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Error Correction for Discrete Tomography

Ceko Matthew, Hajdu Lajos, Tijdeman Rob

Fundamenta Informaticae

2022

Vol. 189, nr 2

91--112

Discrete tomography focuses on the reconstruction of functions from their line sums in a finite number d of directions. In this paper we consider functions f : A → R where A is a finite subset of ℤ2 and R an integral domain. Several reconstruction methods have been introduced in the literature. Recently Ceko, Pagani and Tijdeman developed a fast method to reconstruct a function with the same line sums as f. Up to here we assumed that the line sums are exact. Some authors have developed methods to recover the function f under suitable conditions by using the redundancy of data. In this paper we investigate the case where a small number of line sums are incorrect as may happen when discrete tomography is applied for data storage or transmission. We show how less than d/2 errors can be corrected and that this bound is the best possible. Moreover, we prove that if it is known that the line sums in k given directions are correct, then the line sums in every other direction can be corrected provided that the number of wrong line sums in that direction is less than k/2. [

Liouville theorem with parameters: asymptotics of certain rational integrals in differential fields

Stawiska M.

Commentationes Mathematicae

2010

Vol. 50, [Z] 2

155-159

We study asymptotics of integrals of certain rational functions that depend on parameters in a field K of characteristic zero. We use formal power series to represent the integral and prove certain identities about coefficients of this series following from the generalized Vandermonde determinant expansion. Our result can be viewed as a parametric version of a classical theorem of Liouville. We also give some applications.

The resultants and discriminants of real polynomials and quasipolynomials

Górecki H.

Automatyka / Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie

2006

T. 10, z. 2

167-175

The paper presents the connections between resultants of the algebraic and transcendental polynomials. The resultants give information about the problems of stability and the maximal errors in linear stationary dynamic systems.

W pracy podano związki analityczne między wyróżnikami i rugownikami dla wielomianów i kwaziwielomianów. Formuły te pozwalają na ustalenie warunków istnienia pierwiastków wielokrotnych, rzeczywistych oraz zespolonych w równaniach algebraicznych oraz przestępnych, a także warunków stabilności i stabilności aperiodycznej.