Gaussian Correlation Research Articles

Locally Optimum Nonlinearities for DCT Watermark Detection

The issue of copyright protection of digital multimedia data has attracted a lot of attention during the last decade. An efficient copyright protection method that has been gaining popularity is watermarking, i.e., the embedding of a signature in a digital document that can be detected only by its rightful owner. Watermarks are usually blindly detected using correlating structures, which would be optimal in the case of Gaussian data. However, in the case of DCT-domain image watermarking, the data is more heavy-tailed and the correlator is clearly suboptimal. Nonlinear receivers have been shown to be particularly well suited for the detection of weak signals in heavy-tailed noise, as they are locally optimal. This motivates the use of the Gaussian-tailed zero-memory nonlinearity, as well as the locally optimal Cauchy nonlinearity for the detection of watermarks in DCT transformed images. We analyze the performance of these schemes theoretically and compare it to that of the traditionally used Gaussian correlator, but also to the recently proposed generalized Gaussian detector, which outperforms the correlator. The theoretical analysis and the actual performance of these systems is assessed through experiments, which verify the theoretical analysis and also justify the use of nonlinear structures for watermark detection. The performance of the correlator and the nonlinear detectors in the presence of quantization is also analyzed, using results from dither theory, and also verified experimentally.

Wave reflection and transmission from a thin film with one-dimensional disorder

This paper deals with a TE plane wave reflection and transmission from a thin film with one-dimensional disorder by means of the stochastic functional approach. The relative permittivity of the thin film is written by a Gaussian random field in the horizontal direction with infinite extent, and is uniform in the vertical direction with finite thickness. Arandomwavefield is obtained in terms of a Wiener–Hermite expansion representation with approximate expansion coefficients (Wiener kernels) under a small fluctuation case. For a SiC thin film and a glass thin film having one-dimensional disorder with Gaussian correlation or an exponential correlation, numerical examples of the first-order incoherent scattering cross section and the optical theorem are illustrated in the figures. It is then found that ripples and four major peaks appear in angular distributions of the incoherent scattering. Such four peaks may occur in the directions of forward scattering, specular reflection, backscattering and in the symmetrical direction of forward scattering with respect to the normal to surface of the thin film. Physical processes that yield such ripples and peaks are discussed.

Transient pumping well flow in weakly heterogeneous formations

The solution of pumping well flow serves for identifying aquifer's properties. Most results available in the literature assume homogeneous aquifers or at most stratified structures. This study investigates the unsteady well flow in heterogeneous aquifers by modeling the hydraulic conductivity as a three‐dimensional stationary random function of axisymmetric anisotropy and Gaussian correlations. The main result of the investigation is the mean drawdown in a transient flow toward a fully penetrating well pumping at a constant rate. The solution is obtained in terms of one quadrature over time and is easily calculated. Closed form approximations of the mean head are developed for moderate and large times. The mean drawdowns derived in the study generalize the Theis and Copper and Jacob distributions for aquifers of constant transmissivity. It is shown that the transmissivity, which is identified by using the Cooper and Jacob asymptotic, is precisely the effective conductivity for uniform horizontal flow Khefu. This is compared with the finding that the equivalent conductivity for the steady state flow approaches Khefu with distance from the well [Indelman et al., 1996].

A new approach to turbulent transport of a mean scalar

We develop a simple mean field approach to the transport of a passive scalar for which the fundamental equation is a second order differential equation in the transported quantity, not a first order equation. Triple correlations are included, as they must be for any realistic description of turbulence. No correlation time enters the theory, only an eddy turnover time. The approach is simpler than standard approaches which incorporate triple correlations, but more realistic than Gaussian or short correlation time closures which do not. A similar approach has proven useful in magnetohydrodynamics.

Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media.

We analytically demonstrate the existence of white light solitons in logarithmically saturable noninstantaneous nonlinear media. This incoherent soliton has elliptic Gaussian intensity profile, and elliptic Gaussian spatial correlation statistics. The existence curve of the soliton connects the strength of the nonlinearity, the spatial correlation distance as a function of frequency, and the characteristic width of the soliton. For this soliton to exist, the spatial correlation distance must be smaller for larger temporal frequency constituents of the beam.

