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Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability
In this work we wish to recover an unknown image from a blurry, or noisy-blurry version. We solve this inverse problem by energy
minimization and regularization. We seek a solution of the form $u + v$,
where $u$ is a function of bounded variation (cartoon component), while $v$ is
an oscillatory component (texture), modeled by a Sobolev function with
negative degree of differentiability. We give several results of existence and characterization of minimizers of the proposed optimization problem.
Experimental results show that this
cartoon + texture model better recovers textured details in natural images, by comparison with the more standard models where the unknown is restricted only to the space of functions of bounded variation.