analytical approximation of cuspidal loops using a nonlinear time transformation method abstract in this work we consider cuspidal loops i e homoclinic orbits to cuspidal singular points we develop an iterative procedure founded on the nonlinear time transformation method to estimate such codimension three global bifurcations up to any wanted order not only in the space of parameters but also in the phase plane as far as we know this is the first time in the literature that this theoretical result is achieved for these global connections the existence and uniqueness of the perturbed solution obtained are proved to illustrate the effectiveness of the method we study cuspidal loops in two normal forms of degenerate takens bogdanov bifurcations excellent agreement is found between our analytical predictions and the corresponding numerical continuations

approximation using scattered shifts of a multivariate function the approximation of a general d variate function f by the shifts r d of a fixed functionoccurs in many applications such as data fit ting neural networks and learning theory when hz d is a dilate of the integer lattice there is a rather complete understanding of the approximation problem 6 18 using fourier techniques however in most applications the center set is either given or can be chosen with complete freedom in both of these cases the shift invariant setting is too restrictive this paper studies the approximation problem in the caseis arbitrary it establishes approximation theorems whose error bounds reflect the local density of the points in two different settings are analyzed the first is when the setis prescribed in advance in this case the theorems of this paper show that in analogy with the classical univariate spline ap proximation improved approximation occurs in regions where the density is high the second setting corresponds to the problem of non linear approximation in that setting the setcan be chosen using information about the target function f we discuss how to best make these choices and give estimates for the approximation error

modified adaptive algorithms it is well known that the adaptive algorithm is simple and easy to program but the results are not fully competitive with other nonlinear methods such as free knot spline approximation we modify the algorithm to take full advantages of nonlinear approximation the new algorithms have the same approximation order as other nonlinear methods which is proved by characterizing their approximation spaces one of our algorithms is implemented on the computer with numerical results illustrated by figures and tables

the cyclicity of period annuli of a class of quintic hamiltonian systems abstract in this paper we prove that the cyclicity of period annuli for system d x d t y d y d t a x b x 3 c x 5 under perturbations of polynomials with degree n is not more than 54 n 13 taking into account the multiplicity where a b c r with c 0 such that the unperturbed systems have at least one center

sobolev bounds on functions with scattered zeros with applications to radial basis function surface fitting in this paper we discuss sobolev bounds on functions that vanish at scattered points in a bounded lipschitz domain that satisfies a uniform interior cone condition the sobolev spaces involved may have fractional as well as integer order we then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions rbfs these estimates include situations in which the target function does not belong to the native space of the rbf

besov type spaces of variable smoothness on rough domains abstract the paper puts forward new besov spaces of variable smoothness b p q 0 g t k and b p q r l t k on rough domains a domain g is either a bounded lipschitz domain in r n or the epigraph of a lipschitz function a domain is an e domain these spaces are shown to be the traces of the spaces b p q 0 r n t k and b p q r l r n t k on domains g and respectively the extension operator ext 1 b p q 0 g t k b p q 0 r n t k is linear the operator ext 2 b p q r l t k b p q r l r n t k is nonlinear as a corollary an exact description of the traces of 2 microlocal besov type spaces and weighted besov type spaces on rough domains is obtained

directional short time fourier transform abstract a directionally sensitive variant of the short time fourier transform is introduced which sends functions on r n to those on the parameter space s n 1 r r n this transform which is named directional short time fourier transform dstft uses functions in l r as window and is related to the celebrated radon transform we establish an orthogonality relation for the dstft and explore some operator theoretic aspects of the transform mostly in terms of proving a variant of the hausdorff young inequality the paper is concluded by some reconstruction formulas

design of steerable wavelets to detect multifold junctions we propose a framework for the detection of junctions in images although the detection of edges and key points is a well examined and described area the multiscale detection of junction centers especially for odd orders poses a challenge in pattern analysis the goal of this paper is to build optimal junction detectors based on 2d steerable wavelets that are polar separable in the fourier domain the approaches we develop are general and can be used for the detection of arbitrary symmetric and asymmetric junctions the backbone of our construction is a multiscale pyramid with a radial wavelet function where the directional components are represented by circular harmonics and encoded in a shaping matrix we are able to detect m fold junctions in different scales and orientations we provide experimental results on both simulated and real data to demonstrate the effectiveness of the algorithm

new tight frames of curvelets and optimal representations of objects with piecewise c2 singularities this paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise c 2 edges conceptually the curvelet transform is a multiscale pyramid with many directions and positions at each length scale and needle shaped elements at fine scales these elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets for instance curvelets obey a parabolic scaling relation which says that at scale 2 j each element has an envelope that is aligned along a ridge of length 2 j 2 and width 2 j we prove that curvelets provide an essentially optimal representation of typical objects f that are c 2 except for discontinuities along piecewise c 2 curves such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object for instance the n term partial reconstruction f c n obtained by selecting the n largest terms in the curvelet series obeys f f c n 2 l2 c n 2 log n 3 n this rate of convergence holds uniformly over a class of functions that are c 2 except for discontinuities along piecewise c 2 curves and is essentially optimal in comparison the squared error of n term wavelet approximations only converges as n 1 as n which is considerably worse than the optimal behavior

