Learning PDEs for Image Restoration via Optimal Control
Learning PDEs for Image Restoration via Optimal Control
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DOI：
10.1007/978-3-642-15549-9_9
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发表时间：
2010-09
影响因子：
2.4
通讯作者：
Risheng Liu;Zhouchen Lin;Wayne Zhang;Zhixun Su
中科院分区：
文献类型：
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作者：
Risheng Liu;Zhouchen Lin;Wayne Zhang;Zhixun Su
偏微分方程组已经成功地应用于许多计算机视觉和图像处理问题。然而，设计PDE需要很高的数学技能和对问题的良好洞察力。在本文中，我们证明了通过借鉴机器学习的学习策略，可以使PDE的设计变得更容易。在我们的基于学习的偏微分方程(L-偏微分方程)框架中，我们的偏微分方程模型包括两个项：(I)编码图像模型先验知识的正则化项和(Ii)差分不变量的线性组合，它是数据驱动的，能够有效地适应不同的问题和复杂的条件。通过一种最优控制技术，从训练样本的输入/输出对中学习L偏微分方程。通过图像去噪和修复两个应用实例，验证了L-偏微分方程组在图像复原中的有效性，其中偏微分方程组容易获得，所产生的结果与传统的精心设计的偏微分方程组相当或更好。
Partial differential equations (PDEs) have been successfully applied to many computer vision and image processing problems. However, designing PDEs requires high mathematical skills and good insight into the problems. In this paper, we show that the design of PDEs could be made easier by borrowing thelearning strategyfrom machine learning. In our learning-based PDE (L-PDE) framework for image restoration, there are two terms in our PDE model: (i) a regularizer which encodes the prior knowledge of the image model and (ii) a linear combination of differential invariants, which is data-driven and can effectively adapt to different problems and complex conditions. The L-PDE is learnt from some input/output pairs of training samples via an optimal control technique. The effectiveness of our L-PDE framework for image restoration is demonstrated with two exemplary applications: image denoising and inpainting, where the PDEs are obtained easily and the produced results are comparable to or better than those of traditional PDEs, which were elaborately designed.