Unfolding Sphere Size Distributions with a Density Estimator Based on Tikhonov Regularization

1997 ◽  
Vol 138 (2) ◽  
pp. 331-353 ◽  
Author(s):  
J. Weese ◽  
E. Korat ◽  
D. Maier ◽  
J. Honerkamp
Keyword(s):  
Tikhonov Regularization ◽  
Density Estimator ◽  
Size Distributions ◽  
Sphere Size

scholarly journals A Software Package to Simplify Tikhonov Regularization with Examples for Matrix-Based Inversion of SMPS and HTDMA Data

2021 ◽  
Author(s):  
Markus D. Petters
Keyword(s):  
Tikhonov Regularization ◽  
Software Package ◽  
Large Scale ◽  
Regularization Parameter ◽  
Low Cost ◽  
Measurement Techniques ◽  
Numerical Algorithms ◽  
Department Of Energy ◽  
Design Matrix ◽  
Size Distributions

Abstract. Tikhonov regularization is a tool for reducing noise amplification during data inversion. This work introduces RegularizationTools.jl, a general-purpose software package to apply Tikhonov regularization to data. The package implements well-established numerical algorithms and is suitable for systems of up to ~1000 equations. Included is an abstraction to systematically categorize specific inversion configurations and their associated hyperparameters. A generic interface translates arbitrary linear forward models defined by a computer function into the corresponding design matrix. This obviates the need to explicitly write out and discretize the Fredholm integral equation, thus facilitating fast prototyping of new regularization schemes associated with measurement techniques. Example applications include the inversion involving data from scanning mobility particle sizers (SMPS) and humidified tandem differential mobility analyzers (HTDMA). Inversion of SMPS size distributions reported in this work builds upon the freely-available software DifferentialMobilityAnalyzers.jl. The speed of inversion is improved by a factor of ~200, now requiring between 2 and 5 ms per SMPS scan when using 120 size bins. Previously reported occasional failure to converge to a valid solution is reduced by switching from the L-curve method to generalized cross-validation as the metric to search for the optimal regularization parameter. Higher-order inversions resulting in smooth, denoised reconstructions of size distributions are now included in DifferentialMobilityAnalyzers.jl. This work also demonstrates that an SMPS-style matrix-based inversion can be applied to find the growth factor frequency distribution from raw HTDMA data, while also accounting for multiply-charged particles. The outcome of the aerosol-related inversion methods is showcased by inverting multi-week SMPS and HTDMA datasets from ground-based observations, including SMPS data obtained at Bodega Bay Marine Laboratory during the Calwater 2/ACAPEX campaign, and co-located SMPS and HTDMA data collected at the U.S. Department of Energy observatory located at the Southern Great Plains site in Oklahoma, U.S.A. Results show that the proposed approaches are suitable for unsupervised, nonparametric inversion of large-scale datasets as well as inversion in real-time during data acquisition on low-cost reduced-instruction-set architectures used in single-board computers. The included software implementation of Tikhonov regularization is freely-available, general, and domain-independent, and thus can be applied to many other inverse problems arising in atmospheric measurement techniques and beyond.

scholarly journals Automated SEM Image Analysis of the Sphere Diameter, Sphere-Sphere Separation, and Opening Size Distributions of Nanosphere Lithography Masks

2021 ◽  
pp. 1-11
Author(s):  
Thomas Riedl ◽  
Jörg K.N. Lindner
Keyword(s):  
Detection Efficiency ◽  
Size Variation ◽  
Cost Effective ◽  
Automated Analysis ◽  
Sphere Diameter ◽  
Size Distributions ◽  
Sem Images ◽  
Detection Techniques ◽  
Sem Image ◽  
Sphere Size

