finite analytic method for 2d steady fluid flows in heterogeneous porous media with unstructured grids the finite analytic method is developed to solve the two dimensional steady fluid flows in heterogeneous porous media with full tensor permeability on unstructured grids the proposed fam is constructed based upon the power law analytic nodal solution in the angular domain with arbitrary shape when approaching the grid node joining the subdomains three different flow patterns may exist power law flow linear flow or the stagnant flow based on the nodal analytic solution the triangle based fam is proposed numerical examples show that the proposed numerical scheme makes the convergences much quickly than the traditional methods typically the weighted harmonic mean method under the cell refinement in practical applications the grid refinement parameter n 2 or n 3 is recommended and the relative error of the calculated equivalent permeability will below 4 independent of the heterogeneity in contrast when using the traditional numerical scheme the refinement ratio for the grid cell needs to increase dramatically to get an accurate result especially for strong heterogeneous porous medium

an optimal recovery approach to interpolation a filter class is defined as a ball in an inner product space and some standard results on inner product spaces are applied to filter classes the filter design problem is addressed the theory of optimal recovery is reviewed and the interpolation problem is examined within an optimal recovery context it is shown that the interpolation problem can be reduced to studying a hypercircle in an inner product space the notion of the chebyshev center of a set is introduced and it is noted that the solution to the interpolation problem from the optimal recovery viewpoint is to find the chebyshev center of the hypercircle the interpolation filter is then the operator that transforms the vector of known samples into the center of the hypercircle some auxiliary results such as the linearity and time invariance of the interpolation filter are deduced it is then shown that the estimation of an unknown sample is the same as the problem of approximating the representer of the unknown sample by a linear combination of the representers of the known samples hence the interpolation problem is equivalent to minimizing the l sup 2 norm of the error frequency response from the filter design point of view

a basic analog of a theorem of polya we derive a q analog of a theorem of george polya 1918 0 q 1 concer ning the zeros of basic cosine and sine transforms the results are established without further restrictions on q

on the approximation power of convolution based least squares versus interpolation there are many signal processing tasks for which convolution based continuous signal representations such as splines and wavelets provide an interesting and practical alternative to the more traditional sine based methods the coefficients of the corresponding signal approximations are typically obtained by direct sampling interpolation or quasi interpolation or by using least squares techniques that apply a prefilter prior to sampling we compare the performance of these approaches and provide quantitative error estimates that can be used for the appropriate selection of the sampling step h specifically we review several results in approximation theory with a special emphasis on the strang fix 1971 conditions which relate the general o h sup l behavior of the error to the ability of the representation to reproduce polynomials of degree n l 1 we use this theory to derive pointwise error estimates for the various algorithms and to obtain the asymptotic limit of the l sub 2 error as h tends to zero we also propose a new improved l sub 2 error bound for the least squares case in the process we provide all the relevant bound constants for polynomial splines some of our results suggest the existence of an intermediate range of sampling steps where the least squares method is roughly equivalent to an interpolator with twice the order we present experimental examples that illustrate the theory and confirm the adequacy of our various bound and limit determinations

the error amended sharp edge ease scheme for image zooming this paper proposes a new interpolation method called the error amended sharp edge ease scheme which is a modified bilinear method in order to remove reduce interpolation artifacts such as image blur and the checkerboard effect ringing ease tries to amend the interpolation error by employing the classical interpolation error theorem in an edge adaptive fashion ease is applied for image zooming by both integer and noninteger magnification factors the new interpolation scheme has proved to result in high resolution images having clearer and sharper edges than linear interpolation methods for all synthetic and natural images we have tested ease can be implemented with ease it turns out to be similarly efficient as cubic interpolation schemes

regularity preserving image interpolation assumptions about image continuity lead to oversmoothed edges in common image interpolation algorithms a wavelet based interpolation method that imposes no continuity constraints is introduced the algorithm estimates the regularity of edges by measuring the decay of wavelet transform coefficients across scales and preserves the underlying regularity by extrapolating a new subband to be used in image resynthesis the algorithm produces visibly sharper edges than traditional techniques and exhibits an average peak signal to noise ratio psnr improvement of 2 5 db over bilinear and bicubic techniques

