TABLE 10.1: Standard Notation for Geometrical Quantitiesįinite set of objects N number of objectsĬurved surface M,K integral of mean curvatureįinite set of points I,P number of pointsįinite set of objects N,Q number of objects Timated from the number of points where the curve hits a section plane.Ĭertain quantities related to curvature can also be estimated [9,214. The length of a curve in space can be es. Surface in three-dimensional space can be estimated from the length of its Length of a plane curve can be estimated from the number of crossing pointsīetween the curve and a grid of paraHel lines. Similar tricks exist for estimating other geometrical quantities. The result wiD be unbiased, typically accurate to about Meanwhile estimate the area stereologicallyīy counting the 5 mm crossing points that are visible on the paper, and Ask someone to determine the area of the cutout byĬounting all the ~ mm squares. Ruled with (say) thin lines every 1 mm and thick lines every 5 mm. In both cases the estimators are unbiased.ĭemonstrate this with a "party trick." Take a sheet of graph paper Grid of points over the section plane, the area fraction AA can be estimatedįrom the proportion Pp of grid points that "hit" (lie over) the mineral phase. This was simplified evenįurther by Glagoleff who showed that if we superimpose a rectangular Lengths of the mineral phases on the line grid. Then areaįractions 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 detailsĭelesse'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-ĭimensional 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"Īnd 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 ofĪ 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 methodįor measuring the mineral content in a sample of rock [223. Section 10.3 describes newer discov-ġ0.2.1 Information from Lower-Dimensional Samples Section 10.2 isĪ more detailed statistical treatment. Stereological methods are almost "assumption free," and are applicable inĪpplications and general concepts are described in §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 volumeĪnd 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, mineralogyĪnd 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-ĭimensional 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-ĭimensional 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.Ĭentre for Mathen~atics anti Computer Science
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