Submodular functions can be exactly minimized in polynomial time, and the special case that graph cuts solve with max flow KZ:PAMI04 has had significant impact in computer vision BVZ:PAMI01,Kwatra:SIGGRAPH03,Rother:GrabCut04 . In this paper we address the important class of sum-of-submodular (SoS) functions Arora:ECCV12,Kolmogorov:DAM12 , which can be efficiently minimized via a variant of max flow called submodular flow 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 Joachims et al 09a,Tsochantaridis et al 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 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 Rother:GrabCut04 , which uses hand-tuned parameters instead of machine learning. On a standard dataset 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.
Many learning problems in computer vision can be cast as structured output prediction, which allows learning outputs with spatial coherence. Among the most popular generic methods for structured output learning are Conditional Random Fields (CRFs) trained by maximum conditional likelihood @cite_23 , Maximum-Margin Markov Networks (M3N) @cite_9 , and Structural Support Vector Machines (SVM-struct) @cite_14 @cite_29 . A key advantage of M3N and SVM-struct over CRFs is that training does not require computation of the partition function. Among the two large-margin approaches M3N and SVM-struct, we follow the SVM-struct methodology since it allows the use of efficient inference procedures during training.
@cite_29 @cite_9 @cite_14 @cite_23
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In this paper, we survey the current state-of-art models for structured learning problems, including Hidden Markov Model (HMM), Conditional Random Fields (CRF), Averaged Perceptron (AP), Structured SVMs (SVMstruct), Max Margin Markov Networks (M3N), and an integration of search and learning algorithm (SEARN). With all due tuning efforts of various parameters of each model, on the data sets we have applied the models to, we found that SVMstruct enjoys better performance compared with the others. In addition, we also propose a new method which we call the Structured Learning Ensemble (SLE) to combine these structured learning models. Empirical results show that our SLE algorithm provides more accurate solutions compared with the best results of the individual models. Large amounts of available training data and increasing computing power have led to the recent success of deep convolutional neural networks (CNN) on a large number of applications. In this paper, we propose an effective semantic pixel labelling using CNN features, hand-crafted features and Conditional Random Fields (CRFs). Both CNN and hand-crafted features are applied to dense image patches to produce per-pixel class probabilities. The CRF infers a labelling that smooths regions while respecting the edges present in the imagery. The method is applied to the ISPRS 2D semantic labelling challenge dataset with competitive classification accuracy. Many computer vision problems can be formulated in a Bayesian framework with Markov Random Field (MRF) or Conditional Random Field (CRF) priors. Usually, the model assumes that a full Maximum A Posteriori (MAP) estimation will be performed for inference, which can be really slow in practice. In this paper, we argue that through appropriate training, a MRF CRF model can be trained to perform very well on a suboptimal inference algorithm. The model is trained together with a fast inference algorithm through an optimization of a loss function on a training set containing pairs of input images and desired outputs. A validation set can be used in this approach to estimate the generalization performance of the trained system. We apply the proposed method to an image denoising application, training a Fields of Experts MRF together with a 1-4 iteration gradient descent inference algorithm. Experimental validation on unseen data shows that the proposed training approach obtains an improved benchmark performance as well as a 1000-3000 times speedup compared to the Fields of Experts MRF trained with contrastive divergence. Using the new approach, image denoising can be performed in real-time, at 8 fps on a single CPU for a 256 × 256 image sequence, with close to state-of-the-art accuracy. Conditional random fields (CRF) are widely used for predicting output variables that have some internal structure. Most of the CRF research has been done on structured classification where the outputs are discrete. In this study we propose a CRF probabilistic model for structured regression that uses multiple non-structured predictors as its features. We construct features as squared prediction errors and show that this results in a Gaussian predictor. Learning becomes a convex optimization problem leading to a global solution for a set of parameters. Inference can be conveniently conducted through matrix computation. Experimental results on the remote sensing problem of estimating Aerosol Optical Depth (AOD) provide strong evidence that the proposed CRF model successfully exploits the inherent spatio-temporal properties of AOD data. The experiments revealed that CRF are more accurate than the baseline neural network and domain-based predictors.