Image reconstruction from structured subsampled 2D Fourier data

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.