the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

karhunen loeve expansion of a set of rotated templates we propose a novel method for efficiently calculating the eigenvectors of uniformly rotated images of a set of templates as we show the images can be optimally approximated by a linear series of eigenvectors which can be calculated without actually decomposing the sample covariance matrix

signal flow graph approach to efficient and forward stable dst algorithms abstract in this paper signal flow graphs are mainly addressed for the efficient and forward stable discrete sine transform dst algorithms having sparse scaled orthogonal rotational rotational reflection and butterfly matrices in electrical engineering theoretical computer science control theory system engineering etc we often use signal flow graphs as a modeling tool which interconnects the system components or represents the realization of a system as electronic devices the objective in this paper is to establish the connection between algebraic operations used in sparse and scaled orthogonal factorizations of dst i iv matrices with the signal flow graph building blocks this paper elaborates signal flow graphs for the foundational frameworks with n 8 and n 16 and use these together with the digital structures to describe the dst algorithms having a n 1 point signal flow graph for dst i and n point signal flow graphs for dst ii iv the presented dst algorithms are completely recursive and solely based on corresponding matrices dst i iv these dst algorithms have low arithmetic complexity especially the low number of multiplications and significant speed improvement factor as opposed to most existing algorithms finally this paper establishes that the presented algorithms are forward stable dst algorithms

tree simplification and the plateaux phenomenon of graph laplacian eigenvalues we developed a procedure of reducing the number of vertices and edges of a given tree which we call the tree simplification procedure without changing its topological information our motivation for developing this procedure was to reduce computational costs of graph laplacian eigenvalues of such trees when we applied this procedure to a set of trees representing dendritic structures of retinal ganglion cells of a mouse and computed their graph laplacian eigenvalues we observed two plateaux i e two sets of multiple eigenvalues in the eigenvalue distribution of each such simplified tree in this article after describing our tree simplification procedure we analyze why such eigenvalue plateaux occur in a simplified tree explain such plateaux can occur in a more general graph if it satisfies a certain condition and identify these two eigenvalues specifically as well as the lower bound to their multiplicity

algebraic signal processing theory 1 d space in our paper titled ldquoalgebraic signal processing theory foundation and 1 d timerdquo appearing in this issue of the ieee transactions on signal processing we presented the algebraic signal processing theory an axiomatic and general framework for linear signal processing the basic concept in this theory is the signal model defined as the triple a m phi where a is a chosen algebra of filters m an associated a module of signals and phi is a generalization of the z transform each signal model has its own associated set of basic sp concepts including filtering spectrum and fourier transform examples include infinite and finite discrete time where these notions take their well known forms in this paper we use the algebraic theory to develop infinite and finite space signal models these models are based on a symmetric space shift operator which is distinct from the standard time shift we present the space signal processing concepts of filtering or convolution ldquoz transform rdquo spectrum and fourier transform for finite length space signals we obtain 16 variants of space models which have the 16 discrete cosine and sine transforms dcts dsts as fourier transforms using this novel derivation we provide missing signal processing concepts associated with the dcts dsts establish them as precise analogs to the dft get deep insight into their origin and enable the easy derivation of many of their properties including their fast algorithms

an effective vns for the capacitated p median problem in the capacitated p median problem cpmp a set of n customers is to be partitioned into p disjoint clusters such that the total dissimilarity within each cluster is minimized subject to constraints on maximum cluster capacity dissimilarity of a cluster is the sum of the dissimilarities between each customer who belongs to the cluster and the median associated with the cluster an effective variable neighbourhood search heuristic for this problem is proposed the heuristic is characterized by the use of easily computed lower bounds to assess whether undertaking computationally expensive calculation of the worth of moves within the neighbourhood search is necessary the small proportion of moves that need to be assessed fully are then evaluated by an exact solution of a relatively small subproblem computational results on five standard sets of benchmark problem instances show that the heuristic finds all the best known solutions for one instance the previously best known solution is improved if only marginally

on a set of matrix algebras related to discrete hartley type transforms a set of fast real transforms including the well known hartley transform is fully investigated mixed radix splitting properties of hartley type transforms are examined in detail the matrix algebras diagonalized by the hartley type matrices are expressed in terms of circulant and 1 circulant matrices

jointly optimized spatial prediction and block transform for video and image coding this paper proposes a novel approach to jointly optimize spatial prediction and the choice of the subsequent transform in video and image compression under the assumption of a separable first order gauss markov model for the image signal it is shown that the optimal karhunen loeve transform given available partial boundary information is well approximated by a close relative of the discrete sine transform dst with basis vectors that tend to vanish at the known boundary and maximize energy at the unknown boundary the overall intraframe coding scheme thus switches between this variant of the dst named asymmetric dst adst and traditional discrete cosine transform dct depending on prediction direction and boundary information the adst is first compared with dct in terms of coding gain under ideal model conditions and is demonstrated to provide significantly improved compression efficiency the proposed adaptive prediction and transform scheme is then implemented within the h 264 avc intra mode framework and is experimentally shown to significantly outperform the standard intra coding mode as an added benefit it achieves substantial reduction in blocking artifacts due to the fact that the transform now adapts to the statistics of block edges an integer version of this adst is also proposed

an uncertainty principle for finite frames abstract we consider the notion of uncertainty for finite frames using a difference operator inspired by the gauss hermite differential equation we obtain a time frequency measure for finite frames we then find the minimizer of the measure over all equal norm parseval frames dependent on the dimension of the space and the number of elements in the frame next we show that given a frame one can find the dual frame that minimizes this time frequency measure generalizing some work of daubechies landau and landau to the finite case and extending some recent work on sobolev duals for finite frames

decimation in time and space of finite difference time domain schemes standard isotropic lossless model finite difference time domain fdtd schemes permit changes in the grid density on selected regions of the wave propagation domain which can reduce the computational load of the simulations one possible alternative to varying the spatial density is to change the simulation temporal rate this idea looks attractive when the wave signals exhibit pronounced bandwidth fluctuations across time this is particularly true in sound synthesis where a physically based acoustic resonator can be conveniently modeled using such schemes to overcome the computational constraints that must be met by real time distributed resonator models this paper deals with the decimation in time and space of isotropic lossless finite difference time domain schemes holding conventional nyquist shannon limits on the bandwidth of the wave signals formulas for the reconstruction of these signals at runtime over the interpolated grid are provided for both the 1d and 2d orthogonal case depending on the ideal boundary conditions either neumann or dirichlet holding at each side of the grid in connection with the domain side lengths either even or odd together the boundaries and size determine the type of discrete cosine transform used in the corresponding interpolation formula numerical artifacts arising as a consequence of decimating in space in 2d are discussed in terms of dispersion error and aliasing considerations concerning the temporal reconstruction of components lying at the decimated nyquist frequency are addressed in the conclusion

a new approach to constrained total least squares image restoration abstract recently there has been a growing interest and progress in using total least squares tls methods for solving blind deconvolution problems arising in image restoration here the true image is to be estimated using only partial information about the blurring operator or point spread function psf which is subject to error and noise in this paper we present a new iterative regularized and constrained tls image restoration algorithm neumann boundary conditions are used to reduce the boundary artifacts that normally occur in restoration processes preliminary numerical tests are reported on some simulated optical imaging problems in order to illustrate the effectiveness of the approach as well as the fast convergence of our iterative scheme

