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

Abstract

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. 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 an iterative algorithm for practical implementations.

Generalized Active Contour Model in an Anisotropic Variable Exponent Sobolev Space

“…So, at the first step, we have realized the denoising and debluring procedure (see the right panel in Fig. 5.1) following the variational approach that has been recently proposed in [15]. As Figure 5.2 indicates, the histogram of the smoothed image has a strongly marked compactly localized spectrum that can be considered as a 'good option' for its piesewise constant approximation.…”

Variational Model with Nonstandard Growth Condition in Image Restoration and Contrast Enhancement

“…In this section, we present a novel variational problem for the denoising and contrast enhancement of non-smooth RGB images which can be viewed as an improved version of the variational model recently proposed in Ref. [17]. R 2 ) be a vector field of unit normals to the topographic map of the spectral channel…”

“…Proposition 7 [17] Let p k,ε k∈N be a sequence of exponents that satisfies all preconditions of Lemma 1.…”

“…3, we present a novel variational problem for the denoising and contrast enhancement of non-smooth RGB images which can be viewed as an improved version of the variational model that has been recently proposed in Ref. [17]. Section 4 is devoted to the derivation of optimality conditions to the original problem and their substantiation.…”

Solvability Issues for Some Noncoercive and Nonmonotone Parabolic Equations Arising in the Image Denoising Problems

“…Moreover, the above mentioned problem can be considered as a model for the deblurring and denoising of multi-spectral images. In particular, this model has been proposed in [19,30] in order to avoid the blurring of edges and other localization problems presented by linear diffusion models in images processing. We also refer to [28], where the authors study some optimal control problems associated with a special case of the model (1.1)-(1.3) and show the given class of optimal control problems is well posed.…”