Variational Approaches for Despeckling and Deblurring of Images

Abstract

Speckle reduction is an inevitable pre-processing activity in some of the medical and satellite imaging modalities. This work is dedicated to study the behaviour of speckles and reduce them in medical ultrasound images and synthetic aperture radar images. Three novel methods have been proposed in this thesis to despeckle and deblur the input data. The first two models being proposed are variational frameworks, where a constrained optimization problem is derived with an appropriate objective functional and a set of constraints. The behaviour of the objective functional deeply influences the restoration process. A non-local total bound variational prior is designed in the first place to restore the images from their speckled observations. This objective functional designed using this prior, duly respects the gradient oscillations due to edges in the images while despeckling them. The theory behind the design of the constrained optimization problem lies in the Bayesian maximum a posteriori estimation process, which is re-designed to suit the optimization problem under consideration. The noise distribution plays a vital role in the design of the functional, optimizing which leads to the desired solution eventually. In the second model, the energy (optimizing) functional is designed as mentioned earlier, however, the objective functional is a modified version of the well known Mumford Shah model. Though, Mumford Shah level-set model has been extensively used for image segmentation, its capacity to restore the data is being duly analyzed in this thesis. A controlled evolution of the level-sets under a well designed data-fidelity, duly despeckles and deblurrs the data in the course of the evolution. These two variational regularization models are theoretically analyzed to study the conditions for existence of unique solution. The third model, is based on the Gradient fidelity of the image function and it duly alleviates speckles and blur from images while checking the possibility of a piecewise linear approximations which leads to visual discrepancies in the restored data. All models described in the thesis are experimentally verified using a large set of input data from ultrasound and synthetic radar imaging applications. Furthermore, the performance of these models along with the ones in the literature are statistically quantified. The required mathematical preliminaries, definitions and derivations have been incorporated in the Appendix for a seamless reading of this thesis.