adaptive filtering in subbands using a weighted criterion transform domain adaptive algorithms have been proposed to reduce the eigenvalue spread of the matrix governing their convergence thus improving the convergence rate however a classical problem arises from the conflicting requirements between algorithm improvement requiring rather long transforms and the need to keep the input output delay as small as possible thus imposing short transforms this dilemma has been alleviated by the so called short block transform domain algorithms but is still apparent this paper proposes an adaptive algorithm compatible with the use of rectangular orthogonal transforms e g critically subsampled lossless perfect reconstruction filter banks thus allowing better tradeoffs between algorithm improvement arithmetic complexity and input output delay the method proposed makes a direct connection between the minimization of a specific weighted least squares criterion and the convergence rate of the corresponding stochastic gradient algorithm this method leads to improvements in the convergence rate compared with both lms and classical frequency domain algorithms

tridiagonal commuting matrices and fractionalizations of dct and dst matrices of types i iv v and viii in this paper we first establish new relationships in matrix forms among discrete fourier transform dft generalized dft gdft and various types of discrete cosine transform dct and discrete sine transform dst matrices two new independent tridiagonal commuting matrices for each of dct and dst matrices of types i iv v and viii are then derived from the existing commuting matrices of dft and gdft with these new commuting matrices the orthonormal sets of hermite like eigenvectors for dct and dst matrices can be determined and the discrete fractional cosine transform dfrct and the discrete fractional sine transform dfrst are defined the relationships among the discrete fractional fourier transform dfrft fractional gdft and various types of dfrct and dfrst are developed to reduce computations for dfrft and fractional gdft

tabu search for the multilevel generalized assignment problem abstract the multilevel generalized assignment problem mgap differs from the classical gap in that agents can perform tasks at more than one efficiency level important manufacturing problems such as lot sizing can be formulated as mgaps however the large number of variables in the related 0 1 integer program makes the use of commercial optimization packages impractical in this paper we present a heuristic approach to the solution of the mgap which consists of a novel application of tabu search ts our ts method employs neighborhoods defined by ejection chains that produce moves of greater power without significantly increasing the computational effort

a hybrid tabu search branch bound approach to solving the generalized assignment problem a new approach for solving the generalized assignment problem gap is proposed that combines the exact branch bound approach with the heuristic strategy of tabu search ts to produce a hybrid algorithm for solving gap the algorithm described uses commercial software to solve sub problems generated by the ts guiding strategy the ts approach makes use of the concept of referent domain optimisation and introduces novel add drop strategies in addition the linear programming relaxation of gap that forms part of the branch bound approach is itself helpful in suggesting which variables might take binary values computational results on benchmark test instances are presented and compared with results obtained by the standard branch bound approach and also several other heuristic approaches from the literature the results show the new algorithm performs competitively against the alternatives and is able to find some new best solutions for several benchmark instances

a tabu search heuristic for the generalized assignment problem abstract this paper considers the generalized assignment problem gap it is a well known np hard combinatorial optimization problem that is interesting in itself and also appears as a subproblem in other problems of practical importance a tabu search heuristic for the gap is proposed the algorithm uses recent and medium term memory to dynamically adjust the weight of the penalty incurred for violating feasibility the most distinctive features of the proposed algorithm are its simplicity and its flexibility these two characteristics result in an algorithm that compared to other well known heuristic procedures provides good quality solutions in competitive computational times computational experiments have been performed in order to evaluate the behavior of the proposed algorithm

tight wavelet frames on multislice graphs we present a framework for the design of wavelet transforms tailored to data defined on multislice graphs i e multiplex or dynamic graphs graphs with multiple types of interactions are ubiquitous in real life motivating the extension of wavelets to these complex domains our framework generalizes the recently proposed spectral graph wavelet transform sgwt d hammond p vandergheynst and r gribonval wavelets on graphs via spectral graph theory appl comput harmon anal vol 30 pp 129 150 mar 2011 which is designed in the spectral frequency domain of an arbitrary finite weighted graph we extend the sgwt to form a tight frame which conserves energy in the wavelet domain and define the relationship between conventional and spectral graph wavelets we then propose a design for multislice graphs that is based on the higher order singular value decomposition hosvd a powerful tool from multilinear algebra in particular the multiple adjacency matrices are stacked to form a tensor and the hosvd decomposition provides information about its third order structure analogous to that provided by matrix factorizations we obtain a set of eigennetworks and from these graph wavelets which exploit the variability across the graphs we demonstrate the feasibility of our method 1 by capturing different orientations of a gray scale image and 2 by decomposing brain signals from functional magnetic resonance imaging we show its effectiveness to identify variability across graph edges and provide meaningful decompositions

tensor tensor products with invertible linear transforms abstract research in tensor representation and analysis has been rising in popularity in direct response to a the increased ability of data collection systems to store huge volumes of multidimensional data and b the recognition of potential modeling accuracy that can be provided by leaving the data and or the operator in its natural multidimensional form in recent work 1 the authors introduced the notion of the t product a generalization of matrix multiplication for tensors of order three which can be extended to multiply tensors of arbitrary order 2 the multiplication is based on a convolution like operation which can be implemented efficiently using the fast fourier transform fft the corresponding linear algebraic framework from the original work was further developed in 3 and it allows one to elegantly generalize all classical algorithms from numerical linear algebra in this paper we extend this development so that tensor tensor products can be defined in a so called transform domain for any invertible linear transform in order to properly motivate this transform based approach we begin by defining a new tensor tensor product alternative to the t product we then show that it can be implemented efficiently using dcts and that subsequent definitions and factorizations can be formulated by appealing to the transform domain using this new product as our guide we then generalize the transform based approach to any invertible linear transform we introduce the algebraic structures induced by each new multiplication in the family which is that of c algebras and modules finally in the spirit of 4 we give a matrix algebra based interpretation of the new family of tensor tensor products and from an applied perspective we briefly discuss how to choose a transform we demonstrate the convenience of our new framework within the context of an image deblurring problem and we show the potential for using one of these new tensor tensor products and resulting tensor svd for hyperspectral image compression

a stochastic model for a pseudo affine projection algorithm this paper presents a statistical analysis of a pseudo affine projection pap algorithm obtained from the affine projection algorithm ap for a step size alpha 1 and a scalar error signal in the weight update deterministic recursive equations are derived for the mean weight and for the mean square error mse for a large number of adaptive taps n compared to the order p of the algorithm simulations are presented which show good to excellent agreement with the theory in the transient and steady states the pap learning behavior is of special interest in applications where tradeoffs are necessary between convergence speed and steady state misadjustment

square smoothing regularization matrices with accurate boundary conditions this paper is concerned with the solution of large scale linear discrete ill posed problems the determination of a meaningful approximate solution of these problems requires regularization we discuss regularization by the tikhonov method and by truncated iteration the choice of regularization matrix in tikhonov regularization may significantly affect the quality of the computed approximate solution the present paper describes the construction of square regularization matrices from finite difference equations with a focus on the boundary conditions the regularization matrices considered have a structure that makes them easy to apply in iterative methods including methods based on the arnoldi process numerical examples illustrate the properties and effectiveness of the regularization matrices described

