Imaging technologies such as Magnetic Resonance Imaging (MRI) and Synthetic Aperture Radar (SAR) collect Fourier data and then process the data to form images. Because images are piecewise smooth, the Fourier partial sum (i.e. direct inversion of the Fourier data) yields a poor approximation, with spurious oscillations forming at the interior edges of the image and reduced accuracy overall. This is the well known Gibbs phenomenon and many attempts have been made to rectify its effects. Previous algorithms exploited the sparsity of edges in the underlying image as a constraint with which to optimize for a solution with reduced spurious oscillations. While the sparsity enforcing algorithms are fairly effective, they are sensitive to several issues, including undersampling and noise. Because of the piecewise nature of the underlying image, we theorize that projecting the solution onto the wavelet basis would increase the overall accuracy. Thus in this investigation we develop an algorithm that continues to exploit the sparsity of edges in the underlying image while also seeking to represent the solution using the wavelet rather than Fourier basis. Our method successfully decreases the effect of the Gibbs phenomenon and provides a good approximation for the underlying image. The primary advantages of our method is its robustness to undersampling and perturbations in the optimization parameters.

High-order methods are known for their accuracy and computational performance when applied to solving partial differential equations and have widespread use

in representing images compactly. Nonetheless, high-order methods have difficulty representing functions containing discontinuities or functions having slow spectral decay in the chosen basis. Certain sensing techniques such as MRI and SAR provide data in terms of Fourier coefficients, and thus prescribe a natural high-order basis. The field of compressed sensing has introduced a set of techniques based on $\ell^1$ regularization that promote sparsity and facilitate working with functions having discontinuities. In this dissertation, high-order methods and $\ell^1$ regularization are used to address three problems: reconstructing piecewise smooth functions from sparse and and noisy Fourier data, recovering edge locations in piecewise smooth functions from sparse and noisy Fourier data, and reducing time-stepping constraints when numerically solving certain time-dependent hyperbolic partial differential equations.