Indirect Fourier transform in the context of statistical inference

Inferring structural information from the intensity of a small-angle scattering (SAS) experiment is an ill-posed inverse problem. Thus, the determination of a solution is in general non-trivial. In this work, the indirect Fourier transform (IFT), which determines the pair distance distribution function from the intensity and hence yields structural information, is discussed within two different statistical inference approaches, namely a frequentist one and a Bayesian one, in order to determine a solution objectively From the frequentist approach the cross-validation method is obtained as a good practical objective function for selecting an IFT solution. Moreover, modern machine learning methods are employed to suppress oscillatory behaviour of the solution, hence extracting only meaningful features of the solution. By comparing the results yielded by the different methods presented here, the reliability of the outcome can be improved and thus the approach should enable more reliable information to be deduced from SAS experiments.

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Maximum-entropy method as a routine tool for determination of particle size distributions by small-angle scattering

Journal of Applied Crystallography ◽

10.1107/s0021889803023069 ◽

2004 ◽

Vol 37 (1) ◽

pp. 32-39 ◽

Cited By ~ 10

Author(s):

Dragomir Tatchev ◽

Rainer Kranold

Keyword(s):

Particle Size ◽

Fourier Transform ◽

Small Angle ◽

Maximum Entropy Method ◽

Entropy Method ◽

Size Distributions ◽

Particle Size Distributions ◽

Angle Scattering ◽

The Fourier Transform

Several aspects of the application of the maximum-entropy method (MEM) to the determination of particle size distributions by small-angle scattering (SAS) are discussed. The `historic' version of the MEM produces completely satisfying results. Limiting the data error from below (i.e.imposing a minimal relative error) is proposed as a solution of some convergence problems. The MEM is tested against the Fourier transform technique. The size distribution of Pb particles in an Al–Pb alloy is determined by the MEM and the Fourier transform technique. The size distributions obtained by transmission electron microscopy (TEM) and SAXS show partial agreement.

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Automated Contamination Detection in Single-Cell Sequencing

10.1101/020859 ◽

2015 ◽

Cited By ~ 1

Author(s):

Markus Lux ◽

Barbara Hammer ◽

Alexander Sczyrba

Keyword(s):

Single Cell ◽

Automated Detection ◽

Single Cell Sequencing ◽

Machine Learning Methods ◽

Automatic Clustering ◽

Ill Posed ◽

Foreign Dna ◽

Subspace Projections ◽

Oligonucleotide Signatures ◽

Modern Machine

Novel methods for the sequencing of single-cell DNA offer tremendous opportunities. However, many techniques are still in their infancy and a major obstacle is given by sample contamination with foreign DNA. In this contribution, we present a pipeline that allows for fast, automated detection of contaminated samples by the use of modern machine learning methods. First, a vectorial representation of the genomic data is obtained using oligonucleotide signatures. Using non-linear subspace projections, data is transformed to be suitable for automatic clustering. This allows for the detection of one vs. more genomes (clusters) in a sample. As clustering is an ill-posed problem, the pipeline relies on a thorough choice of all involved methods and parameters. We give an overview of the problem and evaluate techniques suitable for this task.

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Polyelectrolytes and small angle scattering

EPJ Web of Conferences ◽

10.1051/epjconf/201818803001 ◽

2018 ◽

Vol 188 ◽

pp. 03001 ◽

Cited By ~ 2

Author(s):

Jérôme Combet

Keyword(s):

Neutron Scattering ◽

Small Angle ◽

Structural Information ◽

Structure Functions ◽

Partial Structure ◽

Isotopic Labelling ◽

Angle Scattering ◽

Dispersion State ◽

Polyelectrolyte Solutions

We present an introduction to the application of small angle Xray and neutron scattering to the study of polyelectrolyte solutions. We aim to give a simple overview of the structural information that can be gained with these techniques. In particular, we show how neutron scattering associated to isotopic labelling enables the determination of the different partial structure functions as well as the dispersion state and the average conformation of polyions.

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ATSAS 3.0: expanded functionality and new tools for small-angle scattering data analysis

Journal of Applied Crystallography ◽

10.1107/s1600576720013412 ◽

2021 ◽

Vol 54 (1) ◽

Author(s):

Karen Manalastas-Cantos ◽

Petr V. Konarev ◽

Nelly R. Hajizadeh ◽

Alexey G. Kikhney ◽

Maxim V. Petoukhov ◽

...

Keyword(s):

