Laguerre Expansion Research Articles

Time-resolved Fluorescence Spectra of Arterial Fluorescent Compounds: Reconstruction with the Laguerre Expansion Technique

Time-resolved Fluorescence Spectra of Arterial Fluorescent Compounds: Reconstruction with the Laguerre Expansion Technique

The moments method for multi‐modal multi‐component aerosols as applied to the coagulation‐type equation

AbstractThe idea of self‐preserving functions as solutions to a coagulation‐type equation for aerosol is investigated. The aerosol spectrum is assumed to be well represented by a finite but not limited sum of self‐preserving functions (modes). An original set of aerosol dynamics equations for an unlimited number of modes is formulated for self‐preserving functions in the form of the log‐normal distribution, gamma distribution and Laguerre expansion. Additional equations governing the composition of the aerosol are derived. The system of the obtained ordinary integro–differential equations is solved using the Runge–Kutta method with Gaussian quadratures to estimate integrals. The solutions are then successfully tested against direct high resolution numerical solutions of the coagulation‐type equation. Some physical examples illustrating the use of the developed numerical method are also presented. Copyright © 2007 Royal Meteorological Society

Error analysis of a Collocation method for numerically inverting a Laplace transform in case of real samples

In [S. Cuomo, L. D’Amore, A. Murli, M.R. Rizzardi, Computation of the inverse Laplace transform based on a collocation method which uses only real values, J. Comput. Appl. Math., 198 (1) (2007) 98–115] the authors proposed a Collocation method (C-method) for real inversion of Laplace transforms (Lt), based on the truncated Laguerre expansion of the inverse function: f N ( x ) = e σ x ∑ k = 0 N - 1 c k e - bx L k ( 2 bx ) , where σ , b are parameters and c k , k ∈ N , are the MacLaurin coefficients of a function depending on the Lt. The computational kernel of a C-method is the solution of a Vandermonde linear system, where the right hand side is obtained evaluating the Lt on the real axis. The Bjorck Pereira algorithm has been used for solving the Vandermonde linear system, providing a computable componentwise error bound on the solution. For an inversion problem on discrete data F is known on a pre-assigned set of points (we refer to these points as samples of F) only and the major challenge is to deal with a significative loss of information. A natural approach to overcome this intrinsic difficulty is to construct a suitable fitting model that approximates the given data. In this case, we show that such approach leads to a C-method with perturbed right hand side, and then we use again the Bjorck Pereira algorithm. Starting from the error introduced by the fitting model, we study its propagation in order to determine the maximum attainable accuracy on f N . Moreover we derive a computable error bound that allows to get the suitable value of the parameter N that gives the maximum attainable accuracy.

A sharp weighted transplantation theorem for Laguerre function expansions

We find the sharp range of boundedness for transplantation operators associated with Laguerre function expansions in L p spaces with power weights. Namely, the operators interchanging { L k α } and { L k β } are bounded in L p ( y δ p ) if and only if − ρ 2 − 1 p < δ < 1 − 1 p + ρ 2 , where ρ = min { α , β } . This improves a previous partial result by Stempak and Trebels, which was only sharp for ρ ⩽ 0 . Our approach is based on new multiplier estimates for Hermite expansions, weighted inequalities for local singular integrals and a careful analysis of Kanjin's original proof of the unweighted case. As a consequence we obtain new results on multipliers, Riesz transforms and g-functions for Laguerre expansions in L p ( y δ p ) .

New Methods for Time-resolved Fluorescence Spectroscopy Data Analysis Based on the Laguerre Expansion Technique

A new deconvolution method for the analysis of time-resolved laser-induced fluorescence spectroscopy (TR-LIFS) data is introduced and applied for tissue diagnosis. The intrinsic TR-LIFS decays are expanded on a Laguerre basis, and the computed Laguerre expansion coefficients (LEC) are used to characterize the sample fluorescence emission. The method was applied for the diagnosis of atherosclerotic vulnerable plaques. At a first stage, using a rabbit atherosclerotic model, 73 TR-LIFS in-vivo measurements from the normal and atherosclerotic aorta segments of eight rabbits were taken. The Laguerre deconvolution technique was able to accurately deconvolve the TR-LIFS measurements. More interesting, the LEC reflected the changes in the arterial biochemical composition and provided discrimination of lesions rich in macrophages/foam-cells with high sensitivity (> 85%) and specificity (> 95%). At a second stage, 348 TR-LIFS measurements were obtained from the explanted carotid arteries of 30 patients. Lesions with significant inflammatory cells (macrophages/foam-cells and lymphocytes) were detected with high sensitivity (> 80%) and specificity (> 90%), using LEC-based classifiers. This study has demonstrated the potential of using TR-LIFS information by means of LEC for in vivo tissue diagnosis, and specifically for detecting inflammation in atherosclerotic lesions, a key marker of plaque vulnerability.