a pinned sine transform image coder the pinned sine transform pst coder described is a practical approximation to the pinned klt coder the image is partitioned into two fields a boundary field which depends only on the coded block boundaries and a pinned field which vanishes at the boundaries and is subsequently sine transformed and compressed the reconstructed image is continuous across the block boundaries and blocking effects are hardly visible simulation results are given for fixed and adaptive compression schemes with fair image quality at low bit rates down to 0 3 bits pel

two dimensional symmetric and antisymmetric generalizations of exponential and cosine functions the properties of the four families of the recently introduced special functions of two real variables denoted here by e and cos are studied the superscripts and refer to the symmetric and antisymmetric functions respectively the functions are considered in all details required for their exploitation in fourier expansions of digital data sampled on square grids of any density and for general position of the grid in the real plane relative to the lattice defined by the underlying group theory the quality of continuous interpolation resulting from the discrete expansions is studied exemplified and compared for some model functions

curl constrained gradient estimation for image recovery from highly incomplete spectral data in this paper we introduce new gradient based methods for image recovery from a small collection of spectral coefficients of the fourier transform which is of particular interest for several scanning technologies such as magnetic resonance imaging since gradients of a medical image are much more sparse or compressible than the corresponding image classical ell 1 minimization methods have been used to recover these relative differences the image values can then be obtained by integration algorithms imposing boundary constraints compared with classical gradient recovery methods we propose two new techniques that improve reconstruction first we cast the gradient recovery problem as a compressed sensing problem taking into account that the curl of the gradient field should be zero second inspired by the emerging field of signal processing on graphs we formulate the gradient recovery problem as an inverse problem on graphs iteratively reweighted ell 1 recovery methods are proposed to recover these relative differences and the structure of the similarity graph once the gradient field is estimated the image is recovered from the compressed fourier measurements using least squares estimation numerical experiments show that the proposed approach outperforms the state of the art image recovery methods

bees algorithm for generalized assignment problem bees algorithm ba is a new member of meta heuristics ba tries to model natural behavior of honey bees in food foraging honey bees use several mechanisms like waggle dance to optimally locate food sources and to search new ones this makes them a good candidate for developing new algorithms for solving optimization problems in this paper a brief review of ba is first given afterwards development of a ba for solving generalized assignment problems gap with an ejection chain neighborhood mechanism is presented gap is a np hard problem many meta heuristic algorithms were proposed for its solution so far ba is generally applied to continuous optimization in order to investigate the performance of ba on a complex integer optimization problem an attempt is made in this paper an extensive computational study is carried out and the results are compared with several algorithms from the literature

enhanced cross component linear model for chroma intra prediction in video coding cross component linear model cclm for chroma intra prediction is a promising coding tool in the joint exploration model jem developed by the joint video exploration team jvet cclm assumes a linear correlation between the luma and chroma components in a coding block with this assumption the chroma components can be predicted by the linear model lm mode which utilizes the reconstructed neighboring samples to derive parameters of a linear model by linear regression this paper presents three new methods to further improve the coding efficiency of cclm first we introduce a multi model cclm mm cclm approach which applies more than one linear model to a coding block with mm cclm reconstructed neighboring luma and chroma samples of the current block are classified into several groups and a particular set of linear model parameters is derived for each group the reconstructed luma samples of the current block are also classified to predict the associated chroma samples with the corresponding linear model second we propose a multi filter cclm mf cclm technique which allows the encoder to select the optimal down sampling filter for the luma component with the 4 2 0 color format third we present an lm angular prediction method which synthesizes the angular intra prediction and the mm cclm intra prediction into a new chroma intra coding mode simulation results show that the bd rate savings of 0 55 4 66 and 5 08 on average for y cb and cr components respectively are achieved in all intra configurations with the proposed three methods mm cclm and mf cclm have been adopted into the jem by jvet

a family of adaptive filter algorithms with decorrelating properties although the normalized least mean square nlms algorithm is robust it suffers from low convergence speed if driven by highly correlated input signals one method presented to overcome this problem is the ozeki umeda 1984 affine projection ap algorithm the algorithm applies update directions that are orthogonal to the last p input vectors and thus allows decorrelation of an ar p input process speeding up the convergence this article presents a simple approach to show this property which furthermore leads to the construction of new algorithms that can handle other kinds of correlations such as ma and arma processes a statistical analysis is presented for this family of algorithms similar to the ap algorithm these algorithms also suffer a possible increase in the noise energy caused by their pre whitening filters

dct dst alternate transform image coding an image coding method for low bit rates is proposed it is based on alternate use of the discrete cosine transform dct and the discrete sine transform dst on image blocks this procedure achieves the removal of redundancies in the correlation between neighboring blocks as well as the preservation of continuity across the block boundaries an outline of the mathematical justification of the method assuming a certain first order gauss markov model is given the resulting coding method is then adapted to nonstationary real images by locally adapting the model parameters and improving the block classification technique simulation results are shown and compared with the performance of related previous methods namely adaptive dct and fast karhunen loeve transform fklt

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 tabu search approach to the constraint satisfaction problem as a general problem solver many combinatorial problems including a variety of combinatorial optimization problems can be naturally formulated as a constraint satisfaction problem csp we develop in this paper a tabu search based algorithm for the csp as a foundation for a general problem solver in addition to the basic components of tabu search we develop a number of elaborations such as an automatic control mechanism for the tabu tenure modification of the penalty function to handle objective functions and enlargement of the neighborhood by allowing swap operations computational results with our algorithm are reported for various problems selected from a wide range of applications i e graph coloring generalized assignment set covering timetabling and nurse scheduling our results appear to be competitive with those of existing algorithms specially developed for the respective problem domains

principles of scatter search scatter search is an evolutionary method that has been successfully applied to hard optimization problems the fundamental concepts and principles of the method were first proposed in the 1970s based on formulations dating back to the 1960s for combining decision rules and problem constraints in contrast to other evolutionary methods like genetic algorithms scatter search is founded on the premise that systematic designs and methods for creating new solutions afford significant benefits beyond those derived from recourse to randomization it uses strategies for search diversification and intensification that have proved effective in a variety of optimization problems this paper provides the main principles and ideas of scatter search and its generalized form path relinking we first describe a basic design to give the reader the tools to create relatively simple implementations more advanced designs derive from the fact that scatter search and path relinking are also intimately related to the tabu search ts metaheuristic and gain additional advantage by making use of ts adaptive memory and associated memory exploiting mechanisms capable of being tailored to particular contexts these and other advanced processes described in the paper facilitate the creation of sophisticated implementations for hard problems that often arise in practical settings due to their flexibility and proven effectiveness scatter search and path relinking can be successfully adapted to tackle optimization problems spanning a wide range of applications and a diverse collection of structures as shown in the papers of this volume