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 spatial version of the ito stratonovich correction we consider a class of stochastic pdes of burgers type in spatial dimension 1 driven by space time white noise even though it is well known that these equations are well posed it turns out that if one performs a spatial discretization of the nonlinearity in the wrong way then the sequence of approximate equations does converge to a limit but this limit exhibits an additional correction term this correction term is proportional to the local quadratic cross variation in space of the gradient of the conserved quantity with the solution itself this can be understood as a consequence of the fact that for any fixed time the law of the solution is locally equivalent to wiener measure where space plays the role of time in this sense the correction term is similar to the usual ito stratonovich correction term that arises when one considers different temporal discretizations of stochastic odes

nonlocal hierarchical dictionary learning using wavelets for image denoising exploiting the sparsity within representation models for images is critical for image denoising the best currently available denoising methods take advantage of the sparsity from image self similarity pre learned and fixed representations most of these methods however still have difficulties in tackling high noise levels or noise models other than gaussian in this paper the multiresolution structure and sparsity of wavelets are employed by nonlocal dictionary learning in each decomposition level of the wavelets experimental results show that our proposed method outperforms two state of the art image denoising algorithms on higher noise levels furthermore our approach is more adaptive to the less extensively researched uniform noise

perturbation from an elliptic hamiltonian of degree four iii global centre abstract the paper deals with lienard equations of the form x y y p x yq x with p and q polynomials of degree respectively 3 and 2 attention goes to perturbations of the hamiltonian vector fields with an elliptic hamiltonian of degree four exhibiting a global centre it is proven that the least upper bound of the number of zeros of the related elliptic integral is four and this is a sharp one this result permits to prove the existence of lienard equations of type 3 2 with a quadruple limit cycle with both a triple and a simple limit cycle with two semistable limit cycles with one semistable and two simple limit cycles or with four simple limit cycles

representation complexity of adaptive 3d distance fields in this paper we study the representation complexity of a kind of data structure that stores the information necessary to compute the distance from a point to a geometric body these data structures called adaptive splitting based on cubature distance fields asbcdf are binary search trees generated by the adaptive splitting based on cubature asbc algorithm that adaptively subdivides the space surrounding the body into tetrahedra their representation complexity is measured by the number of nodes in the tree two times the number of tetrahedra in the resulting tessellation in the case of convex polyhedra we prove that this quantity remains bounded as the number of vertices of the polyhedra increases to infinity experimental results show that the number of tetrahedra in the tessellations is almost independent of the combinatorial complexity of the polyhedra this means that the average compute time of the distance to arbitrary convex polyhedra is almost constant therefore asbcdfs are especially suitable for real time applications involving rapidly changing environments modelized with complex polyhedra

perturbations from an elliptic hamiltonian of degree four abstract the paper deals with lienard equations of the form x y y p x yq x with p and q polynomials of degree respectively 3 and 2 attention goes to perturbations of the hamiltonian vector field with an elliptic hamiltonian of degree 4 exhibiting a cuspidal loop it is proven that the least upper bound for the number of zeros of the related elliptic integral is four and this upper bound is a sharp one this permits to prove the existence of lienard equations of type 3 2 with at least four limit cycles the paper also contains a complete result on the respective number of small and large limit cycles

continuity properties of solutions to some degenerate elliptic equations abstract we consider a nonlinear possibly degenerate elliptic operator l v div a v b x v where the field a and the function b are unnecessarily strictly monotonic and a satisfies a very mild ellipticity assumption for a given boundary datum we prove the existence of the maximum and the minimum of the solutions and formulate a haar rado type result namely a continuity property for these solutions that may follow from the continuity of in the homogeneous case we formulate some generalizations of the bounded slope condition and use them to obtain the lipschitz or local lipschitz regularity of solutions to l u 0 we prove the global holder regularity of the solutions in the case where is lipschitz

multiscale analysis in sobolev spaces on bounded domains we study a multiscale scheme for the approximation of sobolev functions on bounded domains our method employs scattered data sites and compactly supported radial basis functions of varying support radii at scattered data sites the actual multiscale approximation is constructed by a sequence of residual corrections where different support radii are employed to accommodate different scales convergence theorems for the scheme are proven and it is shown that the condition numbers of the linear systems at each level are independent of the level thereby establishing for the first time a mathematical theory for multiscale approximation with scaled versions of a single compactly supported radial basis function at scattered data points on a bounded domain

embeddings for a weakly differentiable functions on domains abstract we prove that the inhomogeneous estimate of vector fields on balls in r n b d k 1 u n n 1 d x n 1 n c b a u u d x for all u c b r n holds if and only if the linear constant coefficient differential operator a of order k has finite dimensional null space fdn this generalizes the gagliardo nirenberg sobolev inequality on domains and provides the local version of the analogous homogeneous embedding in full space r n d k 1 u n n 1 d x n 1 n c r n a u d x for all u c c r n r n proved by van schaftingen precisely for elliptic and cancelling ec operators building on fundamental l 1 estimates from the works of bourgain and brezis we prove that fdn strictly implies ec and discuss the contrast between homogeneous and inhomogeneous estimates on both algebraic and analytic level

bandlimited shearlet type frames with nice duals the present paper constructs a frame dual frame pair of shearlet type such that both frames possess the distinctive time frequency localization properties needed in establishing their desirable approximation properties our construction is based on a careful pasting together of two bandlimited shearlet parseval frames associated with two different frequency cones inspired by domain decomposition methods used primarily for the solution of pdes

