We present a solver for the 2D high-frequency Helmholtz equation in heterogeneous acoustic media, with online parallel complexity that scales optimally as @math , where @math is the number of volume unknowns, and @math is the number of processors, as long as @math grows at most like a small fractional power of @math . The solver decomposes the domain into layers, and uses transmission conditions in boundary integral form to explicitly define "polarized traces", i.e., up- and down-going waves sampled at interfaces. Local direct solvers are used in each layer to precompute traces of local Green's functions in an embarrassingly parallel way (the offline part), and incomplete Green's formulas are used to propagate interface data in a sweeping fashion, as a preconditioner inside a GMRES loop (the online part). Adaptive low-rank partitioning of the integral kernels is used to speed up their application to interface data. The method uses second-order finite differences. The complexity scalings are empirical but motivated by an analysis of ranks of off-diagonal blocks of oscillatory integrals. They continue to hold in the context of standard geophysical community models such as BP and Marmousi 2, where convergence occurs in 5 to 10 GMRES iterations.
Finally, an earlier version of the work presented in this paper can be found in @cite_20 , in which the notion of polarized traces using Green's representation formula was first introduced, obtaining favorable scalings;
@cite_20
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In this paper we consider the localization of a collection of small, three-dimensional bounded homogeneous inclusions via time-harmonic electromagnetic means, typically using arrays of electric or magnetic dipole transmitters and receivers with given polarization(s) at some distance from the collection, possibly also lying in the far field. The inclusions, somewhat apart or closely spaced, are buried within a homogeneous medium, and are of arbitrary contrast of permittivity, conductivity, and permeability vis-ag-vis this embedding medium. The problem is formulated as an inverse scattering problem for the full Maxwell equations and it involves a robust asymptotic modeling of the multistatic response matrix. No specific application is studied at this stage, but characterization of obstacles in subsoils, nondestructive evaluation of man-made structures, and medical imaging are primary fields of application envisaged. The proposed approach uses a MUSIC (multiple signal classification)-type algorithm, and it yields fast numbering, accurate localization, and estimates of the electromagnetic and geometric parameters (polarization tensors) of the inclusions. The mathematical machinery is detailed first, some specific attention being given to triaxial ellipsoidal inclusions and degenerate spherical shapes (for the latter known results are retrieved). Then, the viability of this algorithm—which would be easily extended to planarly layered environments by introduction of their Green’s functions—is documented by a variety of numerical results from synthetic, noiseless, and severely noisy field data. The recent reformulation of the coupled-wave method by Lalanne and Morris [ J. Opt. Soc. Am. A13, 779 ( 1996)] and by Granet and Guizal [ J. Opt. Soc. Am. A13, 1019 ( 1996)], which dramatically improves the convergence of the method for met allic gratings in TM polarization, is given a firm mathematical foundation in this paper. The new formulation converges faster because it uniformly satisfies the boundary conditions in the grating region, whereas the old formulations do so only nonuniformly. Mathematical theorems that govern the factorization of the Fourier coefficients of products of functions having jump discontinuities are given. The results of this paper are applicable to any numerical work that requires the Fourier analysis of products of discontinuous periodic functions. In the present paper we analyze a class of tensor-structured preconditioners for the multidimensional second-order elliptic operators in ℝ d , d≥2. For equations in a bounded domain, the construction is based on the rank-R tensor-product approximation of the elliptic resolvent ℬ R ≈(ℒ−λ I)−1, where ℒ is the sum of univariate elliptic operators. We prove the explicit estimate on the tensor rank R that ensures the spectral equivalence. For equations in an unbounded domain, one can utilize the tensor-structured approximation of Green’s kernel for the shifted Laplacian in ℝ d , which is well developed in the case of nonoscillatory potentials. For the oscillating kernels e −i κ‖x‖ ‖x‖, x∈ℝ d , κ∈ℝ+, we give constructive proof of the rank-O(κ) separable approximation. This leads to the tensor representation for the discretized 3D Helmholtz kernel on an n×n×n grid that requires only O(κ |log e|2 n) reals for storage. Such representations can be applied to both the 3D volume and boundary calculations with sublinear cost O(n 2), even in the case κ=O(n). We consider the change in polarization P which occurs upon making an adiabatic change in the Kohn-Sham Hamiltonian of the solid. A simple expression for P is derived in terms of the valence-band wave functions of the initial and final Hamiltonians. We show that physically P can be interpreted as a displacement of the center of charge of the Wannier functions. The formulation is successfully applied to compute the piezoelectric tensor of GaAs in a first-principles pseudopotential