对偶Rudin-Osher-Fatemi模型的快速非重叠块Jacobi方法
Fast Nonoverlapping Block Jacobi Method for the Dual Rudin--Osher--Fatemi Model
摘要：
本文研究了Rudin-Osher-Fatemi~（ROF）模型的非重叠区域分解方法。将图像域划分为矩形子区域，并行求解子区域内的局部问题。局部问题可以采用ROF模型的现有最先进的求解器。我们表明，非重叠松弛块Jacobi方法的ROF模型的对偶制定的能量泛函的$O(1/n)$收敛速度，其中$n$是迭代次数。此外，通过利用该方法的前向后分裂结构，我们提出了一个加速版本，其收敛速度为$O(1/n^2)$。该方法在理论上和实际应用中都比现有的区域分解方法收敛得更快，而每次迭代的主要计算量保持不变。我们还提供了块Jacobi方法的收敛速度对图像大小和子域数的依赖性。与现有的方法进行比较的数值结果。
英文摘要：
We consider nonoverlapping domain decomposition methods for the Rudin--Osher--Fatemi~(ROF) model, which is one of the standard models in mathematical image processing. The image domain is partitioned into rectangular subdomains and local problems in subdomains are solved in parallel. Local problems can adopt existing state-of-the-art solvers for the ROF model. We show that the nonoverlapping relaxed block Jacobi method for a dual formulation of the ROF model has the $O(1/n)$ convergence rate of the energy functional, where $n$ is the number of iterations. Moreover, by exploiting the forward-backward splitting structure of the method, we propose an accelerated version whose convergence rate is $O(1/n^2)$. The proposed method converges faster than existing domain decomposition methods both theoretically and practically, while the main computational cost of each iteration remains the same. We also provide the dependence of the convergence rates of the block Jacobi methods on the image size and the number of subdomains. Numerical results for comparisons with existing methods are presented.
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