Unfolding Particle Size Distributions

Nicholson, W. L.; Merckx, K.R.

doi:10.1080/00401706.1969.10490733

Cited by 14 publications

(8 citation statements)

References 10 publications

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“…The latter standard error estimates are strongly assumptiondepcndent, however, and it is advisable to calculate alternative estimates using estimated between-section variances. The statistical approach is well represented by the papers of Meisner (1967) and Nicholson & Merckx (1969). An important statistically-oriented paper is that of Nicholson (1970), who discusses the distribution-free estimation of linear properties of sphere size distributions, namcly of integrals of the form EI/I = J,;c/l(x)dG(x).…”

Section: Distribution-free Numerical Approach: Finite Difference Methodsmentioning

confidence: 99%

Distribution‐free estimation of sphere size distributions from slabs showing overprojection and truncation, with a review of previous methods

Cruz‐Orive

1983

Journal of Microscopy

181

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K E Y w O R D S.Particle size distributions, linear size properties, slab section, overprojection, resolution threshold, capping angle, unfolding algorithm, stereology. S U M M A R YA distribution-free method is presented for unfolding the size distribution of an aggregate of disjoint opaque spherical particles in a translucent medium, from the circular profiles observed in the orthogonal projection of sections of a certain thickness through the medium. This situation pertains especially to light and transmission electron microscopy.The model accounts for overprojection (or 'Holmes effect') due to section thickness, and for two forms of truncation (or 'missing profiles' mechanisms), simultaneously. The profile data may be grouped into an arbitrary class system, chosen by the user, and the procedure can be programmed in a medium-size desk calculator.In order to incorporate the above features, however, we had to choose a finite-difference unfolding algorithm, the advantages and disadvantages of which are discussed early in the paper in the context of the main alternative unfolding procedures hitherto available.The performance of the method is tested against three sets of real data previously analysed by a maximum likelihood method (Keiding er al., 1972).

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Section: Distribution-free Numerical Approach: Finite Difference Methodsmentioning

confidence: 99%

Distribution‐free estimation of sphere size distributions from slabs showing overprojection and truncation, with a review of previous methods

Cruz‐Orive

1983

Journal of Microscopy

181

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“…The paper by NICHOLSON and MERCKX (1969) has results relating to thin slice models when there is a lower resolution limit, and is important for its consideration of statistical methods for controlling measurement errors.…”

Section: Introductionmentioning

confidence: 99%

The Sizes of Spheres from Profiles in a Thin Slice II. Transparent Spheres

Coleman

1983

Biometrical J

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Formulae are derived for obtaining the size distribution of transparent spherical particles which are embedded in an opaque material. The data are the sizes of the circular profiles made by particles in a random thin slice of a specimen when it is viewed orthogonally. The formulae incorporate resolution constraints which cause various profile size ranges to be unobservable. Approximate solutions are obtained under specific resolution constraints when the slice is very thin. The corresponding case of opaque spheres in a transparent material was the subject of Part I (Coleman, 1982).

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“…Several models, many of great refinement, have been produced for the solution of this problem (Abercrombie, 1946;Bach, 1967;Coupland, 1968;Hilliard, 1962;Nicholson & Merckx, 1967) but in general their use requires elaborate mathematical procedures. Nicholson & Merckx (1967) demonstrated that the application of matrix algebra to the general problem of unfolding particle size distributions from data obtained by stereological sampling techniques greatly simplifies many of the mathematical procedures.…”

Section: Introductionmentioning

confidence: 99%

“…By using an approach similar to those of Abercrombie (1946), Coupland (1968) and Nicholson & Merckx (1967) it is shown in this paper that the equations relating the particle size distribution and population density to the frequency distribution of observed diameters can be conveniently presented as a set of conversion tables, the use of which require only elementary arithmetical procedures. Besides incorporating the section thickness, the tables also represent improvements over those previously published by relying on the more realistic definition of the diameter of a 0 1980 The Royal Microscopical Society 135 subpopulation of spheres as being the midpoint value of each class rather than the upper class-limit.…”

Section: Introductionmentioning

confidence: 99%

Improved tables for the evaluation of sphere size distributions including the effect of section thickness

Rose¹

1980

Journal of Microscopy

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The problem of determining the size distribution and population density of a population of spherical particles randomly distributed throughout a transparent matrix from data obtained by transmission microscopy on parallel plane-faced sections of known field area and thickness may be readily solved by the use of matrix algebra. The solution presented in this paper may be used to construct conversion matrices whereby the particle size distribution and the numerical density can be calculated directly from the frequency distribution of the observed diameters; the only mathematical processes necessary being mulitplication and addition. Specimen conversion matrices valid for a wide range of conditions are presented in the appended tables.

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Unfolding Particle Size Distributions

Cited by 14 publications

References 10 publications

Distribution‐free estimation of sphere size distributions from slabs showing overprojection and truncation, with a review of previous methods

Distribution‐free estimation of sphere size distributions from slabs showing overprojection and truncation, with a review of previous methods

The Sizes of Spheres from Profiles in a Thin Slice II. Transparent Spheres

Improved tables for the evaluation of sphere size distributions including the effect of section thickness

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