we reach a reduction of the error by at each step where is the number of applications of the first scheme and gt 1 is a constant independent of n in the second part we consider the prolongation operator p with a special antidiagonal structure and we show that the multigrid method for toeplitz matrices 1 is an example of a method of this class in a third part we extend the multigrid method to block toeplitz matrices obtaining a block antidiagonal structure for toeplitz and block toeplitz cases the choice of p which maximizes convergence speed simply depends on the coefficients of matrix a the numerical results on 1 d and 2 d elliptic differential problems confirm the effectiveness of this multi iterative method in fact its spectral radius is very small and independent of the mesh spacing h

a new modification of the rojo method for solving symmetric circulant five diagonal systems of linear equations abstract a new effective and stable modification of the rojo method 1 for solving of real symmetric circulant five diagonal systems of linear equations is proposed this special kind of system appears in many applications spline approximation difference solution of partial differential equations etc the method presented in this paper a very efficient and stable method

non symbolic algorithms for the inversion of tridiagonal matrices a representation for the entries of the inverse of general tridiagonal matrices is based on the determinants of their principal submatrices it enables us to introduce through the linear recurrence relations satisfied by such determinants a simple algorithm for the entries of the inverse of any tridiagonal nonsingular matrix reduced as well as unreduced the numerical approach is preserved here without invoking the symbolic computation for tridiagonal diagonally dominant matrices a scaling transformation on the recurrences allows us to give another algorithm to avoid overflow and underflow

stability of invariant maximal semidefinite subspaces i abstract we study stability in classes of subspaces which are invariant under a self adjoint matrix in an indefinite inner product and have various maximality and semidefiniteness properties with respect to this indefinite inner product descriptions of all subspaces in such a class for which these properties are stable in one or another way are obtained

on the existence of a positive definite solution of the matrix equation x atx 1a i the question is raised under which conditions on the real square matrix a the matrix equation x atx 1a i has a real symmetric positive definite solution x both necessary and sufficient solvability conditions on a are derived moreover we give an algorithm to calculate the solution for a number of special cases we also present an analytic solution

a scale space filtering approach for visual feature extraction this paper presents a new integrated approach for detecting visual features which include corners ends arcs and lines the effect of scale space filtering on visual features is studied in detail as it forms the theoretical basis of our work in this approach the outline of the object is first extracted and it is then smoothed by scale space filtering at different scale levels subsequently the local extreme curvature points extracted from the smoothed curve and end candidates are determined to guide the termination of the filtering process information about the curvature of each point at the largest scale level is used to detect the different kinds of visual features several algorithms are proposed to determine corners ends arcs and lines experimental results show that our approach is robust to translation rotation and scaling of the object as well as noise corruption in addition efficient visual features can also be successfully extracted with this approach

the rate of convergence of toeplitz based pcg methods for second order nonlinear boundary value problems in previous works 21 23 we proposed the use of tau 5 and band toeplitz based preconditioners for the solution of 1d and 2d boundary value problems bvp by means of the preconditioned conjugate gradient pcg methods as tau and band toeplitz linear systems can be solved 4 by using fast sine transforms 8 these methods become especially attractive in a parallel environment of computation in this paper we extend this technique to the nonlinear nonsymmetric case and in addition we prove some clustering properties for the spectra of the preconditioned matrices showing why these methods exhibit a convergence speed which results to be more than linear therefore these methods work much finer than those based on separable preconditioners 18 45 on incomplete lu factorizations 36 13 27 and on circulant preconditioners 9 30 35 since the latter two techniques do not assure a linear rate of convergence on the other hand the proposed technique has a wider range of application since it can be naturally used for nonlinear nonsymmetric problems and for bvp in which the coefficients of the differential operator are not strictly positive and only piecewise smooth finally the several numerical experiments performed here and in 22 23 confirm the effectiveness of the theoretical analysis

superlinear convergence for pcg using band plus algebra preconditioners for toeplitz systems the paper studies fast and efficient solution algorithms for nxn symmetric ill conditioned toeplitz systems t n f x b where the generating function f is known a priori real valued nonnegative and has isolated roots of even order the preconditioner that we propose is a product of a band toeplitz matrix and matrices that belong to a certain trigonometric algebra the basic idea behind the proposed scheme is to combine the advantages of all components of the product that are well known when every component is used as a stand alone preconditioner as a result we obtain a flexible preconditioner which can be applied to the system t n f x b infusing superlinear convergence to the pcg method the important feature of the proposed technique is that it can be extended to cover the 2d case i e ill conditioned block toeplitz matrices with toeplitz blocks we perform many numerical experiments whose results confirm the theoretical analysis and effectiveness of the proposed strategy

