| In this thesis, robust methods are analyzed and applied to several areas. We propose a novel algorithm for image demosaicking. The principal idea is that by identifying a number of replicas in the filter bank transfer domain, demosaicking schemes can be represented by robust regression frameworks. This approach is very useful since it provides mathematically reliable solutions without relying on a specific color filter array pattern. It also gives us a new way to understand the fundamental theory of demosaicking and directs us toward better color filter array design. For adaptive beamforming, it is well known that performance may degrade in the presence of steering errors. Many robust algorithms have been proposed to overcome this problem, and diagonal loading techniques are one of the most popular. We start from the discussion of loading techniques with careful study of the underline facts to robustify the system and exploit potential improvements. Two new approaches of robust beamforming algorithms---Super Gaussian loading and variable loading techniques---are proposed. Super Gaussian loading is a generalization of traditional diagonal loading in that an ℓ2 norm restriction is replaced by an ℓ p norm one. This follows from a consideration that in beamforming, both the covariance matrix and the steering vector are estimated with uncertainties. Although diagonal loading is optimal in the case that only steering vector error is considered, our proposed Super Gaussian loading technique appears more reasonable in practice. We develop methods to choose the parameter p, and design an online implementation to update the beamformer. Similarly, we propose a variable loading technique after thorough analysis of the eigenvalue structure of sample covariance matrices. Variable loading provides better approximation of the covariance matrix inverse, which leads to more robust beamformer performance. Bayesian beamforming under a minimum mean square error criterion is also developed, using an importance sampling technique. Inspired by robust beamforming, we also apply the proposed loading technique to the control variates method of variance reduction in Monte Carlo estimation. We describe a number of experiments in each topic of study, and show the improvement that we have made. |
