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Error Correction for Discrete Tomography

M. Ceko, L. Hajdu, R. Tijdeman

Abstract

Discrete tomography focuses on the reconstruction of functions $f: A \to \mathbb{R}$ from their line sums in a finite number $d$ of directions, where $A$ is a finite subset of $\mathbb{Z}^2$. Consequently, the techniques of discrete tomography often find application in areas where only a small number of projections are available. In 1978 M.B. Katz gave a necessary and sufficient condition for the uniqueness of the solution. Since then, several reconstruction methods have been introduced. Recently Pagani and Tijdeman developed a fast method to reconstruct $f$ if it is uniquely determined. Subsequently Ceko, Pagani and Tijdeman extended the method to the reconstruction of a function with the same line sums of $f$ in the general case. Up to here we assumed that the line sums are exact. 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.

Error Correction for Discrete Tomography

Abstract

Discrete tomography focuses on the reconstruction of functions from their line sums in a finite number of directions, where is a finite subset of . Consequently, the techniques of discrete tomography often find application in areas where only a small number of projections are available. In 1978 M.B. Katz gave a necessary and sufficient condition for the uniqueness of the solution. Since then, several reconstruction methods have been introduced. Recently Pagani and Tijdeman developed a fast method to reconstruct if it is uniquely determined. Subsequently Ceko, Pagani and Tijdeman extended the method to the reconstruction of a function with the same line sums of in the general case. Up to here we assumed that the line sums are exact. 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 errors can be corrected and that this bound is the best possible.
Paper Structure (12 sections, 9 theorems, 34 equations)

This paper contains 12 sections, 9 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. An example which shows that Theorem \ref{['mainth']} cannot be improved
  4. Vandermonde equations with variable coefficients
  5. A Vandermonde-related determinant
  6. The line sum relation lemma
  7. Error correction of line sums in one direction
  8. Detection of directions with wrong line sums
  9. An error correction algorithm
  10. An example
  11. Complexity
  12. Concluding remarks

Key Result

Theorem 2.1

Let $d,m,n$ be positive integers and let $A$ and $D$ be as defined above. Let $f : A \to \mathbb{R}$ be an unknown function such that for $h=1$, $2$, $\dots$, $d$ the line sums $\ell^*_{h,t}$ in the direction of $(a_h,b_h)$ are measured with in total less than $d/2$ wrong line sums. Then the correct

Theorems & Definitions (9)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Lemma 5.1
  • Corollary 5.2
  • Lemma 6.1