an adaptive reduction procedure for the piecewise linear approximation of digitized curves a new algorithm is presented for the piecewise linear approximation of two dimensional digitized curves against a square grid the algorithm utilizes an adaptive reduction procedure in two approximation phases to select the critical points of a digitized curve such that the deviation from the digitized curve to its final approximated curve is bounded by a uniform error tolerance the time complexity of this algorithm is o m sup 2 rather than o n sup 2 on the theoretical plane in the experiments of fixing the initial and the final processing points the performance of the algorithm has been compared to those of three prominent other algorithms regarding the required number of critical points and the total execution time of the program of the four algorithms compared the present algorithm consistently has the shortest execution time of the program and it tends most to require as few critical points as the optimum algorithm that was tested

a direct method for solving circulant tridiagonal block systems of linear equations this paper presents a modification of rojo s algorithm comput math appl 20 1990 61 to solve block circulant tridiagonal systems of linear equations which are toeplitz and hermitian this new approach gives us a general direct algorithm for solving the problem we will show how to choose a block matrix as a parameter to describe the method we employ the factorization of block toeplitz tridiagonal matrices as the product of two block toeplitz subdiagonal and superdiagonal matrices the algorithm is based on obtaining the solution of the nonlinear matrix equation email protected b c 1b finally some numerical results will be given

two level toeplitz preconditioning approximation results for matrices and functions large 2 level toeplitz systems arise in a variety of applications see e g r h chan and m ng siam rev 38 1996 pp 427 482 for which efficient numerical methods for their solution are required some successful numerical techniques need the explicit knowledge of the generating function f of the considered system t n f rm x b an assumption that usually is not fulfilled in real applications in this paper we analyze and complete the procedure proposed in d noutsas s serra capizzano and p vassalos numer linear algebra appl 12 2005 pp 231 239 for the 2 level case in such a way from the knowledge of the coefficients of t n f we determine optimal preconditioning strategies for the solution of our systems finally some numerical experiments are performed and discussed in connection with our theoretical analysis

new polygonal approximation schemes for object shape representation abstract object shape representation is an important issue in pattern recognition and computer vision applications a set of new polygonal approximation schemes namely the internal maximum area polygonal approximation imapa the external minimum area polygonal approximation emapa and the minimum area deviation polygonal approximation madpa are proposed for object shape representation the area deviation between the initial object shape and its approximation polygon is defined as the cost function instead of outputting a unique approximation polygon each proposed scheme outputs a sequence of approximation polygons with different numbers of line segments and different area deviations for various application situations for a given area deviation bound and an initial object shape each proposed scheme can give a corresponding optimal approximation polygon with a minimum number of line segments a performance comparison among the proposed schemes is included the proposed schemes are compared with two other existing schemes some experimental results show the feasibility of the proposed approaches

feature oriented multiple description wavelet based image coding we address the problem of resilient image coding over error prone networks where packet losses occur recent literature highlights the multiple description coding mdc as a promising approach to solve this problem in this paper we introduce a novel wavelet based multiple description image coder referred to as the feature oriented mdc fo mdc the proposed multiple description md coder exploits the statistics of the wavelet coefficients and identifies the subsets of samples that are sensitive to packet loss a joint optimization between tree pruning and quantizer selection in the rate distortion sense is used in order to allocate more bits to these sensitive coefficients when compared with the state of the art md scalar quantization coder the proposed fo mdc yields a more efficient central side distortion tradeoff control mechanism furthermore it proves to be more robust for image transmission even with high packet loss ratios which makes it suitable for protecting multimedia streams over packet erasure channels

the conditioning of fd matrix sequences coming from semi elliptic differential equations abstract in this paper we are concerned with the study of spectral properties of the sequence of matrices a n a coming from the discretization using centered finite differences of minimal order of elliptic or semielliptic differential operators l a u of the form 1 d d x a x d d x u x f x on 0 1 dirichlet b c on where the nonnegative bounded coefficient function a x of the differential operator may have some isolated zeros in more precisely we state and prove the explicit form of the inverse of a n a and some formulas concerning the relations between the orders of zeros of a x and the asymptotic behavior of the minimal eigenvalue condition number of the related matrices as a conclusion and in connection with our theoretical findings first we extend the analysis to higher order semi elliptic differential operators and then we present various numerical experiments showing that similar results must hold true in 2d as well

an improvement to multiple description transform coding a multiple description transform coding mdtc method has been reported previously the redundancy rate distortion rrd performance of this coding scheme for the independent and identically distributed i i d two dimensional 2 d gaussian source has been analyzed using mean squared error mse as the distortion measure at the small redundancy region the mdtc scheme can achieve excellent rrd performance because a small increase in redundancy can reduce the single description distortion at a rate faster than exponential but the performance of mdtc becomes increasingly poor at larger redundancies this paper describes a generalization of the mdtc gmdtc scheme which introduces redundancy both by transform and through correcting the error resulting from a single description its rrd performance is closer to the theoretical bound in the entire range of redundancy analysis both for a single pair of variables and for multiple variables is presented

