Autocorrelation Equations Research Articles

Spatial autocorrelation equation based on Moran’s index

Moran’s index is an important spatial statistical measure used to determine the presence or absence of spatial autocorrelation, thereby determining the selection orientation of spatial statistical methods. However, Moran’s index is chiefly a statistical measurement rather than a mathematical model. This paper is devoted to establishing spatial autocorrelation models by means of linear regression analysis. Using standardized vector as independent variable, and spatial weighted vector as dependent variable, we can obtain a set of normalized linear autocorrelation equations based on quadratic form and vector inner product. The inherent structure of the models’ parameters are revealed by mathematical derivation. The slope of the equation gives Moran’s index, while the intercept indicates the average value of standardized spatial weight variable. The square of the intercept is negatively correlated with the square of Moran’s index, but omitting the intercept does not affect the estimation of the slope value. The datasets of a real urban system are taken as an example to verify the reasoning results. A conclusion can be reached that the inner product equation of spatial autocorrelation based on Moran’s index is effective. The models extend the function of spatial analysis, and help to understand the boundary values of Moran’s index.

Statistical Lyapunov Theory Based on Bifurcation Analysis of Energy Cascade in Isotropic Homogeneous Turbulence: A Physical–Mathematical Review

This work presents a review of previous articles dealing with an original turbulence theory proposed by the author and provides new theoretical insights into some related issues. The new theoretical procedures and methodological approaches confirm and corroborate the previous results. These articles study the regime of homogeneous isotropic turbulence for incompressible fluids and propose theoretical approaches based on a specific Lyapunov theory for determining the closures of the von Kármán–Howarth and Corrsin equations and the statistics of velocity and temperature difference. While numerous works are present in the literature which concern the closures of the autocorrelation equations in the Fourier domain (i.e., Lin equation closure), few articles deal with the closures of the autocorrelation equations in the physical space. These latter, being based on the eddy–viscosity concept, describe diffusive closure models. On the other hand, the proposed Lyapunov theory leads to nondiffusive closures based on the property that, in turbulence, contiguous fluid particles trajectories continuously diverge. Therefore, the main motivation of this review is to present a theoretical formulation which does not adopt the eddy–viscosity paradigm and summarizes the results of the previous works. Next, this analysis assumes that the current fluid placements, together with velocity and temperature fields, are fluid state variables. This leads to the closures of the autocorrelation equations and helps to interpret the mechanism of energy cascade as due to the continuous divergence of the contiguous trajectories. Furthermore, novel theoretical issues are here presented among which we can mention the following ones. The bifurcation rate of the velocity gradient, calculated along fluid particles trajectories, is shown to be much larger than the corresponding maximal Lyapunov exponent. On that basis, an interpretation of the energy cascade phenomenon is given and the statistics of finite time Lyapunov exponent of the velocity gradient is shown to be represented by normal distribution functions. Next, the self–similarity produced by the proposed closures is analyzed and a proper bifurcation analysis of the closed von Kármán–Howarth equation is performed. This latter investigates the route from developed turbulence toward the non–chaotic regimes, leading to an estimate of the critical Taylor scale Reynolds number. A proper statistical decomposition based on extended distribution functions and on the Navier–Stokes equations is presented, which leads to the statistics of velocity and temperature difference.

Higher‐order phase shift reconstruction approach

Biological soft tissues encountered in clinical and preclinical imaging mainly consists of atoms of light elements with low atomic numbers and their elemental composition is nearly uniform with little density variation. Hence, x-ray attenuation contrast is relatively poor and cannot achieve satisfactory sensitivity and specificity. In contrast, x-ray phase-contrast provides a new mechanism for soft tissue imaging. The x-ray phase shift of soft tissues is about a thousand times greater than the x-ray absorption over the diagnostic x-ray energy range, yielding a higher signal-to-noise ratio than the attenuation contrast counterpart. Thus, phase-contrast imaging is a promising technique to reveal detailed structural variation in soft tissues, offering a high contrast resolution between healthy and malignant tissues. Here the authors develop a novel phase retrieval method to reconstruct the phase image on the object plane from the intensity measurements. The reconstructed phase image is a projection of the phase shift induced by an object and serves as input to reconstruct the 3D refractive index distribution inside the object using a tomographic reconstruction algorithm. Such x-ray refractive index images can reveal structural features in soft tissues, with excellent resolution differentiating healthy and malignant tissues. A novel phase retrieval approach is proposed to reconstruct an x-ray phase image of an object based on the paraxial Fresnel-Kirchhoff diffraction theory. A primary advantage of the authors' approach is higher-order accuracy over that with the conventional linear approximation models, relaxing the current restriction of slow phase variation. The nonlinear terms in the autocorrelation equation of the Fresnel diffraction pattern are eliminated using intensity images measured at different distances in the Fresnel diffraction region, simplifying the phase reconstruction to a linear inverse problem. Numerical experiments are performed to demonstrate the accuracy and stability of the proposed approach. The proposed reconstruction formula is a generalization of the transport of intensity equation (TIE). It has the second-order accuracy compared to the linear model used in the conventional phase retrieval approach. The numerical experiments demonstrate that the accuracy and stability of the proposed phase reconstruction method outperforms the TIE-based reconstruction method. A novel approach has been proposed to retrieve an x-ray phase shift image induced by an object from intensity images measured at different distances in the Fresnel diffraction region. The authors' approach has the second-order accuracy and is able to retrieve the phase shift of an object stably, overcoming the restriction of slow phase variation assumed by the conventional phase retrieval techniques.

