Potts model, parametric maxflow and k-submodular functions

The problem of minimizing the Potts energy function frequently occurs in computer vision applications. One way to tackle this NP-hard problem was proposed by Kovtun [19,20]. It identifies a part of an optimal solution by running $k$ maxflow computations, where $k$ is the number of labels. The number of "labeled" pixels can be significant in some applications, e.g. 50-93% in our tests for stereo. We show how to reduce the runtime to $O(\log k)$ maxflow computations (or one {\em parametric maxflow} computation). Furthermore, the output of our algorithm allows to speed-up the subsequent alpha expansion for the unlabeled part, or can be used as it is for time-critical applications. To derive our technique, we generalize the algorithm of Felzenszwalb et al. [7] for {\em Tree Metrics}. We also show a connection to {\em $k$-submodular functions} from combinatorial optimization, and discuss {\em $k$-submodular relaxations} for general energy functions.Comment: Accepted to ICCV 201

Color Separation for Image Segmentation

Image segmentation is a fundamental problem in computer vision that has drawn intensive research attention during the past few decades, resulting in a variety of segmentation algorithms. Segmentation is often formulated as a Markov random field (MRF) and the solution corresponding to the maximum a posteriori probability (MAP) is found using energy minimiza- tion framework. Many standard segmentation techniques rely on foreground and background appearance models given a priori. In this case the corresponding energy can be efficiently op- timized globally. If the appearance models are not known, the energy becomes NP-hard, and many methods resort to iterative schemes that jointly optimize appearance and segmentation. Such algorithms can only guarantee local minimum. Here we propose a new energy term explicitly measuring L1 distance between the object and background appearance models that can be globally maximized in one graph cut. Our method directly tries to minimize the appearance overlap between the segments. We show that in many applications including interactive segmentation, shape matching, segmentation from stereo pairs and saliency segmentation our simple term makes NP-hard segmentation functionals unnecessary and renders good segmentation performance both qualitatively and quantitatively

Optimization for Image Segmentation

Image segmentation, i.e., assigning each pixel a discrete label, is an essential task in computer vision with lots of applications. Major techniques for segmentation include for example Markov Random Field (MRF), Kernel Clustering (KC), and nowadays popular Convolutional Neural Networks (CNN). In this work, we focus on optimization for image segmentation. Techniques like MRF, KC, and CNN optimize MRF energies, KC criteria, or CNN losses respectively, and their corresponding optimization is very different. We are interested in the synergy and the complementary benefits of MRF, KC, and CNN for interactive segmentation and semantic segmentation. Our first contribution is pseudo-bound optimization for binary MRF energies that are high-order or non-submodular. Secondly, we propose Kernel Cut, a novel formulation for segmentation, which combines MRF regularization with Kernel Clustering. We show why to combine KC with MRF and how to optimize the joint objective. In the third part, we discuss how deep CNN segmentation can benefit from non-deep (i.e., shallow) methods like MRF and KC. In particular, we propose regularized losses for weakly-supervised CNN segmentation, in which we can integrate MRF energy or KC criteria as part of the losses. Minimization of regularized losses is a principled approach to semi-supervised learning, in general. Our regularized loss method is very simple and allows different kinds of regularization losses for CNN segmentation. We also study the optimization of regularized losses beyond gradient descent. Our regularized losses approach achieves state-of-the-art accuracy in semantic segmentation with near full supervision quality

Maximum Persistency in Energy Minimization

We consider discrete pairwise energy minimization problem (weighted constraint satisfaction, max-sum labeling) and methods that identify a globally optimal partial assignment of variables. When finding a complete optimal assignment is intractable, determining optimal values for a part of variables is an interesting possibility. Existing methods are based on different sufficient conditions. We propose a new sufficient condition for partial optimality which is: (1) verifiable in polynomial time (2) invariant to reparametrization of the problem and permutation of labels and (3) includes many existing sufficient conditions as special cases. We pose the problem of finding the maximum optimal partial assignment identifiable by the new sufficient condition. A polynomial method is proposed which is guaranteed to assign same or larger part of variables than several existing approaches. The core of the method is a specially constructed linear program that identifies persistent assignments in an arbitrary multi-label setting.Comment: Extended technical report for the CVPR 2014 paper. Update: correction to the proof of characterization theore

Combinatorial persistency criteria for multicut and max-cut

In combinatorial optimization, partial variable assignments are called persistent if they agree with some optimal solution. We propose persistency criteria for the multicut and max-cut problem as well as fast combinatorial routines to verify them. The criteria that we derive are based on mappings that improve feasible multicuts, respectively cuts. Our elementary criteria can be checked enumeratively. The more advanced ones rely on fast algorithms for upper and lower bounds for the respective cut problems and max-flow techniques for auxiliary min-cut problems. Our methods can be used as a preprocessing technique for reducing problem sizes or for computing partial optimality guarantees for solutions output by heuristic solvers. We show the efficacy of our methods on instances of both problems from computer vision, biomedical image analysis and statistical physics

A Multi-Plane Block-Coordinate Frank-Wolfe Algorithm for Training Structural SVMs with a Costly max-Oracle

Structural support vector machines (SSVMs) are amongst the best performing models for structured computer vision tasks, such as semantic image segmentation or human pose estimation. Training SSVMs, however, is computationally costly, because it requires repeated calls to a structured prediction subroutine (called \emph{max-oracle}), which has to solve an optimization problem itself, e.g. a graph cut. In this work, we introduce a new algorithm for SSVM training that is more efficient than earlier techniques when the max-oracle is computationally expensive, as it is frequently the case in computer vision tasks. The main idea is to (i) combine the recent stochastic Block-Coordinate Frank-Wolfe algorithm with efficient hyperplane caching, and (ii) use an automatic selection rule for deciding whether to call the exact max-oracle or to rely on an approximate one based on the cached hyperplanes. We show experimentally that this strategy leads to faster convergence to the optimum with respect to the number of requires oracle calls, and that this translates into faster convergence with respect to the total runtime when the max-oracle is slow compared to the other steps of the algorithm. A publicly available C++ implementation is provided at http://pub.ist.ac.at/~vnk/papers/SVM.html

Accuracy of MAP segmentation with hidden Potts and Markov mesh prior models via Path Constrained Viterbi Training, Iterated Conditional Modes and Graph Cut based algorithms

In this paper, we study statistical classification accuracy of two different Markov field environments for pixelwise image segmentation, considering the labels of the image as hidden states and solving the estimation of such labels as a solution of the MAP equation. The emission distribution is assumed the same in all models, and the difference lays in the Markovian prior hypothesis made over the labeling random field. The a priori labeling knowledge will be modeled with a) a second order anisotropic Markov Mesh and b) a classical isotropic Potts model. Under such models, we will consider three different segmentation procedures, 2D Path Constrained Viterbi training for the Hidden Markov Mesh, a Graph Cut based segmentation for the first order isotropic Potts model, and ICM (Iterated Conditional Modes) for the second order isotropic Potts model. We provide a unified view of all three methods, and investigate goodness of fit for classification, studying the influence of parameter estimation, computational gain, and extent of automation in the statistical measures Overall Accuracy, Relative Improvement and Kappa coefficient, allowing robust and accurate statistical analysis on synthetic and real-life experimental data coming from the field of Dental Diagnostic Radiography. All algorithms, using the learned parameters, generate good segmentations with little interaction when the images have a clear multimodal histogram. Suboptimal learning proves to be frail in the case of non-distinctive modes, which limits the complexity of usable models, and hence the achievable error rate as well. All Matlab code written is provided in a toolbox available for download from our website, following the Reproducible Research Paradigm