Computational Imaging (CI) is the field concerned with reconstructing images of some quantity of interest from a finite number of measurements. It is pervasive in science and engineering problems by enabling the reconstruction of images from a variety of measurements that are often indirect or non-traditional.

This thesis explores numerical methods to efficiently compute operators that appear in CI problems, specifically non-uniform Fast Fourier Transforms (NUFFT) and X-ray Transforms (XRT). They appear across imaging modalities such as X-ray tomography, radio interferometry, acoustics and optics, where they relate the quantity of interest to measurable data through forward models.

The thesis is structured into two parts. In the first part, we develop novel NUFFT methods capable of handling structured non-uniformity in the spatial and/or spectral knots, a phenomenon that appears in CI problems where measurements, or target images, exhibit patterned sparsity. We also develop XRT methods that can compute projections of arbitrary scan geometries fast.

In the second part, we explore two CI problems related to wide-field imaging in radio astronomy and acoustics. In the radio case, we demonstrate the utility of sparsity-aware NUFFTs for interferometric imaging via the HVOX method, a map synthesis technique that leverages the type-3 NUFFT to achieve high-accuracy imaging on any spherical tessellation without the need for post-reconstruction interpolation.

In the acoustic case, we investigate the problem of real-time imaging using a machine learning approach for image production, specifically using unrolled iterative model-based reconstruction networks. We design DeepWave, an interpretable and compact architecture that can reconstruct high-resolution acoustic intensity maps from raw microphone samples in real-time.

Overall, this thesis aims to augment the computational methods available in the scientist's toolbox to tackle CI problems where current methods are limited or inadequate.