nmr flow velocity mapping in random percolation model objects evidence for a power law dependence of the volume averaged velocity on the probe volume radius two or three dimensional lacunar objects were fabricated using computer simulated random sitepercolation networks as templates the flow of water pumped through the pore space was studied with the aid of nuclear magnetic resonance nmr microscopy modified for mapping of the velocity vector field the percolation backbones of the objects were determined by exclusion of all pixels or voxels of the spin density images with velocities below the noise level an evaluation procedure was established which reliably renders the fractal dimensions of the whole cluster and of its backbone the volume averaged velocity magnitude as a function of the probe volume radius r was found to obey a power lawv v r 2l in the range a r jv where a is the voxel edge length and jv the velocity correlation length the exponents turned out to be l50 3260 04 and l50 8260 03 for the two and three dimensional objects respectively in order to test the time dependence of the mean squared displacement on fractals r 2 5at 2 dw expected for random walks of a fractal dimension dw self diffusion of gaseous methane was examined in the pore spaces of the same objects with the aid of field gradient nmr diffusometry the results are in accordance with the theoretical predictions for anomalous diffusion on percolation clusters this finding is supported by studies of incoherent water flow in the percolation network s1063 651x 96 06510 5

convex sets with semidefinite representation we provide a sufficient condition on a class of compact basic semialgebraic sets bf k subset mathbb r n for their convex hull co k to have a semidefinite representation sdr this sdr is explicitly expressed in terms of the polynomials g j that define k examples are provided we also provide an approximate sdr that is for every fixed epsilon 0 there is a convex set bf k epsilon such that rm co bf k subseteq bf k epsilon subseteq rm co bf k epsilon bf b where b is the unit ball of mathbb r n and bf k epsilon has an explicit sdr in terms of the g j s for convex and compact basic semi algebraic sets k defined by concave polynomials we provide a simpler explicit sdr when the nonnegative lagrangian l f associated with k and any linear f in mathbb r x is a sum of squares we also provide an approximate sdr specific to the convex case

matrix sum of squares relaxations for robust semi definite programs we consider robust semi definite programs which depend polynomially or rationally on some uncertain parameter that is only known to be contained in a set with a polynomial matrix inequality description on the basis of matrix sum of squares decompositions we suggest a systematic procedure to construct a family of linear matrix inequality relaxations for computing upper bounds on the optimal value of the corresponding robust counterpart with a novel matrix version of putinar s sum of squares representation for positive polynomials on compact semi algebraic sets we prove asymptotic exactness of the relaxation family under a suitable constraint qualification if the uncertainty region is a compact polytope we provide a new duality proof for the validity of putinar s constraint qualification with an a priori degree bound on the polynomial certificates finally we point out the consequences of our results for constructing relaxations based on the so called full block s procedure which allows to apply recently developed tests in order to computationally verify the exactness of possibly small sized relaxations

iterated rings of bounded elements and generalizations of schmudgen s positivstellensatz let a be a commutative r algebra of finite transcendence degree d 2 n we investigate the relationship between the subring of geometrically bounded elements

finite difference spectral approximations for the fractional cable equation fujian nsf s0750017 national nsf of china 10531080 ministry of education of china 973 high performance scientific computation research program 2005cb321703

nonlinear wavelet transforms for image coding via lifting we investigate central issues such as invertibility stability synchronization and frequency characteristics for nonlinear wavelet transforms built using the lifting framework the nonlinearity comes from adaptively choosing between a class of linear predictors within the lifting framework we also describe how earlier families of nonlinear filter banks can be extended through the use of prediction functions operating on a causal neighborhood of pixels preliminary compression results for model and real world images demonstrate the promise of our techniques

