Advances in Total Variation Image Restoration: Blur Estimation, Parameter Estimation and Efficient Optimization

This thesis addresses total variation (TV) image restoration and blind image deconvolution. Classical image processing problems, such as deblurring, call for some kind of regularization. Total variation is among the state-of-the-art regularizers, as it provides a good balance between the ability to describe piecewise smooth images and the complexity of the resulting algorithms. In this thesis, we propose a minimization algorithm for TV-based image restoration that belongs to the majorization-minimization class (MM). The proposed algorithm is similar to the known iterative re-weighted least squares (IRSL) approach, although it constitutes an original interpretation of this method from the MM perspective. The problem of choosing the regularization parameter is also addressed in this thesis. A new Bayesian method is introduced to automatically estimate the parameter, by assigning it a non-informative prior, followed by integration based on an approximation of the associated partition function. The proposed minimization problem, also addressed using the MM framework, results on an update rule for the regularization parameter, and can be used with any TV-based image deblurring algorithm. Blind image deconvolution is the third topic of this thesis. We consider the case of linear motion blurs. We propose a new discretization of the motion blur kernel, and a new estimation algorithm to recover the motion blur parameters (orientation and length) from blurred natural images, based on the Radon transform of the spectrum of the blurred images.