The Navier-Stokes-Voight model for image inpainting

Abstract

In this paper we investigate the use of the 2D Navier-Stokes-Voight (NSV) model for use in algorithms and explore its limits in the context of image inpainting. We begin by giving a brief review of the work of Bertalmio et. al. in 2001 on exploiting an analogy between the image intensity function for the image inpainting problem and the stream function in 2D incompressible fluid. An approximate solution to the inpainting problem was then obtained by numerically approximating the steady state solution of the 2D Navier-Stokes 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 (CFD). 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. We give some background on the NSV mathematical model, describe why it is a good candidate sub-grid-scale turbulence model, and propose this model as an alternative for image inpainting. We describe an implementation of inpainting use the NSV model, and present numerical results comparing the resulting images when using the NSE and NSV for inpainting. Our results show that the NSV model allows for a larger time step to converge to the steady state solution, yielding a more efficient numerical process when automating the inpainting process. We compare quality of the resulting images using subjective measure (human evaluation) and objected measure (by calculating the peak signal-to-noise ratio (PSNR), also known as peak signal-to-reconstructed measure). 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. These theoretical and numerical studies shed some light on what can be expected from this category of approach when automating the inpainting problem. CONTENTS