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

subpixel edge localization and the interpolation of still images in this paper we present a nonlinear interpolation scheme for still image resolution enhancement the algorithm is based on a source model emphasizing the visual integrity of detected edges and incorporates a novel edge fitting operator that has been developed for this application a small neighborhood about each pixel in the low resolution image is first mapped to a best fit continuous space step edge the bilevel approximation serves as a local template on which the higher resolution sampling grid can then be superimposed where disputed values in regions of local window overlap are averaged to smooth errors the result is an image of increased resolution with noticeably sharper edges and in all tried cases lower mean squared reconstruction error than that produced by linear techniques

positive polynomials and sequential closures of quadratic modules let s x n f 1 x 0 f s x 0 be a basic closed semi algebraic set in n and let po f 1 f s be the corresponding preordering in x 1 x n we examine for which polynomials f there exist identities f eq po f 1 f s for all e 0 these are precisely the elements of the sequential closure of po f 1 f s with respect to the finest locally convex topology we solve the open problem from kuhlmann marshall and schwartz 2002 2005 whether this equals the double dual cone po f 1 f s vv by providing a counterexample we then prove a theorem that allows us to obtain identities for polynomials as above by looking at a family of fibre preorderings constructed from bounded polynomials these fibre preorderings are easier to deal with than the original preordering in general for a large class of examples we are thus able to show that either every polynomial f that is nonnegative on s admits such representations or at least the polynomials from po f 1 fs vv do the results also hold in the more general setup of arbitrary commutative algebras and quadratic modules instead of preorderings

orthogonal and biorthogonal fir hexagonal filter banks with sixfold symmetry recently hexagonal image processing has attracted attention the hexagonal lattice has several advantages in comparison with the rectangular lattice the conventionally used lattice for image sampling and processing for example a hexagonal lattice needs less sampling points it has better consistent connectivity it has higher symmetry and its structure is plausible to human vision systems the multiresolution analysis method has been used for hexagonal image processing since the hexagonal lattice has high degree of symmetry it is desirable that the hexagonal filter banks designed for multiresolution hexagonal image processing also have high order of symmetry which is pertinent to the symmetry structure of the hexagonal lattice the orthogonal or prefect reconstruction pr hexagonal filter banks that are available in the literature have only threefold symmetry in this paper we investigate the construction of orthogonal and pr finite impulse response fir hexagonal filter banks with sixfold symmetry we obtain block structures of 7 size refinement seven channel two dimensional orthogonal and pr fir hexagonal filter banks with sixfold rotational symmetry radic7 refinement orthogonal and biorthogonal wavelets based on these block structures are constructed in this paper we also consider fir hexagonal filter banks with axial line symmetry and we present a block structure of fir hexagonal filter banks with pseudo sixfold axial symmetry

color image coding using morphological pyramid decomposition presents a new algorithm that utilizes mathematical morphology for pyramidal coding of color images the authors obtain lossy color image compression by using block truncation coding at the pyramid levels to attain reduced data rates the pyramid approach is attractive due to low computational complexity simple parallel implementation and the ability to produce acceptable color images at moderate data rates in many applications the progressive transmission capability of the algorithm is very useful the authors show experimental results for color images at data rates of 1 89 bits pixel

an explicit equivalent positive semidefinite program for nonlinear 0 1 programs we consider the general nonlinear optimization problem in 0 1 variables and provide an explicit equivalent positive semidefinite program in 2n 1 variables the optimal values of both problems are identical from every optimal solution of the former one easily finds an optimal solution of the latter and conversely from every solution of the latter one may construct an optimal solution of the former for illustration the equivalent positive semidefinite program is explicated for quadratic 0 1 programs and max cut in mathbb r 3 for unconstrained 0 1 programs a special representation in terms of a weighted sum of squares is provided

a new family of nonredundant transforms using hybrid wavelets and directional filter banks we propose a new family of nonredundant geometrical image transforms that are based on wavelets and directional filter banks we convert the wavelet basis functions in the finest scales to a flexible and rich set of directional basis elements by employing directional filter banks where we form a nonredundant transform family which exhibits both directional and nondirectional basis functions we demonstrate the potential of the proposed transforms using nonlinear approximation in addition we employ the proposed family in two key image processing applications image coding and denoising and show its efficiency for these applications

