Image processing with multiscale stochastic models

In this thesis, we develop image processing algorithms and applications for a particular class of multiscale stochastic models. First, we provide background on the model class, including a discussion of its relationship to wavelet transforms and the details of a two-sweep algorithm for estimation. A multiscale model for the error process associated with this algorithm is derived. Next, we illustrate how the multiscale models can be used in the context of regularizing ill-posed inverse problems and demonstrate the substantial computational savings that such an approach offers. Several novel features of the approach are developed including a technique for choosing the optimal resolution at which to recover the object of interest. Next, we show that this class of models contains other widely used classes of statistical models including I-D Markov processes and 2-D Markov random fields, and we propose a class of multiscale models for approximately representing Gaussian Markov random fields. These results, coupled with those illustrating the computational efficiencies that the multiscale models lead to, suggest that the multiscale framework is a powerful paradigm for image processing both because of the efficient algorithms it admits and because of the rich class of phenomena it can be used to describe. This motivates us in the final section of this thesis to pursue further algorithmic development for the multiscale models. In particular, we develop an efficient likelihood calculation algorithm for multiscale models and demonstrate an application' of the algorithm in the area of texture discrimination. The thesis concludes with a review of our main results and with a discussion of a few of the many open problems and promising directions for further research and application. Thesis Supervisor: Alan S. Willsky Title: Professor of Electrical Engineering