Image Reconstruction from Partial Fourier Measurements via Curl Constrained Sparse Gradient Estimation

Abstract

In this paper, we propose new gradient-based methods for image reconstruction from partial Fourier measurements, which are commonly used in magnetic resonance imaging (MRI) or synthetic aperture radar. Compared to classical gradient recovery methods, a key improvement is obtained by formulating the gradient recovery problem as a compressed sensing problem with the additional constraint that the curl of the gradient field must be zero. Moreover, we formulate the image recovery problem as an inverse problem on graphs. Iteratively reweighted ℓ₁ recovery methods are proposed to recover these relative differences and the structure of the similarity graph. Finally, the image is recovered from the compressed Fourier measurements using least squares estimation. Numerical experiments demonstrate that the proposed approach outperforms the state-of-the-art image recovery methods.