representing images using points on image surfaces this paper presents a new approach to represent an image by verge points which are defined as high curvature points on the image surface this representation offers a compact and reversible way to preserve the essence of the original image various applications such as compression edge detection image enhancement and image editing can be achieved based on this representation in this paper the whole procedure for verge point representation is presented based on these verge points image reconstruction can be easily achieved via iterative linear interpolation these extracted verge points with compatible properties are further linked into verge curves to offer more compact representation progressive representation is also developed based on a multiscale extraction scheme some potential applications are then presented to demonstrate the versatility of this representation

finite difference time domain dispersion reduction schemes the finite difference time domain fdtd although recognized as a flexible robust and simple to implement method for solving complex electromagnetic problems is subject to numerical dispersion errors in addition to the traditional ways for reducing dispersion i e increasing sampling rate and using higher order degrees of accuracy a number of schemes have been proposed recently in this work an unified methodology for deriving new difference schemes is presented it is based on certain modifications of the characteristic equation that accompanies any given discretized version of the wave equation the method is duly compared with existing schemes and verified numerically

an fdtd scheme on a face centered cubic fcc grid for the solution of the wave equation a method is proposed to improve the numerical dispersion characteristics for simulations of the scalar wave equation in 3d using the fdtd method the improvements are realized by choosing a face centered cubic fcc grid instead of the typical cartesian yee grid which exhibits non physical distortions of the wavefront due to the fd stencil fcc grids are the logical extension of hexagonal grids in 2d and have been shown previously to provide optimal sampling of space based on close packing of spheres highest density the difference equations are developed for the wave equation on this alternative grid and the dispersion relationship and stability for grids of equal and non equal aspect ratios are derived a comparison is made between fcc and cartesian formulations based upon having an equal volume density of gridpoints in each method i e the computational storage requirements of each method would be the same for the same simulated space the comparison shows that the fcc grid exhibits a much more isotropic dispersion relation than the cartesian grid of equivalent density furthermore for an equivalent density the fcc method has a more relaxed stability criterion by a factor of approximately 1 35 resulting in a further reduction in computational resources

fdtd on face centered cubic fcc grids for maxwell s equations the standard fdtd method for maxwell s equations is reformulated on face centered cubic fcc grids as an alternative to the standard cartesian yee grid dispersion and stability for the formulation are also presented and a comparison is made with the standard fdtd formulation using the metric of equal volume density of gridpoints although the new formulation has four times as many lattice points per unit voxel in comparison with the standard formulation the new formulation shows much better grid isotropy furthermore the new formulation has a stability criterion that is approximately 1 4 times more relaxed than that of the standard formulation

a discretization method with the h_ mathrm div inner product for electric field integral equations a discretization method with the h mathrm div inner product for the electric field integral equation efie is proposed the efie with the conventional galerkin discretization shows bad accuracy for problems with a small frequency a problem known as the low frequency breakdown the discretization method proposed in this paper utilizes the h mathrm div scalar product with a scalar coefficient for the galerkin discretization and overcomes the low frequency problem with an appropriately chosen coefficient as regards the preconditioning we find that a naive use of the widely used calderon preconditioning is not efficient for reducing the computational time with the new discretization we therefore propose a new preconditioning which can accelerate the computation successfully the efficiency of the proposed discretization and preconditioning is verified through some numerical examples

the spectral order of accuracy a new unified tool in the design methodology of excitation adaptive wave equation fdtd schemes we define a new concept termed the spectral order of accuracy sooa which is the spectral domain analogue of the familiar order of accuracy ooa the sooa is pivotal in a refined version of a recently introduced methodology for formulating excitation adaptive wave equation fdtd we fdtd schemes described below this concept is the basis for a unified classification for both existing and new schemes both one and two dimensional cases are presented for boundless source free homogeneous isotropic and lossless media the 1 d and 2 d cases are developed in detail for the 3 2m 1 temporal spatial and 3 3 2 d stencils respectively stability analysis is built into the methodology in terms of either analytical conditions or stability maps defined herein the methodology is seen as a generalization of many existing schemes that also provides a unified tool for a systematical design of we fdtd schemes subject to specific requirements in terms of the spectral content of the excitation the computational efficiency for all schemes remains the same for a given stencil since the core of the fdtd code is unchanged between schemes the difference being only in the values of scheme coefficients

self similarity properties of natural images resemble those of turbulent flows we show that the statistics of an edge type variable in natural images exhibits self similarity properties which resemble those of local energy dissipation in turbulent flows our results show that selfsimilarity and extended self similarity hold remarkably for the statistics of the local edge variance and that the very same models can be used to predict all of the associated exponents these results suggest using natural images as a laboratory for testing more elaborate scaling models of interest for the statistical description of turbulent flows the properties we have exhibited are relevant for the modeling of the early visual system they should be included in models designed for the prediction of receptive fields s0031 9007 97 05190 9

a low dispersion 3 d second order in time fourth order in space fdtd scheme m3d sub 24 a second order in time fourth order in space modified finite difference time domain fdtd scheme for three dimensional electromagnetic problems m3d sub 24 is presented the algorithm enables the numerical phase error to be minimized so that it leads to high accuracy with low resolution grids the advantage of this method is demonstrated by considering the long distance propagation of the wave radiated from a time harmonic elementary dipole using a low resolution grid and comparing the results with other fdtd schemes

