Submodular functions can be exactly minimized in polynomial time, and the special case that graph cuts solve with max flow \cite{KZ:PAMI04} has had significant impact in computer vision \cite{BVZ:PAMI01,Kwatra:SIGGRAPH03,Rother:GrabCut04}. In this paper we address the important class of sum-of-submodular (SoS) functions \cite{Arora:ECCV12,Kolmogorov:DAM12}, which can be efficiently minimized via a variant of max flow called submodular flow \cite{Edmonds:ADM77}. SoS functions can naturally express higher order priors involving, e.g., local image patches; however, it is difficult to fully exploit their expressive power because they have so many parameters. Rather than trying to formulate existing higher order priors as an SoS function, we take a discriminative learning approach, effectively searching the space of SoS functions for a higher order prior that performs well on our training set. We adopt a structural SVM approach \cite{Joachims/etal/09a,Tsochantaridis/etal/04} and formulate the training problem in terms of quadratic programming; as a result we can efficiently search the space of SoS priors via an extended cutting-plane algorithm. We also show how the state-of-the-art max flow method for vision problems \cite{Goldberg:ESA11} can be modified to efficiently solve the submodular flow problem. Experimental comparisons are made against the OpenCV implementation of the GrabCut interactive segmentation technique \cite{Rother:GrabCut04}, which uses hand-tuned parameters instead of machine learning. On a standard dataset \cite{Gulshan:CVPR10} our method learns higher order priors with hundreds of parameter values, and produces significantly better segmentations. While our focus is on binary labeling problems, we show that our techniques can be naturally generalized to handle more than two labels.

本文介绍了一种基于判别学习的方法来寻找高阶先验，该方法采用了结构化SVM和用于求解子模流问题的延伸割算法，说明了如何更好地利用子模函数的表达能力，实现更好的交互式分割技术。

结构化学习总和子模高阶能量函数