Abel type integral equations in stereology. II. Computational methods of solution and the random spheres approximation

Abstract

SUMMARYAs mentioned in Part I of this paper, Part II is mainly concerned with the construction of stable computational methods for the solution of integral equations of Abel type which occur in stereology. However, in order to develop a rationale for the types of methods which we shall propose, it is first necessary to review the different types of methods which have been suggested to date in the stereological literature. This is done in section 2. From the review, we conclude that the construction of stable methods can be based on the use of spectral differentiation and product integration to evaluate appropriate inversion formulae, when they are known. Using the Random Spheres Model as exemplification, we examine the nature of such methods in section 3. The potential of the proposed methods is illustrated in section 4 by examining the reliability of the random spheres approximation—an unsolved problem in stereology proposed by Moran (1972). We conclude from this examination that the random spheres approximation is quite reliable for particles which approximate spheres.