In most microscopy applications, the sample is a thin slice or surface through a three-dimensional specimen, and the proper interpretation of the structure measurements is based on stereological rules. Modern stereological procedures emphasize the efficient and unbiased sampling of the specimen, and make use of relatively simple measurement or counting procedures, often using grids placed on the sample, to obtain the desired results. The metric properties of a structure such as the volume fraction, surface area, length, and curvature, can all be determined by the examination of representative section planes. Topological properties such as the number of discrete objects and the connectivity of networks require at a minimum the comparison of two parallel sections.

The Volume Fraction Measurement interactive tutorial illustrates procedures for measuring the volume fraction, in this case of the dark-stained organelles seen in TEM images. Thresholding the digitized images does not delineate the structures well, but a morphological opening and closing correct this. The total area of the organelles can be determined by counting pixels. Assuming that the images are representative of the specimen, the volume fraction is measured by the area fraction.

	Interactive Tutorial
Volume Fraction Measurement Measure volume fraction by area fraction or point count.

However, a preferred method for estimating the volume fraction is carried out by placing a sparse grid of points on the image, as shown, and counting the fraction of those points that “hit” or fall on the structures of interest. This might seem like a less accurate measurement, but it has several advantages. The most important is inherent in changing the procedure from one of measurement to one of counting. The statistics of counting independent events provides a direct estimate of the measurement precision.

Counting a few points on multiple fields of view is very quick, and by repeating the procedure until (for instance) the total number of hits reaches 400, an overall precision of 5 percent would be obtained. This is because the square root of 400 is 20, which is 5 percent of 400. For a precision of 10 percent only 100 hits would be needed, while 1000 hits would produce a precision of 3 percent.

Note in the example that the estimates of volume fraction for each field of view obtained by the area measurement and the grid point count are not too dissimilar, while the variation from field to field is considerable. It is important to examine enough fields of view to obtain a representative sample of the specimen, and the point count method generally insures this. Also note that while the area is measured by counting pixels, this is not the same as the use of the grid because the pixels are close together. For the square root of the number of hits to be a valid estimate of the precision of the count, the points must be independent samples of the structure, meaning that the grid must be sparse enough that the points rarely fall on the same feature.

The surface area of contact between different structures is also an important measure in many instances. In section images, the surfaces present in three dimensions appear as boundary lines. In the example, the boundaries of the white phase in the metal are delineated as contour lines. Measuring the length of the contour lines allows the calculation of the surface area per unit volume of the sample, according to the relationship shown in Equation 1. Note that unlike the volume fraction measurement above, which produces a dimensionless number, the surface area per unit volume measurement requires knowing the image magnification. In this case the area of the region in each image is 567 square µm.

As for the volume fraction, using a counting procedure is preferred to a measurement. In this case a grid of lines is drawn on the images. For the case of random section placement and orientation in the structure, any grid of lines can be used and the square grid is convenient. In the Surface Area Measurements interactive tutorial, the total length of the grid lines is 114 µm. Counting the number of “hits” that the contour lines make with the grid lines also allows calculation of the surface area per unit volume.

(1)

	Interactive Tutorial
Surface Area Measurements Measure surface area by boundary length or intercept count.

As for the volume fraction measurement, note that variation from one field of view to another is much greater than the differences between the results from the grid count and contour length measurements for each region.

The length of structures can also be measured by a counting procedure. The Length Measurement interactive tutorial shows a transmission light microscope image of dendritic processes viewed in a section of known thickness. Measuring the length of the structures seen in the image would not take into account their three-dimensional wanderings up and down in the section. However, any grid lines drawn on the image represent surfaces that extend vertically downwards through the section, and counting the intersections made by the structures with those lines can provide a correct measurement of the total length per unit volume as shown in Equation 2.

(2)

	Interactive Tutorial
Length Measurement Measuring length per unit volume by counting intercepts.

In this example the sections were cut using a specific orienting protocol which, together with the cycloid-shaped grids shown, produces unbiased isotropic sampling of the structure even if the structure itself has preferred orientation. The total length of the grid lines in the example is 326 µm and the section thickness is 4 µm.

In all of these examples, the generation of the appropriate grids of points or lines is conveniently performed by the computer. Counting may be performed manually, but in most cases combining the grid lines with the binary image representing the structure using a Boolean AND, and then automatically counting the number of hits, provides a more efficient way to handle the multiple fields of view that must be measured to obtain meaningful results.

The most straightforward way to count the number of features per unit volume requires the comparison of two section planes a known distance apart (and close enough together that features cannot “hide” in the space between them). Aligning images from serial sections is difficult, but the confocal light microscope simplifies this procedure by obtaining optical sections. Any feature that is seen in the bottom section and is absent in the upper section must have a unique uppermost point within the volume between the sections. Counting those tops provides an unambiguous and unbiased value for the number in the volume defined by the image area and the section spacing.

In the Counting Features Per Unit Volume interactive tutorial, fluorescence images of oil droplets from two sections are thresholded, and the features separated with a watershed. Placing the resulting images into the red and green channels of a color image shows yellow wherever the two colors overlap. Since features can change size from one section to the other, the presence of yellow within a feature means that it continues through the two sections, and therefore should not be counted. Those features that are entirely green are ones that appear in the lower section but not the upper, and are isolated and counted to obtain the result.

	Interactive Tutorial
Counting Features Per Unit Volume Explore unbiased counting using serial sections.

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