Superresolution of Fourier transform spectroscopy data by the maximum entropy method

The maximum entropy method (MEM) is applied to the interferogram data obtained using the technique of Fourier transform spectroscopy for estimating its spectrum with a resolution far exceeding the value set by the spectrometer. For emission line data, the MEM process is directly used with the interferogram data in place of the regular Fourier transformation process required in Fourier transform spectroscopy. It produces a spectral estimate with an enhanced resolution. For absorption data with a broad background spectrum, the method is applied to a modified interferogram which corresponds to the Fourier transform of the absorptance spectrum. Two results are presented to demonstrate the power of the technique: for the visible emission spectrum of a spectral calibration lamp and for the infrared chloroform absorption spectrum. Included in the paper is a discussion of the problems associated with practical use of the MEM.     

Bayesian maximum entropy (two-dimensional) lifetime distribution reconstruction from time-resolved spectroscopic data

Time-resolved spectroscopy is often used to monitor the relaxation processes (or reactions) of physical, chemical, and biochemical systems after some fast physical or chemical perturbation. Time-resolved spectra contain information about the relaxation kinetics, in the form of macroscopic time constants of decay and their decay associated spectra. In the present paper we show how the Bayesian maximum entropy inversion of the Laplace transform (MaxEnt-iLT) can provide a lifetime distribution without sign-restrictions (or two-dimensional (2D)-lifetime distribution), representing the most probable inference given the data. From the reconstructed (2D) lifetime distribution it is possible to obtain the number of exponentials decays, macroscopic rate constants, and exponential amplitudes (or their decay associated spectra) present in the data. More importantly, the obtained (2D) lifetime distribution is obtained free from pre-conditioned ideas about the number of exponential decays present in the data. In contrast to the standard regularized maximum entropy method, the Bayesian MaxEnt approach automatically estimates the regularization parameter, providing an unsupervised and more objective analysis. We also show that the regularization parameter can be automatically determined by the L-curve and generalized cross-validation methods, providing (2D) lifetime reconstructions relatively close to the Bayesian best inference. Finally, we propose the use of MaxEnt-iLT for a more objective discrimination between data-supported and data-unsupported quantitative kinetic models, which takes both the data and the analysis limitations into account. All these aspects are illustrated with realistic time-resolved Fourier transform infrared (FT-IR) synthetic spectra of the bacteriorhodopsin photocycle.     

An efficient algorithm to generate maximum entropy distributions

The paper describes an algorithm for generating maximum entropy distributions for probabilistic data. The central moments of the data form constraint equations developed from Jaynes' formalism, which are solved by mathematical programming. Criteria are presented for selecting starting points and scaling parameters, upon which the accuracy and efficiency of the algorithm depends. Results are given of tests in which maximum entropy distributions are generated from the moment information of numerous analytical distributions. They show that four moments are generally required to produce good agreement and that maximum entropy distributions can represent most data populations very well. The notation used in the paper is defined in Appendix I.     

Deconvolution of two-dimensional NMR spectra by fast maximum likelihood reconstruction: application to quantitative metabolomics

We have developed an algorithm called fast maximum likelihood reconstruction (FMLR) that performs spectral deconvolution of 1D-2D NMR spectra for the purpose of accurate signal quantification. FMLR constructs the simplest time-domain model (e.g., the model with the fewest number of signals and parameters) whose frequency spectrum matches the visible regions of the spectrum obtained from identical Fourier processing of the acquired data. We describe the application of FMLR to quantitative metabolomics and demonstrate the accuracy of the method by analysis of complex, synthetic mixtures of metabolites and liver extracts. The algorithm demonstrates greater accuracy (0.5-5.0% error) than peak height analysis and peak integral analysis with greatly reduced operator intervention. FMLR has been implemented in a Java-based framework that is available for download on multiple platforms and is interoperable with popular NMR display and processing software. Two-dimensional (1)H-(13)C spectra of mixtures can be acquired with acquisition times of 15 min and analyzed by FMLR in the range of 2-5 min per spectrum to identify and quantify constituents present at concentrations of 0.2 mM or greater.     

