An improved nonlinear anisotropic PDE with p(x)-growth conditions applied to image restoration and enhancement

Abstract

This work proposes a novel nonlinear parabolic equation with p(x)-growth conditions for image restoration and enhancement. Based on the generalized Lebesgue and Sobolev spaces with variable exponent, we demonstrate the well-posedness of the proposed model. As a first result, we prove the existence of a weak solution to our model when the reaction term is bounded by a suitable function. Secondly, we use the approximations method to establish the existence of a nonnegative weak SOLA solution (Solution Obtained as Limit of Approximations) to the proposed model. Finally, numerical experiments illustrate that the proposed model performs better for image enhancement and denoising.

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

“…The appearance of the staircase effect is a notable drawback of the classical TV model. However, the non-convex model (1) did not gain significant attention for a long period due to its high numerical complexity and the absence of a rigorous mathematical substantiation of its consistency. Only particular solutions to this problem have been derived for a smoothed version of the integrand, using a weak notion of solution (see, for instance, Ref.…”

“…A recently developed alternative variant is the TV-like method [33] (see also Refs. [1,12]), which computes the variable exponent p in an offline step and keeps it as a fixed parameter in the final optimization problem. This approach allows the exponent to vary based on the spatial location, enabling users to locally select whether to preserve edges or smooth intensity variations.…”

Anisotropic equation based on fractional diffusion tensor for image noise removal

“…For image denoising, it can be reformulated mathematically, by finding an approximation of $$$$ {u}_0=u+\mu, $$$$ where $$$ u $$$ is the original image from a degraded or noisy image $$$ u $$$, and $$$ \mu $$$ is the Gaussian noise. In the past few decades, many methods have been proposed and developed to estimate $$$ u $$$, including wavelet transforms [13], total variation approaches [5, 14], Fourier transforms [15], partial differential equations [16, 17]. In our paper, we will concentrate on the last technique, on using a partial differential equations (PDEs), on images corrupted by white Gaussian noise.…”

Asymptotic Behavior of Some Parabolic Equations and Application in Image Restoration

“…The authors presented a novel model for image denoising and compared the results with the model of Perona-Malik and the method of the total variation (see [18]). And, in [3] the authors proposed a novel parabolic equations for image restoration and enhacement.They proved the existence of solution, established a nonnegative weak solution obtained as limit of approximation and give some application in image processing. Also, in [15] we find some optimal control problem for the Perona-Malik equation.…”