A Parallel Douglas-Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds
A Parallel Douglas-Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds
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DOI：
10.1137/15m1052858
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发表时间：
2015-12
期刊：
影响因子：
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通讯作者：
Ronny Bergmann;Johannes Persch;G. Steidl
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文献类型：
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作者：
Ronny Bergmann;Johannes Persch;G. Steidl
我们感兴趣的恢复图像中的对称阿达玛流形的值通过最小化的功能与二次数据项和总变差的正则化项。为了解决凸极小化问题，我们将道格拉斯-Rachford算法及其并行版本推广到对称Hadamard流形。道格拉斯-Rachford算法的核心是反映的功能涉及的功能，以最小化。在欧几里德背景下，凸下连续函数的反射是非扩张的。因此，Krasnoselski-Mann迭代的收敛结果暗示了道格拉斯-Rachford算法的收敛性。不幸的是，这些一般结果不能推广到阿达玛流形，在那里适当的凸下连续函数可以有扩张反射。然而，分裂我们的恢复功能，以适当的方式，我们只需要处理特殊的功能-即，几个距离一样的功能和一个指标功能。
We are interested in restoring images having values in a symmetric Hadamard manifold by minimizing a functional with a quadratic data term and a total variation--like regularizing term. To solve the convex minimization problem, we extend the Douglas--Rachford algorithm and its parallel version to symmetric Hadamard manifolds. The core of the Douglas--Rachford algorithm is reflections of the functions involved in the functional to be minimized. In the Euclidean setting the reflections of convex lower semicontinuous functions are nonexpansive. As a consequence, convergence results for Krasnoselski--Mann iterations imply the convergence of the Douglas--Rachford algorithm. Unfortunately, these general results do not carry over to Hadamard manifolds, where proper convex lower semicontinuous functions can have expansive reflections. However, splitting our restoration functional in an appropriate way, we have only to deal with special functions---namely, several distance-like functions and an indicator function ...