# On double-resolution imaging and discrete tomography

## TL;DR

This paper investigates the computational complexity of double-resolution imaging for binary objects, showing polynomial solvability under perfect data and NP-hardness with noisy data, highlighting the impact of noise on algorithmic feasibility.

## Contribution

It introduces a formal analysis of the complexity of double-resolution imaging, revealing polynomial solvability with reliable data and NP-hardness when data contains errors.

## Key findings

- Polynomial-time solvable with reliable gray levels.
- NP-hard when gray levels have errors of ±1 or more.
- Noise affects both image quality and computational tractability.

## Abstract

Super-resolution imaging aims at improving the resolution of an image by enhancing it with other images or data that might have been acquired using different imaging techniques or modalities. In this paper we consider the task of doubling, in each dimension, the resolution of grayscale images of binary objects by fusion with double-resolution tomographic data that have been acquired from two viewing angles. We show that this task is polynomial-time solvable if the gray levels have been reliably determined. The problem becomes $\mathbb{N}\mathbb{P}$-hard if the gray levels of some pixels come with an error of $\pm1$ or larger. The $\mathbb{N}\mathbb{P}$-hardness persists for any larger resolution enhancement factor. This means that noise does not only affect the quality of a reconstructed image but, less expectedly, also the algorithmic tractability of the inverse problem itself.