Binary Image Fast Perfect Recovery from Sparse 2D-DFT Coefficients

We investigate the reconstruction of a binary image from its sparse 2D discrete Fourier transform (2D-DFT) coefficients. Specifically, we focus on the case where the image height is equal to its width and is a prime number. We derive a lower bound on the number of coefficients required for perfect image recovery and propose a reconstruction algorithm. In our experiment, we demonstrate that the lower bound can be achieved when the height is less than 20. Consequently, we can efficiently reconstruct a 19×19 binary image using only 21 out of the total 361 2D-DFT coefficients, which accounts for approximately 5.81% of the coefficients..