Variable Exponent, Linear Growth Functionals in Image Restoration

Abstract

We study a functional with variable exponent, 1 ≤ p(x) ≤ 2, which provides a model for image denoising, enhancement, and restoration. The diffusion resulting from the proposed model is a combination of total variation (TV)-based regularization and Gaussian smoothing. The existence, uniqueness, and long-time behavior of the proposed model are established. Experimental results illustrate the effectiveness of the model in image restoration.

Entire solutions of multivalued nonlinear Schrödinger equations in Sobolev spaces with variable exponent

“…For deep results in weighted Sobolev spaces with applications to partial differential equations and nonlinear analysis we refer to the excellent monographs by Drabek, Kufner and Nicolosi [13], Hyers, Isac and Rassias [21], Kufner and Persson [25], and Precup [34]. We also refer to the recent works by Diening [11], Ruzicka [36] and Chen, Levine and Rao [8] for applications of Sobolev spaces with variable exponent in the study of electrorheological fluids or in image restoration.…”

Nonlinear diffusion equations driven by the p(·)-Laplacian

“…A mathematical analysis was also done for such problems by Acerbi and Mingione [1,2] and by Acerbi et al [3]. In [27], some nonlinear parabolic problem proposed by [17] was studied in a weak formulation and an existence result for weak solutions was established. Antontsev and Shmarev studied parabolic equations involving anisotropic p(·, ·)-Laplace operators with log-Hölder continuous (x, t)-dependent exponents and proved the existence, uniqueness, extinction in finite time and blow-up of solutions in [7][8][9][10].…”

“…The constant exponent p(·) ≡ p is particularly known to have a threshold between two drastically different types of nonlinear diffusion, and moreover, the limit of p → ∞ exhibits a peculiar phenomena called fast/slow diffusion. Parabolic equations involving the p(·)-Laplacian have been proposed in the study of image restoration (see [17]) as well as in some model of electrorheological fluids (see [19,21,34]). A mathematical analysis was also done for such problems by Acerbi and Mingione [1,2] and by Acerbi et al [3].…”

Diffusion-Steered Super-Resolution Image Reconstruction

“…Hence, regularization through TV promotes recovery of edges, which appear as "jumps" or discontinuous parts of the image, and effective noise removal. But studies have revealed that TV formulations favor piecewise-constant solutions, a consequence that generates staircase effects and introduces false edges [16]. Also, TV regularization tends to lower contrast even in noise-free or flat image regions [17].…”