calculation. Let λλ be the second largest eigenvalue in absolute value of a uniform random dd-regular graph on nn vertices. It was famously conjectured by Alon and proved by Friedman that if dd is fixed independent of nn, then λ=2d−1−−−−√+o(1)λ=2d−1+o(1) with high probability. In the present work, we show that λ=O(d−−√)λ=O(d) continues to hold with high probability as long as d=O(n2 3)d=O(n2 3), making progress toward a conjecture of Vu that the bound holds for all 1≤d≤n 21≤d≤n 2. Prior to this work the best result was obtained by Broder, Frieze, Suen and Upfal (1999) using the configuration model, which hits a barrier at d=o(n1 2)d=o(n1 2). We are able to go beyond this barrier by proving concentration of measure results directly for the uniform distribution on dd-regular graphs. These come as consequences of advances we make in the theory of concentration by size biased couplings. Specifically, we obtain Bennett-type tail estimates for random variables admitting certain unbounded size biased couplings. In 1999 Kannan, Tet ali and Vempala proposed a MCMC method to uniformly sample all possible realizations of a given graphical degree sequence and conjectured its rapidly mixing nature. Recently their conjecture was proved affirmative for regular graphs (by Cooper, Dyer and Greenhill, 2007), for regular directed graphs (by Greenhill, 2011) and for half-regular bipartite graphs (by Miklos, Erdős and Soukup, 2013). Several heuristics on counting the number of possible realizations exist (via sampling processes), and while they work well in practice, so far no approximation guarantees exist for such an approach. This paper is the first to develop a method for counting realizations with provable approximation guarantee. In fact, we solve a slightly more general problem; besides the graphical degree sequence a small set of forbidden edges is also given. We show that for the general problem (which contains the Greenhill problem and the Miklos, Erdős and Soukup problem as special cases) the derived MCMC process is rapidly mixing. Further, we show that this new problem is self-reducible therefore it provides a fully polynomial randomized approximation scheme (a.k.a. FPRAS) for counting of all realizations. In this study, Green’s function of the two-dimensional radiative transfer equation is derived for an infinitely extended anisotropically scattering medium, which is illuminated by a unidirectional source distribution. In the steady-state domain, the final results, which are based on eigenvalues and eigenvectors, are given analytically apart from the eigenvalues. For the time-dependent case an additional numerical inverse Fourier transform is required. The obtained solutions were successfully validated with another exact analytical solution in the time domain for isotropically scattering and with the Monte Carlo method for anisotropically scattering media. In this paper we provide for a first time, to our knowledge, a mathematical model for imaging an anisotropic, orthotropic medium with polarization-sensitive optical coherence tomography. The imaging problem is formulated as an inverse scattering problem in three dimensions for reconstructing the electrical susceptibility of the medium using Maxwell’s equations. Our reconstruction method is based on the second-order Born-approximation of the electric field. In this paper, we propose a novel graph model, called weighted sparse representation regularized graph, to learn a robust object representation using multispectral (RGB and thermal) data for visual tracking. In particular, the tracked object is represented with a graph with image patches as nodes. This graph is dynamically learned from two aspects. First, the graph affinity (i.e., graph structure and edge weights) that indicates the appearance compatibility of two neighboring nodes is optimized based on the weighted sparse representation, in which the modality weight is introduced to leverage RGB and thermal information adaptively. Second, each node weight that indicates how likely it belongs to the foreground is propagated from others along with graph affinity. The optimized patch weights are then imposed on the extracted RGB and thermal features, and the target object is finally located by adopting the structured SVM algorithm. Moreover, we also contribute a comprehensive dataset for RGB-T tracking purpose. Comparing with existing ones, the new dataset has the following advantages: 1) Its size is sufficiently large for large-scale performance evaluation (total frame number: 210K, maximum frames per video pair: 8K). 2) The alignment between RGB-T video pairs is highly accurate, which does not need pre- and post-processing. 3) The occlusion levels are annotated for analyzing the occlusion-sensitive performance of different methods. Extensive experiments on both public and newly created datasets demonstrate the effectiveness of the proposed tracker against several state-of-the-art tracking methods. We address the problem of recovering a scene recorded through a semireflecting medium (i.e. planar lens), with a virtual reflected image being superimposed on the image of the scene transmitted through the semirefelecting lens. Recent studies propose imaging through a linear polarizer at several orientations to estimate the reflected and the transmitted components in the scene. In this study we extend the sparse ICA (SPICA) technique and apply it to the problem of separating the image of the scene without having any a priori knowledge about its structure or statistics. Recent novel advances in the SPICA approach are discussed. Simulation and experimental results demonstrate the efficacy of the proposed methods.