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Volume Fraction Measurement

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.

This 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.

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 tutorial initializes with one of four regions appearing in the Specimen Image window. The Choose A Specimen pull-down menu provides a selection of images of four different regions of the same specimen. For each, the Original button shows the original grayscale image, and the Thresholded Binary button shows the result of thresholding this to produce a binary image. The Morphological Result button shows the result of applying erosion and dilation to remove noise pixels and smooth boundaries to better represent the structure, and also displays the area fraction of the image that consists of white pixels. The Grid Overlay button superimposes a point grid on the image and also displays the fraction of the points that lie on the structure. The table below summarizes the results.

Area Fraction
Point Fraction
3/25 = 12%
2/25 = 8%
2/25 = 8%
0/25 = 0%
7/100 = 7%

Contributing Authors

John C. Russ - Materials Science and Engineering Dept., North Carolina State University, Raleigh, North Carolina, 27695.

Matthew Parry-Hill, and Michael W. Davidson - National High Magnetic Field Laboratory, 1800 East Paul Dirac Dr., The Florida State University, Tallahassee, Florida, 32310.



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