design of entropy constrained multiple description scalar quantizers the problem of entropy constrained multiple description scalar quantizer design is posed as an optimization problem necessary conditions for optimality are derived and an iterative design algorithm is presented performance results are presented for a gaussian source along with comparisons to the multiple description rate distortion bound and a reference system

robust and optimal multi iterative techniques for iga galerkin linear systems we consider fast solvers for the large linear systems coming from the isogeometric analysis iga collocation approximation based on b splines of full elliptic d dimensional partial differential equations pdes we are interested in designing iterative algorithms which are optimal and robust the former property implies that the computational cost is linear with respect to the number of degrees of freedom i e the matrix size the latter property means that the convergence rate is completely independent of or only mildly dependent on all the relevant parameters in our setting we can mention the coefficients of the pde the dimensionality d the geometric map g describing the physical domain the matrix size related to the fineness parameters and the spline degrees associated with the iga approximation order our approach is based on the spectral symbol which describes the global eigenvalue behavior of the iga collocation matrices it is precisely the spectral information contained in the symbol that is exploited for the design of an optimal and robust multi iterative solver of multigrid type several numerical experiments are presented and discussed in view of recent theoretical findings concerning the symbol and its properties

multiple description wavelet based image coding we consider the problem of coding images for transmission over error prone channels the impairments we target are transient channel shutdowns as would occur in a packet network when a packet is lost or in a wireless system during a deep fade when data is delivered it is assumed to be error free but some of the data may never reach the receiver the proposed algorithms are based on a combination of multiple description scalar quantizers with techniques successfully applied to the construction of some of the most efficient subband coders a given image is encoded into multiple independent packets of roughly equal length when packets are lost the quality of the approximation computed at the receiver depends only on the number of packets received but does not depend on exactly which packets are actually received when compared with previously reported results on the performance of robust image coders based on multiple descriptions on standard test images our coders attain similar psnr values using typically about 50 60 of the bit rate required by these other state of the art coders while at the same time providing significantly more freedom in the mechanism for allocation of redundancy among descriptions

multigrid methods for symmetric positive definite block toeplitz matrices with nonnegative generating functions in this paper we introduce a generalized multigrid method for solving linear systems bf t n m bf x bf b where bf t n m in re nm times nm is a symmetric block toeplitz matrix with symmetric toeplitz blocks we use a special choice of the projection operator whose coefficients simply depend on the generating functions associated with the proposed class of matrices this choice leads to iterative methods with convergence rates independent of the euclidean condition number kappa 2 bf t n m and of the dimension of the involved matrices the total arithmetic cost is o nm log nm for dense matrices and o nm for band matrices with band blocks in the pram model only o log nm parallel steps are required this algorithm is therefore competitive with the preconditioned conjugate gradient methods proposed by chan and jin siam j sci statist comput 13 1992 pp 1218 1235 ku and kuo siam j sci statist comput 13 1992 pp 948 966 di benedetto siam j sci comput 17 1996 and serra bit 34 1994 pp 579 594 for dense matrices and improves those results for band block toeplitz matrices with band toeplitz blocks

asymptotic performance of multiple description transform codes the multiple description transform coder is introduced for sources with memory and an asymptotic analysis is presented for the squared error distortion for stationary gaussian sources the optimal transform and the optimal bit allocation for the multiple description coder are identical to those for the single description coder

polygonal approximation of digital curves based on the principles of perceptual organization abstract a new approach to the problem of polygonal approximation of digital shapes is presented in this paper in this approach the approximation process is divided into three stages based on the principles of perceptual organization and three algorithms are developed accordingly the linking merging algorithm is firstly proposed for low level processing the smoothing algorithm is then followed for further approximation the high level rule based algorithm is finally applied to preserve the visual feature points while reducing the number of segments in the approximated polygon as much as possible the effectiveness of the proposed algorithm is demonstrated by testing both synthetic shapes and scanned in shapes

polygonal approximation using a competitive hopfield neural network abstract polygonal approximation plays an important role in pattern recognition and computer vision in this paper a parallel method using a competitive hopfield neural network chnn is proposed for polygonal approximation based on the chnn the polygonal approximation is regarded as a minimization of a criterion function which is defined as the arc to chord deviation between the curve and the polygon the chnn differs from the original hopfield network in that a competitive winner take all mechanism is imposed the winner take all mechanism adeptly precludes the necessity of determining the values for the weighting factors in the energy function in maintaining a feasible result the proposed method is compared to several existing methods by the approximation error norms l2 and l with the result that promising approximation polygons are obtained