statistics of natural images scaling in the woods we study the statistics of an ensemble of images taken in the woods distributions of local quantities such as contrast are scale invariant and have nearly exponential tails power spectra exhibit scaling with a nontrivial exponent these data limit the information content of natural images and point to the importance of gain control strategies in visual processing

a comprehensive new methodology for formulating fdtd schemes with controlled order of accuracy and dispersion numerical dispersion errors in the wave equation finite difference time domain we fdtd method have been treated by higher order schemes coefficient modification schemes dispersion relation preserving and non standard schemes in this work a unified methodology is formulated for the systematic generation of we fdtd schemes tailored to the spectrum of the excitation the methodology enables the scheme designer to gradually trade order of accuracy ooa for lower dispersion errors in a controlled manner at the cost of sacrificing low frequency behavior that is not deemed critical for this type of excitation the methodology is shown to encompass both existing and new schemes stability analysis is carried out concurrently with the generation of each scheme using a stencil size of 3 and 5 temporal and spatial samples respectively long term errors of a scheme designed for a specific pulse are compared with the standard 4 4 scheme that has the same computational complexity via simulation of a modulated pulse that propagates over a million time steps

microcanonical multifractal formalism a geometrical approach to multifractal systems part i singularity analysis multifractal formalism in the microcanonical framework has proved to be a valuable approach to understand and analyze complex signals typically associated with natural phenomena in scale invariant systems in this paper we discuss the multifractal microcanonical formalism in a comprehensive unified way including new theoretical proofs and validation tests on real signals so completing some known gaps in the foundations of this theory we also review the latest advances and describe the present perspectives in this field some technical details on the implementation of involved algorithms and relevant open issues are also discussed

the multifractal structure of contrast changes in natural images from sharp edges to textures we present a formalism that leads naturally to a hierarchical description of the different contrast structures in images providing precise definitions of sharp edges and other texture components within this formalism we achieve a decomposition of pixels of the image in sets the fractal components of the image such that each set contains only points characterized by a fixed strength of the singularity of the contrast gradient in its neighborhood a crucial role in this description of images is played by the behavior of contrast differences under changes in scale contrary to naive scaling ideas where the image is thought to have uniform transformation properties field 1987 each of these fractal components has its own transformation law and scaling exponents a conjecture on their biological relevance is also given

singularity spectrum of fractal signals from wavelet analysis exact results the multifractal formalism for singular measures is revisited using the wavelet transform for bernoulli invariant measures of some expanding markov maps the generalized fractal dimensions are proved to be transition points for the scaling exponents of some partition functions defined from the wavelet transform modulus maxima the generalization of this formalism to fractal signals is established for the class of distribution functions of these singular invariant measures it is demonstrated that the hausdorff dimensiond h of the set of singularities of holder exponenth can be directly determined from the wavelet transform modulus maxima the singularity spectrum so obtained is shown to be not disturbed by the presence in the signal of a superimposed polynomial behavior of ordern provided one uses an analyzing wavelet that possesses at leastn n vanishing moments however it is shown that ac behavior generally induces a phase transition in thed h singularity spectrum that somewhat masks the weakest singularities this phase transition actually depends on the numbern of vanishing moments of the analyzing wavelet its observation is emphasized as a reliable experimental test for the existence of nonsingular behavior in the considered signal these theoretical results are illustrated with numerical examples they are likely to be valid for a large class of fractal functions as suggested by recent applications to fractional brownian motions and turbulent velocity signals

edges transitions and criticality in this article various notions of edges encountered in digital image processing are reviewed in terms of compact representation or completion we show that critical exponents defined in statistical physics lead to a much more coherent definition of edges consistent across the scales in acquisitions of natural phenomena such as high resolution natural images or turbulent acquisitions edges belong to the multiscale hierarchy of an underlying dynamics they are understood from a statistical perspective well adapted to fit the case of natural images numerical computation methods for the evaluation of critical exponents in the non ergodic case are recalled which apply for the vast majority of natural images we study the framework of reconstructible systems in a microcanonical formulation show how it redefines edge completion and how it can be used to evaluate and assess quantitatively the adequation of edges as candidates for compact representations we study with particular attention the case of turbulent data in which edges in the classical sense are particularly challenged tests are conducted and evaluated on a standard database for natural images we test the newly introduced compact representation as an ideal candidate for evaluating turbulent cascading properties of complex images and we show better reconstruction performance than the classical tested methods highlightsclassical edge detection algorithms are reviewed in terms of completion a new edge detection scheme based on multiscale hierarchical classification assessing the consistency of edges across scales evaluating edge performance through the framework of reconstructible systems comparing the performance with classical edge operators