a hierarchy of near optimal policies for multistage adaptive optimization in this paper we propose a new tractable framework for dealing with linear dynamical systems affected by uncertainty applicable to multistage robust optimization and stochastic programming we introduce a hierarchy of near optimal polynomial disturbance feedback control policies and show how these can be computed by solving a single semidefinite programming problem the approach yields a hierarchy parameterized by a single variable the degree of the polynomial policies which controls the trade off between the optimality gap and the computational requirements we evaluate our framework in the context of three classical applications two in inventory management and one in robust regulation of an active suspension system in which very strong numerical performance is exhibited at relatively modest computational expense

ranging through gabor logons a consistent hierarchical approach in this work the correspondence problem in stereo vision is handled by matching two sets of dense feature vectors inspired by biological evidence these feature vectors are generated by a correlation between a bank of gabor sensors and the intensity image the sensors consist of two dimensional gabor filters at various scales spatial frequencies and orientations which bear close resemblance to the receptive field profiles of simple v1 cells in visual cortex a hierarchical stochastic relaxation method is then used to obtain the dense stereo disparities unlike traditional hierarchical methods for stereo feature based hierarchical processing yields consistent disparities to avoid false matchings due to static occlusion a dual matching based on the imaging geometry is used

image enhancement based on a nonlinear multiscale method an image enhancement method that reduces speckle noise and preserves edges is introduced the method is based on a new nonlinear multiscale reconstruction scheme that is obtained by successively combining each coarser scale image with the corresponding modified interscale image simulation results are included to demonstrate the performance of the proposed method

a general representation theorem for partially ordered commutative rings an extension of the kadison dubois representation theorem is proved this extends both the classical version 3 and the preordering version given by jacobi in 5 it is then shown how this can be used to sharpen the results on representations of strictly positive polynomials given by jacobi and prestel in 6

detail enhanced exposure fusion in a typical processing chain of image enhancement an exposure fusion scheme can be used to synthesize a more detailed low dynamic range ldr image directly from a set of differently exposed ldr images without generation of an intermediate high dynamic range image in this brief we introduce a new quadratic optimization based method to extract fine details from a vector field the new method extracts fine details from a set of differently exposed ldr images simultaneously the extracted fine details are then added to an intermediate ldr image which is fused by simply using an existing exposure fusion scheme with this the proposed scheme can enhance fine details to produce sharper images

optimization of polynomials on compact semialgebraic sets a basic closed semialgebraic subset s of r n is defined by simultaneous polynomial inequalities g 1 ge 0 dotsc g m ge 0 we give a short introduction to lasserre s method for minimizing a polynomial f on a compact set s of this kind it consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs we give a new short proof for the convergence of the optimal values of these relaxations to the infimum f ast of f on s which is constructive and elementary in the case where f possesses a unique minimizer x ast we prove that every sequence of nearly optimal solutions of the successive relaxations gives rise to a sequence of points in r n converging to x ast

on the equivalence of soft wavelet shrinkage total variation diffusion total variation regularization and sides soft wavelet shrinkage total variation tv diffusion tv regularization and a dynamical system called sides are four useful techniques for discontinuity preserving denoising of signals and images in this paper we investigate under which circumstances these methods are equivalent in the one dimensional case first we prove that haar wavelet shrinkage on a single scale is equivalent to a single step of space discrete tv diffusion or regularization of two pixel pairs in the translationally invariant case we show that applying cycle spinning to haar wavelet shrinkage on a single scale can be regarded as an absolutely stable explicit discretization of tv diffusion we prove that space discrete tv diffusion and tv regularization are identical and that they are also equivalent to the sides system when a specific force function is chosen afterwards we show that wavelet shrinkage on multiple scales can be regarded as a single step diffusion filtering or regularization of the laplacian pyramid of the signal we analyze possibilities to avoid gibbs like artifacts for multiscale haar wavelet shrinkage by scaling the thresholds finally we present experiments where hybrid methods are designed that combine the advantages of wavelets and pde variational approaches these methods are based on iterated shift invariant wavelet shrinkage at multiple scales with scaled thresholds