A Method to Reduce the Oscillations of the Solution of Time Domain Integral Equation Using Laguerre Polynomials

In this paper, the reason of oscillations appeared in late time low energy region of time domain electric flled integral equation using laguerre polynomials is analyzed. And then, we propose an approach to reduce the oscillations by combining frequency and time domain. According to the Fourier transform relation between frequency responses and time responses, the solution of TDIE method with Laguerre expansion is absolutely convergent. Replace the spectrum of oscillation parts of time domain data with accurate frequency data by MOM. The stable time domain data can be obtained by performing IFFT to the revised frequency data. Numerical results are presented to illustrate the e-ciency of this approach. 1. INTRODUCTION For solving the time domain integral equation, the marching on in time (MOT) method (1) is usually employed. However, as pointed by many researchers, some of the TD-EFIE formulations associated with the MOT method sufiers from late-time instability, which usually takes the form of an expo- nentially increasing high frequency oscillation. In order to eliminate the instability of the MOT methods, the approximate averaging method (2) and the implicit scheme based on backward flnite difierence (3) approximation or central flnite difierence (4) methodology are proposed. Even though employing these methods, the solution obtained by using MOT has still exponentially increased late-time oscillations but only extend the span of the stable region. Young-seek Chung and Sarkar (5{7) proposed a marching on in order method of using the expansion by weighted Laguerre poly- nomials for temporal coe-cients of Hertz vector. The advantages of using the weighted Laguerre polynomials are the solution is independent of the time discretization, and all the computations are carried out using spatial variables. So the exponentially increasing solution will not emerge in late time. But there still exist some oscillations in late-time low energy region due to oscilla- tions characteristic of weighted Laguerre polynomials. Oscillation phenomenon efiects on obtaining other electromagnetic characteristic from the late time scattering data. In this paper, the reason of oscillations appeared in late time low energy region of time domain electric flled integral equation using laguerre polynomials is analyzed, and then, we consider absolute convergence of the time so- lution and the Fourier transform relation between temporal response and frequency response. An approach based on discrete Fourier transforms to improve the stabilities in late time is proposed.

On ladder height densities and Laguerre series in the study of stochastic functionals. I. Basic methods and results

In this paper we develop methods for reducing the study, the computation, and the construction of stochastic functionals to those of standard concepts such as the moments of the pertinent random variables. Principally, our methods are based on the notion of ladder height densities and their Laguerre expansions, and our results provide a unifying framework for the distinct approaches of Dufresne (2000) and Schröder (2005).

Fractional differential equations through Laguerre expansions in abstract spaces: error estimates

An approximation procedure by the means of expansion with respect to Laguerre orthogonal basis of the space is given. It is applied to solutions of a class of convolution equations, by transforming them to corresponding systems of algebraic equations for the coefficients. Especially, Volterra-type integral and integro-differential equations and system of equations with fractional derivatives modeling viscoelastic rod are solved. Error estimates depending on the order of approximation are derived.

Laguerre expansion on the Heisenberg group and Fourier-Bessel transform on ℂ n

Given a principal value convolution on the Heisenberg group H n = ℂ n × ℝ, we study the relation between its Laguerre expansion and the Fourier-Bessel expansion of its limit on ℂ n . We also calculate the Dirichlet kernel for the Laguerre expansion on the group H n .

Principal dynamic mode analysis of action potential firing in a spider mechanoreceptor

The encoding of mechanical stimuli into action potentials in two types of spider mechanoreceptor neurons is modeled by use of the principal dynamic modes (PDM) methodology. The PDM model is equivalent to the general Wiener-Bose model and consists of a minimum set of linear dynamic filters (PDMs), followed by a multivariate static nonlinearity and a threshold function. The PDMs are obtained by performing eigen-decomposition of a matrix constructed using the first-order and second-order Volterra kernels of the system, which are estimated by means of the Laguerre expansion technique, utilizing measurements of pseudorandom mechanical stimulation (input signal) and the resulting action potentials (output signal). The static nonlinearity, which can be viewed as a measure of the probability of action potential firing as a function of the PDM output values, is computed as the locus of points of the latter that correspond to output action potentials. The performance of the model is assessed by computing receiver operating characteristic (ROC) curves, akin to the ones used in decision theory and quantified by computing the area under the ROC curve. Three PDMs are revealed by the analysis. The first PDM exhibits a high-pass characteristic, illustrating the importance of the velocity of slit displacement in the generation of action potentials at the mechanoreceptor output, while the second and third PDMs exhibit band-pass and low-pass characteristics, respectively. The corresponding three-input nonlinearity exhibits asymmetric behavior with respect to its arguments, suggesting directional dependence of the mechanoreceptor response on the mechanical stimulation and the PDM outputs, in agreement to our findings from a previous study (Ann Biomed Eng 27:391-402, 1999). Differences between the Type A and B neurons are observed in the zeroth-order Volterra kernels (related to the average firing), as well as in the magnitudes of the second and third PDMs that perform band-pass and low-pass processing of the input signal, respectively.