A Levinson Algorithm Based on an Isometric Transformation of Durbin's

Starting from the Durbin algorithm in polynomial space with an inner product defined by the signal autocorrelation matrix, an isometric transformation is defined that maps this vector space into another one where the Levinson algorithm is performed. Alternatively, for iterative algorithms such as discrete all-pole (DAP), an efficient implementation of a Gohberg-Semencul (GS) relation is developed for the inversion of the autocorrelation matrix which considers its centrosymmetry. In the solution of the autocorrelation equations, the Levinson algorithm is found to be less complex operationally than the procedures based on GS inversion for up to a minimum of five iterations at various linear prediction (LP) orders.

Continuity of solutions to the autocorrelation equation

Using Fourier techniques and the inner–outer factorization of holomorphic functions a complete description of the various solutions to the autocorrelation equation on a finite interval is given. The Fourier transform of the solutions admits an representation involving Cauchy integrals and Blaschke products. Using this representation a general efficient criterion for the continuity of the solutions is obtained. Further, an explicit formula for the values of the (canonical) solution at the endpoints of the interval is derived. Copyright © 2005 John Wiley & Sons, Ltd.

On the autocorrelation and the triple correlation equation for complex‐valued signals

AbstractThe paper deals with the autocorrelation equation on a finite interval and on the half‐axis and with the triple correlation equation in the case of complex‐valued signals. Via methods of Fourier transformation the equations are reduced to boundary value and continuation problems for holomorphic functions in the upper half‐plane and to functional equations in two independent variables, respectively, and in this way solved in closed form. Thereby, on the one side, Cauchy integrals and Paley‐Wiener conditions and, on the other side, the solution of a known functional equation in a single variable are applied.

The auto‐correlation equation on the finite interval

AbstractApplying the Fourier cosine transformation, the quadratic auto‐correlation equation on the finite interval [0,T] of the positive real half‐axis ℝ+ is reduced to a problem for the modulus of the finite complex Fourier transform of the solution. From the solutions of this problem L2‐solutions of the auto‐correlation equation are obtained in closed form. Moreover, as in the case of the equation on ℝ+ a Lavrent'ev regularization procedure for the auto‐correlation equation is suggested. Copyright © 2003 John Wiley & Sons, Ltd.

Relaxation procedure for phase retrieval of nonnegative signals

The paper presents an iterative procedure called the relaxation of autocorrelation equations (RAE) for solving the phase retrieval problem for nonnegative signals. First, the phase retrieval problem is formulated in the spatial domain as a set of polynomial equations with autocorrelations as known data and signal values as unknowns. Then, the RAE procedure solves these equations by recognizing one unknown at a time. While other unknowns are held constant at previously estimated values, a single unknown is varied inside the nonnegative region to globally minimize the sum of squared residuals of the equations with respect to the unknown. In every iteration, this procedure is repeated for each signal value. Since the sum of squared residuals is nonincreasing, the algorithm will either converge to a solution or stagnate; ways to overcome stagnation are suggested. The key feature of the RAE procedure is that unlike iterative transform algorithms, it allows direct control over bounding values of the signal at all times. Several numerical examples illustrate the RAE procedure.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Design of bar code systems for accurate alignment: a new method

A method is proposed on the basis of a choice of the appropriate solution of simultaneous equations obtained in the course of linearization of the autocorrelation equation of a bar code system that is used for the purpose of accurate alignment. Numerical examples are presented to show the usefulness of this new method.

Pseudolinear phase retrieval

A new algorithm is given for inverting an autocorrelation function for its input. In the autocorrelation operation, the first member of each product pair is replaced by a previous approximation to the solution. The resulting set of autocorrelation equations is linear, and in echelon form, for convenient solution. The output is made the next input to the algorithm. The algorithm is cycled, until a bistable solution state is attained. The geometric mean of the bistable solutions is then very close to the required solution; further cycling attains the solution. The approach works for general one-dimensional problems, but only for special types of two-dimensional inputs.