Statistical fluctuation of universal mobility curves in sub‐100 nm MOSFETs due to random oxide interface

AbstractWe explore the behavior of universal mobility in sub‐100 nm Si MOSFETs, using a novel 3D statistical simulation approach. Our approach is based on 3D Brownian dynamics in devices with realistic Si/SiO2 interfaces reconstructed from a Gaussian or exponential correlation function. In this approach carrier trajectories in the bulk are treated via 3D Brownian dynamics, while the carrier‐interface roughness scattering is treated using a novel empirical model. Owing to the high efficiency of the transport kernel, effective mobility in 3D MOSFETs with realistic Si/SiO2 can be simulated in a statistical manner. We first demonstrate a practical calibration procedure for the interface mobility and affirm the universal behaviour in the long channel limit, using single atomic steps and a correlation length of 6 nm. Next, effective mobility in ensembles of MOSFETs with gate length down to 10 nm is investigated. It is found that random‐discrete nature of the Si/SiO2 interface leads to a distribution of carrier mobility below the interface, which can deviate considerably from universal mobility curves when Lgate < 6Λ, where Λ is the correlation length for the SiO2 interface. Based on recent data Λ = 2.8 nm, our simulations indicate that universal‐mobility curves should be reliable for MOSFET designs down to a gate length of 17 nm. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Gravitational wave chirp search: no-signal cumulative distribution of the maximum likelihood detection statistic

The cumulative distribution of the supremum of a set (bank) of correlators is investigated in the context of maximum likelihood detection of gravitational wave chirps from coalescing binaries with unknown parameters. Accurate (lower-bound) approximants are introduced based on a suitable generalization of previous results by Mohanty. Asymptotic properties (in the limit where the number of correlators goes to infinity) are highlighted. The validity of numerical simulations made on small-size banks is extended to banks of any size, via a Gaussian correlation inequality.

On the probabilistic approach for Gaussian Berezin integrals

We present a novel approach to Gaussian Berezin correlation functions. A formula well-known in the literature expresses these quantities in terms of submatrices of the inverse matrix appearing in the Gaussian action. By using a recently proposed method to calculate Berezin integrals as an expectation of suitable functionals of Poisson processes, we obtain an alternative formula which allows one to skip the calculation of the inverse of the matrix. This formula, previously derived using different approaches (in particular by means of the Jacobi identity for the compound matrices), has computational advantages which grow rapidly with the dimension of the Grassmann algebra and the order of correlation. By using this alternative formula, we establish a mapping between two fermionic systems, not necessarily Gaussian, with short and long range interaction, respectively.

The trispectrum of the cosmic microwave background on subdegree angular scales: an analysis of the BOOMERanG data

The trispectrum of the cosmic microwave background can be used to assess the level of non-Gaussianity on cosmological scales. It probes the fourth-order moment, as a function of angular scale, of the probability distribution function of fluctuations and has been shown to be sensitive to primordial non-Gaussianity, secondary anisotropies (such as the Ostriker–Vishniac effect) and systematic effects (such as astrophysical foregrounds). In this paper we develop a formalism for estimating the trispectrum from high-resolution sky maps that incorporates the impact of finite sky coverage. This leads to a series of operations applied to the data set to minimize the effects of contamination due to the Gaussian component and correlations between estimates at different scales. To illustrate the effect of the estimation process, we apply our procedure to the BOOMERanG data set and show that it is consistent with Gaussianity. This work presents the first estimation of the cosmic microwave background trispectrum on subdegree scales.

A Gaussian correlation inequality and its applications to the existence of small ball constant

Let X 1,…, X n be jointly Gaussian random variables with mean zero. It is shown that ∀ x>0 and ∀1⩽ k< n P max 1⩽i⩽n |X i|⩽x ⩽(1/ρ)P max 1⩽i⩽k |X i|⩽x P max k<i⩽n |X i|⩽x , P max 1⩽i⩽n |X i|⩽x ⩾ρP max 1⩽i⩽k |X i|⩽x P max k<i⩽n |X i|⩽x and P max 1⩽i⩽n |X i|⩽x ⩾2 − min(k,n−k)/2 P max 1⩽i⩽k |X i|⩽x P max k<i⩽n |X i |⩽x , where ρ=(| Σ|/(| Σ 11| | Σ 22|)) 1/2, Σ, Σ 11 and Σ 22 are the covariance matrices of (X 1,…,X n), (X 1,…,X k) and ( X k+1 ,…, X n ), respectively. In particular, for fractional Brownian motion { X( t), t⩾0} of order α (0< α<1), there exists d α >0 such that P sup 0⩽s⩽a |X(t)|⩽x, sup a⩽t⩽b |X(t)−X(a)|⩽y ⩾d αP sup 0⩽s⩽a |X(t)|⩽x P sup a⩽t⩽b |X(t)−X(a)|⩽y for any 0<a<b, x>0 and y>0. As an application, it is proved that the small ball constant for the fractional Brownian motion of order α exists.

