Research Article: A Rough Set Bounded Spatially Constrained Asymmetric Gaussian Mixture Model for Image Segmentation

Date Published: January 3, 2017

Publisher: Public Library of Science

Author(s): Zexuan Ji, Yubo Huang, Quansen Sun, Guo Cao, Yuhui Zheng, Kristin J. Al-Ghoul.

http://doi.org/10.1371/journal.pone.0168449

Abstract

Accurate image segmentation is an important issue in image processing, where Gaussian mixture models play an important part and have been proven effective. However, most Gaussian mixture model (GMM) based methods suffer from one or more limitations, such as limited noise robustness, over-smoothness for segmentations, and lack of flexibility to fit data. In order to address these issues, in this paper, we propose a rough set bounded asymmetric Gaussian mixture model with spatial constraint for image segmentation. First, based on our previous work where each cluster is characterized by three automatically determined rough-fuzzy regions, we partition the target image into three rough regions with two adaptively computed thresholds. Second, a new bounded indicator function is proposed to determine the bounded support regions of the observed data. The bounded indicator and posterior probability of a pixel that belongs to each sub-region is estimated with respect to the rough region where the pixel lies. Third, to further reduce over-smoothness for segmentations, two novel prior factors are proposed that incorporate the spatial information among neighborhood pixels, which are constructed based on the prior and posterior probabilities of the within- and between-clusters, and considers the spatial direction. We compare our algorithm to state-of-the-art segmentation approaches in both synthetic and real images to demonstrate the superior performance of the proposed algorithm.

Partial Text

As one of the classical problems in image processing, image segmentation has been extensively studied, which can be treated as a classification problem [1–5] for the target image. Various image segmentation algorithms have been developed such as active contour models [6, 7], graph based methods [8, 9] and clustering techniques [10–12]. Over the last decades, model-based techniques [13, 14] have been widely used in image segmentation, where the standard Gaussian mixture model (GMM) [15, 16] is a well-known method because of its simplicity and ease of implementation [17]. The parameters involved in GMM can be efficiently estimated by expectation maximization (EM) algorithm [18]. However, the standard GMM still suffers from the following limitations: sensitivity to noise, less flexibility to fit the shape of the data and unbounded distributions [19].

The notations used throughout this paper are as follows. The target image is denoted as X = {xi, i = 1, 2, …, N}, where xi with dimension D is the intensity values for the ith pixel. The neighborhood of the ith pixel is denoted as ∂i, and the labels are denoted as (Ω1, Ω2, …, ΩK). In order to segment an image with N pixels into K labels, the density function of the finite mixture model [42] is given by:
p(xi|Π,Θ)=∑k=1Kπikp(xi|Ωk),(1)
where Π = {πik}, i = {1, 2, …, N}, k = {1, 2, …, K} are the prior probabilities, and satisfy the constraints 0 ≤ πik ≤ 1 and ∑k=1Kπik=1.

In order to fit different data shapes, Nguyen et al [19] defined a new distribution p(xi|Ωk) to model the component density. Motivated by the bounded asymmetric distribution, in this paper we modify distribution p(xi|Ωk) to allow the model to easily incorporate the spatial information, which can be defined as:
p(xi|Ωk)=∑l=1LηiklΨ(xi|μkl,Σkl),(9)
where L is the number of bounded multivariate Gaussian distribution Ψ(xi|μkl, Σkl), and ηikl is the weighting factor and satisfies the constraints 0 ≤ ηikl ≤ 1 and ∑l=1Lηikl=1. The bounded Gaussian distribution Ψ(xi|μkl, Σkl) is defined as:
Ψ(xi|μkl,Σkl)=Φ(xi|μkl,Σkl)H(xi|Ωk)∫∂ΩkΦ(x|μkl,Σkl)dx,(10)
where Φ(xi|μkl, Σkl) is the Gaussian distribution defined in Eq (2) and H(xi|Ωk) is the indicator function for the bounded support region defined in Eq (3). ∫∂ΩkΦ(x|μkl, Σkl)dx is the normalization constant.

In this paper, we compare the proposed algorithm with four algorithms, i.e., a Bayesian bounded asymmetric mixture model (BAMM) [19], a bounded generalized Gaussian mixture model (BGGMM) [37], a spatially constrained generative model and EM algorithm (SCGM-EM) [26], and a fast and robust spatially constrained GMM (FRSCGMM) [23].

To overcome the limitations involved in most GMM-based algorithms, in this paper, we proposed a rough set bounded asymmetric Gaussian mixture model with spatial constraint for image segmentation. Based on the rough set theory, a new bounded indicator function was proposed to determine the bounded support regions of the observed data. The bounded indicator and posterior probability of a pixel that belongs to each sub-region were estimated based on the rough regions. The within- and between-cluster spatial constraints were introduced by incorporating the spatial information with adaptively selected direction in order to reduce over-smoothness for segmentations. Experimental results demonstrated that the proposed algorithm is flexible to fit the data shapes, and robust to noise, which makes our method be capable of producing more accurate segmentation results comparing with several state-of-the-art algorithms. Future work will be devoted to reducing the complexity of the proposed algorithm.

 

Source:

http://doi.org/10.1371/journal.pone.0168449