ΣΠ-Approximations and Data Compression

  • Anatoly Yu. Bezhaev
  • Vladimir A. Vasilenko
Chapter

Abstract

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
$$\sum\limits_{k = 1}^S {{\Phi _k}(x){\Psi _k}(y)} $$
(7.1)
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.

Keywords

Eigenvalue Problem Data Compression Variational Theory Cholesky Decomposition Generalize Eigenvalue Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

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Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Anatoly Yu. Bezhaev
  • Vladimir A. Vasilenko
    • 1
  1. 1.Institute of Computational Mathematics and Mathematical GeophysicsNovosibirskRussia

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