In many practical imaging scenarios, including computed tomography and magnetic resonance imaging (MRI), the goal is to reconstruct an image from few of its Fourier domain samples. Many state-of-the-art reconstruction techniques, such as total variation minimization, focus on discrete ‘on-the-grid” modelling of the problem both in spatial domain and Fourier domain. While such discrete-to-discrete models allow for fast algorithms, they can also result in sub-optimal sampling rates and reconstruction artifacts due to model mismatch. Instead, this thesis presents a framework for “off-the-grid”, i.e. continuous domain, recovery of piecewise smooth signals from an optimal number of Fourier samples. The main idea is to model the edge set of the image as the level-set curve of a continuous domain band-limited function. Sampling guarantees can be derived for this framework by investigating the algebraic geometry of these curves. This model is put into a robust and efficient optimization framework by posing signal recovery entirely in Fourier domain as a structured low-rank (SLR) matrix completion problem. An efficient algorithm for this problem is derived, which is an order of magnitude faster than previous approaches for SLR matrix completion. This SLR approach based on off-the-grid modeling shows significant improvement over standard discrete methods in the context of undersampled MRI reconstruction.