measurable kac cohomology for bicrossed products we study the kac cohomology for matched pairs of locally compact groups this cohomology theory arises from the extension theory of locally compact quantum groups we prove a measurable version of the kac exact sequence and provide methods to compute the cohomology we give explicit calculations in several examples using results of moore and wigner

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

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

dynamical noether invariants for time dependent nonlinear systems dynamical invariants are derived for time dependent systems with nonlinear equations of motion including nonharmonic damped systems the concept of a dynamical algebra is discussed and its utility for the construction of dynamical invariants for nonharmonic systems is demonstrated finally we show the existence of dynamical invariants for some nonlinear quantum systems

hopf algebra extensions arising from semi direct products of groups hopf algebra extensions arising from a matched pair of groups were intro duced by g i kac kac he established an exact sequence the kac sequence that connects group cohomology and hopf algebra extensions this was re cently considered and generalized by other authors for instance a masuoka in mal ma2

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

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

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

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

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

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

lie families theory and applications we analyze the families of non autonomous systems of first order ordinary differential equations admitting a common time dependent superposition rule i e a time dependent map expressing any solution of each of these systems in terms of a generic set of particular solutions of the system and some constants we next study the relations of these families called lie families with the theory of lie and quasi lie systems and apply our theory to provide common time dependent superposition rules for certain lie families

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

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

feynman propagator for time dependent lagrangians possessing an invariant quadratic in momentum a class of time dependent classical lagrangians possessing an invariant quadratic in momentum is considered from a quantal point of view quantum mechanics is introduced through the feynman propagator defined as a path integral involving the classical action it is shown without carrying out an explicit path integration that the propagator for such a time dependent system is related to the propagator of an associated time independent problem the expansion of the propagator in terms of the eigenfunctions of the invariant operator is derived and the equivalence of the present theory to that of lewis and reisenfeld 1969 is discussed explicit analytic forms of propagators are obtained for some cases to illustrate the application of the present approach

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

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

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

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

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

classifying complements for hopf algebras and lie algebras abstract let a e be a given extension of hopf respectively lie algebras we answer the classifying complements problem ccp which consists of describing and classifying all complements of a in e if h is a given complement then all the other complements are obtained from h by a certain type of deformation we establish a bijective correspondence between the isomorphism classes of all complements of a in e and a cohomological type object ha 2 h a where is the matched pair associated to h the factorization index e a f is introduced as a numerical measure of the ccp for two n th roots of unity we construct a 4 n 2 dimensional hopf algebra whose factorization index over the group algebra is arbitrary large

hopf bimodules coquasibialgebras and an exact sequence of kac abstract based on the ideas of tannaka krein reconstruction we present a categorical construction that assigns to any cleft hopf algebra inclusion k h a coquasibialgebra having k as a hopf subalgebra as a special case the construction gives an intrinsic connection between the bismash product k q and the double cross product q k constructed from the same combinatorial data a cocommutative coquasibialgebra is the same as a cocommutative bialgebra equipped with a sweedler three cocycle thus our construction assigns to every bicrossproduct or hopf algebra extension of a commutative and a cocommutative factor a corresponding cocommutative double crossproduct equipped with a sweedler three cocycle based on this observation we use the construction to prove generalizations of kac s exact sequence for the group of hopf algebra extensions of a group algebra by a dual group algebra

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

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

dissipative quantum dynamics solution of the generalized von neumann equation for the damped harmonic oscillator a recently proposed generalized von neumann equation describing dissipative time evolution of quantum systems is applied to a damped and driven harmonic oscillator utilizing lie algebraic methods the nonlinear operator equation is solved and from this solution the behaviour of the mean value x is extracted including both amplitude and phaseshift relative to the driving force a differential equation for x is derived and the damping constant is identified

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

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

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

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

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

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

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

cohomology of abelian matched pairs and the kac sequence the purpose of this paper is to introduce a cohomology theory for abelian matched pairs of hopf algebras and to explore its relationship to sweedler cohomology to singer cohomology and to extension theory an exact sequence connecting these cohomology theories is obtained for a general abelian matched pair of hopf algebras generalizing those of kac and masuoka for matched pairs of finite groups and finite dimensional lie algebras the morphisms in the low degree part of this sequence are given explicitly enabling concrete computations

nonlinear schrodinger type field equation for the description of dissipative systems iii frictionally damped free motion as an example for an aperiodic motion a new theory for the description of dissipative systems by nonlinear schrodinger type field equations nlse s with logarithmic nonlinearity which has been recently developed by the authors is applied to investigate the frictionally damped free motion and similar spatially unrestricted aperiodic problems wave packet solutions as well as time dependent wave function solutions are derived and discussed in the limit of vanishing friction friction constant 0 these solutions turn into the well known solutions of the respective linear schrodinger field equation the same applies to the mean values of position momentum and energy as well as to the uncertainty product of position and momentum for 0 however interesting new effects appear in contrast to the linear theory the uncertainty product of position and momentum does not diverge any more for infinitely long times t but asymptotically approaches a definite constant value which depends on characteristic parameters of the system like its ma