a sampling theory approach for continuous arma identification the problem of estimating continuous domain autoregressive moving average processes from sampled data is considered the proposed approach incorporates the sampling process into the problem formulation while introducing exponential models for both the continuous and the sampled processes we derive an exact evaluation of the discrete domain power spectrum using exponential b splines and further suggest an estimation approach that is based on digitally filtering the available data the proposed functional which is related to whittle s likelihood function exhibits several local minima that originate from aliasing the global minimum however corresponds to a maximum likelihood estimator regardless of the sampling step experimental results indicate that the proposed approach closely follows the cramer rao bound for various aliasing configurations

locally adaptive wavelet based image interpolation we describe a spatially adaptive algorithm for image interpolation the algorithm uses a wavelet transform to extract information about sharp variations in the low resolution image and then implicitly applies interpolation which adapts to the image local smoothness singularity characteristics the proposed algorithm yields images that are sharper compared to several other methods that we have considered in this paper better performance comes at the expense of higher complexity

a multipoint flux mixed finite element method we develop a mixed finite element method for single phase flow in porous media that reduces to cell centered finite differences on quadrilateral and simplicial grids and performs well for discontinuous full tensor coefficients motivated by the multipoint flux approximation method where subedge fluxes are introduced we consider the lowest order brezzi douglas marini bdm mixed finite element method a special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell centered system for the pressures theoretical and numerical results indicate second order convergence for pressures at the cell centers and first order convergence for subedge fluxes second order convergence for edge fluxes is also observed computationally if the grids are sufficiently regular

on the role of exponential splines in image interpolation a sobolev reproducing kernel hilbert space approach to image interpolation is introduced the underlying kernels are exponential functions and are related to stochastic autoregressive image modeling the corresponding image interpolants can be implemented effectively using compactly supported exponential b splines a tight l2 upper bound on the interpolation error is then derived suggesting that the proposed exponential functions are optimal in this regard experimental results indicate that the proposed interpolation approach with properly tuned signal dependent weights outperforms currently available polynomial b spline models of comparable order furthermore a unified approach to image interpolation by ideal and nonideal sampling procedures is derived suggesting that the proposed exponential kernels may have a significant role in image modeling as well our conclusion is that the proposed sobolev based approach could be instrumental and a preferred alternative in many interpolation tasks

a family of mpfa finite volume schemes with full pressure support for the general tensor pressure equation on cell centered triangular grids a new family of cell centered finite volume schemes is presented for solving the general full tensor pressure equation of subsurface flow in porous media on arbitary unstructured triangulations the new schemes are flux continuous and have full pressure support fps over each subcell with continuous pressure imposed across each control volume sub interface in contrast to earlier formulations the earlier methods are point wise continuous in pressure and flux with triangle pressure support tps which leads to a more limited quadrature range an m matrix analysis identifies bounding limits for the schemes to posses a local discrete maximum principle conditions for the schemes to be positive definite are also derived a range of computational examples are presented for unstructured triangular grids including highly irregular grids and the new fps schemes are compared against the earlier pointwise continuous tps formulations the earlier pointwise tps methods can induce strong spurious oscillations for problems involving strong full tensor anisotropy where the m matrix conditions are violated and can lead to decoupled solutions in such cases unstructured cell centered decoupling is investigated in contrast to tps the new fps formulation leads to well resolved solutions that are essentially free of spurious oscillations a substantial degree of improved convergence behavior for both pressure and velocity is also observed in all convergence tests this is particularly important for problems involving high anisotropy ratios also the new formulation proves to be highly beneficial for an upscaling example where enhancement of convergence is highly significant for certain quadrature points clearly demonstrating further advantages of the new formulation

kernel methods in machine learning we review machine learning methods employing positive definite kernels these methods formulate learning and estimation problems in a reproducing kernel hilbert space rkhs of functions defined on the data domain expanded in terms of a kernel working in linear spaces of function has the benefit of facilitating the construction and analysis of learning algorithms while at the same time allowing large classes of functions the latter include nonlinear functions as well as functions defined on nonvectorial data we cover a wide range of methods ranging from binary classifiers to sophisticated methods for estimation with structured data

cardinal exponential splines part i theory and filtering algorithms causal exponentials play a fundamental role in classical system theory starting from those elementary building blocks we propose a complete and self contained signal processing formulation of exponential splines defined on a uniform grid we specify the corresponding b spline basis functions and investigate their reproduction properties green function and exponential polynomials we also characterize their stability riesz bounds we show that the exponential b spline framework allows an exact implementation of continuous time signal processing operators including convolution differential operators and modulation by simple processing in the discrete b spline domain we derive efficient filtering algorithms for multiresolution signal extrapolation and approximation extending earlier results for polynomial splines finally we present a new asymptotic error formula that predicts the magnitude and the nth order decay of the l sub 2 approximation error as a function of the knot spacing t

