John Hilliard's view of stereology is very general and has interesting parallels with Fourier analysis. He said that any sort of statistical geometric probe interacting with any sort of structure to yield data about that structure constituted stereology. Stereology forms the basis for many analytical procedures involving imaging in anatomy, pathology, histology, metallurgy and materials science.
 
"Stereology was invented because metallurgical specimens were opaque. Then the mathematicians got a hold of stereology and stereology became opaque"-- Robert DeHoff at the Third International Conference on Microstructology (2005).

One of the goals of publishing this book is to mitigate the confusion that has resulted from the collision of stereology with pure mathematics. John Hilliard's stereology book is a book for all seasons. He wrote the book formost as a textbook and tested it as course notes during its development.
 
• It contains detailed derivations for all the major stereology formulas as well as other important formulas used in geostatistics, photogrammetry and imaging sciences related to stereology.
• It teaches the subject from the beginning without assuming a great deal of mathematical background and provides a handy reference to this background material.
• It contains many interesting mathematical problems accessible to the amateur. These appear in the exercises and commentary. There remain many puzzles with potentially valuable results, such as finding intercept distributions for common shapes, that are unsolved to this day.
• It treats many subjects in the literature, such as shape parameters and photogrammetric microscopy, that have not appeared in other compendia but which the student may find exciting to apply to some practical problem.
 

Historically stereology arose to deal with problems of estimating properties of a three-dimensional solid from measurements made on two-dimensional slices as in microscopy. These problems were solved by regarding some measurement taken on the slice as a probe. An averaging process over what amounts to a Grassmann manifold of such probes provides the basis for estimating properties from several random applications of the probe. Volume fractions, area and length per unit volume, curvature and similar quantities could be determined. The latter 20th Century brought two new developments into stereology: First, averaging formulas could be written in almost any number of dimensions -- not just to estimate three dimensional properties from samples in two or one dimension. The practical importance of this discovery will come as we become more aware of the higher-dimensional aspects of our universe. Second, geostatisticians such as Matheron became concerned with quantifying not just the average but the dispersion of geometric measurements. This has linked stereology to geostatistics.