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In image processing, problems of separation and reconstruction of missingpixels from incomplete digital images have been far more advanced in pastdecades. Many empirical results have produced very good results, however,providing a theoretical analysis for the success of algorithms is not an easytask, especially, for inpainting and separating multi-component signals. Inthis paper, we propose two main algorithms based on $l_1$ constrained andunconstrained minimization for separating $N$ distinct geometric components andsimultaneously filling-in the missing part of the observed image. We thenpresent a theoretical guarantee for these algorithms using compressed sensingtechnique, which is based on a principle that each component can be sparselyrepresented by a suitably chosen dictionary. Those sparsifying systems areextended to the case of general frames instead of Parseval frames which havebeen typically used in the past. We finally prove that the method does indeedsucceed in separating point singularities from curvilinear singularities andtexture as well as inpainting the missing band contained in curvilinearsingularities and texture.

Multi-component separation, inpainting and denoising with recovery guarantees