sum of squares approximation of polynomials nonnegative on a real algebraic set wih every real polynomial f we associate a family f epsilon r epsilon r of real polynomials in explicit form in terms of f and the parameters epsilon 0 r in n and such that vert f f epsilon r vert 1 to 0 as epsilon to 0 let v subset r n be a real algebraic set described by finitely many polynomials equations g j x 0 j in j and let f be a real polynomial nonnegative on v we show that for every epsilon 0 there exist nonnegative scalars lambda j epsilon j in j such that for all r sufficiently large f epsilon r sum j in j lambda j epsilon g j 2 quad mbox is a sum of squares this representation is an obvious certificate of nonnegativity of f epsilon r on v and very specific in terms of the g j that define the set v in particular it is valid with no assumption on v in addition this representation is also useful from a computation point of view as we can define semidefinite programming relaxations to approximate the global minimum of f on a real algebraic set v or a semialgebraic set k and again with no assumption on v or k

a sum of squares approximation of nonnegative polynomials we show that every real nonnegative polynomial f can be approximated as closely as desired in the l 1 norm of its coefficient vector by a sequence of polynomials f epsilon that are sums of squares the novelty is that each f epsilon has a simple and explicit form in terms of f and epsilon

wavelet transforms associated with finite cyclic groups multiresolution analysis via decomposition on wavelet bases has emerged as an important tool in the analysis of signals and images when these objects are viewed as sequences of complex or real numbers an important class of multiresolution decompositions are the laplacian pyramid schemes in which the resolution is successively halved by recursively low pass filtering the signal under analysis and decimating it by a factor of two in general the principal framework within which multiresolution techniques have been studied and applied is the same as that used in the discrete time fourier analysis of sequences of complex numbers an analogous framework is developed for the multiresolution analysis of finite length sequences of elements from arbitrary fields attention is restricted to sequences of length 2 sup n for n a positive integer so that the resolution may be recursively halved to completion as in finite length fourier analysis a cyclic group structure of the index set of such sequences is exploited to characterize the transforms of interest for the particular cases of complex and finite fields

image representations using multiscale differential operators differential operators have been widely used for multiscale geometric descriptions of images the efficient computation of these differential operators is always desirable moreover it has not been clear whether such representations are invertible for certain applications it is usually required that such representations should be invertible so that one can facilitate the processing of information in the transform domain and then recover it in this paper such problems are studied we consider multiscale differential representations of images using different types of operators such as the directional derivative operators and laplacian operators in particular we provide a general approach to represent images by their multiscale and multidirectional derivative components for practical implementation efficient pyramid like algorithms are derived using the spline technique for both the the composition and reconstruction of images it is shown that using these representations various meaningful geometric information of images can be extracted at multiple scales therefore these representations can be used for edge based image processing purposes furthermore the intrinsic relationships of the proposed representations with the compact wavelet models and some classical multiscale approaches are also elucidated in the paper

a strict positivstellensatz for the weyl algebra let c be an element of the weyl algebra open image in new window which is given by a strictly positive operator in the schrodinger representation it is shown that under some conditions there exist certain elements b1 bd from open image in new window such that dj 1bjcb j is a finite sum of squares

moment problems for operator polynomials abstract haviland s theorem states that given a closed subset k in r n each functional l r x r positive on pos k p r x p k 0 admits an integral representation by a positive borel measure schmudgen proved that in the case of compact semialgebraic set k it suffices to check positivity of l on a preordering t having k as the non negativity set further he showed that the compactness of k is equivalent to the archimedianity of t the aim of this paper is to extend these results from functionals on the usual real polynomials to operators mapping from the real matrix or operator polynomials into r m n r or b k