wave atoms and time upscaling of wave equations we present a new geometric strategy for the numerical solution of hyperbolic wave equations in smoothly varying two dimensional time independent periodic media the method consists in representing the time dependent green s function in wave atoms a tight frame of multiscale directional wave packets obeying a precise parabolic balance between oscillations and support size namely wavelength diameter 2 wave atoms offer a uniquely structured representation of the green s function in the sense that the resulting matrix is universally sparse over the class of c coefficients even for large times the matrix has a natural low rank block structure after separation of the spatial indices the parabolic scaling is essential for these properties to hold as a result it becomes realistic to accurately build the full matrix exponential in the wave atom frame using repeated squaring up to some time typically of the form delta t sim sqrt delta x which is bigger than the standard cfl timestep once the expensive precomputation of the green s function has been carried out it can be used to perform unusually large upscaled cheap time steps the algorithm is relatively simple in that it does not require an underlying geometric optics solver we prove accuracy and complexity results based on a priori estimates of sparsity and separation ranks on a n by n grid the expensive precomputation takes somewhere between o n 3log n and o n 4log n steps depending on the separability of the acoustic medium the complexity of upscaled timestepping however beats the o n 3log n bottleneck of pseudospectral methods on an n by n grid for a wide range of physically relevant situations in particular we show that a naive version of the wave atom algorithm provably runs in o n 2 operations for arbitrarily small but for the final algorithm we had to slightly increase the exponent in order to reduce the large constant as a result we get estimates between o n 2 5 log n and o n 3 log n for upscaled timestepping we also show several numerical examples in practice the current wave atom solver becomes competitive over a pseudospectral method in regimes where the wave equation should be solved hundreds of times with different initial conditions as in reflection seismology in academic examples of accurate propagation of bandlimited wavefronts if the precomputation step is factored out then the wave atom solver is indeed faster than a pseudospectral method by a factor of about 3 5 at n 512 and a factor 10 20 at n 1024 for the same accuracy very similar gains are obtained in comparison versus a finite difference method

extending sobolev functions with partially vanishing traces from locally e δ domains and applications to mixed boundary problems we prove that given any k n for each open set rn and any closed subset d of such that is locally an e domain near d there exists a linear and bounded extension operator ek d mapping for each p 1 the space wdk p into wdk p rn here with o denoting either or rn the space wdk p o is defined as the completion in the classical sobolev space wk p o of restrictions to o of functions from cc rn whose supports are disjoint from d in turn this result is used to develop a functional analytic theory for the class wdk p including intrinsic characterizations boundary traces and extensions results interpolation theorems among other things which is then employed in the treatment of mixed boundary value problems formulated in locally e domains finally we also prove extension results on the scales of besov and bessel potential spaces on e domains with partially vanishing traces on ahlfors regular sets and explore some of the implications of such extension results

adaptive finite element methods with convergence rates adaptive finite element methods for numerically solving elliptic equations are used often in practice only recently 12 17 have these methods been shown to converge however this convergence analysis says nothing about the rates of convergence of these methods and therefore does in principle not guarantee yet any numerical advantages of adaptive strategies versus non adaptive strategies the present paper modifies the adaptive method of morin nochetto and siebert 17 for solving the laplace equation with piecewise linear elements on domains in 2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm which is equivalent to the h1 norm namely it is shown that whenever s 0 and the solution u is such that for each n 1 it can be approximated to accuracy o n s in the energy norm by a continuous piecewise linear function on a triangulation with n cells using complete knowledge of u then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process moreover the number of arithmetic computations in the proposed method is also of order o n for each n 1 the construction and analysis of this adaptive method relies on the theory of nonlinear approximation

two dimensional nonlinear boundary value problems for elliptic equations on considere la regularite limite des solutions du probleme aux valeurs limites non lineaire f x u du d 2 u 0 dans g x u du 0 sur pour des domaines a 2 dimensions f est uniformement elliptique et g depend de du on demontre des estimations de continuite pour des derivees premieres et secondes de u sous des hypotheses faibles pour la regularite de f g et

wedgelets nearly minimax estimation of edges we study a simple horizon model for the problem of recovering an image from noisy data in this model the image has an edge with fi holder regularity adopting the viewpoint of computational harmonic analysis we develop an overcomplete collection of atoms called wedgelets dyadically organized indicator functions with a variety of locations scales and orientations the wedgelet representation provides nearly optimal representations of objects in the horizon model as measured by minimax description length we show how to rapidly compute a wedgelet approximation to noisy data by flnding a special edgelet decorated recursive partition which minimizes a complexity penalized sum of squares this estimate using su cient sub pixel resolution achieves nearly the minimax mean squared error in the horizon model in fact the method is adaptive in the sense that it achieves nearly the minimax risk for any value of the unknown degree of regularity of the horizon 1 fi 2 wedgelet analysis and de noising may be used successfully outside the horizon model we study images modelled as indicators of star shaped sets with smooth bound aries and show that complexity penalized wedgelet partitioning achieves nearly the minimax risk in that setting also

compressed sensing image restoration based on data driven multi scale tight frame it has been shown that redundant signal representation e g tight frame plays important role in compressed sensing image restoration in order to get a good sparse representation one has made enduring efforts to pursue tight frames although there are some tight frames under which a type of images has a good sparse approximation another type of images may not have sparse approximation because of the images great difference in structure this paper presents a novel compressed sensing image restoration method based on data driven multi scale tight frame this method derives a discrete multi scale tight frame system adaptive to the original image from the input compressed sensing image such an adaptive tight frame construction scheme is applied to compressed sensing image restoration the experimental results show our approach s efficiency

analysis of moving least squares approximation revisited in this article the error estimation of the moving least squares approximation is provided for functions in fractional order sobolev spaces the analysis presented in this paper extends the previous estimations and explains some unnoticed mathematical details an application to galerkin method for partial differential equations is also supplied

the curvelet transform for image denoising we describe approximate digital implementations of two new mathematical transforms namely the ridgelet transform and the curvelet transform our implementations offer exact reconstruction stability against perturbations ease of implementation and low computational complexity a central tool is fourier domain computation of an approximate digital radon transform we introduce a very simple interpolation in the fourier space which takes cartesian samples and yields samples on a rectopolar grid which is a pseudo polar sampling set based on a concentric squares geometry despite the crudeness of our interpolation the visual performance is surprisingly good our ridgelet transform applies to the radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain our curvelet transform uses our ridgelet transform as a component step and implements curvelet subbands using a filter bank of a spl grave trous wavelet filters our philosophy throughout is that transforms should be overcomplete rather than critically sampled we apply these digital transforms to the denoising of some standard images embedded in white noise in the tests reported here simple thresholding of the curvelet coefficients is very competitive with state of the art techniques based on wavelets including thresholding of decimated or undecimated wavelet transforms and also including tree based bayesian posterior mean methods moreover the curvelet reconstructions exhibit higher perceptual quality than wavelet based reconstructions offering visually sharper images and in particular higher quality recovery of edges and of faint linear and curvilinear features existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems the empirical results reported here are in encouraging agreement

