sample adaptive product quantization asymptotic analysis and examples vector quantization vq is an efficient data compression technique for low bit rate applications however the major disadvantage of vq is that its encoding complexity increases dramatically with bit rate and vector dimension even though one can use a modified vq such as the tree structured vq to reduce the encoding complexity it is practically infeasible to implement such a vq at a high bit rate or for large vector dimensions because of the huge memory requirement for its codebook and for the very large training sequence requirement to overcome this difficulty a structurally constrained vq called the sample adaptive product quantizer sapq has recently been proposed we extensively study the sapq that is based on scalar quantizers in order to exploit the simplicity of scalar quantization through an asymptotic distortion result we discuss the achievable performance and the relationship between distortion and encoding complexity we illustrate that even when sapq is based on scalar quantizers it can provide vq level performance we also provide numerical results that show a 2 3 db improvement over the lloyd max 1982 1960 quantizers for data rates above 4 b point

spatio temporal covariance and cross covariance functions of the great circle distance on a sphere abstractin this article we propose stationary covariance functions for processes that evolve temporally over a sphere as well as cross covariance functions for multivariate random fields defined over a sphere for such processes the great circle distance is the natural metric that should be used to describe spatial dependence given the mathematical difficulties for the construction of covariance functions for processes defined over spheres cross time approximations of the state of nature have been proposed in the literature by using the euclidean based on map projections and the chordal distances we present several methods of construction based on the great circle distance and provide closed form expressions for both spatio temporal and multivariate cases a simulation study assesses the discrepancy between the great circle distance chordal distance and euclidean distance based on a map projection both in terms of estimation and prediction in a space time and a bivariate spatial setting where t

hermite birkhoff interpolation on scattered data on the sphere and other manifolds abstract the hermite birkhoff interpolation problem of a function given on arbitrarily distributed points on the sphere and other manifolds is considered each proposed interpolant is expressed as a linear combination of basis functions the combination coefficients being incomplete taylor expansions of the interpolated function at the interpolation points the basis functions depend on the geodesic distance are orthonormal with respect to the point evaluation functionals and have all derivatives equal zero up to a certain order at the interpolation points a remarkable feature of such interpolants which belong to the class of partition of unity methods is that their construction does not require solving linear systems numerical tests are given to show the interpolation performance

classified vector quantization of images vector quantization vq provides many attractive features for image coding with high compression ratios however initial studies of image coding with vq have revealed several difficulties most notably edge degradation and high computational complexity we address these two problems and propose a new coding method classified vector quantization cvq which is based on a composite source model blocks with distinct perceptual features such as edges are generated from different subsources i e belong to different classes in cvq a classifier determines the class for each block and the block is then coded with a vector quantizer designed specifically for that class we obtain better perceptual quality with significantly lower complexity with cvq when compared to ordinary vq we demonstrate with cvq visual quality which is comparable to that produced by existing coders of similar complexity for rates in the range 0 6 1 0 bits pixel

weighted centroid neural network for edge preserving image compression an edge preserving image compression algorithm based on an unsupervised competitive neural network is proposed the proposed neural network the called weighted centroid neural network wcnn utilizes the characteristics of image blocks from edge areas the mean residual vector quantization m rvq scheme is utilized in this proposed approach as the framework of the proposed algorithm the edge strength of image block data is utilized as a tool to allocate the proper code vectors in the proposed wcnn the wcnn successfully allocates more code vectors to the image block data from edge area while it allocates less code vectors to the image black data from shade or non edge area when compared to conventional neural networks based on vq algorithm as a result a simple application of wcnn to an image compression problem gives improved edge characteristics in reconstructed images over conventional neural network based on vq algorithms such as self organizing map som and adaptive som

error estimates for scattered data interpolation on spheres we study sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the n sphere the interpolation knots are scattered our approach partly follows the general theory of golomb and weinberger and related estimates these error estimates are then based on series expansions of smooth functions in terms of spherical harmonics the markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics

a compressed domain scheme for classifying block edge patterns detecting and classifying edge components in images find a wide range of applications from perceptually improved coding schemes to content based searches a fast and systematic scheme is proposed to classify the edge orientation of each block in discrete cosine transform dct compressed images derived from a simple but acceptable pixel domain algorithm the proposed scheme performs directly on dct coefficient domain and thus saves the arithmetic operations by a considerable amount the performance of the proposed method is validated by the experiments against various natural images