the zeros of basic bessel functions the functions jv ax x and associated orthogonal polynomials abstract the modified lommel polynomials are generalized and their orthogonality relation is obtained as a byproduct we prove that the non zero roots of jv ax x are real simple and interlace with those of jv ax 1 x the q lommel polynomials are shown to play in the theory of basic bessel function the role played by the lommel polynomials in the bessel function theory the q lommel polynomials are proved to be orthogonal with respect to a purely discrete measure with bounded support this is then used to prove that the prositive zeros of x j 2 v x q are real simple and interlace with the zeros of x 1j 2 v 1 x q when 1 we also establish the complete monotonicity of 1x 1 2 j v 1 2 i x q j v 2 i x q

cardinal exponential splines part ii think analog act digital by interpreting the green function reproduction property of exponential splines in signal processing terms we uncover a fundamental relation that connects the impulse responses of allpole analog filters to their discrete counterparts the link is that the latter are the b spline coefficients of the former which happen to be exponential splines motivated by this observation we introduce an extended family of cardinal splines the generalized e splines to generalize the concept for all convolution operators with rational transfer functions we construct the corresponding compactly supported b spline basis functions which are characterized by their poles and zeros thereby establishing an interesting connection with analog filter design techniques we investigate the properties of these new b splines and present the corresponding signal processing calculus which allows us to perform continuous time operations such as convolution differential operators and modulation by simple application of the discrete version of these operators in the b spline domain in particular we show how the formalism can be used to obtain exact discrete implementations of analog filters finally we apply our results to the design of hybrid signal processing systems that rely on digital filtering to compensate for the nonideal characteristics of real world analog to digital a to d and d to a conversion systems

m matrix flux splitting for general full tensor discretization operators on structured and unstructured grids diagonal tensor flux approximations are commonly used in fluid dynamics this approximation introduces an o 1 error in flux whenever the coordinate system is nonaligned with the principal axes of the tensor which is particularly common when employing curvilinear gridding in general a consistent full tensor flux approximation leads to a significant increase in support and consequent size of the jacobian matrix after decomposition of a general full tensor flux into a diagonal tensor flux together with cross terms time split semi implicit stable full tensor flux approximations are introduced with in a general finite volume formalism enabling the standard diagonal tensor jacobian matrix structure to be retained for single phase flow impes and standard block fully implicit formulations while ensuring spatial consistency of the discretization for both structured and unstructured grids stability of the scheme is proven for constant elliptic coefficients the results presented demonstrate the benefits of the method for multiphase flow within a fully implicit framework on structured and unstructured grids

moms maximal order interpolation of minimal support we consider the problem of interpolating a signal using a linear combination of shifted versions of a compactly supported basis function spl phi z we first give the expression for the cases of spl phi that have minimal support for a given accuracy also known as approximation order this class of functions which we call maximal order minimal support functions moms is made of linear combinations of the b spline of the same order and of its derivatives we provide an explicit form of the moms that maximizes the approximation accuracy when the step size is small enough we compute the sampling gain obtained by using these optimal basis functions over the splines of the same order we show that it is already substantial for small orders and that it further increases with the approximation order l when l is large this sampling gain becomes linear more specifically its exact asymptotic expression is 2 spl pi e l since the optimal functions are continuous but not differentiable for even orders and even only piecewise continuous for odd orders our result implies that regularity has little to do with approximating performance these theoretical findings are corroborated by experimental evidence that involves compounded rotations of images

coarse to fine segmentation and tracking using sobolev active contours recently proposed sobolev active contours introduced a new paradigm for minimizing energies defined on curves by changing the traditional cost of perturbing a curve and thereby redefining gradients associated to these energies sobolev active contours evolve more globally and are less attracted to certain intermediate local minima than traditional active contours and it is based on a well structured riemannian metric which is important for shape analysis and shape priors in this paper we analyze sobolev active contours using scale space analysis in order to understand their evolution across different scales this analysis shows an extremely important and useful behavior of sobolev contours namely that they move successively from coarse to increasingly finer scale motions in a continuous manner this property illustrates that one justification for using the sobolev technique is for applications where coarse scale deformations are preferred over fine scale deformations along with other properties to be discussed the coarse to fine observation reveals that sobolev active contours are in particular ideally suited for tracking algorithms that use active contours we will also justify our assertion that the sobolev metric should be used over the traditional metric for active contours in tracking problems by experimentally showing how a variety of active contour based tracking methods can be significantly improved merely by evolving the active contour according to the sobolev method

