Inversion of band-limited discrete Fourier transforms of binary images: Uniqueness and algorithms

TL;DR

This paper investigates the theoretical limits and algorithms for uniquely recovering binary images, such as QR codes, from band-limited Fourier data, demonstrating effective reconstruction even with significant blurring.

Contribution

It provides new theoretical results on the minimal band limit for unique binary image recovery and proposes efficient algorithms combining integer programming and lattice reduction.

Findings

Proved uniqueness for certain binary image sizes and band limits.

Developed algorithms that outperform naive methods.

Successfully reconstructed 29x29 binary matrices from limited Fourier data.

Abstract

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.