  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.
