用于图像修复的Navier-Stokes-Voight模型
The Navier-Stokes-Voight Model for Image Inpainting
AI总结：
提出将二维Navier-Stokes-Voight模型应用于图像修复，通过数值模拟稳态解并结合能量方法分析离散格式的稳定性条件。
摘要：
2001年，Bertalmio等人将图像修复问题中的图像强度函数与二维不可压缩流体中的流函数进行了类比。通过在待修复区域数值逼近二维NSE涡量输运方程的稳态解，并同时求解涡量与流函数之间的泊松问题，得到了图像修复问题的近似解。这种优雅的方法允许利用计算流体动力学技术来生成图像修复问题的近似解。最近，Cao等人提出了粘弹性流体的三维Navier-Stokes-Voight（NSV）模型，作为三维Navier-Stokes方程（NSE）的无粘正则化。NSV模型被证明是全局适定的，并具有有限维全局吸引子，使其成为数值模拟中具有吸引力的亚网格尺度湍流模型。本文研究了二维NSV模型在图像修复算法中的应用。我们还基于能量方法提出了新的理论结果，比较了两个模型方程离散格式稳定性的充分条件。
英文摘要：
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.
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