Splitting of the +' Space into a Union of Banach Spaces through Laguerre Expansions

AbstractWe split the space of tempered distributions into a union of Banach spaces with respect to the scale of spaces, which allows us to give the approximation of the generalization of function therein.

Laguerre-based method for analysis of time-resolved fluorescence data: application to in-vivo characterization and diagnosis of atherosclerotic lesions.

We report the application of the Laguerre deconvolution technique (LDT) to the analysis of in-vivo time-resolved laser-induced fluorescence spectroscopy (TR-LIFS) data and the diagnosis of atherosclerotic plaques. TR-LIFS measurements were obtained in vivo from normal and atherosclerotic aortas (eight rabbits, 73 areas), and subsequently analyzed using LDT. Spectral and time-resolved features were used to develop four classification algorithms: linear discriminant analysis (LDA), stepwise LDA (SLDA), principal component analysis (PCA), and artificial neural network (ANN). Accurate deconvolution of TR-LIFS in-vivo measurements from normal and atherosclerotic arteries was provided by LDT. The derived Laguerre expansion coefficients reflected changes in the arterial biochemical composition, and provided a means to discriminate lesions rich in macrophages with high sensitivity (>85%) and specificity (>95%). Classification algorithms (SLDA and PCA) using a selected number of features with maximum discriminating power provided the best performance. This study demonstrates the potential of the LDT for in-vivo tissue diagnosis, and specifically for the detection of macrophages infiltration in atherosclerotic lesions, a key marker of plaque vulnerability.

Autonomic Cardiovascular Control Following Transient Arousal From Sleep: A Time-Varying Closed-Loop Model

Recent studies suggest that exposure to repetitive episodes of hypoxia and transient arousal can lead to increased risk for cardiovascular disease in patients with obstructive sleep apnea syndrome (OSAS). To obtain an improved understanding of and to quantitatively characterize the autonomic effects of arousal from sleep, a time-varying closed-loop model was used to determine the interrelationships among respiration, heart rate and blood pressure in 8 normal adults. A recursive least squares algorithm was used in combination with the Laguerre expansion technique to estimate the time-varying impulse responses of the 4 model components. We found that during arousal: 1) respiratory-cardiac coupling gain increases in nonrapid-eye movement (NREM) but not in REM sleep; 2) in both NREM and REM sleep, baroreflex gain shows an initial increase, but this is followed by a more sustained decrease below pre-arousal baseline levels, allowing sympathetic tone to be elevated over a relatively long duration; 3) the gains of other model components show increases with arousal that are consistent with the increased sympathetic modulation of systemic vascular resistance and contractility of the heart. These findings establish a normative database against which further measurements of cardiovascular arousal responses in OSAS may be compared.

Distribution of a sum of weighted noncentral chi-square variables

We derive Laguerre expansions for the density and distribution functions of a sum of positive weighted noncentral chi-square variables. The procedure that we use is based on the inversion of Laplace transforms. The formulas so obtained depend on certain parameters, which adequately chosen will give some expansions already known in the literature and some new ones. We also derive precise bounds for the truncation error.

Laguerre-Model Blind System Identification: Cardiovascular Dynamics Estimated From Multiple Peripheral Circulatory Signals

This paper presents a method for comparing multiple circulatory waveforms measured at different locations to improve cardiovascular parameter estimation from these signals. The method identifies the distinct vascular dynamics that shape each waveform signal, and estimates the common cardiac flow input shared by them. This signal-processing algorithm uses the Laguerre function series expansion for modeling the hemodynamics of each arterial branch, and identifies unknown parameters in these models from peripheral waveforms using multichannel blind system identification. An effective technique for determining the Laguerre base pole is developed, so that the Laguerre expansion captures and quickly converges to the intrinsic arterial dynamics observed in the two circulatory signals. Furthermore, a novel deconvolution method is developed in order to stably invert the identified dynamic models for estimating the cardiac output (CO) waveform from peripheral pressure waveforms. The method is applied to experimental swine data. A mean error of less than 5% with the measured peripheral pressure waveforms has been achieved using the models and excellent agreement between the estimated CO waveforms and the gold standard measurements have been obtained.

