Abel type integral equations in stereology: I. General discussion

Abstract

SUMMARYIn developing numerical methods for the solution of computational problems of an involved and pragmatic nature, as occur for example in stereology, the approach should be first to classify the distinct types of numerical problems which arise. If available, appropriate results from numerical analysis can then be used to develop stable numerical processes for their solution. In fact, many of the results relating to the construction of basic numerical methods, such as spectral methods for the stable differentiation of experimental data and product integration methods for the evaluation of integrals with oscillatory and singular integrands, have been resolved. Thus, we are now in a position to apply the above approach to the more complex computational problems which arise in stereology.Many of these problems belong to one of the following two classes: (i) the solution of integral equations of Abel type; and (ii) the solution of some numerical differentiation problem.In Part I of this paper, we illustrate the generality in stereology of the Abel type integral equation and numerical differentiation formulations. As well, we derive the specific properties required in Part II which is mainly concerned with the construction of stable computational methods for (i). Included among these properties are the explicit inversion formulae which are known for general Abel type integral equations. It is these inversion formulae which we use to construct stable computational methods.Often, when estimates for linear properties of the solutions of (i) and (ii) are required, the numerical solution of (i) and (ii) can be circumvented by estimating the linear properties directly from the given observational data. In deriving such estimates, use of the properties of the Abel type integral equation and differentiation formulations plays an essential role. Because of the close connection between such estimation problems and (i) and (ii), the estimation of linear properties from truncated observational data is also examined in Part I.