An alternative formulation for the recursive solution of the covariance and autocorrelation equations

The present paper gives an alternative formulation for solving autocorrelation and covariance equations, which is different from that based on a scalar product definition. A simple direct matrix formulation, which leads to recursive algorithms for both covariance and autocorrelation equations is given. For the covariance method k, α and β parameters are defined. Some useful definitions for ARMA models are given.

Updating of speech transmission parameters obtained from nonuniform analysis segments in a linear predictive speech compression system

A linear predictive speech compression system based on stationary autocorrelation formulation is described. This system employs pitch synchronous analysis of 4-kHz bandlimited input speech sampled at 10-kHz rate. An all-pole speech production model has been assumed and a 10th-order linear predicter is used. Quantization properties of k parameters [Markel and Gray, “On Autocorrelation Equations as Applied to Speech Analysis,” IEEE Trans. Audio Electroacoust. (April 1973)], pitch and energy from a large speech data base, and phonetically balanced (PB) words and sentences are discussed. This paper emphasizes the updating of speech transmission parameters based on relative changes in normalized energy between two consecutive analysis intervals. The transmission rate is assumed to be constant but the frame size, for which the transmission parameters are used, is time variant. Thresholds have been defined for relative changes in normalized energy and changes in analysis interval size between two consecutive analysis intervals. Before compression, parameters are extracted every pitch period for voiced speech and every 8.5 msec for unvoiced speech. However, the parameters are used for transmission purpose if and only if the above-mentioned thresholds are exceeded. The parameters are forcefully updated when frame size becomes equal to maximum frame size, a preset value, even though the threshold criteria are satisfied. The transmitted parameters are pitch synchronously interpolated at the receiver end. Results of synthesized PB sentences at different bit rates are reported.

Multiplication reduction in short-term autocorrelation

Expansion of the basic short-term autocorrelation equation into sum-of-products form shows that common factors occur in a predictable order. When the data are assumed to have zero value outside the interval of interest and the number of autocorrelation coefficients is much less than the number of data points, factoring can reduce the number of multiplications by up to 46 percent.

On autocorrelation equations as applied to speech analysis

The mathematical theory underlying one of the techniques currently being used in linear prediction of speech is developed from an inner product formulation. This formulation produces a unified framework for studying the properties of autocorrelation equations. In particular, the solution and stability of the auto-correlation equations are studied in detail. Experimental stability results for finite word length analysis of speech are also presented.

Low Bit-Rate Coding from the Autocorrelation Method of Linear Prediction

The term “autocorrelation method” encompasses the inverse filter, maximum likelihood, and PARCOR method. This method has a significant computational advantage over Atal's covariance method since the analysis does not require polynominal root testing programs and associated modifications, and is thus much easier to implement. Informal listening tests indicate that for low bitrate coding, the resulting synthetic speech is very similar for both methods with fixed frame analysis. Directly from the autocorrelation equation solution, the filter parameters, k parameters (also referred to as reflection coefficients and PARCOR parameters) are obtained, along with the synthesizer gain parameter. Low bit-rate coding based upon the k parameters has been obtained by performing the following steps: (1) prefilter sharply at the half-sampling frequency; (2) prewhiten and Hamm before analysis; (3) transform the parameter values to [ −0.7, 0.7]; (4) quantize with rounding; and (5) interpolate synthesis parameters pitch synchronously. These operations are all very fast and easily implemented. Good quality synthesis over a 3200-Hz input bandwidth is obtainable with [ki BITS, i = 0,1, ⋯,7,] = [6,6,3, ⋯,3]. Examples coded at 2100 and 1400 bits/sec (50-and 33-Hz frame rates, respectively) are presented.

Typical solutions of the differential equations of statistical theories of flow through porous media

The description of the miscible displacement process in porous media (such as occurs, for example, during the pollution of ground‐water resources with wastes) by methods of statistical mechanics has met with considerable success. The method usually employed is that of considering the motion of each particle through the medium as a random‐walk process. Recently, however, it has been suggested that autocorrelation should also be taken into account and a differential equation allowing for this effect to take place has been arrived at. The purpose of the present paper is to give solutions of the autocorrelation equation for some typical cases and to compare the result with that obtained for the random walk model without autocorrelation. These cases refer to the passage of a thin slug of ‘tagged’ fluid through a porous medium and to the intrusion of a sharp concentration front into a medium. It is shown that autocorrelation has the effect that a concentration‐discontinuity proceeds at a finite speed into the medium which cuts off the infinite ‘tail’ of the solution without autocorrelation. Graphs to illustrate the situation are given.