ΣΠ-Approximations and Data Compression

The problem of ΣΠ-approximation in a simple form is the following: let f(x, y) be a real function of two real variables x and y; we want to replace this function by the finite sum of products of one-variable functions 7.1 $$\sum\limits_{k = 1}^S {{\Phi _k}(x){\Psi _k}(y)} $$ and to provide some given accuracy of approximation. This problem is important in various applications, like data compression in digital image processing, in decomposition of two-dimensional digital filters into the one-dimensional filters and so on. In the beginning of the last century E. Schmidt (1907) considered this problem in the analytical form and found the connection between optimal ΣΠ-approximation and singular values of the integral operator with the kernel f(x, y) . After that many mathematicians became interested in this problem, but usually in the analytical form without using numerical algorithms. In this chapter, we consider the so-called finite dimensional ΣΠ-approximations in the general form and in the examples, and give the numerical algorithm for them.