Ultrafast Method for the Analysis of Fluorescence Lifetime Imaging Microscopy Data Based on the Laguerre Expansion Technique.

We report a new deconvolution method for fluorescence lifetime imaging microscopy (FLIM) based on the Laguerre expansion technique. The performance of this method was tested on synthetic and real FLIM images. The following interesting properties of this technique were demonstrated. 1) The fluorescence intensity decay can be estimated simultaneously for all pixels, without a priori assumption of the decay functional form. 2) The computation speed is extremely fast, performing at least two orders of magnitude faster than current algorithms. 3) The estimated maps of Laguerre expansion coefficients provide a new domain for representing FLIM information. 4) The number of images required for the analysis is relatively small, allowing reduction of the acquisition time. These findings indicate that the developed Laguerre expansion technique for FLIM analysis represents a robust and extremely fast deconvolution method that enables practical applications of FLIM in medicine, biology, biochemistry, and chemistry.

General methodology for nonlinear modeling of neural systems with Poisson point-process inputs

This paper presents a general methodological framework for the practical modeling of neural systems with point-process inputs (sequences of action potentials or, more broadly, identical events) based on the Volterra and Wiener theories of functional expansions and system identification. The paper clarifies the distinctions between Volterra and Wiener kernels obtained from Poisson point-process inputs. It shows that only the Wiener kernels can be estimated via cross-correlation, but must be defined as zero along the diagonals. The Volterra kernels can be estimated far more accurately (and from shorter data-records) by use of the Laguerre expansion technique adapted to point-process inputs, and they are independent of the mean rate of stimulation (unlike their P–W counterparts that depend on it). The Volterra kernels can also be estimated for broadband point-process inputs that are not Poisson. Useful applications of this modeling approach include cases where we seek to determine (model) the transfer characteristics between one neuronal axon (a point-process ‘input’) and another axon (a point-process ‘output’) or some other measure of neuronal activity (a continuous ‘output’, such as population activity) with which a causal link exists.

Introduction to the Besov Spaces and Triebel-Lizorkin Spaces for Hermite and Laguerre expansions and some applications

We introduced new definitions of Besov spaces and Triebel-Lizorkin spaces associated with multidimensional Hermite expansions and multidimensional Laguerre expansions. We showed that the set of p-integrable functions is a Triebel-Lizorkin space with respect to the Gaussian measure and similarly, with respect to the probabilistic Gamma measure. Also, we showed that the Gaussian Sobolev spaces and Laguerre Sobolev spaces are Triebel-Lizorkin spaces, associated with Hermite and Laguerre expansions respectively. We defined Carleson measures with respect to the Gaussian measure and probabilistic Gamma measure. By using maximal functions, related to the Ornstein Uhlenbeck semigroup and Laguerre semigroup, we studied these measures, giving a version of Fefferman

Elucidating nonlinear baroreflex and respiratory contributions to heart rate variability in obstructive sleep apnea syndrome.

Using a Volterra-Wiener model and the Laguerre expansion technique, we estimated in a previous study the parameters that characterize linear and the second order effects of respiration ("RSA") and arterial blood pressure ("ABR") on heart rate. RSA and ABR gains were significantly lower in Obstructive Sleep Apnea (OSA) patients than in normal subjects. During sleep, ABR gain increased in normals but remained unchanged in OSA. In the present work, we investigated the physiological interpretation of the nonlinear components of the described model of heart rate variability, by means of simulation on the computed linear and nonlinear kernels. Our results indicate that the 2ndorder kernels reflect specific characteristics of the RSA and ABR mechanisms, such as a RSA frequency response dependence upon tidal volume, saturation in the ABR-Blood Pressure relation, and respiratory modulation of ABR.

FORMAL LINEARIZATION OF NONLINEAR TIME-VARYING DYNAMIC SYSTEMS USING CHEBYSHEV AND LAGUERRE POLYNOMIALS

This paper is concerned with a formal linearization problem for a general class of nonlinear time-varying dynamic systems. To a given system, a linearization function is made up of Chebyshev polynomials about its state variables. The nonlinear time-varying system is transformed into a linear time-varying system in terms of the linearization function using Chebyshev interpolation to state variables and Laguerre expansion to time variable. An error bound formula of this linearization which is derived in this paper explains that the accuracy of this algorithm is improved as the order of Chebyshev and Laguerre polynomials increases. As its application, a nonlinear observer is designed to demonstrate the usefulness of this formal linearization approach.