Comparison of subtractive Kramers-Kronig analysis and maximum entropy model in resolving phase from finite spectral range reflectance data

The maximum entropy model (MEM) and Kramers-Kronig (K-K) analysis were compared with the aim of phase retrieval from reflectance. The object was to test two different phase-retrieval methods when reflectance is known at a finite frequency range and data fitting is not performed beyond the finite frequency band. In addition, it was assumed that the phase is known only at one or two anchor points. As an example, we study the terahertz reflection spectrum related to a semiconductor and an optical spectrum of potassium chloride. It is shown that the MEM resolves the complex refractive index of a medium, in the vicinity of initial and final points of the spectra, better than singly and doubly subtractive K-K relations. Both methods give only satisfactory results in the event of one anchor point, but in the case of two anchor points, the MEM is better than doubly subtractive K-K. It is proposed that the MEM should be used instead of K-K analysis, for a priori information of phase at two anchor points, for the purpose of resolving the complex refractive index of a medium from reflectance with high accuracy.     

Practical aspects of the maximum entropy inversion of the laplace transform for the quantitative analysis of multi-exponential data

The number, position, area, and width of the bands in a lifetime distribution give the number of exponentials present in time-resolved data and their time constants, amplitudes, and heterogeneities. The maximum entropy inversion of the Laplace transform (MaxEnt-iLT) provides a lifetime distribution from time-resolved data, which is very helpful in the analysis of the relaxation of complex systems. In some applications both positive and negative values for the lifetime distribution amplitudes are physical, but most studies to date have focused on positive-constrained solutions. In this work, we first discuss optimal conditions to obtain a sign-unrestricted maximum entropy lifetime distribution, i.e., the selection of the entropy function and the regularization value. For the selection of the regularization value we compared four methods: the chi2 criterion and Bayesian inference (already used in sign-restricted MaxEnt-iLT), and the L-curve and the generalized cross-validation methods (not yet used in MaxEnt-iLT to our knowledge). Except for the frequently used chi2 criterion, these methods recommended similar regularization values, providing close to optimum solutions. However, even when an optimal entropy function and regularization value are used, a MaxEnt lifetime distribution will contain noise-induced errors, as well as systematic distortions induced by the entropy maximization (regularization-induced errors). We introduce the concept of the apparent resolution function in MaxEnt, which allows both the noise and regularization-induced errors to be estimated. We show the capability of this newly introduced concept in both synthetic and experimental time-resolved Fourier transform infrared (FT-IR) data from the bacteriorhodopsin photocycle.     

Reconstruction of random material microstructures: patterns of maximum entropy

This work is focused on the reconstruction of the probability distributions for the basic (topological and metric) characteristics of polycrystalline, random space-filling aggregates of grains (cells). The method is inspired by the Gibbsian ensemble method known in classical statistical physics/mechanics. Among all possible distributions of the microstructural parameters we are looking for the distributions which are compatible with the constraints representing the available macro or mezoscopic information. Specifically, we show how the idea of Gibbs ensembles can be used for random polyhedral space-filling models, such as random tessellation. The microstructural configurations are characterized by a number of parameters and their mutual relations in the aggregate (the consistency requirements). These parameters define a microstate of the system considered. We characterize the unknown probabilities of the microstates making use of the ‘macro’—information given by the appropriate average values defined on microstates, and then—using the method of maximum information entropy. In particular, some specific microstructures are treated numerically and illustrated graphically.     

On estimating distributions with the maximum entropy principle

The possibility to use the maximum entropy principle to estimate distributions from measurements with known resolution functions has been considered. The general analytical form of the distribution estimate has been obtained. The statistical properties of this estimate, i.e. the error matrix and bias, have been analyzed. The method is generalized for the case when the unknown distribution is considered to be close to a certain known one. The proposed method is illustrated by a number of numerical experiments. The results are compared with those obtained by other methods.     

