Sampling Theory for Opaque Spatial Specimens

Davy, Pamela J.; Miles, R. E.

doi:10.1111/j.2517-6161.1977.tb01605.x

Cited by 32 publications

(27 citation statements)

References 11 publications

(7 reference statements)

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“…Taking g(B) = 1{B = ∅} yields the expression (13) for β, taking g(B) = λ 2 (B) yields (14) and taking g(B) = 2J 2 1 (B) yields (15), using (6) in the latter case to evaluate the integral over F (3, 2).…”

Section: Is Obtained From By Restricting Attention To Thosementioning

confidence: 99%

Estimation of mean particle volume using the set covariance function

Cabo

Baddeley

2003

Advances in Applied Probability

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Our aim is to estimate the volume-weighted mean of the volumes of three-dimensional ‘particles’ (compact, not-necessarily-convex subsets) from plane sections of the particle population. The standard stereological technique is to place test lines in the plane section, and measure cubed intercept lengths with the two-dimensional particle profiles. This paper discusses more efficient estimators obtained by integrating over all possible placements of the test line. We prove that these estimators have smaller variance than the line transect estimators, and indeed are related to them by the Rao-Blackwell process. In the improved estimators, the cubed intercept length is replaced by a moment of the distance between two points in the section profile. This can be computed as a moment of the set covariance function, which in turn is computable using the fast Fourier transform. We also derive an isoperimetric-type inequality between the improved estimator and the area-weighted 3/2th moment of the profile areas. Finally, we present two practical applications to particles of silicon carbide and to synaptic boutons in brain tissue. We estimate the variance of the technique and the gain in efficiency over line transect techniques; the efficiency improvement appears to be as much as one order of magnitude.

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Section: Is Obtained From By Restricting Attention To Thosementioning

confidence: 99%

Estimation of mean particle volume using the set covariance function

Cabo

Baddeley

2003

Advances in Applied Probability

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“…In the example, micrographs B2 and A5 were excluded by the above procedure, * The procedure is a modification for a posterior use of the a priori unbiased ratio estimators developed by Lahiri and Midzuno (see Cochran, 1963). but since for the next one, B4, its n (Frames) = 8 > RN3 = 5 we calculate the unbiased There are other unbiased estimators of N A (Davy & Miles, 1977; Miles L ? Davy, 1976), but even the predecessor of this one has rightfully been called: 'This beautiful result, due to Midzuno .…”

Section: Ifmentioning

confidence: 99%

Estimation of tubule or cylinder L_V, S_V and V_V on thick sections

Jørgen¹,

Gundersen²

1979

Journal of Microscopy

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“…If a 3D sample is observed in transmission using one camera then the resulting image is a 2D projection of a 3D sample. This phenomenon is known as a stereology effect [11], specifically, known as over projection [12]. This manuscript proposes a method to relate observed events in the 2D projection to actual events in the 3D sample in real-time during phase change by solidification (polycrystalline nucleation and growth).…”

Section: Introductionmentioning

confidence: 99%

Theoretical analysis to interpret projected image data from in-situ 3-dimensional equiaxed nucleation and growth

Mooney

McFadden

2017

Journal of Crystal Growth

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In-situ observation of crystal growth in transparent media allows us to observe solidification phase change in real-time. These systems are analogous to opaque systems such as metals. The interpretation of transient 2-dimensional area projections from 3-dimensional phase change phenomena occurring in a bulky sample is problematic due to uncertainty of impingement and hidden nucleation events; in stereology this problem is known as overprojection. This manuscript describes and demonstrates a continuous model for nucleation and growth using the well-established Johnson-Mehl-Avrami-Kolmogorov model, and provides a method to relate 3-dimensional volumetric data (nucleation events, volume fraction) to observed data in a 2-dimensional projection (nucleation count, area fraction). A parametric analysis is performed; the over-projection phenomenon is shown to be significant in cases where nucleation is occurring continuously with a relatively large variance. In general, area fraction on a projection plane will overestimate the volume fraction within the sample and the nuclei count recorded on the projection plane will underestimate the number of real nucleation events. The statistical framework given in this manuscript provides a methodology to deal with the differences between the observed (projected) data and the real (volumetric) measures.

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Sampling Theory for Opaque Spatial Specimens

Cited by 32 publications

References 11 publications

Estimation of mean particle volume using the set covariance function

Estimation of mean particle volume using the set covariance function

Estimation of tubule or cylinder L_V, S_V and V_V on thick sections

Theoretical analysis to interpret projected image data from in-situ 3-dimensional equiaxed nucleation and growth

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Sampling Theory for Opaque Spatial Specimens

Cited by 32 publications

References 11 publications

Estimation of mean particle volume using the set covariance function

Estimation of mean particle volume using the set covariance function

Estimation of tubule or cylinder LV, SV and VV on thick sections

Theoretical analysis to interpret projected image data from in-situ 3-dimensional equiaxed nucleation and growth

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Product

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About

Estimation of tubule or cylinder L_V, S_V and V_V on thick sections