On Variational Problem with Nonstandard Growth Conditions and Its Applications to Image Processing

We propose a new variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to image processing. The characteristic feature of the proposed model is that the variable exponent, which is associated with non-standard growth, is unknown a priori and it depends on a particular function that belongs to the domain of objective functional. So, we deal with a constrained minimization problem that lives in variable Sobolev-Orlicz spaces. Because of this the standard approaches are no longer applicable in its study, especially with respect to the existence of minimizers and the study of their basic properties. It makes this minimization problem rather challenging. In view of this, we discuss the consistency of the proposed model, give the scheme for its regularization, derive the corresponding optimality system, and propose the iterative algorithm for practical implementations. In particular, we apply this approach to the image denosing problem and show that in all prototypical examples we obtain better results when compared to the Total Variation (TV) method