Distribution Function ◽

Small Angle ◽

Small Angle Scattering ◽

Distance Distribution ◽

Scattering Data ◽

Mode Analysis ◽

Distance Distribution Function ◽

Search Volume ◽

Angle Scattering ◽

Pair Distance Distribution Function

The ATSAS software suite encompasses a number of programs for the processing, visualization, analysis and modelling of small-angle scattering data, with a focus on the data measured from biological macromolecules. Here, new developments in the ATSAS 3.0 package are described. They include IMSIM, for simulating isotropic 2D scattering patterns; IMOP, to perform operations on 2D images and masks; DATRESAMPLE, a method for variance estimation of structural invariants through parametric resampling; DATFT, which computes the pair distance distribution function by a direct Fourier transform of the scattering data; PDDFFIT, to compute the scattering data from a pair distance distribution function, allowing comparison with the experimental data; a new module in DATMW for Bayesian consensus-based concentration-independent molecular weight estimation; DATMIF, an ab initio shape analysis method that optimizes the search model directly against the scattering data; DAMEMB, an application to set up the initial search volume for multiphase modelling of membrane proteins; ELLLIP, to perform quasi-atomistic modelling of liposomes with elliptical shapes; NMATOR, which models conformational changes in nucleic acid structures through normal mode analysis in torsion angle space; DAMMIX, which reconstructs the shape of an unknown intermediate in an evolving system; and LIPMIX and BILMIX, for modelling multilamellar and asymmetric lipid vesicles, respectively. In addition, technical updates were deployed to facilitate maintainability of the package, which include porting the PRIMUS graphical interface to Qt5, updating SASpy – a PyMOL plugin to run a subset of ATSAS tools – to be both Python 2 and 3 compatible, and adding utilities to facilitate mmCIF compatibility in future ATSAS releases. All these features are implemented in ATSAS 3.0, freely available for academic users at https://www.embl-hamburg.de/biosaxs/software.html.

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The maximum-entropy method without the positivity constraint – applications to the determination of the distance-distribution function in small-angle scattering

Journal of Applied Crystallography ◽

10.1107/s0021889894000932 ◽

1994 ◽

Vol 27 (4) ◽

pp. 574-580 ◽

Cited By ~ 11

Author(s):

S. Steenstrup ◽

S. Hansen

Keyword(s):

Distribution Function ◽

Maximum Entropy ◽

Small Angle ◽

Maximum Entropy Method ◽

Small Angle Scattering ◽

Entropy Method ◽

Distance Distribution ◽

Distance Distribution Function ◽

Angle Scattering

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Information and estimation in image processing

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100125142 ◽

1987 ◽

Vol 45 ◽

pp. 14-17

Author(s):

B. Roy Frieden

Keyword(s):

Image Processing ◽

Optical System ◽

Large Amplitude ◽

Communication Theory ◽

Image Data ◽

Direct Approach ◽

Ill Posed

Despite the skill and determination of electro-optical system designers, the images acquired using their best designs often suffer from blur and noise. The aim of an “image enhancer” such as myself is to improve these poor images, usually by digital means, such that they better resemble the true, “optical object,” input to the system. This problem is notoriously “ill-posed,” i.e. any direct approach at inversion of the image data suffers strongly from the presence of even a small amount of noise in the data. In fact, the fluctuations engendered in neighboring output values tend to be strongly negative-correlated, so that the output spatially oscillates up and down, with large amplitude, about the true object. What can be done about this situation? As we shall see, various concepts taken from statistical communication theory have proven to be of real use in attacking this problem. We offer below a brief summary of these concepts.

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Rapid determination of fractal structure of bacterial assemblages in wastewater treatment: implications to process optimisation

Water Science & Technology ◽

10.2166/wst.1998.0092 ◽

1998 ◽

Vol 38 (2) ◽

pp. 9-15 ◽

Cited By ~ 3

Author(s):

J. Guan ◽

T. D. Waite ◽

R. Amal ◽

H. Bustamante ◽

R. Wukasch

Keyword(s):

Wastewater Treatment ◽

Fractal Structure ◽

Structural Information ◽

Rapid Determination ◽

Fractal Dimensions ◽

Structure Information ◽

Treatment Processes ◽

On Line ◽

Bacterial Aggregates

A rapid method of determining the structure of aggregated particles using small angle laser light scattering is applied here to assemblages of bacteria from wastewater treatment systems. The structure information so obtained is suggestive of fractal behaviour as found by other methods. Strong dependencies are shown to exist between the fractal structure of the bacterial aggregates and the behaviour of the biosolids in zone settling and dewatering by both pressure filtration and centrifugation methods. More rapid settling and significantly higher solids contents are achievable for “looser” flocs characterised by lower fractal dimensions. The rapidity of determination of structural information and the strong dependencies of the effectiveness of a number of wastewater treatment processes on aggregate structure suggests that this method may be particularly useful as an on-line control tool.

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Pushing the Envelope

10.1093/oso/9780199670871.003.0014 ◽

2018 ◽

Author(s):

Eaton E. Lattman ◽

Thomas D. Grant ◽

Edward H. Snell

Keyword(s):

High Resolution ◽

Electron Density ◽

Structural Information ◽

Hydration Shell ◽

Three Dimensional ◽

Scattering Data ◽

Saxs Data ◽

Electron Density Function ◽

Angle Scattering ◽

Iterative Structure

Direct electron density determination from SAXS data opens up new opportunities. The ability to model density at high resolution and the implicit direct estimation of solvent terms such as the hydration shell may enable high-resolution wide angle scattering data to be used to calculate density when combined with additional structural information. Other diffraction methods that do not measure three-dimensional intensities, such as fiber diffraction, may also be able to take advantage of iterative structure factor retrieval. While the ability to reconstruct electron density ab initio is a major breakthrough in the field of solution scattering, the potential of the technique has yet to be fully uncovered. Additional structural information from techniques such as crystallography, NMR, and electron microscopy and density modification procedures can now be integrated to perform advanced modeling of the electron density function at high resolution, pushing the boundaries of solution scattering further than ever before.

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