semidefinite representations for finite varieties we consider the problem of minimizing a polynomial over a set defined by polynomial equations and inequalities when the polynomial equations have a finite set of complex solutions we can reformulate this problem as a semidefinite programming problem our semidefinite representation involves combinatorial moment matrices which are matrices indexed by a basis of the quotient vector space x 1 xn i where i is the ideal generated by the polynomial equations in the problem moreover we prove the finite convergence of a hierarchy of semidefinite relaxations introduced by lasserre semidefinite approximations can be constructed by considering truncated combinatorial moment matrices rank conditions are given in a grid case that ensure that the approximation solves the original problem to optimality

a space time spectral method for the time fractional diffusion equation in this paper we consider the numerical solution of the time fractional diffusion equation essentially the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term in the former case the first order time derivative is replaced by a fractional derivative making the problem global in time we propose a spectral method in both temporal and spatial discretizations for this equation the convergence of the method is proven by providing a priori error estimate numerical tests are carried out to confirm the theoretical results thanks to the spectral accuracy in both space and time of the proposed method the storage requirement due to the global time dependence can be considerably relaxed and therefore calculation of the long time solution becomes possible

polynomials nonnegative on a grid and discrete optimization we characterize the real valued polynomials on r n that are nonnegative not necessarily strictly positive on a grid k of points of r n in terms of a weighted sum of squares whose degree is bounded and known in advance we also show that the mimimization of an arbitrary polynomial on k a discrete optimization problem reduces to a convex continuous optimization problem of fixed size the case of concave polynomials is also investigated the proof is based on a recent result of curto and fialkow on the k moment problem

multiscale hybrid linear models for lossy image representation in this paper we introduce a simple and efficient representation for natural images we view an image in either the spatial domain or the wavelet domain as a collection of vectors in a high dimensional space we then fit a piece wise linear model i e a union of affine subspaces to the vectors at each downsampling scale we call this a multiscale hybrid linear model for the image the model can be effectively estimated via a new algebraic method known as generalized principal component analysis gpca the hybrid and hierarchical structure of this model allows us to effectively extract and exploit multimodal correlations among the imagery data at different scales it conceptually and computationally remedies limitations of many existing image representation methods that are based on either a fixed linear transformation e g dct wavelets or an adaptive uni modal linear transformation e g pca or a multimodal model that uses only cluster means e g vq we will justify both quantitatively and experimentally why and how such a simple multiscale hybrid model is able to reduce simultaneously the model complexity and computational cost despite a small overhead of the model our careful and extensive experimental results show that this new model gives more compact representations for a wide variety of natural images under a wide range of signal to noise ratios than many existing methods including wavelets we also briefly address how the same hybrid linear modeling paradigm can be extended to be potentially useful for other applications such as image segmentation

optimal pyramidal and subband decompositions for hierarchical coding of noisy and quantized images optimal hierarchical coding is sought for progressive or scalable image transmission by minimizing the variance of the error difference between the original image and its lower resolution renditions the optimal according to the above criterion pyramidal and subband image coders are determined for images subject to corruption by quantization or transmission noise given arbitrary analysis filters and assuming adequate knowledge of the noise statistics optimal synthesis filters are found the optimal analysis filters are subsequently determined leading to formulas for globally optimal structures for pyramidal and subband image decompositions experimental results illustrate the implementation and performance of the optimal coders