quasi interpolating spline models for hexagonally sampled data the reconstruction of a continuous domain representation from sampled data is an essential element of many image processing tasks in particular image resampling until today most image data have been available on cartesian lattices despite the many theoretical advantages of hexagonal sampling in this paper we propose new reconstruction methods for hexagonally sampled data that use the intrinsically 2 d nature of the lattice and that at the same time remain practical and efficient to that aim we deploy box spline and hex spline models which are notably well adapted to hexagonal lattices we also rely on the quasi interpolation paradigm to design compelling prefilters that is the optimal filter for a prescribed design is found using recent results from approximation theory the feasibility and efficiency of the proposed methods are illustrated and compared for a hexagonal to cartesian grid conversion problem

theq laguerre polynomials and related moment problems abstract we study two indeterminate hamburger moment problems associated with q laguerre polynomials the coefficients in their recurrence relations are of exponential growth this completes earlier work started by d moak some of the measures that the polynomials are orthogonal with respect to are found and lead to the evaluation of certain integrals a symmetric set of orthogonal polynomials is also considered

image interpolation by adaptive 2 d autoregressive modeling and soft decision estimation the challenge of image interpolation is to preserve spatial details we propose a soft decision interpolation technique that estimates missing pixels in groups rather than one at a time the new technique learns and adapts to varying scene structures using a 2 d piecewise autoregressive model the model parameters are estimated in a moving window in the input low resolution image the pixel structure dictated by the learnt model is enforced by the soft decision estimation process onto a block of pixels including both observed and estimated the result is equivalent to that of a high order adaptive nonseparable 2 d interpolation filter this new image interpolation approach preserves spatial coherence of interpolated images better than the existing methods and it produces the best results so far over a wide range of scenes in both psnr measure and subjective visual quality edges and textures are well preserved and common interpolation artifacts blurring ringing jaggies zippering etc are greatly reduced

monotone finite volume schemes for diffusion equations on unstructured triangular and shape regular polygonal meshes we consider a non linear finite volume fv scheme for stationary diffusion equation we prove that the scheme is monotone i e it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients the scheme is extended to regular star shaped polygonal meshes and isotropic heterogeneous coefficients

on q analogues of the fourier and hankel transforms for h exton s q analogue of the bessel function going back to w hahn in a special case but different from f h jackson s q bessel functions we derive hansen lommel type orthogonality relations which by a symmetry turn out to be equivalent to orthogonality relations which are q analogues of the hankel integral transform pair these results are implicit in the context of quantum groups in a paper by vaksman and korogodskii as a specialization we get q cosines and q sines which admit q analogues of the fourier cosine and fourier sine transforms

quasi positive families of continuous darcy flux finite volume schemes on structured and unstructured grids new families of flux continuous control volume distributed finite volume schemes are presented for the general full tensor pressure equation arising in porous media and formulated for structured and unstructured grids these schemes offer the practical advantage of being flux continuous while only depending on one degree of freedom per control volume unlike rival approximations such as the mixed finite element method m matrix bounds are presented quasi qm matrices are defined and an optimal quadrilateral scheme is identified anisotropy favoring triangulation is also shown to yield an optimal scheme the new schemes prove to be relatively robust for the cases tested including strongly anisotropic full tensor fields strong oscillations encountered with the earlier formulations are removed or minimized

