arXiv subjects
Publications and source records attributed to Alexandros Matsoukas.
We study a double-phase Neumann problem with non-homogeneous boundary conditions, where the lowest exponent $p$ is equal to 1. The existence of a solution is established as the limit of solutions to corresponding double-phase problems with $p>1$. We also provide a variational characterization of the limit.
Even though more than 30 years have passed since the seminal Rudin--Osher--Fatemi (ROF) paper on total variation (TV) denoising, it remains relevant, in particular in scientific applications such as astronomical imaging. However, it is known to suffer from artifacts such as the staircasing effect. Many variants of the model have been proposed with the aim of countering this. Recently, against the backdrop of immense research output on double-phase problems in the mathematical analysis community, a double-phase type integral functional, comprising of TV and a weighted term of quadratic growth, was suggested as a regularizer for image restoration. Here, we propose an adaptive variant of the ROF denoising model based on that regularizer. It is designed to reduce staircasing with respect to the classical ROF model, while preserving the edges of the image in a similar fashion. We implement the model and test its performance on synthetic and natural images over a range of noise levels. Compared to {established} models {with similar interpretability to ROF}, we observe an improved or similar performance in terms of similarity metrics SSIM, PSNR, {and LPIPS}, while the staircasing effect is visibly reduced.
In this paper we study a double-phase problem involving the 1-Laplacian with non-homogeneous Dirichlet boundary conditions and show the existence and uniqueness of a solution in a suitable weak sense. We also provide a variational characterization of this solution via the corresponding minimization problem.
We consider a class of integral functionals with Musielak-Orlicz type variable growth, possibly linear in some regions of the domain. This includes $p(x)$ power-type integrands with $p(x)\ge 1$ as well as double-phase $p\!-\!q$ integrands with $p=1$. The main goal of this paper is to identify the $L^2$-subdifferential of the functional, including a local characterisation in terms of a variant of the Anzellotti product defined through the Young's inequality. As an application, we obtain the Euler-Lagrange equation for the variant of the Rudin-Osher-Fatemi image denoising problem with variable growth regularising term.
In this paper we prove an existence and uniqueness result for the double phase Dirichlet problem when the lowest exponent is equal to 1. Our solution is a function of bounded variation that simultaneously lies in a suitable weighted Sobolev space and is found as the limit of a sequence of solutions of intermediate double phase Dirichlet problems whose lowest exponent $p$ goes to 1. As a result of that, our approach involves the study of some relevant properties of generalized Orlicz-Sobolev spaces.