Twomey Algorithm Research Articles

Simulation research on improved regularized solution of the inverse problem in spectral extinction measurements

We present further results of the simulation research on the constrained regularized least squares (CRLS) solution of the ill-conditioned inverse problem in spectral extinction (turbidimetric) measurements, which we originally presented in this journal [Appl. Opt. 49, 4591 (2010)]. The inverse problem consists of determining the particle size distribution (PSD) function of a particulate system on the basis of a measured extinction coefficient as a function of wavelength. In our previous paper, it was shown that under assumed conditions the problem can be formulated in terms of the discretized Fredholm integral equation of the first kind. The CRLS method incorporates two constraints, which the PSD sought will satisfy: nonnegativity of the PSD values and normalization of the PSD to unity when integrated over the whole range of particle size, into the regularized least squares (RLS) method. This leads to the quadratic programming problem, which is solved by means of the active set algorithm within the research. The simulation research that is the subject of the present paper is a continuation and extension of the research described in our previous paper. In the present research, the performance of the CRLS method variants is compared not only to the corresponding RLS method variants but also to other regularization techniques: the truncated generalized singular value decomposition and the filtered generalized singular value decomposition, as well as nonlinear iterative algorithms: The Twomey algorithm and the Twomey-Markowski algorithm. Moreover, two methods of selecting the optimum value of the regularization parameter are considered: The L-curve method and the generalized cross validation method. The results of our simulation research provide even stronger proof that the CRLS method performs considerably better with reconstruction of PSD than other inversing methods, in terms of better fidelity and smaller uncertainty.

Investigation of low-angle laser light scattering patterns using the modified Twomey iterative method for particle sizing

We first applied the Twomey iteration method to the low-angle laser light scattering (LALLS) method. The Twomey method is known by the simple iteration algorithm for Fredholm integral equation. However, the algorithm is not applied to the LALLS result itself because the kernel function pattern is not stable. We solved the unstable kernel issue by modifying the Twomey algorithm to fit the LALLS data in this study. The performance of inversion was studied by computer simulation and experimental results from detectors containing 24 elements from angle 0.0005 to 2.5 rad. The computer simulation was carried out for particles of mean sizes from 0.1 to 1000 μm with mono-dispersion and log-normal distributions. A conventional algorithm is also carried out to compare the performance of the Twomey method. The noise effect of the inverse problem was tested by computer simulation. Experimental results were measured with standard polystyrene latex from 0.081 to 1007 μm. All the tests were performed under conditions in which the light from a linearly polarized laser at 633 nm was scattered by a diluted aqueous suspension. The modified Twomey (MT) method and conventional method show good reproduction results for particles with log-normal distributions. However, narrow distribution particles indicate that the MT method shows excellent reproduction results when compared with the conventional method.

Comparison of particle sizes determined with impactor, AFM and SEM

Particles size comparisons were made between conventional aerodynamic and mobility sizing techniques and physical geometric sizes measured by high resolution microscopes. Atmospheric particles were collected during the wet and dry seasons in the Amazonian ecosystems. Individual particles deposited on four stages of the MOUDI (Micro-Orifice Uniform Deposition Impactor) were characterised for particle volumes, projected surface diameters and morphologies with an Atomic Force Microscope (AFM) and a Scanning Electron Microscope (SEM). AFM and SEM size distributions were verified against distributions derived from response functions of individual MOUDI stages as specified by Winklmayr et al. [Winklmayr, W., Wang, H.-C., John, W., 1990. Adaptation of the Twomey algorithm to the inversion of cascade impactor data. Aerosol Science and Technology 13, 322–331.]. Particles indicated inherent discrepancies in sizing techniques. Particle volumes were systematically lower than expected by factors of up to 3.6. Differences were attributed to loss of mass, presumably water adsorbed on particles. Losses were high and could not be accounted for by measured humidity growth factors suggesting significant losses of other volatile compounds as well, particularly on particles that were collected during the wet season. Microscopy results showed that for hygroscopic particles, microscopy sizes depend on the relative humidity history of particles before and after sampling. Changes in relative humidity significantly altered particle morphologies. Depending on when changes occur, such losses will bias not only microscopy particle sizes but also impactor mass distributions and number concentrations derived from collected particles.

Effects of fMRI–EEG mismatches in cortical current density estimation integrating fMRI and EEG: A simulation study

