Directional Multiscale Analysis Using Shearlet Theory and Applications

Shearlets emerged in recent years in applied harmonic analysis as a general framework to provide sparse representations of multidimensional data. This construction was motivated by the need to provide more efficient algorithms for data analysis and processing, overcoming the limitations of traditional multiscale methods. Particularly, shearlets have proved to be very effective in handling directional features compared to ideas based on separable extension, used in multi-dimensional Fourier and wavelet analysis. In order to efficiently deal with the edges and the other directionally sensitive (anisotropic) information, the analyzing shearlet elements are defined not only at various locations and scales but also at various orientations. Many important results about the theory and applications of shearlets have been derived during the past 5 years. Yet, there is a need to extend this approach and its applications to higher dimensions, especially 3D, where important problems such as video processing and analysis of biological data in native resolution require the use of 3D representations. The focus of this thesis is the study of shearlet representations in 3D, including their numerical implementation and application to problems of data denoising and enhancement. Compared to other competing methods like 3D curvelet and surfacelet, our numerical experiments show better Peak Signal to Noise Ratio (abbreviated as PSNR) and visual quality. In addition, to further explore the ability of shearlets to provide an ideal framework for sparse data representations, we have introduced and analyzed a new class of smoothness spaces associated with the shearlet decomposition and their relationship with Besov and curvelet spaces. Smoothness spaces associated to a multi-scale representation system are important for analysis and design of better image processing algorithms.