analysis of preconditioning strategies for collocation linear systems in the past few years several authors have studied the preconditioning of collocation matrices by finite differences fds matrices arising from the associated collocation points here we discuss how to solve in an efficient way nonuniform grid fd linear systems including those related to a generic fd collocation preconditioner the main idea is based on a further step of preconditioning defined in terms of diagonal and toeplitz matrices first we identify the limit spectral distributions of the involved fd collocation matrix sequences and then we prove that the proposed toeplitz based preconditioners assure a clustering at the unity with respect to the eigenvalues in the 1d case in the 2d case the situation is different so that more appropriate strategies are discussed a wide numerical experimentation emphasizing the correctness of the theoretical results is also reported

on the nonlinear matrix equation x a f x a q solutions and perturbation theory abstract in this paper the nonlinear matrix equation x a f x a q is discussed sufficient conditions for the existence and uniqueness of a positive semidefinite solution are derived also conditions are given under which the solution depends continuously on the matrices a and q

band toeplitz preconditioners for block toeplitz systems abstract we consider the solutions of block toeplitz systems with toeplitz blocks by the preconditioned conjugate gradient pcg method here the block toeplitz matrices are generated by nonnegative functions f x y we use band toeplitz matrices as preconditioners the generating functions g x y of the preconditioners are trigonometric polynomials of fixed degree and are determined by minimizing f g f we prove that the condition number of the preconditioned system is o 1 an a priori bound on the number of iterations for convergence is obtained

computing the extremal positive definite solutions of a matrix equation an efficient and numerically stable implementation of a known algorithm is suggested for finding the extremal positive definite solutions of the matrix equation x a x 1 a i if such solutions exist the convergence rate is analyzed a new algorithm that avoids matrix inversion is presented numerical examples are given to illustrate the effectiveness of the algorithms

multiple description coding with prediction compensation a new multiple description coding paradigm is proposed by combining the time domain lapped transform block level source splitting linear prediction and prediction residual encoding the method provides effective redundancy control and fully utilizes the source correlation the joint optimization of all system components and the asymptotic performance analysis are presented image coding results demonstrate the superior performance of the proposed method especially at low redundancies

block band toeplitz preconditioners derived from generating function approximations analysis and applications we are concerned with the study and the design of optimal preconditioners for ill conditioned toeplitz systems that arise from a priori known real valued nonnegative generating functions f x y having roots of even multiplicities our preconditioned matrix is constructed by using a trigonometric polynomial x y obtained from fourier kernel approximations or from the use of a proper interpolation scheme both of the above techniques produce a trigonometric polynomial x y which approximates the generating function f x y and hence the preconditioned matrix is forced to have clustered spectrum as x y is chosen to be a trigonometric polynomial the preconditioner is a block band toeplitz matrix with toeplitz blocks and therefore its inversion does not increase the total complexity of the pcg method preconditioning by block toeplitz matrices has been treated in the literature in several papers we compare our method with their results and we show the efficiency of our proposal through various numerical experiments

a new method for solving symmetric circulant tridiagonal systems of linear equations in this paper a new method for computing the solution of a linear system having a symmetric circulant tridiagonal matrix is presented this special kind of system appears in many applications after an appropriate partition of the system and the elimination of the last unknown we apply the woodbury formula to arrive at a very efficient and stable method which is quite competitive with gaussian elimination and with the modified double sweep method

analysis of a multigrid method as an iterative technique for solving linear systems a general class of iterative methods is introduced for solving symmetric positive definite linear systems these methods use two different approximations to the inverse of the matrix of the problem one of which involves the inverse of a smaller matrix it is shown that the methods of this class reduce the error by a constant factor at each step and that under ideal circumstances this constant is equal to kappa 1 kappa 1 where kappa is the ratio of the largest eigenvalue to the j 1 eigenvalue of the matrix j being the dimension of the smaller matrix involved a multigrid method is presented as an example of a method of this class and it is shown that while the multigrid method does not quite achieve this optimal rate of convergence it does reduce the error at each step by a constant factor independent of the mesh spacing h the size of this constant factor and properties of the differential equation and the discretization that affect it are also discussed

spectral analysis of coupled pdes and of their schur complements via generalized locally toeplitz sequences in 2d spectral analysis of coupled pdes and of their schur complements via generalized locally toeplitz sequences in 2d