fourier analysis of numerical algorithms for the maxwell equations the fourier method is used to analyze the dispersive dissipative and isotropy errors of various spatial and time discretizations applied to the maxwell equations on multi dimensional grids both cartesian grids and non cartesian grids based on hexagons and tetradecahedra are studied and compared the numerical errors are quantitatively determined in terms of phase speed wavenumber propagation direction gridspacings and cfl number the study shows that centered schemes are more efficient and accurate than upwind schemes and the non cartesian grids yield superior isotropy than the cartesian ones for the centered schemes the staggered grids produce less errors than the unstaggered ones a new unstaggered algorithm which has all the best properties is introduced using an optimization technique to determine the nodal weights the new algorithm provides the highest accuracy among all the schemes discussed the study also demonstrates that a proper choice of time discretization can reduce the overall numerical errors due to the spatial discretization

reconstructing images from their most singular fractal manifold real world images are complex objects difficult to describe but at the same time possessing a high degree of redundancy a previous on the statistical properties of natural images reveals that natural images can be viewed through different partitions which are essentially fractal in nature one particular fractal component related to the most singular sharpest transitions in the image seems to be highly informative about the whole scene we show how to decompose the image into their fractal components we see that the most singular component is related to but not coincident with the edges of the objects present in the scenes we propose a new simple method to reconstruct the image with information contained in that most informative component we see that the quality of the reconstruction is strongly dependent on the capability to extract the relevant edges in the determination of the most singular set we discuss the results from the perspective of coding proposing this method as a starting point for future developments

the lobatto cell robust explicit higher order fdtd that handles inhomogeneous media to improve numerical dispersion and second order convergence behavior of the traditional yee finite difference time domain fdtd algorithm various higher order finite difference implementations have been proposed many have difficulty handling interfaces between distinct dielectric media or metallic sheets because the high order taylor expansions on which these codes are based are invalidated by field or derivative discontinuities one sided extensions extrapolations are a common approach to remedy interface discontinuities here another approach based upon whitney form mass lumped finite elements is pursued to develop the ldquolobatto cell rdquo a high order replacement for the yee cell with appropriate continuity behavior at contrasting media interfaces the lobatto cell retains many robust features of the yee method discrete conservation laws a guarantee of conditional stability and straightforward explicit updating but provides higher order interpolation and dispersion error convergence a notable drawback of the method is larger modeling granularity staircasing which is managed here by combining the lobatto cell method with preexisting robust subgridding techniques

multiscaling and information content of natural color images naive scale invariance is not a true property of natural images natural monochrome images possess a much richer geometrical structure which is particularly well described in terms of multiscaling relations this means that the pixels of a given image can be decomposed into sets the fractal components of the image with well defined scaling exponents turiel and parga neural comput 12 763 2000 here it is shown that hyperspectral representations of natural scenes also exhibit multiscaling properties observing the same kind of behavior a precise measure of the informational relevance of the fractal components is also given and it is shown that there are important differences between the intrinsically redundant red green blue system and the decorrelated one defined in ruderman cronin and chiao j opt soc am a 15 2036 1998

empirical evidences of a common multifractal signature in economic biological and physical systems abstract recent developments in microcanonical multifractal formalism have lead to a sensible improvement in the numerical techniques for the determination of the multifractal characteristics of real signals with the aid of these techniques we have found empirical evidence of a common multifractal signature in six very different systems ranging from stock market time series to sea surface temperature records these systems are not only found to be multifractal but their singularity spectra are coincident we propose an explanation of this striking coincidence in terms of a cascade process and analyze its consequences

the construction of low dispersive fdtd on hexagon based on the sampling theorem with arbitrary geometry the spatial discretization can be carried out on hexagonal lattices grids instead of rectangular lattices in the two dimensional case with the concept of the directional derivative in aid a new finite difference time domain fdtd method based on hexagonal lattices h fdtd is introduced the coefficients of the h fdtd method are chosen with aim to obtain the low numerical dispersion final numerical experiment shows that the numerical anisotropy of the h fdtd method is lower than that of the conventional yee fdtd method based on rectangular lattices r fdtd in addition some numerical examples are provided to verify its good performance on the low numerical anisotropy

development of higher order fdtd schemes with controllable dispersion error a new methodology that facilitates the control of the inherent dispersion error in the case of higher order finite difference time domain fdtd schemes is presented in this paper the basic idea is to define suitable algebraic expressions that reflect numerical inaccuracies reliably then finite difference operators are determined via the minimization of the error estimators at selected frequencies in order to apply this procedure an error expansion in terms of cylindrical harmonic functions is performed which also enables accuracy enhancement for all propagation angles the design process produces a set of two dimensional 2 d fdtd algorithms with optimized frequency response contrary to conventional methodologies the proposed techniques adjust their reliability range according to the requirements of the examined problem and can be therefore more efficient in computationally demanding simulations

multifractal geometry in stock market time series it has been recently noticed that time series of returns in stock markets are of multifractal multiscaling character in that context multifractality has been always evidenced by its statistical signature i e the scaling exponents associated to a related variable however a direct geometrical framework much more revealing about the underlying dynamics is possible in this paper we present the techniques allowing the multifractal decomposition we will show that there exists a particular fractal component the most singular manifold msm which contains the relevant information about the dynamics of the series it is possible to reconstruct the series at a given precision from the msm we analyze the dynamics of the msm which shows revealing features about the evolution of this type of series