Conventional inversion of the discrete Fourier transform (DFT) requires all DFT coefficients to be known. When the DFT coefficients of a rasterized image (represented as a matrix) are known only within a pass band, the original matrix cannot be uniquely recovered. In many cases of practical importance, the matrix is binary and its elements can be reduced to either 0 or 1. This is the case, for example, for the commonly used QR codes. The {\it a priori} information that the matrix is binary can compensate for the missing high-frequency DFT coefficients and restore uniqueness of image recovery. This paper addresses, both theoretically and numerically, the problem of recovery of blurred images without any known structure whose high-frequency DFT coefficients have been irreversibly lost by utilizing the binarity constraint. We investigate theoretically the smallest band limit for which unique recovery of a generic binary matrix is still possible. Uniqueness results are proved for images of sizes $N_1 \times N_2$, $N_1 \times N_1$, and $N_1^\alpha\times N_1^\alpha$, where $N_1 \neq N_2$ are prime numbers and $\alpha>1$ an integer. Inversion algorithms are proposed for recovering the matrix from its band-limited (blurred) version. The algorithms combine integer linear programming methods with lattice basis reduction techniques and significantly outperform naive implementations. The algorithm efficiently and reliably reconstructs severely blurred $29 \times 29$ binary matrices with only $11\times 11 = 121$ DFT coefficients.

翻译：离散傅里叶变换（DFT）的传统反演要求已知所有DFT系数。当栅格化图像（表示为矩阵）的DFT系数仅在通带内已知时，原始矩阵无法唯一恢复。在许多实际重要情形中，矩阵为二值矩阵，其元素仅可取0或1，例如常用的QR码即属此类。矩阵为二值这一先验信息可补偿缺失的高频DFT系数，从而恢复图像恢复的唯一性。本文从理论和数值两方面探讨了如何利用二值性约束恢复高频DFT系数已不可逆丢失的模糊图像（无需已知任何结构）。我们从理论上研究了通用二值矩阵仍可唯一恢复的最小带限值。针对尺寸为$N_1 \times N_2$、$N_1 \times N_1$及$N_1^\alpha\times N_1^\alpha$（其中$N_1 \neq N_2$为素数，$\alpha>1$为整数）的图像，证明了唯一性结论。提出了从带限（模糊）版本恢复矩阵的反演算法。该算法将整数线性规划方法与格基约化技术相结合，显著优于朴素实现。算法能够高效可靠地恢复仅具有$11\times 11 = 121$个DFT系数的严重模糊$29 \times 29$二值矩阵。