Abstract Colloidal nanosphere monolayers—used as a lithography mask for site-controlled material deposition or removal—offer the possibility of cost-effective patterning of large surface areas. In the present study, an automated analysis of scanning electron microscopy (SEM) images is described, which enables the recognition of the individual nanospheres in densely packed monolayers in order to perform a statistical quantification of the sphere size, mask opening size, and sphere-sphere separation distributions. Search algorithms based on Fourier transformation, cross-correlation, multiple-angle intensity profiling, and sphere edge point detection techniques allow for a sphere detection efficiency of at least 99.8%, even in the case of considerable sphere size variations. While the sphere positions and diameters are determined by fitting circles to the spheres edge points, the openings between sphere triples are detected by intensity thresholding. For the analyzed polystyrene sphere monolayers with sphere sizes between 220 and 600 nm and a diameter spread of around 3% coefficients of variation of 6.8–8.1% for the opening size are found. By correlating the mentioned size distributions, it is shown that, in this case, the dominant contribution to the opening size variation stems from nanometer-scale positional variations of the spheres.

scholarly journals Revisiting matrix-based inversion of scanning mobility particle sizer (SMPS) and humidified tandem differential mobility analyzer (HTDMA) data

2021 ◽  
Vol 14 (12) ◽  
pp. 7909-7928
Author(s):  
Markus D. Petters
Keyword(s):  
Tikhonov Regularization ◽  
Large Scale ◽  
Regularization Parameter ◽  
Low Cost ◽  
Measurement Techniques ◽  
Department Of Energy ◽  
Design Matrix ◽  
Size Distributions ◽  
Scanning Mobility Particle Sizer ◽  
Differential Mobility

Abstract. Tikhonov regularization is a tool for reducing noise amplification during data inversion. This work introduces RegularizationTools.jl, a general-purpose software package for applying Tikhonov regularization to data. The package implements well-established numerical algorithms and is suitable for systems of up to ~1000 equations. Included is an abstraction to systematically categorize specific inversion configurations and their associated hyperparameters. A generic interface translates arbitrary linear forward models defined by a computer function into the corresponding design matrix. This obviates the need to explicitly write out and discretize the Fredholm integral equation, thus facilitating fast prototyping of new regularization schemes associated with measurement techniques. Example applications include the inversion involving data from scanning mobility particle sizers (SMPSs) and humidified tandem differential mobility analyzers (HTDMAs). Inversion of SMPS size distributions reported in this work builds upon the freely available software DifferentialMobilityAnalyzers.jl. The speed of inversion is improved by a factor of ~200, now requiring between 2 and 5 ms per SMPS scan when using 120 size bins. Previously reported occasional failure to converge to a valid solution is reduced by switching from the L-curve method to generalized cross-validation as the metric to search for the optimal regularization parameter. Higher-order inversions resulting in smooth, denoised reconstructions of size distributions are now included in DifferentialMobilityAnalyzers.jl. This work also demonstrates that an SMPS-style matrixbased inversion can be applied to find the growth factor frequency distribution from raw HTDMA data while also accounting for multiply charged particles. The outcome of the aerosol-related inversion methods is showcased by inverting multi-week SMPS and HTDMA datasets from ground-based observations, including SMPS data obtained at Bodega Marine Laboratory during the CalWater 2/ACAPEX campaign and co-located SMPS and HTDMA data collected at the US Department of Energy observatory located at the Southern Great Plains site in Oklahoma, USA. Results show that the proposed approaches are suitable for unsupervised, nonparametric inversion of large-scale datasets as well as inversion in real time during data acquisition on low-cost reducedinstruction- set architectures used in single-board computers. The included software implementation of Tikhonov regularization is freely available, general, and domain-independent and thus can be applied to many other inverse problems arising in atmospheric measurement techniques and beyond.

A wavelet–vaguelet method for unfolding sphere size distributions

2002 ◽  
Vol 18 (1) ◽  
pp. 79-94 ◽  
Author(s):  
Sylvie Champier ◽  
Laurence Grammont
Keyword(s):  
Size Distributions ◽  
Sphere Size
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