Low‐curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes

Abstract

We consider a class of fourth-order nonlinear diffusion equations motivated by Tumblin and Turk's "low-curvature image simplifiers" for image denoising and segmentation. The PDE for the image intensity u is of the formis a "curvature" threshold and λ denotes a fidelitymatching parameter. We derive a priori bounds for u that allow us to prove global regularity of smooth solutions in one space dimension, and a geometric constraint for finite-time singularities from smooth initial data in two space dimensions. This is in sharp contrast to the second-order Perona-Malik equation (an ill-posed problem), on which the original LCIS method is modeled. The estimates also allow us to design a finite difference scheme that satisfies discrete versions of the estimates, in particular, a priori bounds on the smoothness estimator in both one and two space dimensions. We present computational results that show the effectiveness of such algorithms. Our results are connected to recent results for fourth-order lubrication-type equations and the design of positivitypreserving schemes for such equations. This connection also has relevance for other related fourth-order imaging equations.

High-order fractional partial differential equation transform for molecular surface construction

“…Over the past decade, the mathematical analysis of high-order geometric PDEs has attracted much attention. For example, Bertozzi and Greer have analyzed fourth order nonlinear PDEs in the Sobolev space and proved the existence and uniqueness of the solution to a case with H 1 initial data and a regularized operator [7, 28, 29]. Xu and Zhou [65] showed the well-posedness of the solution of fourth order nonlinear PDEs.…”

Two Enhanced Fourth Order Diffusion Models for Image Denoising

“…Secondly, in numerical experiments, such algorithms tend to produce artifacts at edges, such as staircasing (when false edges are introduced) [22,32] and blocky, cartoonish effects (when smooth edges are sharpened into corners) [34]. The literature abounds in attempts to cure these problems, including regularizing the edge detector |∇u| [1,2,5,8,18], and generalizing the ideas of Perona and Malik to higher order equations [6,14,20,21,25,30,31,34]. Of note is a fourth order generalization of (1.1) proposed by You and Kaveh which is the focus of this paper.…”

Discretization schemes and numerical approximations of PDE impainting models and a comparative evaluation on novel real world MRI reconstruction applications

“…The PeronaMalik equation e.g behaves as a backwards heat equation and instantly creates jumps (i.e. shocks) in unpredictable locations [8]. Such results occur either because of the PDE model which might be not representative of the image dynamical system or because of the discretization and numerical approximation schemes involved.…”

“…Examples include the 'Low Curvature Image Simplifier' (LCIS) equation of Tumblin and Turk [8,9] as well as similar higher order PDE models [10]. Other mainstream such attempts include the imposing of constraints in the diffusion PDE models [11].…”

“…The focus of the herein presentation is, however, the discretization and numerical approximation algorithms involved in these PDE models. Although the mainstream discretization algorithm is, as mentioned above the finite difference method, there are many variations of it [8] that will be herein reviewed as well as the more recent Radial Basis Functions (RBF) discretization schemes that will be analysed in the context of the above PDE models, along with other important discretization and numerical approximation models, including [12]- [15].…”