a quasi positive family of continuous darcy flux finite volume schemes with full pressure support a new family of flux continuous locally conservative finite volume schemes is presented for solving the general tensor pressure equation of subsurface flow in porous media the new schemes have full pressure continuity imposed across control volume faces previous families of flux continuous schemes are point wise continuous in pressure and flux when applying the earlier point wise flux continuous schemes to strongly anisotropic full tensor fields their failure to satisfy a maximum principle as with other fem and finite volume methods can result in loss of local stability for high anisotropy ratios which can cause strong spurious oscillations in the numerical pressure solution an m matrix analysis reveals the upper limits for guaranteeing a maximum principle for general 9 point schemes and aids in the design of schemes that minimize the occurrence of spurious oscillations in the discrete pressure field the full pressure continuity schemes are shown to possess a larger range of flux continuous schemes than the previous point wise counter parts for strongly anisotropic full tensor cases it is shown that the full quadrature range possessed by the new schemes permits these schemes to exploit quadrature points previously out of range that are shown to minimize spurious oscillations in discrete pressure solutions the new formulation leads to a more robust quasi positive family of flux continuous schemes applicable to general discontinuous full tensor fields

on the zeros of basic finite hankel transforms in this article we prove that the basic finite hankel transform whose kernel is the third type jackson q bessel function has only infinitely many real and simple zeros provided that q satisfies a condition additional to the standard one we also study the asymptotic behavior of the zeros the obtained results are applied to investigate the zeros of q bessel functions as well as the zeros of q trigonometric functions a basic analog of a theorem of g polya 1918 on the zeros of sine and cosine transformations is also given

adaptive image resizing based on continuous domain stochastic modeling we introduce an adaptive continuous domain modeling approach to texture and natural images the continuous domain image is assumed to be a smooth function and we embed it in a parameterized sobolev space we point out a link between sobolev spaces and stochastic auto regressive models and exploit it for optimally choosing sobolev parameters from available pixel values to this aim we use exact continuous to discrete mapping of the auto regressive model that is based on symmetric exponential splines the mapping is computationally efficient and we exploit it for maximizing an approximated gaussian likelihood function we account for non gaussian levy type processes by deriving a more robust estimator that is based on the sample auto correlation sequence both estimators use multiple initialization values for overcoming the local minima structure of the fitting criteria experimental image resizing results indicate that the auto correlation criterion can cope better with non gaussian processes and model mismatch our work demonstrates the importance of the auto correlation function in adaptive image interpolation and image modeling tasks and we believe it is instrumental in other image processing tasks as well

an integral representation and some transformation properties of q bessel functions a poisson type integral representation for jackson s q bessel function is obtained by using askey and wilson s q beta integral and nassrallah and rahman s integral formula for an 8 7 series this representation along with some transformtion formulas for basic hypergeometric series help express the q bessel functions as a 3 2 series in base q a 2 2 series in base q2 and a 2 1 series in base qq12 where 0 q 1 as an application q analogues are found for gegenbauer s degenerate addition formulas for bessel functions

generalized smoothing splines and the optimal discretization of the wiener filter we introduce an extended class of cardinal l sup l splines where l is a pseudo differential operator satisfying some admissibility conditions we show that the l sup l spline signal interpolation problem is well posed and that its solution is the unique minimizer of the spline energy functional spl par ls spl par sub l2 sup 2 subject to the interpolation constraint next we consider the corresponding regularized least squares estimation problem which is more appropriate for dealing with noisy data the criterion to be minimized is the sum of a quadratic data term which forces the solution to be close to the input samples and a smoothness term that privileges solutions with small spline energies here too we find that the optimal solution among all possible functions is a cardinal l sup l spline we show that this smoothing spline estimator has a stable representation in a b spline like basis and that its coefficients can be computed by digital filtering of the input signal we describe an efficient recursive filtering algorithm that is applicable whenever the transfer function of l is rational which corresponds to the case of exponential splines we justify these algorithms statistically by establishing an equivalence between l sup l smoothing splines and the minimum mean square error mmse estimation of a stationary signal corrupted by white gaussian noise in this model based formulation the optimum operator l is the whitening filter of the process and the regularization parameter is proportional to the noise variance thus the proposed formalism yields the optimal discretization of the classical wiener filter together with a fast recursive algorithm it extends the standard wiener solution by providing the optimal interpolation space we also present a bayesian interpretation of the algorithm

finite volume scheme for anisotropic diffusion in this paper we apply a special finite volume scheme limited to smooth temperature distributions and cartesian grids to test the importance of connectivity of the finite volumes the area of application is nuclear fusion plasma with field line aligned temperature gradients and extreme anisotropy we apply the scheme to the anisotropic heat conduction equation and compare its results with those of existing finite volume schemes for anisotropic diffusion also we introduce a general model adaptation of the steady diffusion equation for extremely anisotropic diffusion problems with closed field lines