Resonant elastic scattering of light from single quantum wells in semiconductor multilayers

A theory is developed for steady-state elastic scattering of light via quasi-2D excitons from a quantum well (QW) whose interfaces are randomly rough. The study is mainly focused on the angle dependences of radiation giving direct information about static disorder responsible for the elastic scattering. A nonlocal excitonic susceptibility is expressed in terms of random profile functions of QW interfaces. Treated is elastic scattering of light from a disordered QW in the following actual dielectric environments: (i) a uniform background, (ii) a Fabry–Perot film with rough boundaries, and (iii) a semiconductor microcavity. The cross-sections are derived analytically for scattering of linearly polarized light to the lowest (Born's) approximation with arbitrary roughness statistics. The spectral and angle dependencies of scattering intensity are analyzed numerically in the absolute-value scale with Gaussian correlation of interface roughness. The probability ∼10 −2 was found for the exciton-mediated scattering of a photon from a QW interface roughness whose root-mean-square height is on the level of 2×10 −1 nm . This probability is shown to exceed by two orders of magnitude that is typical of resonant scattering from either a single semiconductor surface or rough boundaries of a semiconductor Fabry–Perot film containing the QW. The scattering spectrum of a QW placed in a microcavity is predicted to have a doublet structure whose components are associated with the cavity exciton–polaritons.

Spontaneous emission and elastic scattering of light from quantum-well excitons in a Fabry-Perot microcavity

A theory of spontaneous emission and elastic light scattering by quasi-two-dimensional excitons in a quantum well placed in a Fabry-Perot microcavity is developed. The problem is solved by means of electrodynamic Green’s functions with inclusion of fluctuations of the quantum-well width and cavity wall shape treated as a perturbation. General expressions are found in a zero approximation of perturbation theory (plane interfaces) for the radiative decay rates of quasi-two-dimensional excitons and for their energy shifts in the cavity. The boundary conditions for the electromagnetic field are taken into account through the coefficients of inward light reflection from the cavity walls. Resonance contributions to the scattering cross sections, which differ in the polarizations (p or s) of the incident and scattered waves, are derived in the lowest (Born) approximation in quantum-well width fluctuations. The spectral and angular dependences of elastic light scattering are studied numerically for Gaussian and exponential correlation functions. It is shown that the contribution from quantum-well width fluctuations to light scattering exceeds that due to single interfaces (surfaces) of a heterostructure by two orders of magnitude.

A study of the higher-order small-slope approximation for scattering from a Gaussian rough surface

Results from the first three terms of the small-slope approximation (SSA) for incoherent electromagnetic scattering from a penetrable randomly rough interface are discussed. Surface roughness is characterized as a Gaussian random process with an isotropic Gaussian correlation function. Sample results illustrate parameter spaces for which each correction term is appreciable. Reduction of the SSA to the physical optics theory is also discussed for both perfectly conducting and dielectric surfaces.

Scattering losses from planar waveguides with material inhomogeneity

We consider the propagation of guided waves in a planar waveguide that has a finite-length segment of inhomogeneous media. Except for the inhomogeneous segment, which varies in length from 0 to 160λ, the waveguide is piecewise homogeneous everywhere. The inhomogeneity is modelled by two-dimensional random permittivity fluctuations that are numerically generated from an assumed Gaussian correlation function. In 2D, the Maxwell equations are solved in the frequency domain for both TE and TM polarization by using modal expansion methods, perfectly matched absorbing boundary layers and the R -matrix transfer matrix algorithm. The guided waves are excited by a Gaussian beam incident on the waveguide aperture. For various waveguide design parameters, numerical results are given for waveguide power loss per unit length of waveguide inhomogeneity. The power loss curves are calculated as the average from numerous realizations of the random permittivity and the coefficient of variation is also given.

