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 $N_1^α\times N_1^α$, where $N_1 \neq N_2$ are prime numbers and $α>1$ 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.
A fast methodology for large-scale focusing inversion of gravity and magnetic data using the structured model matrix and the $2D$ fast Fourier transform
Nonuniform Iterative Phasing Framework and Sampling Requirements for 3D Dynamical Inversion from Coherent Surface Scattering Imaging
Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；Advanced Photon Source, Argonne National Laboratory（阿贡国家实验室先进光子源）；University of California, Berkeley（加州大学伯克利分校）
