离散层析成像解的界
Bounds for discrete tomography solutions
摘要：
考虑了在给定方向上的线和的情况下，函数在$\mathbb{Z}^2$的有限子集上的重构问题。实数解形成一个线性流形，整数解形成一个网格.首先，我们提供了一个明确的表达的投影向量从原点到线性解流形的情况下，只有行和列和的有限子集的$\mathbf{Z}^2$。接下来，我们提出了一种方法来估计两个二元解之间的最大距离。随后，我们推导出一个上界的距离从任何给定的实数解到最近的整数解。这使我们能够估计解的稳定性。最后，我们将第一个结果推广到环面情形和连续情形。
英文摘要：
We consider the reconstruction of a function on a finite subset of $\mathbb{Z}^2$ if the line sums in certain directions are prescribed. The real solutions form a linear manifold, its integer solutions a grid. First we provide an explicit expression for the projection vector from the origin onto the linear solution manifold in the case of only row and column sums of a finite subset of $\mathbf{Z}^2$. Next we present a method to estimate the maximal distance between two binary solutions. Subsequently we deduce an upper bound for the distance from any given real solution to the nearest integer solution. This enables us to estimate the stability of solutions. Finally we generalize the first mentioned result to the torus case and to the continuous case.
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