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Efficient Algorithms for Markov Random Fields, Isotonic Regression, Graph Fused Lasso, and Image Segmentation

Abstract

Markov random field (MRF) is a multi-label clustering model with applications in image segmentation, image deblurring, isotonic regression, graph fused lasso, and semi-supervised learning. It is a convex optimization problem on an arbitrary graph of objective function consisting of functions defined on the nodes (called deviation functions) and edges (called separation functions) of the graph. There exists a provably fastest MRF-algorithm for arbitrary graphs and a broad class of objective functions. This MRF-algorithm uses the technique of graph minimum cut. Although MRF of this class of objective functions generalizes isotonic regression and graph fused lasso, this MRF-algorithm has lower time complexity than those specialized algorithms for isotonic regression and graph fused lasso.

Some problems in time series and gene sequence signal processing are special cases of MRF on a path graph. We present three efficient algorithms to solve MRF on a path graph for different classes of objective functions. The algorithms are the fastest algorithms for the respective classes of objective functions. The first algorithm uses graph minimum cut technique inspired by the provably fastest MRF-algorithm. The second algorithm is based on a relationship with a lot-sizing problem in production planning. The third algorithm is based on the technique of Karush-Kuhn-Tucker (KKT) optimality conditions.

MRF is used in image segmentation to identify multiple salient features in an image. The Hochbaum Normalized Cut (HNC) model is a binary clustering model that is also applicable to image segmentation, with the goal to separate the foreground from the background. We compare the empirical performance between the HNC model and the spectral method in image segmentation. Both HNC and the spectral method can be viewed as heuristics for the NP-hard normalized cut criterion, which is another binary clustering criterion often used for image segmentation. We present experimental evidence that the HNC model provides solutions which not only better approximate the optimal normalized cut solution, but also have better visual segmentation quality over those provided by the spectral method. The experimental evidence further suggests that HNC should be the preferred image segmentation criterion over normalized cut.

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