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

h div finite elements based on nonaffine meshes for 3d mixed formulations of flow problems with arbitrary high order accuracy of the divergence of the flux

enhanced cell centered finite differences for elliptic equations on general geometry we present an expanded mixed finite element method for solving second order elliptic partial differential equations on geometrically general domains for the lowest order raviart thomas approximating spaces we use quadrature rules to reduce the method to cell centered finite differences possibly enhanced with some face centered pressures this substantially reduces the computational complexity of the problem to a symmetric positive definite system for essentially only as many unknowns as elements our new method handles general shape elements triangles quadrilaterals and hexahedra and full tensor coefficients while the standard mixed formulation reduces to finite differences only in special cases with rectangular elements as in other mixed methods we maintain the local approximation of the divergence i e local mass conservation in contrast galerkin finite element methods facilitate general element shapes at the cost of achieving only global mass conservation our method is shown to be as accurate as the standard mixed method for a large class of smooth meshes on nonsmooth meshes or with nonsmooth coefficients one can add lagrange multiplier pressure unknowns on certain element edges or faces this enhanced cell centered procedure recovers full accuracy with little additional cost if the coefficients or mesh geometry are piecewise smooth theoretical error estimates and numerical examples are given illustrating the accuracy and efficiency of the methods

a meshless interpolation algorithm using a cell based searching procedure in this paper we propose a fast algorithm for bivariate interpolation of large scattered data sets it is based on the partition of unity method for constructing a global interpolant by blending radial basis functions as local approximants and using locally supported weight functions the partition of unity algorithm is efficiently implemented and optimized by connecting the method with an effective cell based searching procedure more precisely we construct a cell structure which partitions the domain and strictly depends on the dimension of the subdomains thus providing a meaningful improvement in the searching process compared to the nearest neighbour searching techniques presented in allasia et al 2011 and cavoretto and de rossi 2010 2012 in fact this efficient algorithm and in particular the new searching procedure enable us a fast computation also in several applications where the amount of data to be interpolated is often very large up to many thousands or even millions of points analysis of computational complexity shows the high efficiency of the proposed interpolation algorithm this is also supported by numerical experiments

distributions of the two dimensional dct coefficients for images for a two dimensional discrete cosine transform dct image coding system there have been different assumptions concerning the distributions of the transform coefficients this paper presents results of distribution tests that indicate that for many images the statistics of the coefficients are best approximated by a gaussian distribution for the dc coefficient and a laplacian distribution for the other coefficients furthermore from a simulation of the dct coding system it is shown that the assumption that the coefficients are laplacian yields a higher actual output signal to noise ratio and a much better agreement between theory and simulation than the gaussian assumption

a trivariate interpolation algorithm using a cube partition searching procedure in this paper we propose a fast algorithm for trivariate interpolation which is based on the partition of unity method for constructing a global interpolant by blending local radial basis function interpolants and using locally supported weight functions the partition of unity algorithm is efficiently implemented and optimized by connecting the method with an effective cube partition searching procedure more precisely we construct a cube structure which partitions the domain and strictly depends on the size of its subdomains so that the new searching procedure and accordingly the resulting algorithm enable us to efficiently deal with a large number of nodes complexity analysis and numerical experiments show high efficiency and accuracy of the proposed interpolation algorithm

hierarchical high order finite element bases for h div spaces based on curved meshes for two dimensional regions or manifolds the mixed finite element formulation for elliptic problems is characterized by simultaneous calculations of the potential primal variable and of the flux field dual variable this work focuses on new h div conforming finite element spaces which are suitable for flux approximations based on curved meshes of a planar region or a manifold domain embedded in r 3 the adopted methodology for the construction of h div bases consists in using hierarchical h 1 conforming scalar bases multiplied by vector fields that are properly constructed on the master element and mapped to the geometrical elements by the piola transformation followed by a normalization procedure they are classified as being of edge or internal type the normal component of an edge function coincides on the corresponding edge with the associated scalar shape function and vanishes over the other edges and the normal components of an internal shape function vanishes on all element edges these properties are fundamental for the global assembly of h div conforming functions locally defined by these vectorial shape functions for applications to the mixed formulation the configuration of the approximation spaces is such that the divergence of the dual space and the primal approximation space coincides results of verification numerical tests are presented for curved triangular and quadrilateral partitions on circular cylindrical and spherical regions demonstrating stable convergence with optimal convergence rates coinciding for primal and dual variables innovative two dimensional hdiv approximation spaces for curved quadrilateral and triangular elements hdiv bases properly constructed on the master element and mapped to the geometrical elements by the piola transformation followed by a normalization procedure stable convergence with optimal convergence rates coinciding for primal and dual variables study of number of condensed equations as a function of the polynomial order of the shape functions