the divergence equation in rough spaces abstract we aim at extending the existence theory for the equation div v f in a bounded or exterior domain with homogeneous dirichlet boundary conditions to a class of solutions which need not have a trace at the boundary typically the weak solutions that we shall consider will belong to some besov space b p q s with s 1 1 p 1 p after generalizing the notion of a solution for this equation we propose an explicit construction by means of the classical bogovski formula this construction enables us to keep track of a marginal information about the trace of solutions in particular it ensures that the trace is zero if f is smooth enough we expect our approach to be of interest for the study of rough solutions to systems of fluid mechanics

shearlet based total variation diffusion for denoising we propose a shearlet formulation of the total variation tv method for denoising images shearlets have been mathematically proven to represent distributed discontinuities such as edges better than traditional wavelets and are a suitable tool for edge characterization common approaches in combining wavelet like representations such as curvelets with tv or diffusion methods aim at reducing gibbs type artifacts after obtaining a nearly optimal estimate we show that it is possible to obtain much better estimates from a shearlet representation by constraining the residual coefficients using a projected adaptive total variation scheme in the shearlet domain we also analyze the performance of a shearlet based diffusion method numerical examples demonstrate that these schemes are highly effective at denoising complex images and outperform a related method based on the use of the curvelet transform furthermore the shearlet tv scheme requires far fewer iterations than similar competitors

on the importance of combining wavelet based nonlinear approximation with coding strategies this paper provides a mathematical analysis of transform compression in its relationship to linear and nonlinear approximation theory contrasting linear and nonlinear approximation spaces we show that there are interesting classes of functions random processes which are much more compactly represented by wavelet based nonlinear approximation these classes include locally smooth signals that have singularities and provide a model for many signals encountered in practice in particular for images however we also show that nonlinear approximation results do not always translate to efficient compress on strategies in a rate distortion sense based on this observation we construct compression techniques and formulate the family of functions stochastic processes for which they provide efficient descriptions in a rate distortion sense we show that this family invariably leads to besov spaces yielding a natural relationship among besov smoothness linear nonlinear approximation order and compression performance in a rate distortion sense the designed compression techniques show similarities to modern high performance transform codecs allowing us to establish relevant rate distortion estimates and identify performance limits

multiresolution graph fourier transform for compression of piecewise smooth images piecewise smooth pws images e g depth maps or animation images contain unique signal characteristics such as sharp object boundaries and slowly varying interior surfaces leveraging on recent advances in graph signal processing in this paper we propose to compress the pws images using suitable graph fourier transforms gfts to minimize the total signal representation cost of each pixel block considering both the sparsity of the signal s transform coefficients and the compactness of transform description unlike fixed transforms such as the discrete cosine transform we can adapt gft to a particular class of pixel blocks in particular we select one among a defined search space of gfts to minimize total representation cost via our proposed algorithms leveraging on graph optimization techniques such as spectral clustering and minimum graph cuts furthermore for practical implementation of gft we introduce two techniques to reduce computation complexity first at the encoder we low pass filter and downsample a high resolution hr pixel block to obtain a low resolution lr one so that a lr gft can be employed at the decoder upsampling and interpolation are performed adaptively along hr boundaries coded using arithmetic edge coding so that sharp object boundaries can be well preserved second instead of computing gft from a graph in real time via eigen decomposition the most popular lr gfts are pre computed and stored in a table for lookup during encoding and decoding using depth maps and computer graphics images as examples of the pws images experimental results show that our proposed multiresolution gft scheme outperforms h 264 intra by 6 8 db on average in peak signal to noise ratio at the same bit rate

the power of adaption for approximating functions with singularities consider approximating functions based on a finite number of their samples we show that adaptive algorithms are much more powerful than nonadaptive ones when dealing with piecewise smooth functions more specifically let f 1 r be the class of scalar functions f 0 t r whose derivatives of order up to r are continuous at any point except for one unknown singular point we provide an adaptive algorithm a n ad that uses at most n samples of f and whose worst case l p error 1 p oo with respect to reasonable function classes f 1 r f 1 r is proportional to n r on the other hand the worst case error of any nonadaptive algorithm that uses n samples is at best proportional to n 1 p the restriction to only one singularity is necessary for superiority of adaption in the worst case setting fortunately adaption regains its power in the asymptotic setting even for a very general class f r consisting of piecewise c r smooth functions each having a finite number of singular points for any f f r our adaptive algorithm approximates f with error converging to zero at least as fast as n r we also prove that the rate of convergence for nonadaptive methods cannot be better than n 1 p i e is much slower the results mentioned above do not hold if the errors are measured in the l norm since no algorithm produces small l errors for functions with unknown discontinuities however we strongly believe that the l norm is inappropriate when dealing with singular functions and that the skorohod metric should be used instead we show that our adaptive algorithm retains its positive properties when the approximation error is measured in the skorohod metric that is the worst case error with respect to f r 1 equals n r and the convergence in the asymptotic setting for f r is n numerical results confirm the theoretical properties of our algorithms

an optimal geometric condition on domains for boundary differentiability of solutions of elliptic equations abstract in this paper a geometric condition on domains will be given which guarantees the boundary differentiability of solutions of elliptic equations that is the solutions are differentiable at any boundary point we will show that this geometric condition is optimal