ObjectiveMultimodal functional neuroimaging by combining functional magnetic resonance imaging (fMRI) and electroencephalography (EEG) has been studied to achieve high-resolution reconstruction of the spatiotemporal cortical current density (CCD) distribution. However, mismatches between these two imaging modalities may occur due to their different underlying mechanisms. The aim of the present study is to investigate the effects of different types of fMRI–EEG mismatches, including fMRI invisible sources, fMRI extra regions and fMRI displacement, on the fMRI-constrained cortical imaging in a computer simulation based on realistic-geometry boundary-element-method (BEM) model. MethodsTwo methods have been adopted to integrate the synthetic fMRI and EEG data for CCD imaging. In addition to the well-known 90% fMRI-constrained Wiener filter approach (Liu AK, Belliveau JW, Dale AM. PNAS 1998;95:8945–8950.), we propose a novel two-step algorithm (referred to as ‘Twomey algorithm’) for fMRI–EEG integration. In the first step, a ‘hard’ spatial prior derived from fMRI is imposed to solve the EEG inverse problem with a reduced source space; in the second step, the fMRI constraint is removed and the source estimate from the first step is re-entered as the initial guess of the desired solution into an EEG least squares fitting procedure with Twomey regularization. Twomey regularization is a modified Tikhonov technique that attempts to simultaneously minimize the distance between the desired solution and the initial estimate, and the residual errors of fitness to EEG data. The performance of the proposed Twomey algorithm has been evaluated both qualitatively and quantitatively along with the lead-field normalized minimum norm (WMN) and the 90% fMRI-weighted Wiener filter approach, under repeated and randomized source configurations. Point spread function (PSF) and localization error (LE) are used to measure the performance of different imaging approaches with or without a variety of fMRI–EEG mismatches. ResultsThe results of the simulation show that the Twomey algorithm can successfully reduce the PSF of fMRI invisible sources compared to the Wiener estimation, without losing the merit of having much lower PSF of fMRI visible sources relative to the WMN solution. In addition, the existence of fMRI extra sources does not significantly affect the accuracy of the fMRI–EEG integrated CCD estimation for both the Wiener filter method and the proposed Twomey algorithm, while the Twomey algorithm may further reduce the chance of occurring spurious sources in the extra fMRI regions. The fMRI displacement away from the electrical source causes enlarged localization error in the imaging results of both the Twomey and Wiener approaches, while Twomey gives smaller LE than Wiener with the fMRI displacement ranging from 1–2cm. With less than 2cm fMRI displacement, the LEs for the Twomey and Wiener approaches are still smaller than in the WMN solution. ConclusionsThe present study suggests that the presence of fMRI invisible sources is the most problematic factor responsible for the error of fMRI–EEG integrated imaging based on the Wiener filter approach, whereas this approach is relatively robust against the fMRI extra regions and small displacement between fMRI activation and electrical current sources. While maintaining the above advantages possessed by the Wiener filter approach, the Twomey algorithm can further effectively alleviate the underestimation of fMRI invisible sources, suppress fMRI spurious sources and improve the robustness against fMRI displacement. Therefore, the Twomey algorithm is expected to improve the reliability of multimodal cortical source imaging against fMRI–EEG mismatches. SignificanceThe proposed method promises to provide a useful alternative for multimodal neuroimaging integrating fMRI and EEG.

Estimation of actinic flux and photolysis rate constant of NO 2 from aerosol size data

Aerosol mass measurements made using Andersen cascade impactor at different locations in an urban area were used to study the variation in power-law exponent in the Junge-type number distribution. The discrete mass size data were inverted by modified Twomey algorithm to obtain continuous mass size distribution from which number distributions were derived. These were used for evaluating actinic fluxes and hence NO 2 photolysis rate constants employing a radiative transfer model with delta-Eddington approximation. Results of the spatial variation of Junge power-law exponent over the urban region are presented. The relative magnitude of photolysis rate constants over regions with differing aerosol emission characteristics are illustrated. Parametric studies indicated the importance of considering fractional elemental carbon content of the aerosols in the atmosphere. Results of a limited validation study of the model are also presented using quartz crystal microbalance impactor data at one of the sites.

On Markowski's Article “Improving Twomey's Algorithm for Inversion of Aerosol Measurement Data”

Please click here to find the Original Article to which this Letter refers: http://dx.doi.org/10.1080/02786828708959153

Adaptation of the Twomey Algorithm to the Inversion of Cascade Impactor Data

The Twomey nonlinear iterative algorithm for inverting aerosol size distribution data has been adapted for use with the Berner cascade impactor. Key factors affecting the performance of the algorithm have been identified. A procedure producing accurate inversions without artifacts includes establishing a set of continuous and smooth kernel functions based on measurements of the impactor cutoff characteristics, using modified weighting functions, placing constraints on the first-guess distribution, treating zero values and the ends of the distribution, and setting stopping criteria based on experimental errors. This general approach should be useful in solving inversion problems for other cascade impactors. The routine was extensively tested with numerically created test distributions and a large data base of ambient aerosol samples. An extended version of the program separates the inverted size distribution into a spectrum of lognormal distributions. Lognormal parameters from 44 ambient nitrate size distributions have been compared to the results of the inversion program of Dzubay and Hasan.

Improving Twomey's Algorithm for Inversion of Aerosol Measurement Data

Inversion of aerosol size distribution data is often needed because many aerosol measurement instruments have sufficiently broad responses that assuming sharp channel boundaries when reducing their raw data may result in substantial errors. Several routines are available to do reasonable inversions on cascade impactor and optical particle counter data. However, these routines often give mediocre results when used with Electrical Aerosol Analyzer data. Twomey's algorithm has been widely used, but it depends to some extent on the initial guess and often produces oscillatory results. Adding smoothing to the Twomey algorithm and an intelligent initial guess and stopping point greatly reduce these problems. Negative results do not occur as with unconstrained inversions. If fitting tolerances cannot be met, an approximate solution is always obtained, whereas other inversion techniques may sometimes stop due to lack of convergence in a subroutine. The performance of the improved inversion algorithm is comparable to the better routines for impactor data and offers important advantages over other routines for inversion of Electrical Aerosol Analyzer data.