image compression using vq btc vector quantization vq and block truncation coding btc are successful image compression techniques however a reproduced image using vq or btc suffers from edge degradation a new technique that combines the advantages of both vq and btc to combat this degradation is presented and is referred to as vq btc in vq btc a low detail block is encoded using vq for a high detail block a modification of btc is used to determine the locations of the relatively lighter and relatively darker pixels inside the block and vq is then used to encode each vq btc provides improved edge reproduction and much lower bit rates than those obtained by btc

three dimensional hierarchical mixed finite element approximations with enhanced primal variable accuracy abstract there are different possibilities of choosing balanced pairs of approximation spaces for dual flux and primal pressure variables to be used in discrete versions of the mixed finite element method for elliptic problems arising in fluid simulations three cases shall be studied and compared for discretized three dimensional formulations based on tetrahedral hexahedral and prismatic meshes the principle guiding the constructions is the property that the divergence of the dual space and the primal approximation space should coincide while keeping the same order of accuracy for the flux variable and varying the accuracy order of the primal variable there is the classic case of b d m k spaces based on tetrahedral meshes and polynomials of total degree k for the dual variable and k 1 for the primal variable showing stable simulations with optimal convergence rates of orders k 1 and k respectively another case is related to r t k and b d f m k 1 spaces for hexahedral and tetrahedral meshes respectively but holding for prismatic elements as well it gives identical approximation order k 1 for both primal and dual variables an improvement in accuracy obtained by increasing the degree of primal functions to k and by enriching the dual space with some properly chosen internal shape functions of degree k 1 while keeping degree k for the border fluxes a new type of approximation is proposed by further incrementing the order of some internal flux functions to k 2 and matching primal functions to k 1 higher than the border fluxes of degree k thus higher convergence rate of order k 2 is obtained for the primal variable using static condensation the global condensed system to be solved in all the cases has same dimension and structure which is proportional to the space dimension of the border fluxes for each element geometry illustrating results comparing the three different space configurations are presented for simulations based on hierarchical high order shape functions for h div conforming spaces which are specially constructed for affine tetrahedral hexahedral and prismatic meshes expected convergence rates are obtained for the flux pressure and divergence variables

inverse error diffusion using classified vector quantization this correspondence extends and modifies classified vector quantization cvq to solve the problem of inverse halftoning the proposed process consists of two phases the encoding phase and decoding phase the encoding procedure needs a codebook for the encoder which transforms a halftoned image to a set of codeword indices the decoding process also requires a different codebook for the decoder which reconstructs a gray scale image from a set of codeword indices using cvq the reconstructed gray scale image is stored in compressed form and no further compression may be required this is different from the existing algorithms which reconstructed a halftoned image in an uncompressed form the bit rate of encoding a reconstructed image is about 0 51 b pixel

radial basis functions for the multivariate interpolation of large scattered data sets an efficient method for the multivariate interpolation of very large scattered data sets is presented it is based on the local use of radial basis functions and represents a further improvement of the well known shepard s method although the latter is simple and well suited for multivariate interpolation it does not share the good reproduction quality of other methods widely used for bivariate interpolation on the other band radial basis functions which have proven to be highly useful for multivariate scattered data interpolation have a severe drawback they are unable to interpolate large sets in an efficient and numerically stable way and maintain a good level of reproduction quality at the same time both problems have been circumvented using radial basis functions to evaluate the nodal function of the modified shepard s method this approach exploits the flexibility of the method and improves its reproduction quality the proposed algorithm has been implemented and numerical results confirm its efficiency

efficient assembly of h mathrm div and h mathrm curl conforming finite elements in this paper we discuss how to efficiently evaluate and assemble general finite element variational forms on h mathrm div and h mathrm curl the proposed strategy relies on a decomposition of the element tensor into a precomputable reference tensor and a mesh dependent geometry tensor two key points must then be considered the appropriate mapping of basis functions from a reference element and the orientation of geometrical entities to address these issues we extend here a previously presented representation theorem for affinely mapped elements to piola mapped elements we also discuss a simple numbering strategy that removes the need to contend with directions of facet normals and tangents the result is an automated efficient and easy to use implementation that allows a user to specify finite element variational forms on h mathrm div and h mathrm curl in close to mathematical notation

fast and stable rational rbf based partition of unity interpolation abstract we perform a local computation via the partition of unity pu method of rational radial basis function rbf interpolants we investigate the well posedness of the problem and we provide error bounds the resulting scheme efficiently implemented by means of the deflation accelerated conjugate gradient dacg enables us to deal with huge data sets and thanks to the use of variably scaled kernels vsks it turns out to be stable for functions with steep gradients or discontinuities which are truly common in applications the results show that the new proposed method outperforms the classical and rescaled pu schemes

