Basic Ideas of Geometric Sampling

Most of the material in Chapter 1 is concerned with integral-geometric formulae for motion-invariant measures of the intersection between a compact set and a probe (unbounded, or bounded). No clue is given there, however, of how to implement a sampling design which respects motion invariance and opens the way to the estimation of geometric properties. In this chapter, probability concepts are incorporated to describe sampling strategies, and to obtain mean values and ratio identities for the relevant parameters, known as fundamental equations of stereology. The basic concepts and results are derived first for a single test probe hitting a bounded target set. The corresponding results involving test systems, more common in practical stereology (Chapter 4 ) are deferred to Sections 2.24–2.31. The approach is entirely design-based: the target set is fixed and non-random, whereas the probe is equipped with a probability measure based on the motion-invariant density – principles and methods of model-based stereology, in which the target is modelled by a stationary random set whereas the location of the probe may be fixed, are described in Chapter 3 .