b spline snakes a flexible tool for parametric contour detection we present a novel formulation for b spline snakes that can be used as a tool for fast and intuitive contour outlining we start with a theoretical argument in favor of splines in the traditional formulation by showing that the optimal curvature constrained snake is a cubic spline irrespective of the form of the external energy field unfortunately such regularized snakes suffer from slow convergence speed because of a large number of control points as well as from difficulties in determining the weight factors associated to the internal energies of the curve we therefore propose an alternative formulation in which the intrinsic scale of the spline model is adjusted a priori this leads to a reduction of the number of parameters to be optimized and eliminates the need for internal energies i e the regularization term in other words we are now controlling the elasticity of the spline implicitly and rather intuitively by varying the spacing between the spline knots the theory is embedded into a multiresolution formulation demonstrating improved stability in noisy image environments validation results are presented comparing the traditional snake using internal energies and the proposed approach without internal energies showing the similar performance of the latter several biomedical examples of applications are included to illustrate the versatility of the method

spectral direction splitting methods for two dimensional space fractional diffusion equations a numerical method for a kind of time dependent two dimensional two sided space fractional diffusion equations is developed in this paper the proposed method combines a time scheme based on direction splitting approaches and a spectral method for the spatial discretization the direction splitting approach renders the underlying two dimensional equation into a set of one dimensional space fractional diffusion equations at each time step then these one dimensional equations are solved by using the spectral method based on weak formulations a time error estimate is derived for the semi discrete solution and the unconditional stability of the fully discretized scheme is proved some numerical examples are presented to validate the proposed method

stereo correspondence by surface reconstruction an algorithm that solves the computational stereo correspondence and the surface reconstruction is presented the algorithm integrates the reconstruction process in the correspondence analysis by means of multipass attribute matching and disparity refinement in the matching process the requirement of attribute similarity is relaxed with the pass number while the requirement for agreement between the predicted and the measured disparity is tightened disparity discontinuities and occluded areas are detected by analyzing the partial derivatives of the reconstructed disparity surface results on synthetic and on real stereo image pairs are reported

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

anomalous diffusion as modeled by a nonstationary extension of brownian motion if the mean square displacement of a stochastic process is proportional to t ensuremath beta ensuremath beta ensuremath ne 1 then it is said to be anomalous we construct a family of markovian stochastic processes with independent nonstationary increments and arbitrary but a priori specified mean square displacement we label the family as an extended brownian motion and show that they satisfy a langevin equation with time dependent diffusion coefficient if the time derivative of the variance of the process is homogeneous then by computing the fractal dimension it can be shown that the complexity of the family is the same as that of the brownian motion for two particles initially separated by a distance x the finite size lyapunov exponent fsle measures the average rate of exponential separation to a distance ax an analytical expression is developed for the fsles of the extended brownian processes and numerical examples presented the explicit construction of these processes illustrates that contrary to what has been stated in the literature a power law mean square displacement is not necessarily related to a breakdown in the classical central limit theorem clt caused by for example correlation fractional brownian motion or correlated continuous time random walk schemes or infinite variance levy motion the classical clt coupled with nonstationary increments can and often does give rise to power law moments such as the mean square displacement

significance linked connected component analysis for wavelet image coding the success in wavelet image coding is mainly attributed to a recognition of the importance of data organization and representation there have been several very competitive wavelet coders developed namely shapiro s 1993 embedded zerotree wavelets ezw servetto et al s 1995 morphological representation of wavelet data mrwd and said and pearlman s see ieee trans circuits syst video technol vol 6 p 245 50 1996 set partitioning in hierarchical trees spiht we develop a novel wavelet image coder called significance linked connected component analysis slcca of wavelet coefficients that extends mrwd by exploiting both within subband clustering of significant coefficients and cross subband dependency in significant fields extensive computer experiments on both natural and texture images show convincingly that the proposed slcca outperforms ezw mrwd and spiht for example for the barbara image at 0 25 b pixel slcca outperforms ezw mrwd and spiht by 1 41 db 0 32 db and 0 60 db in psnr respectively it is also observed that slcca works extremely well for images with a large portion of texture for eight typical 256 spl times 256 grayscale texture images compressed at 0 40 b pixel slcca outperforms spiht by 0 16 db 0 63 db in psnr this performance is achieved without using any optimal bit allocation procedure thus both the encoding and decoding procedures are fast