fast and accurate interpolation of large scattered data sets on the sphere in this paper a new efficient algorithm for spherical interpolation of large scattered data sets is presented the solution method is local and involves a modified spherical shepard s interpolant which uses zonal basis functions as local approximants the associated algorithm is implemented and optimized by applying a nearest neighbour searching procedure on the sphere specifically this technique is mainly based on the partition of the sphere in a suitable number of spherical zones the construction of spherical caps as local neighbourhoods for each node and finally the employment of a spherical zone searching procedure computational cost and storage requirements of the spherical algorithm are analyzed moreover several numerical results show the good accuracy of the method and the high efficiency of the proposed algorithm

a new procedure for the construction of hierarchical high order hdiv and hcurl finite element spaces this paper considers a systematic procedure for the construction of a hierarchy of high order finite element approximation for hdiv and hcurl spaces based on triangular and quadrilateral partitions of bidimensional domains the principle is to choose an appropriate set of vectors based on the geometry of each element which are multiplied by an available set of h 1 hierarchical scalar basic functions this strategy produces vector basis functions with continuous normal or tangent components on the elements interfaces properties that characterise functions in hdiv or hcurl respectively we also present a numerical study to evaluate the correct balancedness of the resulting hdiv spaces of degree k and l 2 spaces of degree k 1 on the resolution of the mixed formulation for a steklov eigenvalue problem

mesh free semi lagrangian methods for transport on a sphere using radial basis functions abstract we present three new semi lagrangian methods based on radial basis function rbf interpolation for numerically simulating transport on a sphere the methods are mesh free and are formulated entirely in cartesian coordinates thus avoiding any irregular clustering of nodes at artificial boundaries on the sphere and naturally bypassing any apparent artificial singularities associated with surface based coordinate systems for problems involving tracer transport in a given velocity field the semi lagrangian framework allows these new methods to avoid the use of any stabilization terms such as hyperviscosity during time integration thus reducing the number of parameters that have to be tuned the three new methods are based on interpolation using 1 global rbfs 2 local rbf stencils and 3 rbf partition of unity for the latter two of these methods we find that it is crucial to include some low degree spherical harmonics in the interpolants standard test cases consisting of solid body rotation and deformational flow are used to compare and contrast the methods in terms of their accuracy efficiency conservation properties and dissipation dispersion errors for global rbfs spectral spatial convergence is observed for smooth solutions on quasi uniform nodes while high order accuracy is observed for the local rbf stencil and partition of unity approaches

systematic and generic construction of shape functions for p adaptive meshes of multidimensional finite elements this paper presents a methodology for generating high order shape functions for the complete family of finite elements the geometric entities presented are the point line triangle quadrilateral tetrahedron pyramid prism and hexahedron the shape functions constructed are hierarchical and generate continuous approximation spaces the order of interpolation of the shape function can be determined independently for each edge face and volume the space of interpolation of order p for each element is complete this definition allows for the construction of hybrid meshes with any combination of these elements including elements of distinct dimension and or interpolation order the reader will observe that our systematic way of constructing shape functions can be extended to elements of higher dimension

smooth side match classified vector quantizer with variable block size although the side match vector quantizer smvq reduces the bit rate the image coding quality by smvq generally degenerates as the gray level transition across the boundaries of the neighboring blocks is increasing or decreasing this study presents a smooth side match method to select a state codebook according to the smoothness of the gray levels between neighboring blocks this method achieves a higher psnr and better visual perception than smvq does for the same bit rate moreover to design codebooks a genetic clustering algorithm that automatically finds the appropriate number of clusters is proposed the proposed smooth side match classified vector quantizer ssm cvq is thus a combination of three techniques the classified vector quantization the variable block size segmentation and the smooth side match method experimental results indicate that ssm cvq has a higher psnr and a lower bit rate than other methods furthermore the lena image can be coded by ssm cvq with 0 172 bpp and 32 49 db in psnr

scattered and track data interpolation using an efficient strip searching procedure abstract a new local algorithm for bivariate interpolation of large sets of scattered and track data is presented the method which changes partially depending on the kind of data is based on the partition of the interpolation domain in a suitable number of parallel strips and starting from these on the construction for any data point of a square neighbourhood containing a convenient number of data points then the well known modified shepard s formula for surface interpolation is applied with some effective improvements the proposed algorithm is very fast owing to the optimal nearest neighbour searching and achieves good accuracy computational cost and storage requirements are analyzed moreover the efficiency and reliability of the algorithm are shown by several numerical tests also performed by renka s algorithm for a comparison