how to choose the best iterative strategy for symmetric toeplitz systems in the last decade many efficient iterative solvers for n x n symmetric also hermitian or generic toeplitz systems have been devised r h chan and g strang siam j sci stat comput 10 1989 pp 104 119 t f chan siam j sci stat comput 9 1988 pp 766 771 d bini and f di benedetto proceedings 2nd spaa conference 1990 pp 220 223 d bini and p favati siam j matrix anal appl 14 1993 pp 500 507 t huckle siam j matrix anal appl 13 1992 pp 767 777 f di benedetto g fiorentino and s serra comput math appl 26 1993 pp 35 45 s serra siam j matrix anal appl 17 1996 pp 1007 1019 g fiorentino and s serra calcolo 28 1991 pp 283 305 s serra siam j matrix anal appl 20 1998 pp 31 44 in all these methods it is assumed that the considered systems have the form an f x b where the symbol f the generating function is an l1 function and the entries of an f along the kth diagonal coincide with the kth fourier coefficient of f the proof of convergence and sometimes even the definition of these solvers are derived by using some symbolic or analytic properties of f while this function is often unknown in real applications in this paper by means of the knowledge of the coefficients of an f only we propose and discuss an algorithm to economically determine the minimal features of f allowing us to choose and define the best iterative strategy as a consequence of the analysis of the proposed procedure we find some results on the structure of quasi algebra of the set a n f f in l 1 finally we exhibit several numerical experiments which confirm the effectiveness of the proposed idea

a qr method for computing the singular values via semiseparable matrices the standard procedure to compute the singular value decomposition of a dense matrix first reduces it into a bidiagonal one by means of orthogonal transformations once the bidiagonal matrix has been computed the qr method is applied to reduce the latter matrix into a diagonal one in this paper we propose a new method for computing the singular value decomposition of a real matrix in a first phase an algorithm for reducing the matrix a into an upper triangular semiseparable matrix by means of orthogonal transformations is described a remarkable feature of this phase is that depending on the distribution of the singular values after few steps of the reduction the largest singular values are already computed with a precision depending on the gaps between the singular values an implicit qr method for upper triangular semiseparable matrices is derived and applied to the latter matrix for computing its singular values the numerical tests show that the proposed method can compete with the standard method using an intermediate bidiagonal matrix for computing the singular values of a matrix

iterative solution of two matrix equations we study iterative methods for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a q and x a x 1 a q where q is hermitian positive definite general convergence results are given for the basic fixed point iteration for both equations newton s method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation numerical results are reported to illustrate the convergence behaviour of various algorithms

planar shape matching based on binary tree shape representation abstract in this paper we suggest a new way of representing planar two dimensional shapes and a shape matching method which utilizes the new representation through merging of the neighboring boundary runs a shape can be partitioned into a set of triangles these triangles are inherently connected according to a binary tree structure here we use the binary tree with the triangles as its nodes to represent the shape this representation is found to be insensitive to shape translation rotation scaling and skewing changes due to viewer s location changes or the object s pose changes furthermore the representation is of multiresolution in shape matching we compare the two trees representing two given shapes node by node according to the breadth first tree traversing sequence the comparison is done from top of the tree and moving downward which means that we first compare the lower resolution approximations of the two shapes if the two approximations are different the comparison stops otherwise it goes on and compares the finer details of the two shapes only when the two shapes are very similar will the two corresponding trees be compared entirely thus the matching algorithm utilizes the multiresolution characteristic of the tree representation and appears to be very efficient

optimum uniform piecewise linear approximation of planar curves two dimensional digital curves are often uniformly approximated by polygons or piecewise linear curves several algorithms have been proposed in the literature to find such curves we present an algorithm that finds a piecewise linear curve with the minimal number of segments required to approximate a curve within a uniform error with fixed initial and final points we compare our optimal algorithm to several suboptimal algorithms with respect to the number of linear segments required in the approximation and the execution time of the algorithm

data reduction of large vector graphics fast algorithm for joint near optimal approximation of multiple polygonal curves is proposed it is based on iterative reduced search dynamic programming introduced earlier for the min eproblem of a single polygonal curve the proposed algorithm jointly optimizes the number of line segments allocated to the different individual curves and the approximation of the curves by the given number of segments trade off between time and optimality is controlled by the breadth of the search and by the numbers of iterations applied

positive solutions to x a bx 1 b abstract we study the positive semidefinite solutions to the matrix equation x a bx 1 b under the assumption that a 0 we show that a positive solution exists if and only if a certain block tridiagonal operator is positive in which case the solution is given by the shorted operator or generalized schur complement of that operator

optimal polygonal approximation of digitized curves using the sum of square deviations criterion abstract we address the problem of finding an optimal polygonal approximation of digitized curves several optimal algorithms have already been proposed to determine the minimum number of points that define a polygonal approximation of the curve with respect to a criterion error we present a new algorithm with reasonable complexity to determine the optimal polygonal approximation using the mean square error norm the idea is to estimate the remaining number of segments and to integrate the cost in the a algorithm the solution is optimal in the minimum number of segments

