Variational Models for Restoration of images with blur and noise

In recent years, the interdisciplinary field of imaging science has been experiencing an explosive growth in active research and applications. In this talk I shall present some recent and new work of modeling the inverse problem of removing noise and blur in a given and observed image. Here we assume the Gaussian additive noise is present and the blur is defined by some linear filters. Inverting the filtering process does not lead to unique solutions without suitable regularization. There are several cases to discuss: Firstly I discuss the problem of how to select optimal coupling parameters, given an accurate estimate of the noise level, in a total variation (TV) optimisation model. Secondly I show a new algorithm for imposing the positivity constraint for the TV model for the case of a known blur. Finally I show how to generalise the new idea to the blind deconvolution where the blur operator is unknown and must be restored along with the image. Again the TV regularisers are used. However with the splitting idea, our work can be extended to include other high order regularizers such as the mean curvature. Once an observed image is improved, further tasks such as segmentation and co-registration become feasible. There will be potentially ample applications to follow up. Joint work with B. Williams, J. P. Zhang, Y.Zheng, S. Harding (Liverpool) and E. Piccolomini, F. Zama (Bologna). Other collaborators in imaging in general include T. F. Chan, R. H. Chan, B. Yu, N. Badshah, H. Ali, L. Rada, C. Brito, L. Sun, F. L. Yang, N. Chumchob, M. Hintermuller, Y. Q. Dong, X. C. Tai, etc.