Shearlet based Fourier compressed sensing

This thesis investigates the sample complexity of signal recovery in compressed sensing, understood as the number of measurements required to recover a sparse signal, comparing the performance of sparsifying frames of shearlets and wavelets. Directional representation systems such as shearlets can sparsely represent images with anisotropic features which allows for efficient nonlinear approximations. The central question we address is whether this difference in efficiency allows for a proportional reduction in the number of Fourier measurements needed for successful reconstruction. We analyse how much of the known recovery theory for multiscale systems can extend to shearlets. We study the obstacles encountered when trying to apply standard frame based recovery results to shearlet systems. First, we obtain a local coherence decay bound between Fourier measurements and cone-adapted shearlets that is slower than the corresponding local coherence for wavelets. Second, managing the sparsifying system's redundancy relies on evaluating a localization factor, which requires lower frame bounds that can be exponentially small in the finest scale. Even under optimistic assumptions on the localization factor, the sufficient recovery condition considered in this thesis scales quadratically with the shearlet sparsity, which offers no substantial theoretical reduction over wavelets. We further investigate the underlying lower Riesz bound problem for finite wavelet and shearlet systems. For shearlets, explicit lower bounds can deteriorate rapidly with the number of scale and shear pairs, while a Gram matrix approach reduces the eigenvalue problem for its finest scale submatrix to a collection of lower dimensional eigenvalue problems with a computable approximation error. These theoretical limitations are tested through a series of numerical experiments on a dataset of piecewise regular digital images. Our experiments confirm that shearlets can accurately represent these images using substantially fewer coefficients than wavelets. However, phase diagram analyses indicate that this advantage in sparsity does not yield a proportional reduction in required samples. We conclude that despite the superior nonlinear approximation rates of shearlets, their practical sample complexity in compressed sensing applications with subsampled Fourier measurements remains comparable to that of wavelets.