Properties of the incomplete Abel transform and some of its generalizations.

Abstract

The mathematical model of a two-dimensional linear imaging system is given in terms of an integral operator whose kernel is called the point-spread function (PSF) of the system. Through this model, the PSF completely characterizes the noise-free imaging properties of the system. Therefore, in order for an imaging system to be useful for quantitative work, its PSF must be known. An innovative technique for inferring the PSF of a locally isotropic imaging system from the images of line-segment sources was proposed in a paper by Dallas et al. (J. Opt. Soc. Am. A 4, 2039 (1987)). Although algorithms based on their calculations provided sufficiently accurate results when applied to simulated problems, a rigorous proof of their formal calculations had not been established. In this dissertation, we supply the mathematical details of their formulation. The problem of determining the PSF from the image of a line-segment source can be formulated in a Hilbert space setting as an inverse problem. By applying the proper coordinate transformations, the operator modeling the relationship between the PSF and the image of a line-segment source can be cast into the form of a Wiener-Hopf integral operator of the first kind whose kernel vanishes along the positive axis. We will perform a detailed analysis of this operator, known as the incomplete Abel transform. Instead of limiting attention to this single operator, however, we will generalize it, thereby introducing large class of Wiener-Hopf integral operators, acting in the Hilbert space of square-integrable functions supported along the positive axis. By appealing the theory of Hardy spaces on the real line, necessary and sufficient conditions will be developed for determining when an operator falling into this class is invertible. Furthermore, by using a regularization technique, we will develop explicit expressions for the inverses of a certain subclass of these operators. As a special case, we will present a number of inversion formulas for the incomplete Abel transform. In addition, we will formulate, and test, an efficient algorithm for performing a numerical inversion of the incomplete Abel transform.