显微镜下的二元矩阵：一个层析问题
Binary Matrices under the Microscope: A Tomographical Problem
摘要：
一个二进制矩阵可以通过移动一个固定的矩形窗口（子矩阵）来扫描，就像在显微镜下仔细观察一样。对于每次观察，一个方便的测量方法是窗口中可见的1的数量，这可以被认为是窗口的亮度。二进制矩阵的矩形扫描则是以矩阵形式呈现的这些光度的集合。我们表明，至少在一个光滑的m × n二进制矩阵的技术情况下，它可以重建从其矩形扫描在多项式时间内的参数m和n，其中多项式的次数取决于检查窗口的大小。对于一个任意的二进制矩阵，我们然后扩展这一结果，确定其矩形扫描，排除了矩阵的光滑性的条目。
英文摘要：
A binary matrix can be scanned by moving a fixed rectangular window (submatrix) across it, rather like examining it closely under a microscope. With each viewing, a convenient measurement is the number of 1s visible in the window, which might be thought of as the luminosity of the window. The rectangular scan of the binary matrix is then the collection of these luminosities presented in matrix form. We show that, at least in the technical case of a smooth m x n binary matrix, it canbe reconstructed from its rectangular scan in polynomial time in the parameters m and n, where the degree of the polynomial depends on the size of the window of inspection. For an arbitrary binary matrix, we then extend this result by determining the entries in its rectangular scan that preclude the smoothness of the matrix.
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