This was simplified evenР”Р‡urther by Glagoleff who showed that if we superimpose a rectangular Lengths of the mineral phases on the line grid. Then areaР”Р‡ractions AA can be estimated from length fractions At, i.e., the relative Lines over the plane section, with the aid of a transparent sheet. Suitable random sampling design (not the most obvious one) we give detailsР”Вelesse's method was later simplified by placing a grid of parallel Here E denotes expectation with respect to a Volume of a three-dimensional object is the integral of the areas of its two-Р”Вimensional plane slices. This follows from the basic geometrical fact that the (under the right sampling conditions), without any assumptions about the Astoundingly, AA is an unbiased estimator This is like a survey sampling problem: X represents the 'population'Р”Р„nd X n T the 'sample' from which we want to estimate a population Where A(-) denotes area in the two-dimensional section. That X n T is the plane section of the rock, and Y n T is that part of the Let T denote a plane in three dimensions, so The objective is to estimate the volume fraction Model the rock as a set X C R3 containing a subset Y C X, the mineral Delesse had realized that the proportion by volume ofР”Р„ particular mineral can be estimated from its proportion by area visible in Ing the rock to separate the different minerals, one simply cuts a plane In 1847 the French mineralogist Delesse published a revolutionary methodР”Р‡or measuring the mineral content in a sample of rock [223.

Section 10.3 describes newer discov-Р”РЋ0.2.1 Information from Lower-Dimensional Samples Section 10.2 isР”Р„ more detailed statistical treatment. Stereological methods are almost 'assumption free,' and are applicable inР”Р„pplications and general concepts are described in Р’В§10.1. On fundamental geometric facts and classical sampling theory. Simplicity is the key word the estimation relies only Parametric techniques to estimate 'geometrical parameters' such as volumeР”Р„nd surface area. Further it is usually impossible to model the Typically, only a few sections or samples are taken, and their spa. It is not the aim of stereology to reconstruct an entire three-dimensional Include anatomy, cell biology and pathology materials science, mineralogyР”Р„nd metallurgy botany, ecology and forestry geology and petrology and Sampling a spatial pattern at the points of a rectangular grid. It now embraces all geometrical sampling operations, such as clipping a two-Р”Вimensional image inside a window, taking one-dimensional linear probes, or Information visible on a two-dimensional plane section through the material. Opecl in biology and materials science as a quick way of analyzing three-Р”Вimensional solid materials (such as rock, living tissue, and metals) from Stereology is a spatial version of sampling theory. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.Р”В¬entre for Mathen~atics anti Computer Science Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book.

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