  In image processing, problems of separation and reconstruction of missing
pixels from incomplete digital images have been far more advanced in past
decades. Many empirical results have produced very good results, however,
providing a theoretical analysis for the success of algorithms is not an easy
task, especially, for inpainting and separating multi-component signals. In
this paper, we propose two main algorithms based on $l_1$ constrained and
unconstrained minimization for separating $N$ distinct geometric components and
simultaneously filling-in the missing part of the observed image. We then
present a theoretical guarantee for these algorithms using compressed sensing
technique, which is based on a principle that each component can be sparsely
represented by a suitably chosen dictionary. Those sparsifying systems are
extended to the case of general frames instead of Parseval frames which have
been typically used in the past. We finally prove that the method does indeed
succeed in separating point singularities from curvilinear singularities and
texture as well as inpainting the missing band contained in curvilinear
singularities and texture.
