Astrogeometry, error estimation, and other applications of set-valued analysis

Abstract

In many real-life application problems, we are interested in numbers, namely, in the numerical values of the physical quantities. There are, however, at least two classes of problems, in which we are actually interested in sets:• In image processing (e.g., in astronomy), the desired black-and-white image is, from the mathematical viewpoint, a set.• In error estimation (e.g., in engineering, physics, geophysics, social sciences, etc.), in addition to the estimates x1, ...., xn for n physical quantities, we want to know what can the actual values xi of these quantities be, i.e., the set of all possible vectors x = (x,1, ...., xn).In both cases, we need to process sets. To define a generic set, we need infinitely many parameters; therefore, if we want to represent and process sets in the computer, we must restrict ourselves to finite-parametric families of sets that will be used to approximate the desired sets. The wrong choice of a family can lead to longer computations and worse approximation. Hence, it is desirable to find the family that it is the best in some reasonable sense.A similar problem occurs for random sets. To define a generic set, we need infinitely many parameters; as a result, traditional (finite-parametric) statistical methods are often not easily applicable to random sets. To avoid this difficulty, several researchers (including U. Grenander) have suggested to approximate arbitrary sets by sets from a certain finite-parametric family. As soon as we fix this family, we can use methods of traditional statistics. Here, a similar problem appears: a wrong choice of an approximation family can lead to a bad approximation and/or long computations; so, which family should we choose?In this paper, we show, on several application examples, how the problems of choosing the optimal family of sets can be formalized and solved. As a result of the described general methodology:•for astronomical images, we get exactly the geometric shapes that have been empirically used by astronomers and astrophysicists (thus, we have a theoretical explanation for these shapes), and• for error estimation, we get a theoretical explanation of why ellipsoids turn out to be experimentally the best shapes (and also, why ellipsoids are used in Khachiyan's and Karmarkar's algorithms for linear programming).