on the equivalence of fractional order sobolev semi norms abstract we present various results on the equivalence and mapping properties under affine transformations of fractional order sobolev norms and semi norms of orders between zero and one main results are mutual estimates of the three semi norms of sobolev slobodeckij interpolation and quotient space types in particular we show that the former two are uniformly equivalent under affine mappings that ensure shape regularity of the domains under consideration

perturbation from an elliptic hamiltonian of degree four iv figure eight loop the paper deals with lienard equations of the form x y y p x yq x with p and q polynomials of degree respectively 3 and 2 attention goes to perturbations of the hamiltonian vector fields with an elliptic hamiltonian of degree four exhibiting a figure eight loop it is proved that the least upper bound of the number of zeros of the related elliptic integral is five and this is a sharp bound multiplicity taken into account moreover if restricting to the level curves inside a saddle loop or outside the figure eight loop the sharp upper bound is respectively two or four also the multiplicity of the zeros is at most four this is the last one in a series of papers on this subject the results of this paper together with j differential equations 176 2001 114 j differential equations 175 2001 209 j differential equations to be published largely finish the study of the cubic perturbations of the elliptic hamiltonians of degree four and presumably provide a complete description of the number and the possible configurations of limit cycles for cubic lienard equations with small quadratic damping as a special case we obtain a configuration of four limit cycles surrounding three singularities together with a small limit cycle which surrounds one of the singularities

perturbations of a class of hyper elliptic hamiltonian systems of degree seven with nilpotent singular points in this paper we study the number of isolated zeros of abelian integrals associated to system x y y x 3 x 2 1 2 under the perturbations of 0 1 x 2 2 x 4 3 x 6 y y where 0 1 and i r i 0 1 2 3 we find a special transformation such that four generating elements are reduced to special three ones and then the algebraic method can be used the bounds and sharp bounds are obtained

regularity results for elliptic equations in lipschitz domains abstract we develop a simple variational argument based on the usual nirenberg difference quotient technique to deal with the regularity of the solutions of dirichlet and neumann problems for some linear and quasilinear elliptic equation in lipschitz domains we obtain optimal regularity results in the natural family of sobolev spaces associated with the variational structure of the equations in the linear case we obtain in a completely different way some of the results of d jerison and c e kenig about the laplace equation

interpolation of besov spaces we investigate besov spaces and their connection with dyadic spline approximation in lp q 0 p oo our main results are the determination of the interpolation spaces between a pair of besov spaces an atomic decomposition for functions in a besov space the characterization of the class of functions which have certain prescribed degree of approximation by dyadic splines

on two channel filter banks with directional vanishing moments the contourlet transform was proposed to address the limited directional resolution of the separable wavelet transform one way to guarantee good approximation behavior is to let the directional filters in the contourlet filter bank have sharp frequency response this requires filters with large support size we seek to isolate the key filter property that ensures good approximation in this direction we propose filters with directional vanishing moments dvm these filters we show annihilate information along a given direction we study two channel filter banks with dvm filters we provide conditions under which the design of dvm filter banks is possible a complete characterization of the product filter is thus obtained we propose a design framework that avoids 2 d factorization using the mapping technique the filters designed when used in the contourlet transform exhibit nonlinear approximation comparable to the conventional filters while being shorter and therefore providing better visual quality with less ringing artifacts furthermore experiments show that the proposed filters outperform the conventional ones in image approximation and denoising

nonlinear functionals of wavelet expansions adaptive reconstruction and fast evaluation summary this paper is concerned with the efficient evaluation of nonlinear expressions of wavelet expansions obtained through an adaptive process in particular evaluation covers here the computation of inner products of such expressions with wavelets which arise for instance in the context of galerkin or petrov galerkin schemes for the solution of differential equations the central objective is to develop schemes that facilitate such evaluations at a computational expense exceeding the complexity of the given expansion i e the number of nonzero wavelet coefficients as little as possible the following issues are addressed first motivated by previous treatments of the subject we discuss the type of regularity assumptions that are appropriate in this context and explain the relevance of besov norms the principal strategy is to relate the computation of inner products of wavelets with compositions to approximations of compositions in terms of possibly few dual wavelets the analysis of these approximations finally leads to a concrete evaluation scheme which is shown to be in a certain sense asymptotically optimal we conclude with a simple numerical example

on various approaches to besov type spaces of variable smoothness abstract the paper is concerned with besov spaces of variable smoothness b p q 0 r n t k in which the norms are defined in terms of convolutions with smooth functions a relation is found between the spaces b p q 0 r n t k and the spaces b p q r l r n t k which were introduced earlier by the author as an application an equivalent description of the trace space of the weighted sobolev space with muckenhoupt weight is given

light field reconstruction using shearlet transform in this article we develop an image based rendering technique based on light field reconstruction from a limited set of perspective views acquired by cameras our approach utilizes sparse representation of epipolar plane images epi in shearlet transform domain the shearlet transform has been specifically modified to handle the straight lines characteristic for epi the devised iterative regularization algorithm based on adaptive thresholding provides high quality reconstruction results for relatively big disparities between neighboring views the generated densely sampled light field of a given 3d scene is thus suitable for all applications which require light field reconstruction the proposed algorithm compares favorably against state of the art depth image based rendering techniques and shows superior performance specifically in reconstructing scenes containing semi transparent objects