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

orthogonal modes of frequency modulation and the sturm liouville frequency modulation model sinusoidal signals and complex exponentials play a critical role in lti system theory in that they are eigenfunctions of the lti convolution operator while processing frequency modulated fm waveforms using lti systems restrictive assumptions must be placed on the system so that a quasi eigenfunction approximation holds upon deviation from these assumptions fm waveforms incur significant distortion in this paper a sturm liouville s l model for frequency modulation introduced by the author is extended to a study orthogonal modes of continuous and discrete frequency modulation and b to develop system theoretical underpinnings for fm waveforms these fm modes have the same special connection with respect to the fm s l system operator that complex exponentials have with lti systems and the convolution operator the finite s l fm spectrum or transform that measures the strength of the orthogonal fm modes present in a fm signal analogous to the discrete fourier spectrum for sinusoids is introduced finally similarities between the orthogonal s l fm modes and angular mathieu functions are exposed and a conjecture connecting the two is put forth

a path relinking approach with ejection chains for the generalized assignment problem the generalized assignment problem is a classical combinatorial optimization problem known to be np hard it can model a variety of real world applications in location allocation machine assignment and supply chains the problem has been studied since the late 1960s and computer codes for practical applications emerged in the early 1970s we propose a new algorithm for this problem that proves to be more effective than previously existing methods the algorithm features a path relinking approach which is a mechanism for generating new solutions by combining two or more reference solutions it also features an ejection chain approach which is embedded in a neighborhood construction to create more complex and powerful moves computational comparisons on benchmark instances show that the method is not only effective in general but is especially effective for types d and e instances which are known to be very difficult

kronecker product approximations for image restoration with whole sample symmetric boundary conditions reflexive boundary conditions bcs assume that the array values outside the viewable region are given by a symmetry of the array values inside the reflection guarantees the continuity of the image in fact there are usually two choices for the symmetry symmetry around the meshpoint and symmetry around the midpoint the first is called whole sample symmetry in signal and image processing the second is half sample many researchers have developed some fast algorithms for the problems of image restoration with the half sample symmetric bcs over the years however little attention has been given to the whole sample symmetric bcs in this paper we consider the use of the whole sample symmetric boundary conditions in image restoration the blurring matrices constructed from the point spread functions psfs for the bcs have block toeplitz plus pseudohankel with toeplitz plus pseudohankel blocks structures recently regardless of symmetric properties of the psfs a technique of kronecker product approximations was successfully applied to restore images with the zero bcs half sample symmetric bcs and anti reflexive bcs respectively all these results extend quite naturally to the whole sample symmetric bcs since the resulting matrices have similar structures it is interesting to note that when the size of the true psf is small the computational complexity of the algorithm obtained for the kronecker product approximation of the resulting matrix in this paper is very small it is clear that in this case all calculations in the algorithm are implemented only at the upper left corner submatrices of the big matrices finally detailed experimental results reporting the performance of the proposed algorithm are presented

symmetric convolution and the discrete sine and cosine transforms this paper discusses the use of symmetric convolution and the discrete sine and cosine transforms dsts and dcts for general digital signal processing the operation of symmetric convolution is a formalized approach to convolving symmetrically extended sequences the result is the same as that obtained by taking an inverse discrete trigonometric transform dtt of the product of the forward dtts of those two sequences there are 16 members in the family of dtts each provides a representation for a corresponding distinct type of symmetric periodic sequence the author defines symmetric convolution relates the dsts and dcts to symmetric periodic sequences and then use these principles to develop simple but powerful convolution multiplication properties for the entire family of dsts and dcts symmetric convolution can be used for discrete linear filtering when the filter is symmetric or antisymmetric the filtering will be efficient because fast algorithms exist for all versions of the dtts conventional linear convolution is possible if one first zero pad the input data symmetric convolution and its fast implementation using dtts are now an alternative to circular convolution and the dft

a branch and price algorithm for the generalized assignment problem the generalized assignment problem examines the maximum profit assignment of jobs to agents such that each job is assigned to precisely one agent subject to capacity restrictions on the agents a new algorithm for the generalized assignment problem is presented that employs both column generation and branch and bound to obtain optimal integer solutions to a set partitioning formulation of the problem

factorizations and representations of the backward second order linear recurrences we show the relationships between the determinants and permanents of certain tridiagonal matrices and the negatively subscripted terms of second order linear recurrences also considering how to the negatively subscripted terms of second order linear recurrences can be connected to chebyshev polynomials by determinants of these matrices we give factorizations and representations of these numbers

extended rls lattice adaptive filters this paper solves the problem of developing exact fast weighted rls lattice adaptive filters for input signals induced by general orthonormal filter models the resulting algorithm can be viewed as a counterpart of the extended fast fixed order rls adaptive filters previously derived

extended fast fixed order rls adaptive filters the existing derivations of conventional fast rls adaptive filters are intrinsically dependent on the shift structure in the input regression vectors this structure arises when a tapped delay line fir filter is used as a modeling filter we show unlike what original derivations may suggest that fast fixed order rls adaptive algorithms are not limited to fir filter structures we show that fast recursions in both explicit and array forms exist for more general data structures such as orthonormally based models one of the benefits of working with orthonormal bases is that fewer parameters can be used to model long impulse responses

algebraic signal processing theory sampling for infinite and finite 1 d space we derive a signal processing framework called space signal processing that parallels time signal processing as such it comes in four versions continuous discrete infinite finite each with its own notion of convolution and fourier transform as in time these versions are connected by sampling theorems that we derive in contrast to time however space signal processing is based on a different notion of shift called space shift which operates symmetrically our work rigorously connects known and novel concepts into a coherent framework most importantly it shows that the sixteen discrete cosine and sine transforms are the space equivalent of the discrete fourier transform and hence can be derived by sampling the platform for our work is the algebraic signal processing theory an axiomatic approach and generalization of linear signal processing that we recently introduced

the algebraic approach to the discrete cosine and sine transforms and their fast algorithms it is known that the discrete fourier transform dft used in digital signal processing can be characterized in the framework of the representation theory of algebras namely as the decomposition matrix for the regular module mathbb c z n mathbb c x x n 1 this characterization provides deep insight into the dft and can be used to derive and understand the structure of its fast algorithms in this paper we present an algebraic characterization of the important class of discrete cosine and sine transforms as decomposition matrices of certain regular modules associated with four series of chebyshev polynomials then we derive most of their known algorithms by pure algebraic means we identify the mathematical principle behind each algorithm and give insight into its structure our results show that the connection between algebra and digital signal processing is stronger than previously understood

complete radiation boundary conditions for the helmholtz equation i waveguides we consider the use of complete radiation boundary conditions for the solution of the helmholtz equation in waveguides a general analysis of well posedness convergence and finite element approximation is given in addition methods for the optimization of the boundary condition parameters are considered the theoretical results are illustrated by some simple numerical experiments

