Well-posedness for a fourth order nonlinear equation related to image processing

Abstract

Using the entropy estimates in [14], we establish the global existence and uniqueness of solutions to a fourth order equation related to image processing. Some numerical results on the Lena image are also presented to show the effectiveness of the equation for noise removal.

Entropy estimates of radial symmetric solutions to a fourth order nonlinear degenerate equation in higher dimensions

“…This paper is a sequel to [9,10], we continue to give some theoretical analysis to the following fourth order nonlinear degenerate equation in higher dimensions:…”

“…The equation (1) has backgrounds in image processing: the solution u represents the gray level of a picture and the diffusivity function g(u) = u −n corresponds to the TV diffusivity if n = 1 [4,12], while it is the BFB diffusivity for n = 2 [7]. In [10], We gave some numerical results to show the effectiveness of the model (1) for noise removal, and made some comparison with the second order PM equation and YK equation which are two famous models in image processing. The initial boundary value problem for equation (1) in dimensional one:…”

“…was investigated in [9,10] recently, where Ω is a bounded interval and u 0 is a positive function in H 1 (Ω). In [9], under the assumption of existence of classical solutions to the equation, we discussed the asymptotic behavior of solutions to (2) by using the entropy dissipation method.…”

“…Based on these estimates, we showed that the solution u(x, t) to (2) decays to its mean value uniformly in x, in an order like t − 1 4 for long time. Later, using the approximation method combined with the entropy estimates, we investigated the well-posedness theory of the problem (2) in reference [10], and proved the global existence and uniqueness of classical solution for suitable u 0 .…”

A nonlinear fourth-order PDE for multi-frame image super-resolution enhancement

“…To increase the smoothness near edges, we use a nonlinear fourth-order PDE derived from the proposed one in [15] using Weickert filter [60]. In fact, the weakness of the second-order partial differential equations resulting in the appearance to a class of high-order diffusion models [40,63,62,26,22] that in general outperform the second order ones. Indeed, to restore corners, curvatures and for matching edges across large distances, more regularity is needed in the used PDE.…”