Prediction of Rod-Airfoil Interaction Noise Using the Ffowcs-Williams-Hawkings Analogy

SoundgeneratedatlowMachnumberbyanairfoilin thewakeofarodisinvestigatednumerically.TheGaussian spanwise loss of coherence of the vortex shedding is shown to have a signie cant ine uence on the broadband noise. Spanwise effects are successfully introduced into a time-domain formulation of the Ffowcs-Williams ‐Hawkings analogy, which is applied to aerodynamicdata computed on various contoursaround the sourceregion. Itisshown that a careful choice of these contours is required. The e owe eld is obtained from a two-dimensional Reynolds averaged Navier ‐Stokes calculation. Computed far-e eld spectra compare very well to measurements obtained in an accompanying experiment. Nomenclature Cp = pressure coefe cient c = airfoil chord d = rod diameter f = frequency g = integration surface, 0 k = turbulent kinetic energy Lg = Gaussian correlation length l = span length Mi = Mach number of g D0 Moi = observer Mach number vector Mref = reference Mach number M1 = ine ow Mach number

Signed zeros of Gaussian vector fields density, correlation functions and curvature

We calculate correlation functions of the (signed) density of zeros of Gaussian distributed vector fields. We are able to express correlation functions of arbitrary order through the curvature tensor of a certain abstract Riemann-Cartan or Riemannian manifold. As an application, we discuss one- and two-point functions. The zeros of a two-dimensional Gaussian vector field model the distribution of topological defects in the high-temperature phase of two-dimensional systems with orientational degrees of freedom, such as superfluid films, thin superconductors and liquid crystals.

Onsager Machlup Functional for Stochastic Evolution Equations in a Class of Norms

In this note, we show that the Onsager Machlup functional for stochastic evolution equations on a given separable Hilbert space H, computed in Bardina, Rovira, and Tindel (Onsager Machlup functional for stochastic evolution equations. In Annales Institut Henri Poincaré, 2003, 39, 69–93.), does not depend on the measurable norm considered on the functions from [0, 1] to H, dominating the norm on L 2([0, 1]; H), and satisfying some lower bound conditions for the stochastic convolution associated to our evolution system. The main ingredient of the proof is the extension of some Gaussian correlation inequalities of Hargé (Une inégalité de décorrélation pour la mesure gaussienne, C. R. Acad. Sci. Paris Sér. I Math. 326, no. 11, (1998), 1325–1328) and S˘idak (On multivariate normal probabilities on rectangles (their dependence on correlation), Ann. Math. Statist. 39, (1968), 1425–1434) to the infinite dimensional case.

Shadowing function with single reflection from anisotropic Gaussian rough surface. Application to Gaussian, Lorentzian and sea correlations

In this paper, the monostatic (transmitter and receiver are located at the same place) and bistatic (transmitter and receiver are distinct) statistical shadowing functions from an anisotropic two-dimensional randomly rough surface are presented. This parameter is especially important in the case of grazing angles for computing the bistatic scattering coefficient in optical and microwave frequencies. The objective of this paper is to extend the previous work (Bourlier C, Berginc G and Saillard J 2002 Waves Random Media 12 145–74), valid for a one-dimensional surface, to a two-dimensional anistropic surface by considering a joint Gaussian process of surface slopes and heights separating two points of the surface. The monostatic average (statistical shadowing function average over the statistical variables) shadowing function is then performed in polar coordinates with respect to the incidence angle, the azimuthal direction and the surface height two-dimensional autocorrelation function. In addition, for a bistatic configuration, it depends on the incidence angle and azimuthal direction of the receiver. For Gaussian and Lorentzian correlation profiles and practically important power-type spectra such as the Pierson–Moskowitz sea roughness spectrum, the numerical solution, obtained from generating the surface Gaussian elevations (Monte Carlo method), is compared with the uncorrelated and correlated models. The results show that the correlation underestimates the shadow slightly, whereas the uncorrelated results weakly overpredict the shadow and are close to the numerical solution.

Electromagnetic scattering by partially buried PEC cylinder at the dielectric rough surface interface: TM case

The Gaussian rough surface with the Gaussian correlation function is used to represent the statistical rough surface. Electromagnetic scattering from a partially buried perfectly electrically conducting (PEC) cylinder at the rough surface interface between two dielectric media is solved for the case of TM wave incidence using the method of moments with point matching. The effects of surface roughness and moisture content of the soil on the bistatic scattering cross section of the partially buried PEC objects at the rough surface interface are investigated using the Monte Carlo procedure.

Median correlation for the analysis of gene expression data

Microarray technology is revolutionizing functional genomics research by allowing scientists to measure the expression level of thousands of genes simultaneously from a single sample. However, a standard protocol for microarray data analysis has yet to be established. Many analysis techniques currently rely on linear correlation between pairs of genes. Such analysis only detects relationships in signal components exhibiting Gaussian correlation statistics. This paper focuses on determining the relationships between gene patterns based on a hierarchical clustering of nonlinear correlation measurements. The methods described herein are illustrated with gene expression data from yeast, and from human cancer cell lines. The results indicate that in some cases, improved clustering of genes can be achieved by the use of nonlinear correlation metrics.