论文标题
离散断层扫描的错误校正
Error Correction for Discrete Tomography
论文作者
论文摘要
离散的断层扫描重点介绍了功能的重建$ f：a \ to \ mathbb {r} $从其行总和以有限的数字$ d方向汇总，其中$ a $是$ \ mathbb {z}^2 $的有限子集。因此，离散断层扫描的技术通常在只有少量预测的领域找到应用。 1978年M.B. Katz为溶液的独特性提供了必要和充分的条件。从那时起，已经引入了几种重建方法。最近，Pagani和Tijdeman开发了一种快速的方法来重建$ f $，如果它是唯一确定的。随后，Ceko，Pagani和Tijdeman将方法扩展到了一般情况下具有相同行和$ f $的函数的重建。直到这里，我们假设行总和是精确的。在本文中，我们调查了少量线和不正确的情况，就像将离散断层扫描应用于数据存储或传输时发生的情况一样。我们展示了如何纠正$ d/2 $错误的少于$ d/2 $的错误，并且这种界限是最好的。
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.
