Exact reconstruction of an image from measurements of its Discrete Fourier Transform (DFT) typically requires all DFT coefficients to be available. However, incorporating the prior assumption that the image contains only integer values enables unique recovery from a limited subset of DFT coefficients. This paper develops both theoretical and algorithmic foundations for this problem. We use algebraic properties of the DFT to define a reduction from two-dimensional recovery to several well-chosen one-dimensional recoveries. Our reduction framework characterizes the minimum number and location of DFT coefficients that must be sampled to guarantee unique reconstruction of an integer-valued image. Algorithmically, we develop reconstruction procedures which use dynamic programming to efficiently recover an integer signal or image from its minimal set of DFT measurements. While the new inversion algorithms still involve NP-hard subproblems, we demonstrate how the divide-and-conquer approach drastically reduces the associated search space. To solve the NP-hard subproblems, we employ a lattice-based framework which leverages the LLL approximation algorithm to make the algorithms fast and practical. We provide an analysis of the lattice method, suggesting approximate parameter choices to ensure correct inversion. Numerical results for the algorithms support the parameter analysis and demonstrate successful recovery of large integer images.

翻译：从离散傅里叶变换（DFT）测量值精确重建图像通常需要所有DFT系数。然而，若引入图像仅包含整数值的先验假设，则可通过有限子集的DFT系数实现唯一恢复。本文为该问题建立了理论与算法基础。我们利用DFT的代数性质，将二维恢复问题约简为若干精心选择的一维恢复问题。该约简框架刻画了保证整数值图像唯一重建所需采样的DFT系数最小数量与位置分布。在算法层面，我们开发了基于动态规划的重建方法，能够从最小DFT测量集中高效恢复整数信号或图像。尽管新逆变换算法仍涉及NP难子问题，我们论证了分治策略如何显著缩减相关搜索空间。针对NP难子问题，我们采用基于格的框架，利用LLL近似算法实现快速实用的求解。我们对格方法进行了理论分析，提出了确保正确逆变换的近似参数选择方案。算法数值实验结果验证了参数分析的有效性，并展示了大规模整数图像的成功恢复案例。