The Navier-Stokes-Voight model for image inpainting

摘要

In this paper, we investigate the advantages of the 2D Navier-Stokes-Voight (NSV) turbulence 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 when an elegant analogy between the image intensity function for the image inpainting problem and the stream function in 2D incompressible fluid was established. 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. Recently, the 3D NSV model of viscoelastic fluid was suggested by Cao et al. as an inviscid regularization to the 3D Navier-Stokes equations (NSEs). We give some background on the NSV model, describe why it is a good candidate sub-grid scale turbulence model and then we propose this model as an alternative partial differential equation for image inpainting. We describe an implementation of the inpainting procedure using the NSV model and then present numerical results comparing the images obtained when using the NSE versus the NSV model. 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 the quality of the resulting images using a subjective measure (human evaluation) and an objected measure (by calculating the peak signal-to-noise ratio). We also present some new theoretical results based on energy methods comparing the sufficient conditions for numerical stability 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.