The Navier-Stokes-Voight Model for Image Inpainting

Abstract

In 2001, Bertalmio et. al. drew an analogy between the image intensity function for the image inpainting problem and the stream function in a two-dimensional (2D) incompressible fluid. An approximate solution to the inpainting problem is obtained by numerically approximating the steady state solution of the 2D NSE vorticity transport equation, and simultaneously solving the Poisson problem between the vorticity and stream function, in the region to be inpainted. This elegant approach allows one to produce an approximate solution to the image inpainting problem by using techniques from computational fluid dynamics. Recently, the three-dimensional (3D) Navier-Stokes-Voight (NSV) model of viscoelastic fluid, was suggested by Cao et. al. as an inviscid regularization to the 3D Navier-Stokes equations (NSE). The NSV model is shown to be globally well-posed and has a finite-dimensional global attractor, making it an attractive sub-grid scale turbulence model for purposes of numerical simulation. In this paper we investigate the use of the 2D NSV model for use in algorithms for the inpainting problem. We also present some new theoretical results based on energy methods comparing the sufficient conditions for stability of the discretization scheme for the two model equations.

25 pages, 6 figures. (Version 3 is revised version for publication.)

References in corpus (5)

• Analytical Study of Certain Magnetohydrodynamic-alpha Models
• On the Clark-alpha model of turbulence: global regularity and long--time dynamics
• A study of the Navier-Stokes-alpha model for two-dimensional turbulence
• Gevrey regularity of the global attractor of the 3D Navier-Stokes-Voight equations
• Global regularity for a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations

Cited by in corpus (4)

• Analysis of a General Family of Regularized Navier-Stokes and MHD Models
• Global Well-posedness for The 2D Boussinesq System Without Heat Diffusion and With Either Anisotropic Viscosity or Inviscid Voigt-$\alpha$ Regularization
• Invariant measures for the 3D Navier-Stokes-Voigt equations and their Navier-Stokes limit
• Higher-Order Global Regularity of an Inviscid Voigt-Regularization of the Three-Dimensional Inviscid Resistive Magnetohydrodynamic Equations