Inverse Saltykov analysis for particle-size distributions and their time evolution

Jeppsson, Johan; Mannesson, Karin; Borgenstam, Annika; Ågren, John

doi:10.1016/j.actamat.2010.09.046

Cited by 16 publications

(3 citation statements)

References 6 publications

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“…As a result, the PDF cannot be computed without a regularization procedure, such as linear interpolation or smoothing (Anderssen and Jakeman, 1975). For instance, Jeppsson et al (2011) have used the so-called kernel density estimation to get a continuous description of the empirical PDF. In addition, the estimation of the expectation (E) is not straightforward.…”

Section: Aims Of This Workmentioning

confidence: 99%

Resolution of the Wicksell's equation by Minimum Distance Estimation

Depriester

Kubler

2019

Image Anal Stereol

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The estimation of the grain size in granular materials is usually performed by 2Dobservations. Unfolding the grain size distribution from apparent 2D sizes is commonly referred as the corpuscle problem. For spherical particles, the distribution of the apparent size can be related to that of the actual size thanks to the Wicksell’s equation. The Saltikov method, which is based on Wicksell’s equation, is the most widely used method for resolving corpuscle problems. This method is recursive and works on the finite histogram of the grain size. In this paper, we propose an algorithm based on a minimizing procedure to numerically solve the Wicksell’s equation, assuming a parametric model for the distribution (e.g. lognormal distribution). This algorithm is applied on real material and the results are compared to those found using Saltikov or Saltikov-basedstereology techniques. A criterion is proposed for choosing the number of bins in the Saltikov method. The accuracy of the proposed algorithm, depending on the sample size, is studied.

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Section: Aims Of This Workmentioning

confidence: 99%

Resolution of the Wicksell's equation by Minimum Distance Estimation

Depriester

Kubler

2019

Image Anal Stereol

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“…For any shape more complex than either an elongated or flattened spheroid with bivariate axes, there is no analytical solution to the stereological problem without imposing additional assumptions, such as the requirement of a specific size distribution of particles [16][17][18]. Since particle shapes in materials are neither necessarily uniform nor is the size distribution usually known, the assumption of spherical particles (or at least globally-convex particles which have the average shape of a sphere in rotation) remains common in materials science (e.g., [5,[19][20][21]). In the present work, the unfolding relationship for spheres is re-derived in terms of the histogram of observation areas as inputs (rather than radii or diameters) and a practical method is presented for utilizing the un-physical results that propagate during the calculation for the creation of error bars.…”

Section: Introductionmentioning

confidence: 99%

Revisiting Sphere Unfolding Relationships for the Stereological Analysis of Segmented Digital Microstructure Images

Payton¹

2012

JMMCE

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Sphere unfolding relationships are revisited with a specific focus on the analysis of segmented digital images of microstructures. Since the features of such images are most easily quantified by counting pixels, the required equations are re-derived in terms of the histogram of areas (instead of diameters or radii) as inputs and it is shown that a substitution can be made that simplifies the calculation. A practical method is presented for utilizing negative number fraction bins (which sometimes arise from erroneous assumptions and/or insufficient numbers of observations) for the creation of error bars. The complete algorithm can be implemented in a spreadsheet. The derived unfolding equations are explored using both linear and logarithmic binning schemes, and the pros and cons of both binning schemes are illustrated using simulated data. The effects of the binning schemes on the stereological results are demonstrated and discussed with reference to their consequences for practical materials characterization situations, allowing for the suggestion of guidelines for proper application of this, and other, distribution-free stereological methods.

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“…He then used the empirical data and a histogram estimator to solve his particular problem. This basic stereological model has been applied in a variety of disciplines where it is not possible to obtain full 3D measurements of objects simply by looking at them; this includes biology, geology, astronomy and materials science: [Cruz-Orive and Weibel (1990), Giumelli, Militzer and Hawbolt (1999), Higgins (2000), Jeppsson et al (2011), Miyamoto (1994), Sahagian and Proussevitch (1998), Sen and Woodroofe (2012), Tewari and Gokhale (2001)]. Not surprisingly, the method has also gained considerable attention in the statistics literature.…”

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confidence: 99%

Nonparametric inference in a stereological model with oriented cylinders applied to dual phase steel

McGarrity¹,

Sietsma²,

Jongbloed³

2014

Ann. Appl. Stat.

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Oriented circular cylinders in an opaque medium are used to represent certain microstructural objects in steel. The opaque medium is sliced parallel to the cylinder axes of symmetry and the cut-plane contains the observable rectangular profiles of the cylinders. A oneto-one relation between the joint density of the squared radius and height of the 3D cylinders and the joint density of the squared halfwidth and height of the observable 2D rectangles is established. We propose a nonparametric estimation procedure to estimate the distributions and expectations of various quantities of interest, such as the cylinder radius, height, aspect ratio, surface area and volume from the observed 2D rectangle widths and heights. Also, the covariance between the radius and height of a cylinder is estimated. The asymptotic behavior of these estimators is established to yield point-wise confidence intervals for the expectations and point-wise confidence sets for the distributions of the quantities of interest. Many of these quantities can be linked to the mechanical properties of the material, and are, therefore, useful for industry. We illustrate the mathematical model and estimation procedures using a banded microstructure for which nearly 90 µm of depth have been observed via serial sectioning.

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Inverse Saltykov analysis for particle-size distributions and their time evolution

Cited by 16 publications

References 6 publications

Resolution of the Wicksell's equation by Minimum Distance Estimation

Resolution of the Wicksell's equation by Minimum Distance Estimation

Revisiting Sphere Unfolding Relationships for the Stereological Analysis of Segmented Digital Microstructure Images

Nonparametric inference in a stereological model with oriented cylinders applied to dual phase steel

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