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

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

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

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

extensions of hopf algebras and lie bialgebras let f g be finite dimensional lie algebras over a field of characteristic zero regard f and g the dual lie coalgebra of g as lie bialgebras with zero cobracket and zero bracket respectively suppose that a matched pair f g of lie bialgebras is given which has structure maps then it induces a matched pair u f ug of hopf algebras where u f is the universal envelope of f and ug is the hopf dual of ug we show that the group opext u f ug of cleft hopf algebra extensions associated with u f ug is naturally isomorphic to the group opext f g of lie bialgebra extensions associated with f g an exact sequence involving either of these groups is obtained which is a variation of the exact sequence due to g i kac if g g g there follows a bijection between the set ext u f ug of all cleft hopf algebra extensions of u f by ug and the set ext f g of all lie bialgebra extensions of f by g

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

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

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

ermakov and non ermakov systems in quantum dissipative models via the hydrodynamical formulation of quantum mechanics a unified protocol to treat the quantum time dependent harmonic oscillator with friction is presented described by two different models an explicitly time dependent linear schrodinger equation caldirola kanai model and a logarithmic nonlinear schrodinger equation kostin model for the former model an ermakov system that makes it possible to obtain an invariant of ermakov lewis type is derived for the latter model a non ermakov system is derived instead and it is shown that neither an exact nor an approximate invariant of ermakov lewis type exists

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

extensions of locally compact quantum groups and the bicrossed product construction abstract in the framework of locally compact quantum groups we study cocycle actions we develop the cocycle bicrossed product construction starting from a matched pair of locally compact quantum groups we define exact sequences and establish a one to one correspondence between cocycle bicrossed products and cleft extensions in this way we obtain new examples of locally compact quantum groups

time dependent invariant associated to nonlinear schrodinger langevin equations via the quantum hydrodynamical method a time dependent invariant associated to the quantum dissipative time dependent harmonic oscillator tdho described by two classes of nonlinear schrodinger langevin equations with the following frictional nonlinear terms is constructed i w1 i ln ln which is the schuch chung hartman frictional nonlinear term and ii w2 x x cp 1 c p 1 2 i c which includes the sussmann c 1 the hasse c 1 2 and the albrecht kostin c 0 frictional nonlinear operators the associated invariant found is exact for the schuch chung hartman and hasse models and only approximate for the sussman and albrecht kostin models

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

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

exact invariants for a class of time dependent nonlinear hamiltonian systems a method of generalizing a class of invariants for a time dependent linear oscillator is developed for the motion of a mass point in one dimension with a general time dependent nonlinear potential formulas are derived for the allowable time dependent potentials and for the corresponding invariants the method by which these conclusions are reached is interesting theoretically and is explained in detail

cohomology and coquasi bialgebra extensions associated to a matched pair of bialgebras abstract by using braid diagrams we explicitly reconstruct the cohomology associated to a matched pair of cocommutative bialgebras in order to give a method of constructing coquasi bialgebras which generalize bialgebras and classifying them up to monoidal equivalence of their comodule categories an alternative homological proof is given for schauenburg s generalized kac sequence involving the abelian group opext h k of bialgebra extensions we define an abelian group opext h k of coquasi bialgebra extensions associated to a singer pair h k of bialgebras and prove a variant of schauenburg s sequence which involves the group it is also proved that there is a natural isomorphism opext h 1 k opext h 2 k that preserves monoidal equivalence classes if h 1 k and h 2 k arise from such matched pairs that are shifts of each other

calculations of some groups of hopf algebra extensions w let us recall the theory of group extensions with abelian kernels mc x chap iv let g be a group and m a g module the equivalence classes of group extensions m a e a g giving rise to the g module action on m form a group with respect to the baer sum furthermore this group is 2 isomorphic to the cohomology group h g m the theory is extended to the theory of hopf algebra extensions which w x are abelian in some sense h s we restrict our interests to hopf algebra extensions over a field k which are of the form

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

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

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

isomorphism classes of finite dimensional connected hopf algebras in positive characteristic abstract we classify all finite dimensional connected hopf algebras with large abelian primitive spaces we show that they are hopf algebra extensions of restricted enveloping algebras of certain restricted lie algebras for any abelian matched pair associated with these extensions we construct a cohomology group which classifies all the extensions up to equivalence moreover we present a 1 1 correspondence between the isomorphism classes and a group quotient of the cohomology group deleting some exceptional points where the group respects the automorphisms of the abelian matched pair and the exceptional points represent those restricted lie algebra extensions

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

nonlinear time dependent anharmonic oscillator asymptotic behavior and connected invariants the motion of a particle in a potential decreasing with time as x n is considered different time and space rescaling are considered in order to obtain the asymptotic solutions the validity of adiabatic invariants is discussed the classical critical case corresponds to the obtainment of self similar solutions for the quantum problem

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

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