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

bifurcations of limit cycles for a quintic hamiltonian system with a double cuspidal loop in this work we consider the number of limit cycles that can bifurcate from periodic orbits located inside a double cuspidal loop of the quintic hamiltonian vector field x h y x x 3 x 2 1 y under small perturbations of the form e a bx 2 cx 4 y y where 0 e 1 and a b c are real constants using picard fuchs equations for related abelian integrals asymptotic expansion of these integrals about critical level curves of h and some geometric properties of the curves defined by ratios of two especial integrals we show that the least upper bound for the number of limit cycles appeared in this bifurcation is two

bifurcations of limit cycles from quintic hamiltonian systems with an eye figure loop abstract in this paper we consider lienard equations of the form x y y x 2 x 3 x 5 e x 2 x 4 y where 0 e 1 r 3 and is bounded we prove that the least upper bound for the number of zeros of the related abelian integrals i h h x 2 x 4 y d x is 2 taking into account their multiplicities for h 0 1 6 and this upper bound is a sharp one this implies that the number of limit cycles bifurcated from periodic orbits in the vicinity of the center of the unperturbed system for e 0 inside an eye figure loop is less than or equal to 2

nonlinear approximation and the space bv r2 given a function 2 0 q 0 l 2 and a real number t 0 let u f t infg bv g 11 ll vgig where the infimum is taken over all functions g g bv of bounded variation on this and related extremal problems arise in several areas of mathematics such as interpolation of operators and statistical estimation as well as in digital image processing techniques for finding minimizers g for u f t based on variational calculus and nonlinear partial differential equations have been put forward by several authors dms ro ms cl the main disadvantage of these approaches is that they are numerically intensive on the other hand it is well known that more elementary methods based on wavelet shrinkage solve related extremal problems for example the above problem with bv replaced by the besov space b l i see e g cdll however since bv has no simple description in terms of wavelet coefficients it is not clear that minimizers for u f t can be realized in this way we shall show in this paper that simple methods based on haar thresholding provide near minimizers for u f t our analysis of this extremal problem brings forward many interesting relations between haar decompositions and the space bv 1 introduction nonlinear approximation has recently played an impor tant role in several problems of image processing including compression noise removal and feature extraction we have in mind techniques such as wavelet compression djl wavelet shrinkage or thresholding djkp1 wavelet packets cw and greedy algorithms mz dt there has also been an impressive contribution of techniques based on variational calculus and nonlinear partial dif ferential equations see e g dms ro ms cl especially to the problems of noise removal and image segmentation the common point between these two approaches is their ability to adapt to the composite nature of images edge tex tures and smooth regions should be treated adaptively a requirement which is certainly not fulfilled by the classical linear filtering techniques

exact bound on the number of zeros of abelian integrals for two hyper elliptic hamiltonian systems of degree 4 abstract this paper concerns the exact bound on the number of zeros of abelian integrals associated with two hyper elliptic hamiltonian systems of degree 4 the upper and lower bounds for the two systems have been obtained in several previous works however the sharp bounds are still unknown in this paper we provide a proof to show that the exact bound is 3 for both systems the basic idea of our method is to bound the parameter space in r 3 to obtain two parameter sets which might yield maximal 4 zeros of abelian integrals corresponding to the two sets further we show that the existence of 4 zeros on the two sets is not possible and thus the sharp bound is 3

fusion frames and g frames fusion frames and g frames were considered recently as generalizations of frames in hilbert spaces in this paper we generalize some of the known results in frame theory to fusion frames and g frames we obtain new g frames by considering g frames for its components we also obtain some results about alternate dual g frames excess of g frames and stability of g frames under small perturbations

convergence of the discontinuous finite volume method for elliptic problems with minimal regularity this paper investigates convergence of the discontinuous finite volume method dfvm under minimal regularity assumptions on solutions of second order elliptic boundary value problems conventional analysis requires the solutions to be in sobolev spaces h 1 s s 12 here we assume the solutions are in h 1 s s 0 and employ the techniques developed in gudi 2010 18 20 to derive error estimates in a mesh dependent energy norm and the l 2 norm for dfvm the theoretical estimates are illustrated by numerical results which include problems with corner singularity and intersecting interfaces

recovering edges in ill posed inverse problems optimality of curvelet frames we consider a model problem of recovering a function f x 1 x 2 from noisy radon data the function f to be recovered is assumed smooth apart from a discontinuity along a c 2 curve that is an edge we use the continuum white noise model with noise level varepsilon traditional linear methods for solving such inverse problems behave poorly in the presence of edges qualitatively the reconstructions are blurred near the edges quantitatively they give in our model mean squared errors mses that tend to zero with noise level varepsilon only as o varepsilon 1 2 as varepsilon to 0 a recent innovation nonlinear shrinkage in the wavelet domain visually improves edge sharpness and improves mse convergence to o varepsilon 2 3 however as we show here this rate is not optimal in fact essentially optimal performance is obtained by deploying the recently introduced tight frames of curvelets in this setting curvelets are smooth highly anisotropic elements ideally suited for detecting and synthesizing curved edges to deploy them in the radon setting we construct a curvelet based biorthogonal decomposition of the radon operator and build curvelet shrinkage estimators based on thresholding of the noisy curvelet coefficients in effect the estimator detects edges at certain locations and orientations in the radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain we prove that the curvelet shrinkage can be tuned so that the estimator will attain within logarithmic factors the mse o varepsilon 4 5 as noise level varepsilon to 0 this rate of convergence holds uniformly over a class of functions which are c 2 except for discontinuities along c 2 curves and except for log terms is the minimax rate for that class our approach is an instance of a general strategy which should apply in other inverse problems we sketch a deconvolution example

sharp hodge decompositions maxwell s equations and vector poisson problems on nonsmooth three dimensional riemannian manifolds we solve three basic potential theoretic problems hodge decompositions for vector fields poisson problems for the hodge laplacian and inhomogeneous maxwell equations in arbitrary lipschitz subdomains of a smooth compact three dimensional riemannian manifold in each case we derive sharp estimates on sobolev besov scales and establish integral representation formulas for the solution the proofs rely on tools from harmonic analysis and algebraic topology such as calderon zygmund theory and de rham theory

