The (MAP) assignment for general structure Markov random fields (MRFs) is computationally intractable. In this paper, we exploit tree-based methods to efficiently address this problem. Our novel method, named Tree-based Iterated Local Search (T-ILS) takes advantage of the tractability of tree-structures embedded within MRFs to derive strong local search in an ILS framework. The method efficiently explores exponentially large neighborhood and does so with limited memory without any requirement on the cost functions. We evaluate the T-ILS in a simulation of Ising model and two real-world problems in computer vision: stereo matching, image denoising. Experimental results demonstrate that our methods are competitive against state-of-the-art rivals with a significant computational gain.
The MAP assignment for Markov random fields (MRFs) as a combinatorial search problem has attracted a great amount of research in the past several decades, especially in the area of computer vision @cite_20 and probabilistic artificial intelligence @cite_29 . The problem is known to be NP-hard @cite_15 . For example, in labeling of an image of size @math , the problem space is @math large, where @math is the number of possible labels per pixel.
@cite_29 @cite_20 @cite_15
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Markov Random Fields (MRFs) are ubiquitous in low- level computer vision. In this paper, we propose a new approach to the optimization of multi-labeled MRFs. Similarly to a-expansion it is based on iterative application of binary graph cut. However, the number of binary graph cuts required to compute a labelling grows only logarithmically with the size of label space, instead of linearly. We demonstrate that for applications such as optical flow, image restoration, and high resolution stereo, this gives an order of magnitude speed-up, for comparable energies. Iterations are performed by "fusion" of solutions, done with QPBO which is related to graph cut but can deal with non- submodularity. At convergence, the method achieves optima on a par with the best competitors, and sometimes even exceeds them. Markov Random Fields (MRFs) are a widely used graphical model, but the inference problem is NP-hard. For first-order MRFs with binary labels, Dead End Elimination (DEE) [7] and QPBO [2, 14] can find the optimal labeling for some variables, the much harder case of larger label sets has been addressed by Kovtun [16, 17] and related methods [12, 23, 24, 25], which impose substantial computational overhead. We describe an efficient algorithm to correctly label a subset of the variables for arbitrary MRFs, with particularly good performance on binary MRFs. We propose a sufficient condition to check if a partial labeling is optimal, which is a generalization of DEE's purely local test. We give a hierarchy of relaxations that provide larger optimal partial labelings at the cost of additional computation. Empirical studies were conducted on several benchmarks, using expansion moves [4] for inference. Our algorithm runs in a few seconds, and improves the speed of MRF inference with expansion moves by a factor of 1.5 to 12. The notion of using context information for solving high-level vision problems has been increasingly realized in the field. However, how to learn an effective and efficient context model, together with the image appearance, remains mostly unknown. The current literature using Markov random fields (MRFs) and conditional random fields (CRFs) often involves specific algorithm design, in which the modeling and computing stages are studied in isolation. In this paper, we propose an auto-context algorithm. Given a set of training images and their corresponding label maps, we first learn a classifier on local image patches. The discriminative probability (or classification confidence) maps by the learned classifier are then used as context information, in addition to the original image patches, to train a new classifier. The algorithm then iterates to approach the ground truth. Auto-context learns an integrated low-level and context model, and is very general and easy to implement. Under nearly the identical parameter setting in the training, we apply the algorithm on three challenging vision applications: object segmentation, human body configuration, and scene region labeling. It typically takes about 30 70 seconds to run the algorithm in testing. Moreover, the scope of the proposed algorithm goes beyond high-level vision. It has the potential to be used for a wide variety of problems of multi-variate labeling. We consider the question of computing Maximum A Posteriori (MAP) assignment in an arbitrary pair-wise Markov Random Field (MRF). We present a randomized iterative algorithm based on simple local updates. The algorithm, starting with an arbitrary initial assignment, updates it in each iteration by first, picking a random node, then selecting an (appropriately chosen) random local neighborhood and optimizing over this local neighborhood. Somewhat surprisingly, we show that this algorithm finds a near optimal assignment within n log2 n iterations with high probability for any n node pair-wise MRF with geometry (i.e. MRF graph with polynomial growth) with the approximation error depending on (in a reasonable manner) the geometric growth rate of the graph and the average radius of the local neighborhood - this allows for a graceful tradeoff between the complexity of the algorithm and the approximation error. Through extensive simulations, we show that our algorithm finds extremely good approximate solutions for various kinds of MRFs with geometry.