Image reconstruction from structured subsampled 2D Fourier data

TL;DR

This paper investigates image reconstruction from limited 2D Fourier samples, emphasizing the importance of sampling pattern design and proposing a hybrid algorithm that outperforms traditional methods, especially for natural and cartoon-like images.

Contribution

It introduces a hybrid reconstruction algorithm combining TV minimization and pattern-specific recovery, demonstrating superior performance over non-adaptive methods.

Findings

Effective reconstruction with data reduction rates up to 8 for complex images.

Superior performance on natural and cartoon-like images.

Highlights the importance of sampling pattern choice for optimal results.

Abstract

In this paper we study the performance of image reconstruction methods from incomplete samples of the 2D discrete Fourier transform. Inspired by requirements in parallel MRI, we focus on a special sampling pattern with a small number of acquired rows of the Fourier transformed image. We show the importance of the low-pass set of acquired rows around zero in the Fourier space for image reconstruction. A suitable choice of the width $L$ of this index set depends on the image data and is crucial to achieve optimal reconstruction results. We prove that non-adaptive reconstruction approaches cannot lead to satisfying recovery results. We propose a new hybrid algorithm which connects the TV minimization technique based on primal-dual optimization with a recovery algorithm which exploits properties of the special sampling pattern for reconstruction. Our method shows very good performance for natural images as well as for cartoon-like images for a data reduction rate up to 8 in the complex setting and even 16 for real images.