Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 683176, 9 pages

http://dx.doi.org/10.1155/2015/683176

## Image Segmentation by Edge Partitioning over a Nonsubmodular Markov Random Field

^{1}Division of Computer and Electronic Systems Engineering, Hankuk University of Foreign Studies, Yongin 449-791, Republic of Korea^{2}Department of Electrical and Computer Engineering, College of Engineering, Seoul National University, Seoul 151-744, Republic of Korea

Received 26 July 2015; Revised 16 November 2015; Accepted 3 December 2015

Academic Editor: Costas Panagiotakis

Copyright © 2015 Ho Yub Jung and Kyoung Mu Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Edge weight-based segmentation methods, such as normalized cut or minimum cut, require a partition number specification for their energy formulation. The number of partitions plays an important role in the segmentation overall quality. However, finding a suitable partition number is a nontrivial problem, and the numbers are ordinarily manually assigned. This is an aspect of the general partition problem, where finding the partition number is an important and difficult issue. In this paper, the edge weights instead of the pixels are partitioned to segment the images. By partitioning the edge weights into two disjoints sets, that is, cut and connect, an image can be partitioned into all possible disjointed segments. The proposed energy function is independent of the number of segments. The energy is minimized by iterating the QPBO--expansion algorithm over the pairwise Markov random field and the mean estimation of the cut and connected edges. Experiments using the Berkeley database show that the proposed segmentation method can obtain equivalently accurate segmentation results without designating the segmentation numbers.

#### 1. Introduction

There are numerous approaches and applications for unsupervised image segmentation in computer vision. Many different theories are proposed for varying the roles of the unsupervised segmentation. As a low level vision problem, an image can be simplified by oversegmentation using a number of different approaches, such as mode-seeking mean shift, multilevel thresholding, histogram-based neural networks, superpixel algorithms, and various graph-based methods [1–4]. Conversely, semantic segmentation is attempted for simultaneous detection, recognition, and segmentation [5].

Generally, the role of unsupervised segmentation falls between image simplification and full semantic segmentation, where semantically meaningful segments are expected to be found but not necessarily recognized. Segmentation is posed as an image-coloring problem that minimizes specific energy functions. Energy functions can be optimized using stochastic methods such as deterministic annealing and stochastic clustering [6–10]. For graph theoretic segmentation approaches, the spectral method and graph cut are efficient deterministic optimization methods [11–13]. Another traditional segmentation method is the variational method, which evolves boundary contours in a level set framework [14, 15].

The edge weight-based segmentation methods have evolved together with graph partition problems. When edge weights are all positive, the minimum cut can be found; however, the minimum cut has bias toward smaller cuts. Adding negative edge weights can prevent the problem so the graph becomes nonsubmodular; however, the problem becomes NP-hard [16]. Different algorithms have been introduced to estimate the correlation in clustering problem [17, 18]. In contrast, Shi and Malik normalized nonnegative edge weights so the bias toward smaller cuts was eliminated [11].

For the graph theoretic segmentation and level set methods, the number of segments must be predefined. The segment number choice greatly influences the quality of segmentation, especially for a normalized cut. Nonetheless, there have been attempts to solve this problem. The number of segments can be controlled by setting the threshold value to the recursive normalized cut [11]. For level set approaches, a four-color theorem was used to segment images with an arbitrary number of phases with one or two level set functions [19]. However, these methods are still functions of , the number of segments.

In this paper, transforming the pixel clustering problem into an edge partition problem circumvents the segment number selection problem. Edges among adjacent pixels can represent dissimilarity or similarity weights. Two edge partitions are always sufficient for pixel-partitioning problems. An edge can be in a cut set or connected set, which can then be translated into a unique segmentation, as in Figure 1(c). The cut edges indicate that the two node labels are different, whereas the connected edges indicate that two nodes have the same labels. In most cases, however, the cut or connect assignments on the edges are not enough to define a specific segmentation configuration, as in Figure 1(d). Random cut and connect assignments on the edges may result in contradiction of the node labels. However, under the pixel coloring framework, cut and connect assignments on the edges are defined concurrently with pixel labels, and inconsistencies, such as those in Figure 1(d), are prevented.