image reconstruction by convolution with symmetrical piecewise nth order polynomial kernels the reconstruction of images is an important operation in many applications from sampling theory it is well known that the sine function is the ideal interpolation kernel which however cannot be used in practice in order to be able to obtain an acceptable reconstruction both in terms of computational speed and mathematical precision it is required to design a kernel that is of finite extent and resembles the sinc function as much as possible in this paper the applicability of the sine approximating symmetrical piecewise nth order polynomial kernels is investigated in satisfying these requirements after the presentation of the general concept kernels of first third fifth and seventh order are derived an objective quantitative evaluation of the reconstruction capabilities of these kernels is obtained by analyzing the spatial and spectral behavior using different measures and by using them to translate rotate and magnify a number of real life test images from the experiments it is concluded that while the improvement of cubic convolution over linear interpolation is significant the use of higher order polynomials only yields marginal improvement

image enhancement by nonlinear extrapolation in frequency space a technique for enhancing the perceptual sharpness of an image is described the enhancement algorithm augments the frequency content of the image using shape invariant properties of edges across scale by using a nonlinearity that generates phase coherent higher harmonics the procedure utilizes the laplacian transform and the laplacian pyramid image representation results are presented depicting the power spectra augmentation and the visual enhancement of several images simplicity of computations and ease of implementation allow for real time applications such as high definition television hdtv

signal and image approximation using interval wavelet transform in signal approximation classical wavelet synthesis are known to produce gibbs like phenomenon around discontinuities when wavelet coefficients in the cone of influence of the discontinuities are quantized by analyzing a function in a piecewise manner filtering across discontinuities can be avoided using this principle the interval wavelet transform can generate sparser representations in the vicinity of discontinuities than classical wavelet transforms this work introduces two new constructions of interval wavelets and shows how they can be used for image compression and upscaling

expansion and asymptotic in terms of basic bessel functions abstract this work aims to study the expansion and asymptotic for solutions of q difference equations in terms of the basic bessel functions namely j 2 x q for this purpose we will show that the constructive method introduced by olver f w j olver asymptotics and special functions academic press inc 1974 and exploited early by fitouhi et al h chebli a fitouhi m m hamza expansion in series of bessel functions and transmutations for perturbed bessel operator j math anal appl 181 3 1994 a fitouhi m m hamza uniform expansion for eigenfunction of singular second order differential operator siam j math anal 21 6 1990 a fitouhi m m hamza expansion in series of laguerre functions for solution of perturbed laguerre equations contemp math 183 1995 a fitouhi j el kamel expansion in series of gegenbauer polynomials int trans sp funct 5 3 4 1997 213 226 can be extended in this context as application new expansions of some basic special functions are established in particular these given by ismail y chen m e h ismail k a muttalib asymptotics of basic bessel functions and q laguerre polynomilas j comput appl math 54 1994 263 272 m e h ismail classical and quantum orthogonal polynomials in one variable cambridge university press 2005

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

image zooming method using edge directed moving least squares interpolation based on exponential polynomials this paper presents a nonlinear image interpolation algorithm the suggested method is based on the moving least squares mls projection technique but introduces a fundamental modification the algebraic polynomial based mls methods provide very satisfactory results however the associated approximation space is shift and scale invariant so that it cannot be adjusted according to the characteristic of a given data as a result when upsampling images it has a limitation in producing sharp edges such that edges are often blurred in the magnified images to recover sharper edges we need to reduce smoothing parameter or adapt a new parameter sharpening the edges motivated by this observations we propose a novel mls method governed by a set of exponential polynomials with tension parameters such that they can be tuned to the characteristic of given data moreover for a better match to the local structures around the edges the suggested algorithm uses weights which consider the edge orientation numerical results are presented and compared visually and by using some quantitative fidelity measures psnr epsnr ssim and fsim to the bicubic spline interpolation and other recently developed nonlinear methods the results demonstrate the new algorithm s ability to magnify an image while preserving edge features

