Recovery of Integer Images from Limited DFT Measurements with Lattice Methods

TL;DR

This work proves that integer-valued images can be uniquely recovered from far fewer than all DFT coefficients by exploiting algebraic DFT structure and an integer prior. It develops a 2D-to-1D reduction, a minimal-sampling theorem, and a dynamic-programming inversion framework, later enhanced by a lattice-based SVP approach (via LLL) to solve NP-hard subproblems efficiently. The paper provides a thorough parameter analysis (notably $\beta_0$, $\beta_1$, $\beta_2$, and $K$) and validates the theory with extensive numerical experiments, including large-scale image reconstructions and structured data like QR codes. Practically, the method enables exact recovery of sizeable integer-valued images from a small fraction of Fourier measurements, with performance tied to coefficient precision and problem divisors. This offers a principled pathway for reliable partial-spectrum imaging when integer priors are appropriate and measurement budgets are tight.

Abstract

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.

Figures (13)

• Figure 1: Recursion tree showing subproblem structure when \ref{['alg:1D']} is applied to a length $N = 30$ signal.
• Figure 2: Left: The Hasse diagram for the subgroup lattice of $\mathbb{Z}_{30}$. Two subgroups $H \subsetneq K$ are connected if they are related by inclusion and there is no subgroup $H'$ satisfying $H \subsetneq H' \subsetneq K$. The lattice is arranged into levels where each subgroup on any given level is only connected to subgroups on the levels immediately above and below it. Right: In contrast with \ref{['fig:recursion']}, the subproblem lattice of recovering a length 30 integer signal. The identical lattice structure between the two diagrams suggest how the correspondence between subgroups and subproblems makes a dynamic programming algorithm efficient.
• Figure 3: The Hasse diagram of the cyclic subgroup lattice of $\mathbb{Z}_{4} \times \mathbb{Z}_{6}$. Only the generators of each subgroup are shown explicitly, as all other elements are contained in a proper subgroup lower in the lattice. Color of vertices of $\langle(k, l)\rangle$ indicates iteration of loop in Line \ref{['ln:outer']} of \ref{['alg:memo_2D']} where the $(k, l)$-subsignal is recovered ($D_1=1$ is blue, $D_1=2$ is green, and $D_1=4$ is red).
• Figure 4: For $M = 1, 2, 3$, the smoothed Monte Carlo distribution of guess errors $\lVert{\bf x} - \overline{\bf x}\rVert_2$ (solid blue line), the experimental average error (blue dashed line, value in legend), and the theoretical bound (red dashed line, value in legend). The test set used 1e5 signals of length $N = 30$ with entries distributed as ${\rm binom}(N, 0.5)$.
• Figure 5: For $M = 1,2,3$, the estimated $\beta_2$ from \ref{['eq:beta2_final']} plotted against $\beta_0$ when $N = 30$ and $K$ is computed from the bound in \ref{['eq:K_bound']}.

Theorems & Definitions (19)

• Lemma 2.1: pei2022binary1D
• Lemma 2.2: Special case of pei2022binary1D
• proof
• definition 1
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• definition 2: Subsignal