Application of the maximum entropy method in texture analysis

The problem of estimating the crystallite orientation distribution function (codf) based on the leading texture coefficients is considered. Problems of such a type are called moment problems, which are well known in statistical mechanics and other areas of science. It is shown how the maximum entropy method can be applied to estimate the codf. Special emphasis is given to a coordinate-free formulation of the problem. The codf is represented by a tensorial Fourier series. The equations, which have to be solved for the estimate of the distribution function, are derived for all tensor ranks of the Fourier coefficients. As a numerical example, a model codf is estimated based on a set of discrete crystal orientations given by a full-constrained Taylor type texture simulation.     

Using partial probability weighted moments and partial maximum entropy to estimate quantiles from censored samples

The maximum entropy principle constrained by probability weighted moments is an useful technique for unbiasedly and efficiently estimating the quantile function of a random variable from a sample of complete observations. However, censored or incomplete data are often encountered in engineering reliability and lifetime distribution analysis. This paper presents a new distribution free method for the estimation of the quantile function of a non-negative random variable using a censored sample of data, which is based on the principle of partial maximum entropy (MaxEnt) in which partial probability weighted moments (PPWMs) are used as constraints. Numerical results and practical examples presented in the paper confirm the accuracy and efficiency of the proposed partial MaxEnt quantile function estimation method for censored samples.     

Generalizations of the maximum average correlation height filter

Recently several approaches have been presented in which the shape of the correlation peak is used to distinguish between target and clutter. The well-known maximum average correlation height (MACH) filter was specifically designed to produce similar correlation planes for target variations present in the training set. Results are presented of a study of certain generalizations of the MACH filter intended to enhance the performance in clutter. It is shown that by taking into account the nonoverlapping character of the background noise and focusing the MACH correlation plane similarity requirement to the peak neighborhood, it is possible to simultaneously achieve a small variation in correlation peak shape and high peak-to-sidelobe ratios for cluttered images.     

Dense granular flow at the critical state: maximum entropy and topological disorder

After extensive quasi-static shearing, dense dry granular flows attain a steady-state condition of porosity and deviatoric stress, even as particles are continually rearranged. The paper considers two-dimensional flow and derives the probability distributions of two topological measures of particle arrangement—coordination number and void valence—that maximize topological entropy. By only considering topological dispersion, the method closely predicts the distribution of void valences, as measured in discrete element (DEM) simulations. Distributions of coordination number are also derived by considering packings that are geometrically and kinetically consistent with the particle sizes and friction coefficient. A cross-entropy principle results in a distribution of coordination numbers that closely fits DEM simulations.     

Kernel density maximum entropy method with generalized moments for evaluating probability distributions,including tails,from a small sample of data

In this paper, a novel method to determine the distribution of a random variable from a sample of data is presented. The approach is called generalized kernel density maximum entropy method, because it adopts a kernel density representation of the target distribution, while its free parameters are determined through the principle of maximum entropy (ME). Here, the ME solution is determined by assuming that the available information is represented from generalized moments, which include as their subsets the power and the fractional ones. The proposed method has several important features: (1) applicable to distributions with any kind of support, (2) computational efficiency because the ME solution is simply obtained as a set of systems of linear equations, (3) good trade‐off between bias and variance, and (4) good estimates of the tails of the distribution, in the presence of samples of small size. Moreover, the joint application of generalized kernel density maximum entropy with a bootstrap resampling allows to define credible bounds of the target distribution. The method is first benchmarked through an example of stochastic dynamic analysis. Subsequently, it is used to evaluate the seismic fragility functions of a reinforced concrete frame, from the knowledge of a small set of available ground motions.     

