Abstract We present a convergence rate analysis of the Rudin–Osher–Fatemi (ROF) denoising problem for two different discretizations of the total variation. The first is the standard discretization, which induces blurring in some particular diagonal directions. We prove that in a simplified setting corresponding to such a direction, the discrete ROF energy converges to the continuous one with the rate $h^{2/3}$. The second discretization is based on dual Raviart–Thomas fields and achieves an optimal $O(h)$ convergence rate for the same quantity, for discontinuous solutions with some standard hypotheses.

中文翻译：

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总变异的有限差分近似的误差估计

摘要 我们针对总变化的两种不同离散化提出了 Rudin-Osher-Fatemi (ROF) 去噪问题的收敛速度分析。第一个是标准离散化，它会在某些特定的对角线方向上引起模糊。我们证明了在对应于这样一个方向的简化设置中，离散的 ROF 能量以 $h^{2/3}$ 的速率收敛到连续的能量。第二个离散化基于对偶 Raviart-Thomas 场，对于具有一些标准假设的不连续解，在相同数量下实现了最优的 $O(h)$ 收敛速度。