Complex Wavelet Based Image Analysis and Synthesis

This dissertation investigates the use of complex wavelets in image processing. The limitations of standard real wavelet methods are explained with emphasis on the problem of shift dependence. Complex wavelets can be used for both Bayesian and non-Bayesian processing. The complex wavelets are first used to perform some non-Bayesian processing. We describe how to extract features to characterise textured images and test this characterisation by resynthesizing textures with matching features. We use these features for image segmentation and show how it is possible to extend the feature set to model longerrange correlations in images for better texture synthesis. Second we describe a number of image models from within a common Bayesian framework. This framework reveals the theoretical relations between wavelet and alternative methods. We place complex wavelets into this framework and use the model to address the problems of interpolation and approximation. Finally we show how the model can be extended to cover blurred images and thus perform Bayesian wavelet based image deconvolution. Theoretical results are developed that justify the methods used and show the connections between these methods and alternative techniques. Numerical experiments on the test problems demonstrate the usefulness of the proposed methods, and give examples of the superiority of complex wavelets over the standard forms of both decimated and non-decimated real wavelets.