Many computer vision and image processing problems can be posed as solving
partial differential equations (PDEs). However, designing PDE system usually
requires high mathematical skills and good insight into the problems. In this
paper, we consider designing PDEs for various problems arising in computer
vision and image processing in a lazy manner: \emph{learning PDEs from real
data via data-based optimal control}. We first propose a general intelligent
PDE system which holds the basic translational and rotational invariance rule
for most vision problems. By introducing a PDE-constrained optimal control
framework, it is possible to use the training data resulting from multiple ways
(ground truth, results from other methods, and manual results from humans) to
learn PDEs for different computer vision tasks. The proposed optimal control
based training framework aims at learning a PDE-based regressor to approximate
the unknown (and usually nonlinear) mapping of different vision tasks. The
experimental results show that the learnt PDEs can solve different vision
problems reasonably well. In particular, we can obtain PDEs not only for
problems that traditional PDEs work well but also for problems that PDE-based
methods have never been tried before, due to the difficulty in describing those
problems in a mathematical way.

本文提出了一种从真实数据中学习建立偏微分方程组以解决计算机视觉和图像处理问题的方法，并通过实验表明该方法可以相对良好地解决传统建立方式无法解决的问题。