© 2005 Wiley Periodicals, Inc. © 2005 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 15, 84–91, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002 ima.20042 In this paper, polar codes for the m-user multiple access channel (MAC) with binary inputs are constructed. It is shown that Arikan's polarization technique applied individually to each user transforms independent uses of an m-user binary input MAC into successive uses of extremal MACs. This transformation has a number of desirable properties: 1) the “uniform sum-rate” of the original MAC is preserved, 2) the extremal MACs have uniform rate regions that are not only polymatroids but matroids, and thus, 3) their uniform sum-rate can be reached by each user transmitting either uncoded or fixed bits; in this sense, they are easy to communicate over. A polar code can then be constructed with an encoding and decoding complexity of O(n log n) (where n is the block length), a block error probability of o(exp (- n1 2 - e)), and capable of achieving the uniform sum-rate of any binary input MAC with arbitrary many users. Applications of this polar code construction to channels with a finite field input alphabet and to the additive white Gaussian noise channel are also discussed. Currently sparse signal reconstruction gains considerable interest and is applied in many fields. In this paper, a similarity induced by joint sparse representation is designed to construct the likelihood function of particle filter tracker so that the color visual spectrum and thermal spectrum images can be fused for object tracking. The proposed fusion scheme performs joint sparse representation calculation on both modalities and the resultant tracking results are fused using min operation on the sparse representation coefficients. In addition, a co-learning approach is proposed to update the reference templates of both modality and enhance the tracking robustness. The proposed fusion scheme outperforms state-of-the-art approaches, and its effectiveness is verified using OTCBVS database. A theory of edge detection is presented. The analysis proceeds in two parts. (1) Intensity changes, which occur in a natural image over a wide range of scales, are detected separately at different scales. An appropriate filter for this purpose at a given scale is found to be the second derivative of a Gaussian, and it is shown that, provided some simple conditions are satisfied, these primary filters need not be orientation-dependent. Thus, intensity changes at a given scale are best detected by finding the zero values of ∇ 2 G(x, y) * I(x, y) for image I, where G(x, y) is a two-dimen­sional Gaussian distribution and ∇ 2 is the Laplacian. The intensity changes thus discovered in each of the channels are then represented by oriented primitives called zero-crossing segments, and evidence is given that this representation is complete. (2) Intensity changes in images arise from surface discontinuities or from reflectance or illumination bound­aries, and these all have the property that they are spatially localized. Because of this, the zero-crossing segments from the different channels are not independent, and rules are deduced for combining them into a description of the image. This description is called the raw primal sketch. The theory explains several basic psychophysical findings, and the opera­tion of forming oriented zero-crossing segments from the output of centre-surround ∇ 2 G filters acting on the image forms the basis for a physiological model of simple cells (see Marr & Ullman 1979). Abstract In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is problematic. In this article we develop a fast high accuracy Polar FFT. For a given two-dimensional signal of size N × N , the proposed algorithm's complexity is O ( N 2 log N ) , just like in a Cartesian 2D-FFT. A special feature of our approach is that it involves only 1D equispaced FFT's and 1D interpolations. A central tool in our method is the pseudo-Polar FFT, an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. We describe the concept of pseudo-Polar domain, including fast forward and inverse transforms. For those interested primarily in Polar FFT's, the pseudo-Polar FFT plays the role of a halfway point—a nearly-Polar system from which conversion to Polar coordinates uses processes relying purely on 1D FFT's and interpolation operations. We describe the conversion process, and give an error analysis of it. We compare accuracy results obtained by a Cartesian-based unequally-sampled FFT method to ours, both algorithms using a small-support interpolation and no pre-compensating, and show marked advantage to the use of the pseudo-Polar initial grid. In this study, the two-dimensional radiative transfer equation is solved considering the exact boundary conditions for describing the propagation of particles in bounded anisotropically scattering media. The obtained Green’s functions have an analytical dependence on the spatial and angular variables and, therefore, can be evaluated rapidly and implemented without any computational complexities. The Monte Carlo method was used for a successful verification of the derived solutions. Inspired by the effectiveness of global priors for 2D saliency