Inverse Problems in Mathematical Imaging with Applications in Multiplicative Noise Removal, Remotely Sensed Atmospheric Wind Velocity Estimation, and Turbulent Fluid Simulation

This thesis considers three inverse problems originating in mathematical imaging, remote sensing and simulation. First, it addresses a challenging image processing task: recovering images corrupted by multiplicative noise. Motivated by the success of multiscale hierarchical decomposition methods (MHDM) in image processing, a variety of both classical and new multiplicative noise removing models are adapted to the MHDM form. On the basis of previous work, tight and refined extensions of the multiplicative MHDM process are proposed. Existence and uniqueness of solutions for the proposed models is discussed, in addition to convergence properties. Moreover, the work introduces a discrepancy principle stopping criterion which prevents recovering excess noise in the multiscale reconstruction. The validity of all the proposed models is qualitatively and quantitatively evaluated through comprehensive numerical experiments and comparisons across several images degraded by multiplicative noise. By construction, these multiplicative multiscale hierarchical decomposition methods have the added benefit of recovering many scales of an image, which can provide features of interest beyond image denoising. Second, this work considers an applied mathematical imaging problem in remote sensing. Accurate estimation of atmospheric wind velocity plays an important role in weather forecasting, flight safety assessment and cyclone tracking. Atmospheric data captured by infrared and microwave satellite instruments provide global coverage for weather analysis. Extracting wind velocity fields from such data has traditionally been done through feature tracking, correlation/matching or optical flow means from computer vision. However, these recover either sparse velocity estimates, oversmooth details or are designed for quasi-rigid body motions which over-penalize vorticity and divergence within the often turbulent weather systems. This thesis proposes a texture based optical flow procedure tail