using the sherman morrison woodbury inversion formula for a fast solution of tridiagonal block toeplitz systems abstract a fast numerical algorithm for solving systems of linear equations with tridiagonal block toeplitz matrices is presented the algorithm is based on a preliminary factorization of the generating quadratic matrix polynomial associated with the toeplitz matrix followed by the sherman morrison woodbury inversion formula and solution of two bidiagonal and one diagonal block toeplitz systems tight estimates of the condition numbers are provided for the matrix system and the main matrix systems generated during the preliminary factorization the emphasis is put on rigorous stability analysis to rounding errors of the sherman morrison woodbury inversion numerical experiments are provided to illustrate the theory

matrix algebra preconditioners for multilevel toeplitz matrices are not superlinear let f be a d variate 2 periodic continuous function and let tn f n n n1 n d be the multiindexed sequence of multilevel n n toeplitz matrices n n n i ni generated by f let a an n be a sequence of matrix algebras simultaneously diagonalized by unitary transforms we show that there exist infinitely many linearly independent trigonometric polynomials and continuous nonpolynomial functions f such that rank tn f pn o n n d 1 n 1 i for any matrix sequence p pn a this implies that no superlinear matrix algebra preconditioner exists in the multilevel toeplitz case the above mentioned result improves the analysis of the author and e tyrtyshnikov siam j matrix anal appl 21 2 1999 431 where the same was proved under the assumption that the involved algebras are of circulant type 2002 elsevier science inc all rights reserved

asymptotic analysis of multiple description quantizers a high rate analysis of multiple description quantizers is presented for rth power distortions and general source densities both fixed length and variable length encoding of the quantizer indices are considered optimal companding functions are shown to be the same as for single channel quantizers as compared to the bound of ozarow 1980 a gap of 8 69 db and 3 07 db exists between the entropy constrained and level constrained cases respectively for a memoryless gaussian source and r 2

necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation x a x 1a q abstract we consider the problem of when the matrix equation x a x 1 a q has a positive definite solution here q is positive definite we study both the real and the complex case this equation plays a crucial role in solving a special case of the discrete time riccati equation we present both necessary and sufficient conditions for its solvability this result is obtained by using an analytic factorization approach moreover we present algebraic recursive algorithms to compute the largest and smallest the solution of the equation respectively finally we discuss the number of solutions

some theorems on linear positive operators and functionals and their applications abstract in this paper we prove two theorems concerning linear positive operators and functionals in a very general sense we show how these results can be used to derive new results in a wide range of applications from approximation theory to the numerical treatment of differential equations and to the numerical analysis connected with signals and images

moment based methods for polygonal approximation of digitized curves object shape representation plays an important role in the area of image processing pattern recognition and computer vision in the past two decades many algorithms have been suggested for creating approximated polygons in this study two new polygonal approximation methods based on the geometric moments and the orthogonal moments defined in terms of legendre polynomials are proposed the difference between the moments defined by the initial contour and those of the approximated polygon is taken as the objective function each algorithm provides various polygonal approximation results with different number of line segments for different application situations for a given error bound we can determine the optimal polygon with a minimum number of line segments the procedures are applied to some digital curves and better results are obtained in comparison with some known methods

some algorithms for solving special tridiagonal block toeplitz linear systems this paper is focused on different methods and algorithms for solving tridiagonal block toeplitz systems of linear equations we consider the el sayed method ph d thesis 1996 for such systems and propose several modifications that lead to different algorithms which we discuss in detail our algorithms are then compared with some classical techniques as far as implementation time is concerned number of operations and storage comments and conclusions for computing efficiency of the proposed new algorithms are given numerical experiments corroborating the theoretical results are also presented

equilateral polygon approximation of closed contours numerous algorithms have been suggested for computing polygonal approximations of closed contours we present an algorithm which as a unique feature creates polygons which are equilateral that is whose edges are all of the same length this allows one dimensional shape descriptions to be derived using the interior polygon angles conceptually the algorithm is an optimization framework for finding the minimum energy configuration of a mechanical system of particles and springs that represent the contour points the polygon vertices and their interaction in practice the dominant points are detected on the gaussian smoothed contour and used to seed an initial polygon nonlinear programming is then used to minimize the system energy subject to the constraint that adjacent vertices must be equidistant the objective function is the sum of a curvature weighted distance between each vertex and a set of contour points associated therewith experimental results are given for closed contours obtained from grey scale images

multiple description image coding based on lagrangian rate allocation in this paper a novel multiple description coding technique is proposed based on optimal lagrangian rate allocation the method assumes the coded data consists of independently coded blocks initially all the blocks are coded at two different rates then blocks are split into two subsets with similar rate distortion characteristics two balanced descriptions are generated by combining code blocks belonging to the two subsets encoded at opposite rates a theoretical analysis of the approach is carried out and the optimal rate distortion conditions are worked out the method is successfully applied to the jpeg 2000 standard and simulation results show a noticeable performance improvement with respect to state of the art algorithms the proposed technique enables easy tuning of the required coding redundancy moreover the generated streams are fully compatible with part 1 of the standard