analysis, this paper aims to explore those particular to RGB-D data. To this end, we propose two priors, which are the normalized depth prior and the global-context surface orientation prior, and formulate them in the forms simple for computation. A two-stage RGB-D salient object detection framework is presented. It first integrates the region contrast, together with the background, depth, and orientation priors to achieve a saliency map. Then, a saliency restoration scheme is proposed, which integrates the PageRank algorithm for sampling high confident regions and recovers saliency for those ambiguous. The saliency map is thus reconstructed and refined globally. We conduct comparative experiments on two publicly available RGB-D datasets. Experimental results show that our approach consistently outperforms other state-of-the-art algorithms on both datasets. Digital cameras sample scenes using a color filter array of mosaic pattern (e.g., the Bayer pattern). The demosaicking of the color samples is critical to the image quality. This paper presents a new color demosaicking technique of optimal directional filtering of the green-red and green-blue difference signals. Under the assumption that the primary difference signals (PDS) between the green and red blue channels are low pass, the missing green samples are adaptively estimated in both horizontal and vertical directions by the linear minimum mean square-error estimation (LMMSE) technique. These directional estimates are then optimally fused to further improve the green estimates. Finally, guided by the demosaicked full-resolution green channel, the other two color channels are reconstructed from the LMMSE filtered and fused PDS. The experimental results show that the presented color demosaicking technique outperforms the existing methods both in PSNR measure and visual perception. A method for efficient detection of the dominant local reflectional symmetry in grey level images is described. The general approach is to define a local measure of reflectional symmetry that transforms the symmetry detection problem to an optimization problem, and obtain the symmetric regions by an efficient global optimization algorithm. The symmetry of a 1D function can be measured in the frequency domain as the fraction of its energy that resides in symmetric Fourier basis functions. This approach is extended to two dimensions. Locality can be formally treated in terms of the Gabor decomposition and implemented via soft windowing. The resulting measure is a complicated multimodal function of the location of the center of the supporting region, its size, and the orientation of the symmetry axis. A new probabilistic generic algorithm is applied to the determination of the global maximum of the reflectional symmetry function. Less than one thousand evaluations of the local symmetry measure are typically needed in order to locate the dominant symmetry in natural, wildlife test images. The aim of the present work is to validate a full vectorial electromagnetic inverse scattering algorithm against experimental data. Data were provided courtesy of Institut Fresnel, Marseille, France. These data were carried out in an anechoic chamber and correspond to different canonical targets as well as one mysterious object which is known only by experimentalists who measured the associated scattered field. The inverse algorithm was first developed in the optical domain and is adapted herein to the microwave domain. It is an iterative approach where the parameter of interest, namely the relative permittivity distribution, is updated gradually by minimizing a cost function describing the discrepancy between data and those that would be obtained via a forward solver for the best available estimate of the relative permittivity. The forward solver is based on the coupled dipole method which was introduced in the seventies to study the scattering of light by non-spherical dielectric grains. The forward and inverse schemes are briefly described and various examples are presented that demonstrate the efficiency of the inverse algorithm. We propose a new application of « Shape from Polarization » method to reconstruct surface shapes of specular met allic objects. Studying the polarization state of the reflected light is very useful to get information on the surface normals. After reflection, an unpolarized light wave becomes partially linearly polarized. Such a wave, can be described by its three parameters: intensity, degree of polarization, and angle of polarization. By using the refractive index of the surface, a relationship between the degree of polarization and the reflection angle can be established. Unfortunately, the relation commonly used for dielectrics, cannot be applied since the refractive index of met allic surfaces is complex. To get a similar relation, we apply a usual approximation valid in the visible region. The Fresnel reflectance model can also provide a relationship between the angle of polarization and the incidence plane orientation. Thus, the reflection angle and the incidence plane orientation give the surface normals. The shape is finally computed by integrating the normals with a relaxation algorithm. Applications on met allic objects made by stamping and polishing are also described, and show the efficiency of our system to discriminate shape defects. Future works will consist in integrating the system into an automatic process of defects detection.