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Center for Nonlinear Studies |
In this talk I will present three novel families of convex energy functionals that can be employed to derive regularized solutions of ill-posed inverse imaging problems. One of the main features shared by all the proposed regularizers, and which also differentiates them from existing regularization schemes, is that they depend on matrix-valued differential operators. In particular, the first regularization family depends on the structure tensor operator which can measure the geometry of image structures in a local neighborhood. The second one depends on a novel non-local version of the structure tensor, which has the ability to further capture non-local information. Finally, the third regularization family depends on the Hessian operator which embodies all the available second-order derivative information of the image intensity map. It will be shown that these regularizers can be considered as valid extensions/generalizations of the popular total variation (TV)semi-norm, in the sense that they satisfy the same invariance properties w.r.t affine transformations of the coordinate system. Meanwhile, by taking advantage of additional information about the image structures, they can avoid the staircase effect, a common artifact of TV-based reconstructions, and perform very well for a wide range of imaging applications. The proposed functionals are non-smooth and, thus, their minimization can be a challenging task. To deal with this difficulty, I will further discuss about an efficient iterative algorithm that can be utilized for evaluating their proximal maps. Finally, I will present image reconstruction results for several inverse imaging problems such as denoising, deconvolution, and zooming (super-resolution) that illustrate the potential of the proposed methods in practical applications.
Host: Brendt Wohlberg