convergence analysis of two grid methods for elliptic toeplitz and pdes matrix sequences summary the solution of large toeplitz systems with nonnegative generating functions by multigrid methods was proposed in previous papers 13 14 22 the technique was modified in 6 36 and a rigorous proof of convergence of the tgm two grid method was given in the special case where the generating function has only a zero at x 0 0 of order at most two here by extending the latter approach we perform a complete analysis of convergence of the tgm under the sole assumption that f is nonnegative and with a zero at x 0 0 of finite order an extension of the same analysis in the multilevel case and in the case of finite difference matrix sequences discretizing elliptic pdes with nonconstant coefficients and of any order is then discussed

a direct method for solving block circulant banded system of linear equations abstract a direct method to solve block circulant banded system of linear equations is presented in the paper the method applies the kernel solution of linear equations with circulant banded matrix proposed by fuyong 2011 17 by using the kernel solution of linear equations with block circulant banded matrix we obtain the boundary equations of the linear equations and by solving the boundary equations of the linear systems we finally get the solution of the linear equations the proposed method has great advantage in getting solutions of linear systems with block circulant banded matrix coefficient matrices an example is presented in the paper

multigrid methods for anisotropic bttb systems multigrid methods are highly efficient solution techniques for large sparse multilevel toeplitz systems which are positive definite and ill conditioned in this paper we develop multigrid methods which are especially designed for anisotropic two level toeplitz bttb matrices first a method is described for systems with anisotropy along coordinate axes as a suitable combination of semicoarsening and full coarsening steps although the basic idea is known from the solution of partial differential equations we present it here in a more formal way using generating functions and their level curves this enables us not only to prove the optimal convergence of the two grid method but also to carry over the results to systems with anisotropy in other directions we introduce new coordinates in order to describe these more complicated systems in terms of generating functions this enables us to solve them with the same efficiency for the two level method we present a convergence proof in this more general case

on the spectrum of stiffness matrices arising from isogeometric analysis we study the spectral properties of stiffness matrices that arise in the context of isogeometric analysis for the numerical solution of classical second order elliptic problems motivated by the applicative interest in the fast solution of the related linear systems we are looking for a spectral characterization of the involved matrices in particular we investigate non singularity conditioning extremal behavior spectral distribution in the weyl sense as well as clustering of the eigenvalues to a certain compact subset of mathbb c c all the analysis is related to the notion of symbol in the toeplitz setting and is carried out both for the cases of 1d and 2d problems

a v cycle multigrid for multilevel matrix algebras proof of optimality we analyze the convergence rate of a multigrid method for multilevel linear systems whose coefficient matrices are generated by a real and nonnegative multivariate polynomial f and belong to multilevel matrix algebras like circulant tau hartley or are of toeplitz type in the case of matrix algebra linear systems we prove that the convergence rate is independent of the system dimension even in presence of asymptotical ill conditioning this happens iff f takes the zero value more precisely if the d level coefficient matrix has partial dimension nr at level r with r 1 ldots d then the size of the system is n varvec n prod r 1 d n r varvec n n 1 ldots n d and o n n operations are required by the considered v cycle multigrid in order to compute the solution within a fixed accuracy since the total arithmetic cost is asymptotically equivalent to the one of a matrix vector product the proposed method is optimal some numerical experiments concerning linear systems arising in 2d and 3d applications are considered and discussed

matrix algebra preconditioners for multilevel toeplitz systems do not insure optimal convergence rate in the last decades several matrix algebra optimal and superlinear preconditioners those assuring a strong clustering at the unity have been proposed for the solution of polynomially ill conditioned toeplitz linear systems the corresponding generalizations for multilevel structures are neither optimal nor superlinear see e g contemp math 281 2001 193 concerning the notion of superlinearity it has been recently shown that the proper clustering cannot be obtained in general see linear algebra appl 343 344 2002 303 siam j matrix anal appl 22 1 1999 431 math comput 72 2003 1305 in this paper by exploiting a proof technique previously proposed by the authors see contemp math 323 2003 313 we prove that the spectral equivalence and the essential spectral equivalence up to a constant number of diverging eigenvalues are impossible too in conclusion optimal matrix algebra preconditioners in the multilevel setting simply do not exist in general and therefore the search for optimal iterative solvers should be oriented to different directions with special attention to multilevel multigrid techniques

