Conditional image diffusion
Antoni Buades Bartomeu Coll Jose-Luis Lisani Catalina Sbert
Inverse Problems & Imaging 2007, 1(4): 593-608 doi: 10.3934/ipi.2007.1.593
In this paper, a theoretical framework for the conditional diffusion of digital images is presented. Different approaches have been proposed to solve this problem by extrapolating the idea of the anisotropic diffusion for a grey level images to vector-valued images. Then, the diffusion of each channel is conditioned to a direction which normally takes into account information from all channels. In our approach, the diffusion model assumes the a priori knowledge of the diffusion direction during all the process.
    The consistency of the model is shown by proving the existence and uniqueness of solution for the proposed equation from the viscosity solutions theory. Also a numerical scheme adapted to this equation based on the neighborhood filter is proposed. Finally, we discuss several applications and we compare the corresponding numerical schemes for the proposed model.
keywords: PDEs image filtering. Color image processing
Half-linear regularization for nonconvex image restoration models
Bartomeu Coll Joan Duran Catalina Sbert
Inverse Problems & Imaging 2015, 9(2): 337-370 doi: 10.3934/ipi.2015.9.337
Image restoration is the problem of recovering an original image from an observation of it in order to extract the most meaningful information. In this paper, we study this problem from a variational point of view through the minimization of energies composed of a quadratic data-fidelity term and a nonsmooth nonconvex regularization term. In the discrete setting, existence of minimizer is proved for arbitrary linear operators. For this kind of problems, fully segmented solutions can be found by minimizing objective nonconvex functionals. We propose a dual formulation of the model by introducing an auxiliary variable with a double function. On one hand, it marks the edges and it ensures their preservation from smoothing. On the other hand, it makes the criterion half-linear in the sense that the dual energy depends linearly on the gradient of the image to be recovered. This leads to design an efficient optimization algorithm with wide applicability to several image restoration tasks such as denoising and deconvolution. Finally, we present experimental results and we compare them with TV-based image restoration algorithms.
keywords: nonconvex optimization edge-preserving regularization Restoration segmentation deconvolution denoising duality. inverse problems weighted total variation

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