Image Denoising Using Directional Adaptive Variable Exponents Model

Abstract

In this paper, a new variational image denoising model is proposed. The new model could be seen to be a twostep method. In the first step, structure tensor analysis is used to infer something about the local geometry. The eigenvectors and the eigenvalues of the structure tensor are used in the construction of the denoising energy. In the second step, the actual variational denoising takes place. The steps are coupled in the sense that the energy expression is built using the underlying image, not the data. Two variable exponents are incorporated into the regularizer in order to reduce the staircasing effect, which is often present in the methods based on the first-order partial derivatives, and to increase smoothing along the image boundaries. In addition, two pointwise weight functions try to help to preserve small-scale details. In the theoretical part, the existence of a minimizer of a weak form of the original energy is considered. In the numerical part, an algorithm based on iterative minimization is presented and the numerical experiments demonstrate the possible advantages of the new model over some existing variational and partial differential equations methods.

On some equation defined on the whole euclidean space ℝN and involving the p(u)‐Laplacian

“…However, some numerical techniques applied to the image restoration process lead to the minimization of some functional with growth conditions depending on the unknown solution itself or its gradient; see other works. [6][7][8] Indeed, the observation has proved that the consideration of variable exponents depending on the solution u or its gradient ∇u reduces the noise of the restored image u. In other words, considering the variational problem with exponents depending on the solution has a confirmed denoising effect on digital images.…”

On a class of nonlocal evolution equations with the p[u(x,t)]-Laplace operator

“…Functionals with the growth condition depending on the solution or its gradient are successfully used for denoising of digital images -see, e.g., [5][6][7] for the models based on minimization of functionals with p(|∇u|)-growth and [8] for a discussion of the model of denoising of the image f based on the minimization of the functional…”

Existence results for a class of local and nonlocal nonlinear elliptic problems

“…processing and computer vision [5,6,17]. J. Türola in [17] presented several numerical examples suggesting that the consideration of exponents p = p(u) preserves the edges and reduces the noise of the restored images u.…”

“…processing and computer vision [5,6,17]. J. Türola in [17] presented several numerical examples suggesting that the consideration of exponents p = p(u) preserves the edges and reduces the noise of the restored images u. A numerical example suggesting a reduction of noise in the restored images u when the exponent of the regularization term is p = p(|∇u|) is presented in [5].…”