block based spatial prediction and transforms based on 2d markov processes for image and video compression conventional intraframe coding is performed in two steps first a block of pixels are predicted by copying previously reconstructed neighbor pixels of the block along an angular direction inside the block then the prediction residual block is transform coded with the well known 2d discrete cosine transform dct recently it has been shown that transforming the intraprediction residuals with the odd type 3 discrete sine transform along the prediction direction and the dct along the perpendicular direction improves the compression performance more recently a recursive prediction approach has been proposed to improve intra prediction performance both of these recent approaches utilize markov processes to develop improvements in either the transform or the prediction step but not in both in this paper both the intraprediction and the transform steps are obtained based on 2d markov processes the derived overall intraframe coding approaches can generalize the mentioned two approaches provide improved coding gains and produce less blocking effects at low bitrates

low complexity rls algorithms using dichotomous coordinate descent iterations in this paper we derive low complexity recursive least squares rls adaptive filtering algorithms we express the rls problem in terms of auxiliary normal equations with respect to increments of the filter weights and apply this approach to the exponentially weighted and sliding window cases to derive new rls techniques for solving the auxiliary equations line search methods are used we first consider conjugate gradient iterations with a complexity of operations per sample being the number of the filter weights to reduce the complexity and make the algorithms more suitable for finite precision implementation we propose a new dichotomous coordinate descent dcd algorithm and apply it to the auxiliary equations this results in a transversal rls adaptive filter with complexity as low as multiplications per sample which is only slightly higher than the complexity of the least mean squares lms algorithm multiplications simulations are used to compare the performance of the proposed algorithms against the classical rls and known advanced adaptive algorithms fixed point fpga implementation of the proposed dcd based rls algorithm is also discussed and results of such implementation are presented

weighted inner products for gmres and gmres dr the convergence of the restarted gmres method can be significantly improved for some problems by using a weighted inner product that changes at each restart how does this weighting affect convergence and when is it useful we show that weighted inner products can help in two distinct ways when the coefficient matrix has localized eigenvectors weighting can allow restarted gmres to focus on eigenvalues that otherwise cause slow convergence for general problems weighting can break the cyclic convergence pattern into which restarted gmres often settles the eigenvectors of matrices derived from differential equations are often not localized thus limiting the impact of weighting for such problems incorporating the discrete cosine transform into the inner product can significantly improve gmres convergence giving a method we call w gmres dct integrating weighting with eigenvalue deflation via gmres dr can also give effective solutions

on a constant diagonals matrix in this paper we consider a constant diagonals matrix the matrix was discussed in witula and slota r witula d slota on computing the determinants and inverses of some special type of tridiagonal and constant diagonals matrices appl math comput 189 1 2007 514 527 the authors gave some results on determinant and the inverse of the matrix for some special cases we give lu factorization and then compute determinant of the matrix we determine eigenvalues of the matrix further we obtain some relationships between permanent of the matrix and terms of a certain recurrence relation

the relationship of transform coefficients for differing transforms and or differing subblock sizes jiang and feng have developed a relationship between the discrete cosine transform dct coefficients of a block and those of its subblocks their derivation of this result can be significantly simplified the new derivation also generalizes to all linear invertible transforms and any separable subblock geometry

analysis of linear prediction coding and spectral estimation from subbands linear prediction schemes make a prediction x spl circ sub i of a data sample x sub i using p previous samples it has been shown by woods and o neil 1986 as well as pearlman 1991 that as the order of prediction p spl rarr spl infin there is no gain to be obtained by coding subband samples this paper deals with the less well understood theory of finite order prediction and optimal coding from subbands which are generated by ideal brickwall filtering of a stationary gaussian source we first prove that pth order prediction from subbands is superior to pth order prediction in the fullband when p is finite this fact adduces that optimal vector p tuple coding in the subbands is shown to offer quantifiable gains over optimal fullband p tuple coding again when p is finite the properties of subband spectra are analyzed using the spectral flatness measure these results are used to prove that subband dpcm provides a coding gain over fullband dpcm for finite orders of prediction in addition the proofs provide means of quantifying the subband advantages in linear prediction optimal coding and dpcm coding in the form of gain formulas subband decomposition of a source is shown to result in a whitening of the composite subband spectrum this implies that for any stationary source a pth order prediction error filter pef can be found that is better than the pth order pef obtained by solving the yule walker equations resulting from the fullband data we demonstrate the existence of such a super optimal pef and provide algorithmic approaches to obtaining this pef the equivalence of linear prediction and ar spectral estimation is then exploited to show theoretically and with simulations that ar spectral estimation from subbands offers a gain over fullband ar spectral estimation

finite difference discretization of the extended fisher kolmogorov equation in two dimensions approximate solutions are considered for the extended fisher kolmogorov efk equation in two space dimension with dirichlet boundary conditions by a crank nicolson type finite difference scheme a priori bounds are proved using lyapunov functional further existence uniqueness and convergence of difference solutions with order o h 2 k 2 in the l norm are proved numerical results are also given in order to check the properties of analytical solutions

a unified approach to structured covariances fast generalized sliding window rls recursions for arbitrary basis this paper extends the existing fast rls recursions originally intended to exponentially windowed problems for general models to a generalized sliding window formulation gswrls from a matrix algebra perspective we show explicitly how the displacement rank of the underlying inverse covariance matrix associated to any operator is defined as a function of the number of window breakpoints and how the fast gswrls calculates these rank factors in a fast manner the recursions hold regardless of the first order data structure induced and show that fast fixed order and order recursive rls algorithms can still be obtained for unwindowed data matrices exhibiting a fixed arbitrary relation between successive regressors our approach highlights the existence of a certain degree of freedom inherent to structured data matrices induced by general models showing that efficient representations of their inverse covariances are not limited to factor circulants but rather constructed from any arbitrary operator these bezoutians usually expressed via reproducing kernel relations can be represented exactly in matrix form along with a precise correspondence to variables of a gswrls as a fallout we obtain a vector relation stating the so called minimality property for extended models and windows as opposed to analogous generating function arguments normally seen in original approaches these results pave the way to a more general framework of polynomial vandermonde covariance decompositions which arise naturally via a proper choice of recurrence related polynomials this has further impact on several signal processing applications including superfast realization of equalizers in communications scenarios

dct dst based transform coding for intra prediction in image video coding in this paper we present a dct dst based transform scheme that applies either the conventional dct or type 7 dst for all the video coding intra prediction modes vertical horizontal and oblique our approach is applicable to any block based intra prediction scheme in a codec that employs transforms along the horizontal and vertical direction separably previously han saxena and rose showed that for the intra predicted residuals of horizontal and vertical modes the dst is the optimal transform with performance close to the klt here we prove that this is indeed the case for the other oblique modes the optimal choice of using dct or dst is based on intra prediction modes and requires no additional signaling information or rate distortion search the dct dst scheme presented in this paper was adopted in the hevc standardization in march 2011 further simplifications especially to reduce implementation complexity which remove the mode dependency between dct and dst and simply always use dst for the 4 4 intra luma blocks were adopted in the hevc standard in july 2012 simulation results conducted for the dct dst algorithm are shown in the reference software for the ongoing hevc standardization our results show that the dct dst scheme provides significant bd rate improvement over the conventional dct based scheme for intra prediction in video sequences