relay fusion frames and bridging results for fusion frames abstract we examine some connections among the original fusion frames relay operators relay spaces and so called relay fusion frames r fusion frames we show that a necessary condition for a setting to be an r fusion frame is that the initial subspaces can span the whole ambient space we generalize the notion of redundancy of fusion frames to more general r fusion frames to explain whether the constructed r fusion frame is good in a specific sense also we demonstrate how model sensor relay networks with the r fusion frames further we consider certain bridging problem for fusion frames and provide several solutions to this problem some sufficient conditions for a fixed fusion frame to have an associated r fusion frame will be given we indicate that the process of r fusion frame model can be implemented in both positive and negative directions which is consistent with the requirement that subnets have the same type of network topology in practical applications moreover some examples are also given

finite element approximation of spectral acoustic problems on curved domains this paper deals with the finite element approximation of the displacement formulation of the spectral acoustic problem on a curved non convex two dimensional domain convergence and error estimates are proved for raviart thomas elements on a discrete polygonal domain h inline equation in the framework of the abstract spectral approximation theory similar results have been previously proved only for polygonal domains numerical tests confirming the theoretical results are reported

nonlinear regularized reaction diffusion filters for denoising of images with textures denoising is always a challenging problem in natural imaging and geophysical data processing in this paper we consider the denoising of texture images using a nonlinear reaction diffusion equation and directional wavelet frames in our model a curvelet shrinkage is used for regularization of the diffusion process to preserve important features in the diffusion smoothing and a wave atom shrinkage is used as the reaction in order to preserve and enhance interesting oriented textures we derive a digital reaction diffusion filter that lives on graphs and show convergence of the corresponding iteration process experimental results and comparisons show very good performance of the proposed model for texture preserving denoising

inverse inequalities on non quasi uniform meshes and application to the mortar element method we present a range of mesh dependent inequalities for piecewise constant and continuous piecewise linear finite element functions u defined on locally refined shape regular but possibly non quasi uniform meshes these inequalities involve norms of the form h u w s p for positive and negative s and where h is a function which reflects the local mesh diameter in an appropriate way the only global parameter involved is n the total number of degrees of freedom in the finite element space and we avoid estimates involving either the global maximum or minimum mesh diameter our inequalities include new variants of inverse inequalities as well as trace and extension theorems they can be used in several areas of finite element analysis to extend results previously known only for quasi uniform meshes to the locally refined case here we describe applications to i the theory of nonlinear approximation and ii the stability of the mortar element method for locally refined meshes

shiftable multiscale transforms one of the major drawbacks of orthogonal wavelet transforms is their lack of translation invariance the content of wavelet subbands is unstable under translations of the input signal wavelet transforms are also unstable with respect to dilations of the input signal and in two dimensions rotations of the input signal the authors formalize these problems by defining a type of translation invariance called shiftability in the spatial domain shiftability corresponds to a lack of aliasing thus the conditions under which the property holds are specified by the sampling theorem shiftability may also be applied in the context of other domains particularly orientation and scale jointly shiftable transforms that are simultaneously shiftable in more than one domain are explored two examples of jointly shiftable transforms are designed and implemented a 1 d transform that is jointly shiftable in position and scale and a 2 d transform that is jointly shiftable in position and orientation the usefulness of these image representations for scale space analysis stereo disparity measurement and image enhancement is demonstrated

the perturbations of a class of hyper elliptic hamilton systems with a double homoclinic loop through a nilpotent saddle abstract in this paper we consider the upper bounds of a number of isolated zeros of abelian integrals associated to system x y y x 3 x 2 1 under the perturbations of 0 1 x 2 x 2 3 x 3 4 x 4 y y where 0 1 and i r i 0 4 the unperturbed system has a double homoclinic loop with a nilpotent saddle of order 1 the sharp upper bounds are obtained for each of the cases of 1 4 0 1 3 0 2 3 0 and 3 4 0 when abelian integrals are defined in the bounded period annuli

combined curvelet shrinkage and nonlinear anisotropic diffusion in this paper a diffusion based curvelet shrinkage is proposed for discontinuity preserving denoising using a combination of a new tight frame of curvelets with a nonlinear diffusion scheme in order to suppress the pseudo gibbs and curvelet like artifacts the conventional shrinkage results are further processed by a projected total variation diffusion in which only the insignificant curvelet coefficients or high frequency part of the signal are changed by use of a constrained projection numerical experiments from piecewise smooth to textured images show good performances of the proposed method to recover the shape of edges and important detailed components in comparison to some existing methods

a family of stable nonlinear nonseparable multiresolution schemes in 2d multiresolution representations of data are powerful tools in data compression for a proper adaptation to the edges a good strategy is to consider a nonlinear approach thus one needs to control the stability of these representations in this paper 2d multiresolution processing algorithms that ensure this stability are introduced a prescribed accuracy is ensured by these strategies

the discrete shearlet transform a new directional transform and compactly supported shearlet frames it is now widely acknowledged that analyzing the intrinsic geometrical features of the underlying image is essential in many applications including image processing in order to achieve this several directional image representation schemes have been proposed in this paper we develop the discrete shearlet transform dst which provides efficient multiscale directional representation and show that the implementation of the transform is built in the discrete framework based on a multiresolution analysis mra we assess the performance of the dst in image denoising and approximation applications in image approximations our approximation scheme using the dst outperforms the discrete wavelet transform dwt while the computational cost of our scheme is comparable to the dwt also in image denoising the dst compares favorably with other existing transforms in the literature

