Inversion of the two-dimensional discrete Fourier transform (DFT) typically requires all DFT coefficients to be known. When only band-limited DFT coefficients of a matrix are known, the original matrix can not be uniquely recovered. Using a priori information that the matrix is binary (all elements are either 0 or 1) can overcome the missing high-frequency DFT coefficients and restore uniqueness. We theoretically investigate the smallest pass band that can be applied while still guaranteeing unique recovery of an arbitrary binary matrix. The results depend on the dimensions of the matrix. Uniqueness results are proven for the dimensions $p\times q$, $p\times p$, and $p^\alpha\times p^\alpha$, where $p\neq q$ are primes numbers and $\alpha>1$ an integer. An inversion algorithm is proposed for practically recovering the unique binary matrix. This algorithm is based on integer linear programming methods and significantly outperforms naive implementations. The algorithm efficiently reconstructs $17\times17$ binary matrices using 81 out of the total 289 DFT coefficients.

翻译：将二维离散Fourier变异( DFT) 转换为二维离散 Fourier 变异( DFT) 通常需要所有 DFT 系数才能为人所知。 当只知道一个矩阵的带宽 DFT 系数时, 原始矩阵无法被独一地回收。 使用矩阵二进制( 所有元素均为 0 或 1) 的先验信息可以克服缺失的高频 DFT 系数并恢复独一性。 我们理论上调查了可以应用的最最小的通过带, 同时仍然保证任意的二进制矩阵的独特恢复。 其结果取决于矩阵的尺寸。 当一个矩阵的尺寸为$p\ times q$, $p\ alpha\ times p ⁇ alpha$, 其中$p\ garpha$q q q q q q q 和$\ alpha>1$ an 整数。 为实际恢复独一二进制矩阵提出了一种反演算法。 该算法基于整数线性编程线性编程方法, 大大超出天真性执行。 。 17\ 17\ 177timememets 177$ binmattlemmmates misstlegnations, 总共289 289 DFTggggggggs。