graph based transform for 2d piecewise smooth signals with random discontinuity locations the graph based block transform recently emerged as an effective tool for compressing some special signals such as depth images in 3d videos however in existing methods overheads are required to describe the graph of the block from which the decoder has to calculate the transform via time consuming eigendecomposition to address these problems in this paper we aim to develop a single graph based transform for a class of 2d piecewise smooth signals with similar discontinuity patterns we first consider the deterministic case with a known discontinuity location in each row we propose a 2d first order autoregression 2d ar1 model and a 2d graph for this type of signals we show that the closed form expression of the inverse of a biased laplacian matrix of the proposed 2d graph is exactly the covariance matrix of the proposed 2d ar1 model therefore the optimal transform for the signal are the eigenvectors of the proposed graph laplacian next we show that similar results hold in the random case where the locations of the discontinuities in different rows are randomly distributed within a confined region and we derive the closed form expression of the corresponding optimal 2d graph laplacian the theory developed in this paper can be used to design both pre computed transforms and signal dependent transforms with low complexities finally depth image coding experiments demonstrate that our methods can achieve similar performance to the state of the art method but our complexity is much lower

fast affine projection adaptation algorithms with stable and robust symmetric linear system slovers this paper proposes two noniterative approaches to solve a symmetric linear system associated with the fast affine projection adaptation algorithm the first approach using matrix ldlt factorization can provide an exact solution at a moderate complexity in contrast with the fact that existing stable approaches are all approximate based on a reciprocating recursion scheme the second approach has a very low complexity and gives a good approximate solution steady state and transient properties of the proposed and certain previous fap algorithms are studied in detail being stable and optimal under all step size conditions fast affine projection algorithms incorporating the proposed approaches are promising in telecom and other applications of adaptive filtering

complexity reduction self completely recursive radix 2 dct i iv algorithms abstract this paper proposes fast self completely recursive radix 2 discrete cosine transform dct of types i iv algorithms implementations of these new algorithms are stated through signal flow graphs these algorithms are derived mainly using a matrix factorization technique to factor the dct i iv matrices into sparse and scaled orthogonal matrices this paper fills the gap of addressing the remaining dcts in connection to our former work in the lowest multiplication complexity self recursive radix 2 dct ii iii algorithms there has been no radix 2 dct i algorithm which is self recursive and has sparse and scaled orthogonal factors as proposed in this paper the paper also establishes a novel relationship between dct i and dct iii matrices having sparse factors for any n 2 t where t 1 this enables one to observe all traditional completely recursive dct i algorithms as the self recursive dct i algorithm proposed in this paper apart from the novel dct i algorithm we also establish a novel completely recursive dct iv algorithm which executes recursively with the lowest multiplication complexity self recursive radix 2 dct ii algorithm which was derived in our former paper the proposed dct iv algorithm attains the lowest possible multiplication counts

a new texture generation method based on pseudo dct coefficients in this paper a new method for generating different texture images is presented this method involves a simple transform from a certain one dimensional 1 d signal to an expected two dimensional 2 d image unlike traditional methods the input signal is generated by a simple 1 d function in our work instead of a sample texture we first transform the 1 d input signal into frequency domain using fast fourier transform based on the sufficient analysis in 2 d discrete cosine transform dct domain where each of the coefficients expresses a texture feature in a certain direction the 2 d pseudo dct coefficients are then constructed by appropriately rearranging the fourier coefficients in terms of their frequency components finally the corresponding texture image can be produced by 2 d inverse dct algorithm we applied the proposed method to generate several stochastic textures i e cloud illumination and sand and several structural texture images experimental results indicate the good performance of the proposed method

low latency secondary transforms for intra inter prediction residual in this paper we present a transform scheme where a secondary transform is applied after the conventional dct for intra as well as inter prediction residues our approach is applicable to any block based video codec that employs transforms along the horizontal and vertical direction separably the secondary transform is applied to the lower k k 4 or 8 frequency coefficients of the output of conventional dct at block with dimensions 8 and larger the proposed transform scheme has low complexity as it is applied only to the top left portion of the dct output especially in the context of large blocks such as 32 32 where an alternate non dct 32 32 transform would have a prohibitive implementation hardware cost the proposed technique is single pass and the choice of whether to use the secondary transform is solely based on the prediction direction for intra residue and on transform unit location in the prediction unit for the inter residue the scheme requires no additional signaling information or r d search our simulation results show that the proposed transform scheme provides significant bd rate improvement over the conventional dct based coding scheme finally we also show how to implement the proposed secondary transforms with low latency in hardware

generalized eigenvectors and fractionalization of offset dfts and dcts the offset discrete fourier transform dft is a discrete transform with kernel exp j2 spl pi m a n b n it is more generalized and flexible than the original dft and has very close relations with the discrete cosine transform dct of type 4 dct iv dct viii discrete sine transform dst iv dst viii and discrete hartley transform dht iv in this paper we derive the eigenvectors eigenvalues of the offset dft especially for the case where a b is an integer by convolution theorem we can derive the close form eigenvector sets of the offset dft when a b is an integer we also show the general form of the eigenvectors in this case then we use the eigenvectors eigenvalues of the offset dft to derive the eigenvectors eigenvalues of the dct iv dct viii dst iv dst viii and dht iv after the eigenvectors eigenvalues are derived we can use the eigenvectors decomposition method to derive the fractional operations of the offset dft dct iv dct viii dst iv dst viii and dht iv these fractional operations are more flexible than the original ones and can be used for filter design data compression encryption and watermarking etc

rls laguerre lattice adaptive filtering error feedback normalized and array based algorithms this paper develops several lattice structures for rls laguerre adaptive filtering including a posteriori and a priori based lattice filters with error feedback array based lattice filters and normalized lattice filters all structures are efficient in that their computational cost is proportional to the number of taps albeit some structures require more multiplications or divisions than others the performance of all filters however can differ under practical considerations such as finite precision effects and regularization simulations are included to illustrate these facts

low complexity implementation of the improved multiband structured subband adaptive filter algorithm previously we proposed an improved multiband structured subband adaptive filter imsaf algorithm to accelerate the convergence rate of the msaf algorithm when the projection order and or the number of subbands is increased the convergence rate of the imsaf algorithm improves at the cost of increased complexity thus this paper proposes several approaches to reduce the complexity of the imsaf algorithm both in error vector calculation and matrix inversion operation specifically three approaches are developed to efficiently calculate error vector the first approach gives an approximate filtering whereas the other two approaches can provide a fast exact filtering with or without update of the weight vector explicitly based on a recursive scheme the decorrelation property of imsaf is determined and two simplified variants are developed to reduce the complexity as by products i e the simplified imsaf simsaf and pseudo imsaf algorithms then we discuss the problem of solving a linear system of equations the performance advantages limitations and preferable applications of various methods are analyzed and discussed computer simulations are conducted in the context of system identification to determine the principle and efficiency of the proposed fast algorithms

the equilibrium generalized assignment problem and genetic algorithm abstract the well known generalized assignment problem gap is to minimize the costs of assigning n jobs to m capacity constrained agents or machines such that each job is assigned to exactly one agent this problem is known to be np hard and it is hard from a computational point of view as well in this paper follows from practical point of view in real systems the gap is extended to the equilibrium generalized assignment problem egap and the equilibrium constrained generalized assignment problem ecgap a heuristic equilibrium strategy based genetic algorithm ga is designed for solving the proposed egap finally to verify the computational efficiency of the designed ga some numerical experiments are performed on some known benchmarks the test results show that the designed ga is very valid for solving egap