data compression and harmonic analysis in this paper we review some recent interactions between harmonic analysis and data compression the story goes back of course to shannon s r d theory in the case of gaussian stationary processes which says that transforming into a fourier basis followed by block coding gives an optimal lossy compression technique practical developments like transform based image compression have been inspired by this result in this paper we also discuss connections perhaps less familiar to the information theory community growing out of the field of harmonic analysis recent harmonic analysis constructions such as wavelet transforms and gabor transforms are essentially optimal transforms for transform coding in certain settings some of these transforms are under consideration for future compression standards we discuss some of the lessons of harmonic analysis in this century typically the problems and achievements of this field have involved goals that were not obviously related to practical data compression and have used a language not immediately accessible to outsiders nevertheless through an extensive generalization of what shannon called the sampling theorem harmonic analysis has succeeded in developing new forms of functional representation which turn out to have significant data compression interpretations we explain why harmonic analysis has interacted with data compression and we describe some interesting recent ideas in the field that may affect data compression in the future

besov spaces on domains in rd we study besov spaces 5 lp i2 0 p q a oo on do mains i2 in rd we show that there is an extension operator w which is a bounded mapping from b lp u onto b lp rd this is then used to derive various properties of the besov spaces such as interpolation theorems for a pair of b lp il atomic decompositions for the elements of 5 lp f2 and a description of the besov spaces by means of spline approximation

non smooth atomic decompositions traces on lipschitz domains and pointwise multipliers in function spaces abstract we provide non smooth atomic decompositions for besov spaces b p q s r n s 0 0 p q defined via differences the results are used to compute the trace of besov spaces on the boundary of bounded lipschitz domains with smoothness s restricted to 0 s 1 and no further restrictions on the parameters p q we conclude with some more applications in terms of pointwise multipliers

directional multiscale representations and applications in digital neuron reconstruction abstract recent advances in the field of multiscale representations have spurred the emergence of a new generation of powerful techniques for the efficient analysis of images and other multidimensional data these novel techniques enable the quantification of essential geometric characteristics in complex imaging data resulting in improved algorithms for image restoration feature extraction and classification we discuss the application of these ideas in neuroscience imaging and describe a novel method for the accurate and efficient identification of cellular bodies of neurons in multicellular images this method is instrumental to the design of a novel algorithm for neuronal tracing

a criterion for determining the monotonicity of the ratio of two abelian integrals abstract a new method to determine the monotonicity of the ratio of two abelian integrals is given we find two criterion functions defined directly by the functions which appear in the abelian integrals the monotonicities of the criterion functions imply the monotonicity of the ratio of the abelian integrals by using this criterion several known results can be deduced easier and some new results are obtained

anisotropic smoothness spaces via level sets it has been understood for sometime that the classical smoothness spaces such as the sobolev and besov classes are not satisfactory for certain problems in image processing and nonlinear pdes their deficiency lies in their isotropy functions in these smoothness spaces must be simultaneously smooth in all directions the anisotropic generalizations of these spaces also have the deficiency that they are biased in coordinate directions while they allow different smoothness in certain directions these directions must be aligned to the coordinate axes in the application areas mentioned above it would be desirable to measure smoothness in new ways that would allow one to have more local control over the smoothness directions we introduce one possible approach to this problem based on defining smoothness via level sets we present this approach in the case of functions defined on r d our smoothness spaces depend on two smoothness indices s1 s 2 the first reflects the smoothness of the level sets of the function while the second index reflects how smoothly the level sets themselves are changing as a motivation we start with d d 2 and investigate besov smooth domains c 2007 wiley periodicals inc

divergence free kernel methods for approximating the stokes problem we develop a new discretization scheme for the stokes equations using analytically divergence free approximation spaces our scheme is based upon a meshfree method using matrix valued kernels the scheme works in arbitrary space dimension and can produce arbitrarily smooth and hence high order approximations after deriving the scheme we analyze the discretization error in sobolev spaces and give a numerical example

characterization of embeddings of sobolev type spaces into generalized holder spaces defined by lp modulus of smoothness abstract we prove a sharp estimate for the k modulus of smoothness modelled upon a l p lebesgue space of a function f in w k l p n n k p p where is a domain with minimally smooth boundary and finite lebesgue measure k n n k n and n n k p this sharp estimate is used to establish necessary and sufficient conditions for continuous embeddings of sobolev type spaces into generalized holder spaces defined by means of the k modulus of smoothness general results are illustrated with examples in particular we obtain a generalization of the classical jawerth embeddings

bifurcations of limit cycles from quintic hamiltonian systems with an eye figure loop ii abstract in this paper we consider lienard equations of the form x y y x 2 x 3 x 5 e x 2 x 4 y where 0 e 1 r 3 and is bounded we prove that the least upper bound for the number of zeros of the related abelian integrals i h h x 2 x 4 y d x for h 1 6 is three and for h 0 is four counted with multiplicity for all parameters and this implies that the number of limit cycles that bifurcated from periodic orbits of the unperturbed system for e 0 outside an eye figure loop is less than or equal to three

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics

estimating the intensity of a random measure by histogram type estimators the purpose of this paper is to estimate the intensity of some random measure n on a set mathcal x by a piecewise constant function on a finite partition of mathcal x given a possibly large family mathcal m of candidate partitions we build a piecewise constant estimator histogram on each of them and then use the data to select one estimator in the family choosing the square of a hellinger type distance as our loss function we show that each estimator built on a given partition satisfies an analogue of the classical squared bias plus variance risk bound moreover the selection procedure leads to a final estimator satisfying some oracle type inequality with as usual a possible loss corresponding to the complexity of the family mathcal m when this complexity is not too high the selected estimator has a risk bounded up to a universal constant by the smallest risk bound obtained for the estimators in the family for suitable choices of the family of partitions we deduce uniform risk bounds over various classes of intensities our approach applies to the estimation of the intensity of an inhomogenous poisson process among other counting processes or the estimation of the mean of a random vector with nonnegative components