Transformation of time-resolved spectra to lifetime-resolved spectra by maximum entropy inversion of the laplace transform

We present a method for the analysis of time-resolved spectroscopic data following first-order kinetics. The time traces at all the available spectroscopic channels (e.g., wavelength or wavenumber) are inverse Laplace transformed. The transformation is stabilized by the maximum entropy method generalized for solutions without sign-restriction. In this way, time-resolved spectra can be converted to lifetime-resolved spectra, where bands appear at coordinates corresponding to their spectroscopic maxima and time constant of appearance (negative amplitude) or disappearance (positive amplitude). From the lifetime-resolved spectra, the number of exponentially decaying components, their time constants, and their decay-associated spectra are readily available. Moreover, since bands are spread in two dimensions extra band-resolution is possible. We named this method of transforming time-resolved spectra into lifetime-resolved spectra multi-spectroscopic channel maximum entropy inversion of the Laplace transform (M-MaxEnt-iLT). The basis of M-MaxEnt-iLT is presented in detail and its properties and limitations are thoroughly discussed. We also show how the combination of M-MaxEnt-iLT with spectral smoothing or deconvolution can improve the appearance and/or band resolution of the obtained lifetime-resolved spectra.     

Application of the maximum entropy method to spectral-domain optical coherence tomography for enhancing axial resolution

For the first time we applied the maximum entropy method (MEM) to spectral domain optical coherence tomography to enhance axial resolution (AR). The MEM estimates the power spectrum by fitting. For an onion with optimization of M = 70, the AR of 18.8 microm by discrete Fourier transform (DFT) was improved three times compared with peak widths. The calculation time by the MEM with M = 70 was 20 times longer than that of DFT. However, further studies are needed for practical applications, because the validity of the MEM depends on the sample structures.     

Phase retrieval from intensity-only data by relative entropy minimization

A recursive algorithm, which appears to be new, is presented for estimating the amplitude and phase of a wave field from intensity-only measurements on two or more scan planes at different axial positions. The problem is framed as a nonlinear optimization, in which the angular spectrum of the complex field model is adjusted in order to minimize the relative entropy, or Kullback-Leibler divergence, between the measured and reconstructed intensities. The most common approach to this so-called phase retrieval problem is a variation of the well-known Gerchberg-Saxton algorithm devised by Misell (J. Phys. D6, L6, 1973), which is efficient and extremely simple to implement. The new algorithm has a computational structure that is very similar to Misell's approach, despite the fundamental difference in the optimization criteria used for each. Based upon results from noisy simulated data, the new algorithm appears to be more robust than Misell's approach and to produce better results from low signal-to-noise ratio data. The convergence of the new algorithm is examined.     

Pure absorption-mode spectra from Bayesian maximum entropy analysis of ion cyclotron resonance time-domain signals

Fourier transform ion cyclotron resonance (FT/ICR) mass spectra are normally reported in the (phase-independent) magnitude-mode format. In principle, the absorption-mode format offers spectral resolution enhanced by a factor ranging from square root of 3 to 2 over the corresponding magnitude-mode spectrum obtained by discrete FT of the same unapodized time-domain data. However, an absorption-mode display is generally unsuitable in practice because of the auxiliary spectral peaks (Gibbs oscillations) resulting from the relatively long time delay between excitation and detection. Although the resulting large phase variation (up to 100 pi rad or more across the Nyquist spectral bandwidth) can be corrected exactly for a continuous time-domain signal, phase correction of a discrete time-domain signal results in Gibbs oscillations even for a perfectly phased absorption-mode spectrum. In this paper, we show that a Bayesian "maximum-entropy" analysis of simulated and experimental ion cyclotron resonance time-domain noisy signals can recover a precisely phased absorption-mode frequency-domain spectrum that is devoid of Gibbs oscillations, is less sensitive to noise, and offers improved mass accuracy over that obtained from a conventional magnitude-mode discrete fast Fourier transform (FFT) spectrum.     

微型飞行器数据集成的硬件实现

为了实现微型飞行器的自主飞行控制,需要将多种传感器的数据集成起来,这些传感器包括摄像系统、GPS、高度计、加速度计、风速计、陀螺等.本文兼顾微型飞行器体积小、重量轻的要求,使用LM331作为关键器件使数据易于处理,提高了数据处理的速度和控制的反应速度,并给出了具体的方案和处理结果,证明了方案的可行件和有效性.另外,在研究微型飞行器的动力特性时,利用LM331的F/V转换特性,设计并研制了微型动力测试仪.