lagrangian relaxation guided problem space search heuristics for generalized assignment problems we develop and test a heuristic based on lagrangian relaxation and problem space search to solve the generalized assignment problem gap the heuristic combines the iterative search capability of subgradient optimization used to solve the lagrangian relaxation of the gap formulation and the perturbation scheme of problem space search to obtain high quality solutions to the gap we test the heuristic using different upper bound generation routines developed within the overall mechanism using the existing problem data sets of various levels of difficulty and sizes including the challenging largest instances we observe that the heuristic with a specific version of the upper bound routine works well on most of the benchmark instances known and provides high quality solutions quickly an advantage of the approach is its generic nature simplicity and implementation flexibility

an iterative shrinkage approach to total variation image restoration the problem of restoration of digital images from their degraded measurements plays a central role in a multitude of practically important applications a particularly challenging instance of this problem occurs in the case when the degradation phenomenon is modeled by an ill conditioned operator in such a situation the presence of noise makes it impossible to recover a valuable approximation of the image of interest without using some a priori information about its properties such a priori information commonly referred to as simply priors is essential for image restoration rendering it stable and robust to noise moreover using the priors makes the recovered images exhibit some plausible features of their original counterpart particularly if the original image is known to be a piecewise smooth function one of the standard priors used in this case is defined by the rudin osher fatemi model which results in total variation tv based image restoration the current arsenal of algorithms for tv based image restoration is vast in this present paper a different approach to the solution of the problem is proposed based upon the method of iterative shrinkage aka iterated thresholding in the proposed method the tv based image restoration is performed through a recursive application of two simple procedures viz linear filtering and soft thresholding therefore the method can be identified as belonging to the group of first order algorithms which are efficient in dealing with images of relatively large sizes another valuable feature of the proposed method consists in its working directly with the tv functional rather then with its smoothed versions moreover the method provides a single solution for both isotropic and anisotropic definitions of the tv functional thereby establishing a useful connection between the two formulae finally a number of standard examples of image deblurring are demonstrated in which the proposed method can provide restoration results of superior quality as compared to the case of sparse wavelet deconvolution

the green s function for the huckel tight binding model applications of the huckel tight binding model are ubiquitous in quantum chemistry and solid state physics the matrix representation of this model is isomorphic to an unoriented vertex adjacency matrix of a bipartite graph which is also the laplacian matrix plus twice the identity in this paper we analytically calculate the determinant and when it exists the inverse of this matrix in connection with the green s function g of the n n huckel matrix a corollary is a closed form expression for a harmonic sum eq 12 we then extend the results to d dimensional lattices whose linear size is n the existence of the inverse becomes a question of number theory we prove a new theorem in number theory pertaining to vanishing sums of cosines and use it to prove that the inverse exists if and only if n 1 and d are odd and d is smaller than the smallest divisor of n 1 we corroborate our results by demonstrating the entry patterns of the green s function and discuss applications related to transpo

the discretew transform four different versions of the discretew transform dwt are introduced the dwt may be decomposed into the discrete cosine transform dct and the discrete sine transform dst eight versions of both dct and dst are introduced for the decomposition of the dwt the relationship among different versions of dwt and their relation with the discrete fourier transform dft are given convolution theorems represented by different versions of the dwt are derived

discrete and continuous sine transform generalized to semisimple lie groups of rank two a new continuous group transform together with its discretization over a lattice of any density and admissible symmetry is defined and described for the four compact semisimple lie groups of rank 2 in the case of rank 1 the discrete version is the sine transform properties of the expansion functions of the transform called s functions are studied digital data processing is our motivating application

a unified approach to fast image registration and a new curvature based registration technique abstract image registration is central to many challenges in medical imaging today it has a vast range of applications the purpose of this note is twofold first we review some of the most promising non linear registration strategies currently used in medical imaging we show that all these techniques may be phrased in terms of a variational problem and allow for a unified treatment second we introduce within the variational framework a new non linear registration model based on a curvature type smoother we show that affine linear transformations belong to the kernel of this regularizer as a result the approach becomes more robust against poor initializations of a pre registration step furthermore we develop a stable and fast implementation of the new scheme based on a real discrete cosine transformation we demonstrate the advantages of the new technique for synthetic data sets and present an application of the algorithm for registering mr mammography images

parallel preconditioners for large scale partial difference equation systems we propose a new preconditioner dasp discrete approximate spectral preconditioner based on the existing well known preconditioners and our computational experience parallel preconditioning strategies for large scale partial difference equation systems arising from partial differential equations are investigated numerical results are given to show the efficiency and effectiveness of the new preconditioners for both model problems and real applications in petroleum reservoir simulation

properties of continuous fourier extension of the discrete cosine transform and its multidimensional generalization a versatile method is described for the practical computation of the exact discrete fourier transforms dft both the direct and the inverse ones of a continuous function g given by its values gj at the points of a uniform grid fn generated by conjugacy classes of elements of finite adjoint order n in the fundamental region f of compact semisimple lie groups the present implementation of the method is for the groups su 2 when f is reduced to a one dimensional segment and for su 2 su 2 su 2 in multidimensional cases this simplest case turns out to be a version of the discrete cosine transform dct implementations abbreviated as dgt for discrete group transform based on simple lie groups of higher ranks are to be considered separately dct is often taken to be simply a specific type of the standard dft here we show that the dct is very different from the standard dft when the properties of the continuous extensions of the two inverse discrete transforms are studied the following propertie

stationary signal processing on graphs graphs are a central tool in machine learning and information processing as they allow to conveniently capture the structure of complex datasets in this context it is of high importance to develop flexible models of signals defined over graphs or networks in this paper we generalize the traditional concept of wide sense stationarity to signals defined over the vertices of arbitrary weighted undirected graphs we show that stationarity is expressed through the graph localization operator reminiscent of translation we prove that stationary graph signals are characterized by a well defined power spectral density that can be efficiently estimated even for large graphs we leverage this new concept to derive wiener type estimation procedures of noisy and partially observed signals and illustrate the performance of this new model for denoising and regression

generalized commuting matrices and their eigenvectors for dfts offset dfts and other periodic operations it is well known that some matrices such as dickinson steiglitz s matrix can commute with the discrete fourier transform dft and that one can use them to derive the complete and orthogonal dft eigenvector set recently candan found the general form of the dft commuting matrix in this paper we further extend the previous work and find the general form of the commuting matrix for any periodic quasi periodic and offset quasi periodic operations using the general commuting matrix we can derive the complete and orthogonal eigenvector sets for offset dfts dcts of types 1 4 5 and 8 dsts of types 1 4 5 and 8 discrete hartley transforms of types 1 and 4 the walsh transform and the projection operation the operation that maps a whole vector space into a subspace successfully moreover several novel ways of finding dft eigenfunctions are also proposed furthermore we also extend our theories to the continuous case i e if a continuous transform is periodic quasi periodic or offset quasi periodic such as the ft and some cyclic operations in optics we can find the general form of the commuting operation and then find the complete and orthogonal eigenfunctions set for the continuous transform