on the approximation of l_ 2 inner products from sampled data most signal processing applications are based on discrete time signals although the origin of many sources of information is analog in this paper we consider the task of signal representation by a set of functions focusing on the representation coefficients of the original continuous time signal the question considered herein is to what extent the sampling process keeps algebraic relations such as inner product intact by interpreting the sampling process as a bounded operator a vector like interpretation for this approximation problem has been derived giving rise to an optimal discrete approximation scheme different from the riemann type sum often used the objective of this optimal scheme is in the min max sense and no bandlimitedness constraints are imposed tight upper bounds on this optimal and the riemann type sum approximation schemes are then derived we further consider the case of a finite number of samples and formulate a closed form solution for such a case the results of this work provide a tool for finding the optimal scheme for approximating an l2 inner product and to determine the maximum potential representation error induced by the sampling process the maximum representation error can also be determined for the riemann type sum approximation scheme examples of practical applications are given and discussed

a new orientation adaptive interpolation method we propose an isophote oriented orientation adaptive interpolation method the proposed method employs an interpolation kernel that adapts to the local orientation of isophotes and the pixel values are obtained through an oriented bilinear interpolation we show that by doing so the curvature of the interpolated isophotes is reduced and thus zigzagging artifacts are largely suppressed analysis and experiments show that images interpolated using the proposed method are visually pleasing and almost artifact free

wavelet algorithms for high resolution image reconstruction high resolution image reconstruction refers to the reconstruction of high resolution images from multiple low resolution shifted degraded samples of a true image in this paper we analyze this problem from the wavelet point of view by expressing the true image as a function in cal l bbb r 2 we derive iterative algorithms which recover the function completely in the cal l sense from the given low resolution functions these algorithms decompose the function obtained from the previous iteration into different frequency components in the wavelet transform domain and add them into the new iterate to improve the approximation we apply wavelet packet thresholding methods to denoise the function obtained in the previous step before adding it into the new iterate our numerical results show that the reconstructed images from our wavelet algorithms are better than that from the tikhonov least squares approach extension to super resolution image reconstruction where some of the low resolution images are missing is also considered

an edge guided image interpolation algorithm via directional filtering and data fusion preserving edge structures is a challenge to image interpolation algorithms that reconstruct a high resolution image from a low resolution counterpart we propose a new edge guided nonlinear interpolation technique through directional filtering and data fusion for a pixel to be interpolated two observation sets are defined in two orthogonal directions and each set produces an estimate of the pixel value these directional estimates modeled as different noisy measurements of the missing pixel are fused by the linear minimum mean square error estimation lmmse technique into a more robust estimate using the statistics of the two observation sets we also present a simplified version of the lmmse based interpolation algorithm to reduce computational cost without sacrificing much the interpolation performance experiments show that the new interpolation techniques can preserve edge sharpness and reduce ringing artifacts

edge forming methods for color image zooming this paper introduces edge forming schemes for image zooming of color images by general magnification factors in order to remove reduce artifacts arising in image interpolation such as image blur and the checkerboard effect an edge forming method is suggested to be applied as a postprocess of standard interpolation methods the method is based on nonconvex nonlinear partial differential equations the equations are carefully discretized incorporating numerical schemes of anisotropic diffusion to be able to form reliable edges satisfactorily the alternating direction implicit adi method is employed for an efficient simulation of the model it has been numerically verified that the resulting algorithm can form clear edges in 2 to 3 adi iterations various results are given to show th effectiveness and reliability of the algorithm

survey interpolation methods in medical image processing image interpolation techniques often are required in medical imaging for image generation e g discrete back projection for inverse radon transform and processing such as compression or resampling since the ideal interpolation function spatially is unlimited several interpolation kernels of finite size have been introduced this paper compares 1 truncated and windowed sine 2 nearest neighbor 3 linear 4 quadratic 5 cubic b spline 6 cubic g lagrange and 7 gaussian interpolation and approximation techniques with kernel sizes from 1 spl times 1 up to 8 spl times 8 the comparison is done by 1 spatial and fourier analyses 2 computational complexity as well as runtime evaluations and 3 qualitative and quantitative interpolation error determinations for particular interpolation tasks which were taken from common situations in medical image processing for local and fourier analyses a standardized notation is introduced and fundamental properties of interpolators are derived successful methods should be direct current dc constant and interpolators rather than dc inconstant or approximators each method s parameters are tuned with respect to those properties this results in three novel kernels which are introduced in this paper and proven to be within the best choices for medical image interpolation the 6 spl times 6 blackman harris windowed sinc interpolator and the c2 continuous cubic kernels with n 6 and n 8 supporting points for quantitative error evaluations a set of 50 direct digital x rays was used they have been selected arbitrarily from clinical routine in general large kernel sizes were found to be superior to small interpolation masks except for truncated sine interpolators all kernels with n 6 or larger sizes perform significantly better than n 2 or n 3 point methods p spl lt 0 005 however the differences within the group of large sized kernels were not significant summarizing the results the cubic 6 spl times 6 interpolator with continuous second derivatives as defined in 24 can be recommended for most common interpolation tasks it appears to be the fastest six point kernel to implement computationally it provides eminent local and fourier properties is easy to implement and has only small errors the same characteristics apply to b spline interpolation but the 6 spl times 6 cubic avoids the intrinsic border effects produced by the b spline technique however the goal of this study was not to determine an overall best method but to present a comprehensive catalogue of methods in a uniform terminology to define general properties and requirements of local techniques and to enable the reader to select that method which is optimal for his specific application in medical imaging