the k moment problem with densities given a compact basic semi algebraic set mathbf k subset mathbb r n a rational fraction f mathbb r n to mathbb r and a sequence of scalars y y we investigate when y alpha int mathbf k x alpha f d mu holds for all alpha in mathbb n n i e when y is the moment sequence of some measure fd supported on k this yields a set of convex linear matrix inequalities lmi we also use semidefinite programming to develop a converging approximation scheme to evaluate the integral fd when the moments of are known and f is a given rational fraction numerical expreriments are also provided we finally provide again lmi conditions on the moments of two measures nu mu with support contained in k to have d nu f d mu for some rational fraction f

positivity sums of squares and the multi dimensional moment problem let k be the basic closed semi algebraic set in r n defined by some finite set of polynomials s and t the preordering generated by s for k compact f a polynomial in n variables nonnegative on k and real e 0 we have that f e t in particular the k moment problem has a positive solution in the present paper we study the problem when k is not compact for n 1 we show that the k moment problem has a positive solution if and only if s is the natural description of k see section 1 for n 2 we show that the k moment problem fails if k contains a cone of dimension 2 on the other hand we show that if k is a cylinder with compact base then the following property holds f r x f 0 on k q t such that real 0 f eq t this property is strictly weaker than the one given in schmudgen 1991 but in turn it implies a positive solution to the k moment problem using results of marshall 2001 we provide many noncompact examples in hypersurfaces for which holds finally we provide a list of 8 open problems

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

nonlinear image filtering with edge and corner enhancement the authors describe a model of nonlinear image filtering for noise reduction and edge enhancement using anisotropic diffusion the method is designed to enhance not only edges but corners and t junctions as well the method roughly corresponds to a nonlinear diffusion process with backward heat flow across the strong edges such a process is ill posed making the results depend strongly on how the algorithm differs from the diffusion process two ways of modifying the equations using simulations on a variable grid are studied

local adaptivity to variable smoothness for exemplar based image regularization and representation a novel adaptive and exemplar based approach is proposed for image restoration denoising and representation the method is based on a pointwise selection of similar image patches of fixed size in the variable neighborhood of each pixel the main idea is to associate with each pixel the weighted sum of data points within an adaptive neighborhood we use small image patches e g 7 7 or 9 9 patches to compute these weights since they are able to capture local geometric patterns and texels seen in images in this paper we mainly focus on the problem of adaptive neighborhood selection in a manner that balances the accuracy of approximation and the stochastic error at each spatial position the proposed pointwise estimator is then iterative and automatically adapts to the degree of underlying smoothness with minimal a priori assumptions on the function to be recovered the method is applied to artificially corrupted real images and the performance is very close and in some cases even surpasses to that of the already published denoising methods the proposed algorithm is demonstrated on real images corrupted by non gaussian noise and is used for applications in bio imaging

sos approximations of nonnegative polynomials via simple high degree perturbations we show that every real polynomial f nonnegative on 1 1 n can be approximated in the l1 norm of coefficients by a sequence of polynomials f epsilon r that are sums of squares s o s this complements the existence of s o s approximations in the denseness result of berg christensen and ressel as we provide a very simple and explicit approximation sequence then we show that if the moment problem holds for a basic closed semi algebraic set k s subset mathbb r n with nonempty interior then every polynomial nonnegative on ks can be approximated in a similar fashion by elements from the corresponding preordering finally we show that the degree of the perturbation in the approximating sequence depends on epsilon as well as the degree and the size of coefficients of the nonnegative polynomial f but not on the specific values of its coefficients