filter banks for prediction compensated multiple description coding in this paper a prediction compensated multiple description md coding framework for two band filter banks is proposed in which the coefficients in each subband are split into two descriptions each description also includes the prediction residuals of the data in the other description the designs of the optimal orthogonal and biorthogonal filter banks are formulated in a unified framework and both one level and multiple level decompositions are analyzed contrary to the existing md filter banks in the literature the optimal filter banks in the proposed scheme are quite similar to those in single description coding therefore the method can be applied to systems with single description optimized filter banks and still attain near optimal performance image coding results show that this method achieves better performance and lower complexity than the latest jpeg 2000 based mdc

optimal filter banks for multiple description coding analysis and synthesis multiple description md coding is a source coding technique for information transmission over unreliable networks in md coding the coder generates several different descriptions of the same signal and the decoder can produce a useful reconstruction of the source with any received subset of these descriptions in this paper we study the problem of md coding of stationary gaussian sources with memory first we compute an approximate md rate distortion region for these sources which we prove to be asymptotically tight at high rates this region generalizes the md rate distortion region of el gamal and cover 1982 and ozarow 1980 for memoryless gaussian sources then we develop an algorithm for the design of optimal two channel biorthogonal filter banks for md coding of gaussian sources we show that optimal filters are obtained by allocating the redundancy over frequency with a reverse water filling strategy finally we present experimental results which show the effectiveness of our filter banks in the low complexity low rate regime

preconditioning of block toeplitz matrices by sine transforms the iterative solution of a block toeplitz linear system by the conjugate gradient method is analyzed the preconditioning step being solved by means of a discrete sine transform convergence properties are established and compared to the behavior of the block circulant preconditioner recently proposed in the literature as in the scalar case the new approach takes advantage if the system is ill conditioned

design of multiple description scalar quantizers the design of scalar quantizers for communication systems that use diversity to overcome channel impairments is considered the design problem is posed as an optimization problem and necessary conditions for optimality are derived a design algorithm a generalization of s p lloyd s 1962 algorithm for quantizer design is developed unlike a single channel scalar quantizer the performance of a multiple description scalar quantizer is dependent on the index assignment the problem of index assignment is addressed good index assignments performance results and sample quantizer designs are presented for a memoryless gaussian source comparisons are made with rate distortion bounds for the multiple description problem

2 d object recognition using invariant contour descriptor and projective refinement abstract this paper presents an efficient model based recognition method to recognize 2 d objects and to obtain correspondences between models and scene boundaries with a subpixel positioning error as a shape signature for a contour we propose a descriptor consisting of five point invariants which are used to index a hash table also we propose a projective refinement as a verification method to compute exact correspondences between models and scene contour points this method repeatedly computes projective transformation using a weighted pseudo inverse we present an error model for five point invariants which are used to define a similarity between two descriptors to determine a searching bound in indexing and to obtain the weights in the projective refinement in experiments using seriously distorted images of forty models this method led to the recognition of planar curved objects a transformation using the correspondence between the model and scene points on contours was also obtained

a curve bend function based method to characterize contour shapes in this paper a new curve bend function cbf based method is presented to extract main features of a contour it utilizes two quantities one called a curve bend angle cba and the other known as a type coefficient of curve segment defined on a contour the cba measures the bending degree at each point on the contour while the type coefficient characterizes the properties convexity and concavity of simple curve segments the cbf combines the advantages of the two quantities and provides an overall description of a contour a local maximum or minimum of the cbf corresponds to a critical point of the contour and the peak value is the cosine of the cba hence the problem of locating critical points of a contour is reduced to the problem of finding peak points of the cbf the sign of a peak value indicates the related cba being an inner or outer angle our experimental results and comparisons between the cbf method and rosenfeld johnston s method show that our method is effective and accurate in generating the description of a contour

lapped transform via time domain pre and post filtering this paper presents a general framework of constructing a large family of lapped transforms with symmetric basis functions by adding simple time domain pre and post processing modules onto existing block discrete cosine transform dct based infrastructures a subset of the resulting solutions is closed form fast computable modular near optimal in the energy compaction sense and leads to an elegant boundary handling of finite length data starting from these solutions a general framework for block based signal decomposition with a high degree of flexibility and adaptivity is developed several simplified models are also introduced to approximate the optimal solutions these models are based on cascades of plane rotation operators and lifting steps respectively despite tremendous savings in computational complexity the optimized results of these simplified models are virtually identical to that of the complete solution the multiplierless versions of these pre and post filters when combined with an appropriate multiplierless block transform such as the bindct which is described in an earlier paper by liang and tran see ieee trans signal processing vol 49 p 3032 44 dec 2001 generate a family of very large scale integration vlsi friendly fast lapped transforms with reversible integer to integer mapping numerous design examples with arbitrary number of channels and arbitrary number of borrowed samples are presented