Optical Phase Space Measurements and Applications to 3D Imaging and Light Scattering

4D phase space descriptions of light capture wave-optical and angular information, enabling digitally refocusing, 3D reconstructions and aberration removal. The wave-optical theory includes diffraction and interference effects, making phase space applicable to scales near the wavelength of light (e.g. in microscopy); however, at the cost of making phase space functions more complicated than their ray optics counterparts (light fields). In this thesis, we aim at bridging the gap between the abstract high-dimensional phase space and actual experiments upon which the reconstruction of unknown objects relies. We achieve the aim by 1) providing practical methods of measuring phase-space functions with good resolution in all 4D and 2) developing phase-space theories that we use to computationally mitigate scattering in experimental situations. We extend phase-space measurement schemes from from lenslet arrays to a scanning-based coded aperture method in order to improve information throughput. Theory and experiment for designed coded apertures is proposed that can efficiently capture the entire 4D phase space. Next, we develop a phase-space theory for imaging through scattering and apply it to experimentally imaging point sources through scattering and tracking neural activity in a scattering environment, such as mouse brain tissue. The method utilizes the dimension mismatch between 3D object and 4D phase-space measurements, along with a sparsity prior, to ensure robustness and allow 3D localization of point sources relatively deep into scattering tissue. We develop theory and verify the mathematical phase-space scattering operator, then study how light interacts with scatterers and propose a fast wave-equation solver. This method uses an accelerated gradient descent solver and expands the solution to the wave equation as a series of the gradient solver updates. The method outperforms the first Born approximation and the Rytov approximation in predicting the scattered fie