distinguished representations of non negative polynomials abstract let g 1 g r r x 1 x n such that the set k g 1 0 g r 0 in r n is compact we study the problem of representing polynomials f with f k 0 in the form f s 0 s 1 g 1 s r g r with sums of squares s i with particular emphasis on the case where f has zeros in k assuming that the quadratic module of all such sums is archimedean we establish a local global condition for f to have such a representation vis a vis the zero set of f in k this criterion is most useful when f has only finitely many zeros in k we present a number of concrete situations where this result can be applied as another application we solve an open problem from s kuhlmann et al positivity sums of squares and the multi dimensional moment problem ii adv geometry in press on one dimensional quadratic modules

a representation theorem for certain partially ordered commutative rings let a be a commutative ring with 1 let p subset a be a preordering of higher level i e 0 1 in p 1 not in p p p subset p p cdot p subset p and a 2n subset p for some n in n and let m subset a be an archimedean p module i e 1 in m 1 not in m m m subset m p cdot m subset m and forall a in a exists n in n n a in m we endow x m varphi in rm hom a r mid varphi m subset r with the weak topology with respect to all mappings widehat a x m rightarrow r widehat a varphi varphi a and consider the representation phi m a rightarrow cal c x m r a mapsto widehat a we find that x m is a non empty compact hausdorff space further we prove that

semidefinite programming vs lp relaxations for polynomial programming we consider the global minimization of a multivariate polynomial on a semi algebraic set o defined with polynomial inequalities we then compare two hierarchies of relaxations namely lp relaxations based on products of the original constraints in the spirit of the rlt procedure of sherali and adams 1990 and recent semidefinite programming sdp relaxations introduced by the author the comparison is analyzed in light of recent results in real algebraic geometry on various representations of polynomials positive on a compact semi algebraic set

a semidefinite programming approach to the generalized problem of moments we consider the generalized problem of moments gpm from a computational point of view and provide a hierarchy of semidefinite programming relaxations whose sequence of optimal values converges to the optimal value of the gpm we then investigate in detail various examples of applications in optimization probability financial economics and optimal control which all can be viewed as particular instances of the gpm

local variance controlled forward and backward diffusion for image enhancement and noise reduction in order to improve signal to noise ratio snr and contrast to noise ratio this paper introduces a local variance controlled forward and backward lvcfab diffusion algorithm for edge enhancement and noise reduction in our algorithm an alternative fab diffusion algorithm is proposed the results for the alternative fab algorithm show better algorithm behavior than other existing diffusion fab approaches furthermore two distinct discontinuity measures and the alternative fab diffusion are incorporated into a lvcfab diffusion algorithm where the joint use of the two measures leads to a complementary effect for preserving edge features in digital images this lvc mechanism adaptively modifies the degree of diffusion at any image location and is dependent on both local gradient and inhomogeneity qualitative experiments based on general digital images and magnetic resonance images show significant improvements when the lvcfab diffusion algorithm is used versus the existing anisotropic diffusion and the previous fab diffusion algorithms for enhancing edge features and improving image contrast quantitative analyses based on peak snr confirm the superiority of the proposed lvcfab diffusion algorithm

adaptive nonlocal filtering a fast alternative to anisotropic diffusion for image enhancement nonlinear anisotropic diffusion algorithms provide significant improvement in image enhancement as compared to linear filters however the excessive computational cost of solving nonlinear pdes precludes their use in real time vision applications we show that two orders of magnitude speed improvement is provided by a new image filtering paradigm in which an adaptively determined vector field specifies nonlocal application points for an image filter

polynomial programming lp relaxations also converge we consider the global minimization of a multivariate polynomial on a compact semialgebraic set using a result of krivine j analyse math 12 1964 pp 307 326 c r acad sci paris 258 1964 pp 3417 3418 it is shown that lp relaxations based on products of the original constraints in the spirit of the reformulation linearization technique procedure of sherali and adams converge to the global optimum this extends to compact semialgebraic sets previous results known for polytopes and 0 1 programs the asymptotic behavior of the lp relaxations is also analyzed and it is shown that in important cases finite convergence cannot occur