kronecker markov prior for dynamic 3d reconstruction recovering dynamic 3d structures from 2d image observations is highly under constrained because of projection and missing data motivating the use of strong priors to constrain shape deformation in this paper we empirically show that the spatiotemporal covariance of natural deformations is dominated by a kronecker pattern we demonstrate that this pattern arises as the limit of a spatiotemporal autoregressive process and derive a kronecker markov random field as a prior distribution over dynamic structures this distribution unifies shape and trajectory models of prior art and has the individual models as its marginals the key assumption of the kronecker mrf is that the spatiotemporal covariance is separable into the product of a temporal and a shape covariance and can therefore be modeled using the matrix normal distribution analysis on motion capture data validates that this distribution is an accurate approximation with significantly fewer free parameters using the trace norm we present a convex method to estimate missing data from a single sequence when the marginal shape distribution is unknown the kronecker markov distribution fit to a single sequence outperforms state of the art methods at inferring missing 3d data and additionally provides covariance estimates of the uncertainty

diagonalizing properties of the discrete cosine transforms since its introduction in 1974 by ahmed et al the discrete cosine transform dct has become a significant tool in many areas of digital signal processing especially in signal compression there exist eight types of discrete cosine transforms dcts we obtain the eight types of dcts as the complete orthonormal set of eigenvectors generated by a general form of matrices in the same way as the discrete fourier transform dft can be obtained as the eigenvectors of an arbitrary circulant matrix these matrices can be decomposed as the sum of a symmetric toeplitz matrix plus a hankel or close to hankel matrix scaled by some constant factors we also show that all the previously proposed generating matrices for the dcts are simply particular cases of these general matrix forms using these matrices we obtain for each dct a class of stationary processes verifying certain conditions with respect to which the corresponding dct has a good asymptotic behavior in the sense that it approaches karhunen loeve transform performance as the block size n tends to infinity as a particular result we prove that the eight types of dcts are asymptotically optimal for all finite order markov processes we finally study the decorrelating power of the dcts obtaining expressions that show the decorrelating behavior of each dct with respect to any stationary processes

a generalized grid connectivity based parameterization for subsurface flow model calibration 1 we develop a novel method of parameterization for spatial hydraulic property characterization to mitigate the challenges associated with the nonlinear inverse problem of subsurface flow model calibration the parameterization is performed by the projection of the estimable hydraulic property field onto an orthonormal basis derived from the grid connectivity structure the basis functions represent the modal shapes or harmonics of the grid are defined by a modal frequency and converge to special cases of the discrete fourier series under certain grid geometries and boundary assumptions therefore hydraulic property updates are performed in the spectral domain and merge with fourier analysis in ideal cases dependence on the grid alone implies that the basis may characterize any grid geometry including corner point and unstructured is model independent and is constructed off line and only once prior to flow data assimilation we apply the parameterization in an adaptive multiscale model calibration workflow for three subsurface flow models several different grid geometries are considered in each case the prior hydraulic property model is updated using a parameterized multiplier field that is superimposed onto the grid and assigned an initial value of unity at each cell the special case corresponding to a constant multiplier is always applied through the constant basis function higher modes are adaptively employed during minimization of data misfit to resolve multiscale heterogeneity in the geomodel the parameterization demonstrates selective updating of heterogeneity at locations and spatial scales sensitive to the available data otherwise leaving the prior model unchanged as desired

intra prediction based on markov process modeling of images in recent video coding standards intraprediction of a block of pixels is performed by copying neighbor pixels of the block along an angular direction inside the block each block pixel is predicted from only one or few directionally aligned neighbor pixels of the block although this is a computationally efficient approach it ignores potentially useful correlation of other neighbor pixels of the block to use this correlation a general linear prediction approach is proposed where each block pixel is predicted using a weighted sum of all neighbor pixels of the block the disadvantage of this approach is the increased complexity because of the large number of weights in this paper we propose an alternative approach to intraprediction where we model image pixels with a markov process the markov process model accounts for the ignored correlation in standard intraprediction methods but uses few neighbor pixels and enables a computationally efficient recursive prediction algorithm compared with the general linear prediction approach that has a large number of independent weights the markov process modeling approach uses a much smaller number of independent parameters and thus offers significantly reduced memory or computation requirements while achieving similar coding gains with offline computed parameters

the compressed state kalman filter for nonlinear state estimation application to large scale reservoir monitoring reservoir monitoring aims to provide snapshots of reservoir conditions and their uncertainties to assist operation management and risk analysis these snapshots may contain millions of state variables e g pressures and saturations which can be estimated by assimilating data in real time using the kalman filter kf however the kf has a computational cost that scales quadratically with the number of unknowns m due to the cost of computing and storing the covariance and jacobian matrices along with their products the compressed state kalman filter cskf adapts the kf for solving large scale monitoring problems the cskf uses n preselected orthogonal bases to compute an accurate rank n approximation of the covariance that is close to the optimal spectral approximation given by svd the cskf has a computational cost that scales linearly in m and uses an efficient matrix free approach that propagates uncertainties using n 1 forward model evaluations where n m here we present a generalized cskf algorithm for nonlinear state estimation problems such as co2 monitoring for simultaneous estimation of multiple types of state variables the algorithm allows selecting bases that represent the variability of each state type through synthetic numerical experiments of co2 monitoring we show that the cskf can reproduce the kalman gain accurately even for large compression ratios m n for a given computational cost the cskf uses a robust and flexible compression scheme that gives more reliable uncertainty estimates than the ensemble kalman filter which may display loss of ensemble variability leading to suboptimal uncertainty estimates

schroedinger eigenmaps for the analysis of biomedical data we introduce schroedinger eigenmaps se a new semi supervised manifold learning and recovery technique this method is based on an implementation of graph schroedinger operators with appropriately constructed barrier potentials as carriers of labeled information we use our approach for the analysis of standard biomedical datasets and new multispectral retinal images

the discrete cosine transform each discrete cosine transform dct uses n real basis vectors whose components are cosines in the dct 4 for example the j th component of boldv k is cos j frac 1 2 k frac 1 2 frac pi n these basis vectors are orthogonal and the transform is extremely useful in image processing if the vector boldx gives the intensities along a row of pixels its cosine series sum c k boldv k has the coefficients c k boldx boldv k n they are quickly computed from a fast fourier transform but a direct proof of orthogonality by calculating inner products does not reveal how natural these cosine vectors are we prove orthogonality in a different way each dct basis contains the eigenvectors of a symmetric second difference matrix by varying the boundary conditions we get the established transforms dct 1 through dct 4 other combinations lead to four additional cosine transforms the type of boundary condition dirichlet or neumann centered at a meshpoint or a midpoint determines the applications that are appropriate for each transform the centering also determines the period n 1 or n in the established transforms n frac 1 2 or n frac 1 2 in the other four the key point is that all these eigenvectors of cosines come from simple and familiar matrices