implementation of continuous h p adaptive finite element spaces without limitations on hanging sides and distribution of approximation orders adaptive techniques using h p refinement are known to be one of the most efficient methodologies to accelerate the convergence of finite element algorithms however the implementation of computational tools for the development of h p adaptive algorithms is intricate and depends strongly on the data structure there exist few computational environments available to the scientific finite element community capable to implement h p adaptive approximation spaces for the complete family of finite element topologies and which implement hanging sides this article describes a methodology for the development of continuous h p adaptive finite element approximation spaces without constraints on the refinement strategy concerning the difference of levels and approximation orders between neighboring elements the shape functions are hierarchical and the coefficient constraints associated with hanging sides that can occur in non conformal geometric meshes are defined using l 2 projections the topological and functional aspects of the construction are described in one two and three dimensions for a variety of geometric entities line triangle quadrilateral tetrahedron pyramid prism and hexahedron the implementation is demonstrated in the object oriented scientific computational environment neopz http github com labmec neopz neopz is a general finite element approximation software which incorporates a variety of variational formulations validation of the refinement methodology is demonstrated by two and three dimensional numerical experiments

strictly positive definite functions on spheres in euclidean spaces in this paper we study strictly positive definite functions on the unit sphere of the m dimensional euclidean space such functions can be used for solving a scattered data interpolation problem on spheres since positive definite functions on the sphere were already characterized by schoenberg some fifty years ago the issue here is to determine what kind of positive definite functions are actually strictly positive definite the study of this problem was initiated recently by xu and cheney proc amer math soc 116 1992 977 981 where certain sufficient conditions were derived a new approach which is based on a critical connection between this problem and that of multivariate polynomial interpolation on spheres is presented here the relevant interpolation problem is subsequently analyzed by three different complementary methods the first is based on the de boor ron general least solution for the multivariate polynomial interpolation problem the second which is suitable only for m 2 is based on the connection between bivariate harmonic polynomials and univariate analytic polynomials and reduces the problem to the structure of the integer zeros of bounded univariate exponentials finally the last method invokes the realization of harmonic polynomials as the polynomial kernel of the laplacian thereby exploiting some basic relations between homogeneous ideals and their polynomial kernels

spherical spline interpolation basic theory and computational aspects abstract the purpose of the paper is to adapt to the spherical case the basic theory and the computational method known from surface spline interpolation in euclidean spaces spline functions are defined on the sphere the solution process is made simple and efficient for numerical computation in addition the convergence of the solution obtained by spherical spline interpolation is developed using estimates for legendre polynomials

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

image vector quantizer based on a classification in the dct domain a classification algorithm in the discrete cosine transform dct domain for the classified vector quantization cvq technique is proposed the classifier employs four dct coefficients of 4 4 subblock as edge oriented features the classifier is designed using a cluster seeking algorithm to ensure that the centroid of a set of vectors in a class always belong to that class since the classification is performed in the dct domain this approach can be easily extended to the dct transform coding technique simulation results show that a good visual quality of the coded image at fixed rates in the 0 625 0 813 b pixel bpp range is obtained with comparable complexity the weighted mse wmse analysis in conjunction with the proposed classifier is discussed

hierarchical high order finite element spaces in h div ω h1 ω for a stabilized mixed formulation of darcy problem abstract the classical dual mixed finite element method for flow simulations is based on h div conforming approximation spaces for the flux which guarantees continuous normal components on element interfaces and discontinuous approximations in l 2 for the pressure however stability and convergence can only be obtained for compatible approximation spaces stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance the main purpose of this paper is to present a high order finite element methodology to solve the darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of h div and h 1 the chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables without requiring any compatibility constraint as required in mixed methods for these problems convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with raviart thomas elements

orientation embedded high order shape functions for the exact sequence elements of all shapes a unified construction of high order shape functions is given for all four classical energy spaces h1 h curl h div and l2 and for elements of all shapes segment quadrilateral triangle hexahedron tetrahedron triangular prism and pyramid the discrete spaces spanned by the shape functions satisfy the commuting exact sequence property for each element the shape functions are conforming hierarchical and compatible with other neighboring elements across shared boundaries so they may be used in hybrid meshes expressions for the shape functions are given in coordinate free format in terms of the relevant affine coordinates of each element shape the polynomial order is allowed to differ for each separate topological entity vertex edge face or interior in the mesh so the shape functions can be used to implement local p adaptive finite element methods each topological entity may have its own orientation and the shape functions can have that orientation embedded by a simple permutation of arguments