quantitative fourier analysis of approximation techniques i interpolators and projectors we present a general fourier based method that provides an accurate prediction of the approximation error as a function of the sampling step t our formalism applies to an extended class of convolution based signal approximation techniques which includes interpolation generalized sampling with prefiltering and the projectors encountered in wavelet theory we claim that we can predict the l sup 2 approximation error by integrating the spectrum of the function to approximate not necessarily bandlimited against a frequency kernel e spl omega that characterizes the approximation operator this prediction is easier yet more precise than was previously available our approach has the remarkable property of providing a global error estimate that is the average of the true approximation error over all possible shifts of the input function our error prediction is exact for stationary processes as well as for bandlimited signals we apply this method to the comparison of standard interpolation and approximation techniques our method has interesting implications for approximation theory in particular we use our results to obtain some new asymptotic expansions of the error as t spl rarr 0 as well as to derive improved upper bounds of the kind found in the strang fix 1971 theory we finally show how we can design quasi interpolators that are near optimal in the least squares sense

warped distance for space variant linear image interpolation the problem of image interpolation using linear techniques is dealt with in this paper conventional space invariant methods are revisited and changed into space variant ones by introducing the concept of the warped distance among the pixels of an image a better perceptual rendition of the image details is obtained in this way this effect is proved both via the evaluation of the response to an idealized sigmoidal edge model and with experiments on real world images the computational costs of the proposed approach are very small when compared to those of state of the art nonlinear interpolation operators

on the linearity of bayesian interpolators for non gaussian continuous time ar 1 processes bayesian estimation problems involving gaussian distributions often result in linear estimation techniques nevertheless there are no general statements as to whether the linearity of the bayesian estimator is restricted to the gaussian case the two common strategies for non gaussian models are either finding the best linear estimator or numerically evaluating the bayesian estimator by monte carlo methods in this paper we focus on bayesian interpolation of non gaussian first order autoregressive ar processes where the driving innovation can admit any symmetric infinitely divisible distribution characterized by the levy khintchine representation theorem we redefine the bayesian estimation problem in the fourier domain with the help of characteristic forms by providing analytic expressions we show that the optimal interpolator is linear for all symmetric stable distributions the bayesian interpolator can be expressed in a convolutive form where the kernel is described in terms of exponential splines we also show that the limiting case of levy type ar 1 processes the system of which has a pole at the origin always corresponds to a linear bayesian interpolator made of a piecewise linear spline irrespective of the innovation distribution finally we show the two mentioned cases to be the only ones within the family for which the bayesian interpolator is linear

octobasic laguerre polynomials and permutation statistics abstract a set of orthogonal polynomials with eight independent q s is defined which generalizes the laguerre polynomials the moments of the measure for these polynomials are the generating functions for permutations according to eight different statistics specializing these statistics gives many other well known sets of combinatorial objects and relevant statistics the specializations are studied with applications to classical orthogonal polynomials and equidistribution theorems for statistics

minimax approximation of representation coefficients from generalized samples many sources of information are of analog or continuous time nature however digital signal processing applications rely on discrete data we consider the problem of approximating l2 inner products i e representation coefficients of a continuous time signal from its generalized samples adopting a robust approach we process these generalized samples in a minimax optimal sense specifically we minimize the worst approximation error of the desired representation coefficients by proper processing of the given sample sequence we then extend our results to criteria which incorporate smoothness constraints on the unknown function finally we compare our methods with the piecewise constant approximation technique commonly used for this problem and discuss the possible improvements by the suggested schemes