Coarse–fine surrogate model driven multiobjective evolutionary fuzzy clustering algorithm with dual memberships for noisy image segmentation

Highlights

• Dual fuzzy membership degree functions are first designed.

• Spatial information-guided fitness functions are constructed for image segmentation.

• Coarse–fine surrogate model is employed to assist the evolutionary optimization.

• A dual memberships-driven cluster validity index is designed.

Abstract

To improve the segmentation performance and boost evolutionary efficiency of multiobjective evolutionary clustering algorithms on noisy images, this paper proposes a coarse–fine surrogate model driven multiobjective evolutionary fuzzy clustering algorithm with dual membership functions (CFS-MOEFC). The existing fuzzy clustering validity-based fitness functions generally consider one single fuzzy membership function, which cannot fully process the uncertainty in an image. Moreover, most existing fitness functions used in image segmentation only utilize one type of spatial information derived from the image, which cannot behave robust on images with uncertain types of noise. To deal with the above problems, dual fuzzy membership functions are first designed by utilizing the local and non-local image spatial information. Then, the designed dual membership functions and the two kinds of spatial information are both used to construct spatial information-motivated fitness functions for image segmentation. To promote the time efficiency, the coarse–fine surrogate model is employed to assist the evolutionary optimization process, such as approximating the fitness functions instead of real function evaluations and evolving satisfactory cluster centers for CFS-MOEFC. Besides, a dual memberships-driven cluster validity index combining the local and non-local spatial information is designed for selecting an optimal solution from the final non-dominated solution set of CFS-MOEFC. Experiments on Berkeley and Magnetic Resonance (MR) images indicate that CFS-MOEFC greatly improves the segmentation accuracy on images with multiple types of noise, preserves more significant detailed image information, and behaves well on time cost comparing with state-of-the-art methods.

Introduction

Image segmentation plays an important role in image processing and analysis [1]. It divides an image into several mutually disjoint regions according to features, such as gray level, color, texture, geometric shape and so on, so that the features show similarity in the same region and obvious difference between different regions. Image segmentation techniques include histogram thresholding methods [2], clustering algorithms [3], [4], [5], region growing approaches [6], edge extraction methods [7] and graph-based partitioning algorithms [8], [9], etc. Fuzzy c-means (FCM) [10] clustering algorithm is one of the most important and classic clustering algorithms. FCM combines fuzzy set theory with clustering algorithm and therefore can deal with the ambiguity of data. FCM has been successfully used in the noise-free image segmentation. However, it is sensitive to image noise due to not considering any spatial information derived from the image in the clustering process. To solve the noise sensitivity, many improved fuzzy clustering algorithms have been proposed by utilizing the image spatial information. Ahmed et al. [11] proposed a modified fuzzy c-means clustering algorithm with neighborhood-based spatial constraints (FCM-S). It should be noted that FCM-S consumes too much time for computing the neighborhood-based spatial constraints in each iteration step when compared with FCM. Then Chen and Zhang [12] proposed two improved algorithms of FCM-S, FCM-S1 and FCM-S2, which reduce the time complexity by computing the mean-filtered image and median-filtered image in advance. However, the parameter that controls the role of the spatial terms in FCM-S1 and FCM-S2 is a constant thus inappropriate parameter settings may greatly influence the algorithm performance, and moreover these two algorithms do not make full use of the gray level information and spatial location information of the pixels. Krinidis and Chatzis [4] presented a fuzzy local information c-means clustering algorithm (FLICM). This algorithm introduces a local spatial fuzzy factor into the clustering objective function, which not only utilizes the gray level information and spatial location information of the pixels, but also does not need to set up the spatial parameter. Wang et al. [13] proposed a robust fuzzy c-means clustering algorithm with adaptive spatial & intensity constraint and membership linking. Its main characteristics include: (1) constraints of spatial & intensity and original FCM are adaptively specified and vary between pixels; (2) membership linking is obtained to decrease the number of iteration steps. Above all, the above mentioned fuzzy clustering algorithms are still sensitive to the initialized cluster centers and easy to fall into the local optimum. In addition, these methods only consider a single clustering criterion function and therefore cannot satisfy multiple segmentation demands.

In recent years, multiobjective evolutionary fuzzy clustering algorithms [14], [15], [16] have been proposed and obtained success in image segmentation. These algorithms simultaneously optimize two or more clustering objective functions and can overcome the above mentioned drawbacks of fuzzy clustering algorithms. Mukhopadhyay and Maulik [15] proposed a multiobjective variable string length genetic fuzzy clustering algorithm (MOVGA) and successfully applied MOVGA to segment magnetic resonance (MR) brain images. MOVGA simultaneously considers a global fuzzy compactness function and a fuzzy separation function for image segmentation, and also introduces a cluster validity index to obtain an optimal solution from the final non-dominated solution set. However, MOVGA fails to segment images polluted by noise due to ignorance of the spatial information in images. Therefore, Zhao et al. [16] proposed a multiobjective spatial fuzzy clustering algorithm (MSFCA) for image segmentation, which incorporates the non-local spatial information derived from the image into the constructed objective functions and obtains satisfactory results on images contaminated by noise. However, due to using the non-dominated sorting genetic algorithm II (NSGA-II) [17] as the underlying optimization framework, the above multiobjective evolutionary fuzzy clustering algorithms require expensive function evaluations and therefore consume a long computational time to obtain the segmentation result.

NSGA-II is a well-known dominance-based multiobjective evolutionary algorithm and requires expensive fitness function evaluations for a complex problem. To address this issue, Chugh et al. [18] proposed a Kriging model [19] assisted reference vector guided evolutionary algorithm (K-RVEA). Compared with NSGA-II, K-RVEA utilizes the Kriging surrogate model for predicting fitness functions, which can partly replace the direct calculation of fitness functions. Another example, Zhou et al. developed a two-stage adaptive multi-fidelity surrogate (MFS) model-assisted multi-objective genetic algorithms [20]. In order to improve the time efficiency of the existing multiobjective evolutionary fuzzy clustering algorithms for image segmentation mentioned above, Zhao et al. [21] proposed a Kriging-assisted reference vector guided multi-objective robust spatial fuzzy clustering algorithm (KRV-MRSFC) for image segmentation. KRV-MRSFC not only overcomes the problem of expensive fitness function evaluations, but also improves the robustness to image noise. It should be noted that the performance of surrogate-assisted evolutionary algorithms highly depends on the used model [22]. Multiple surrogates generally yield robust and improved solution quality for the same computational budget compared with one surrogate model [23]. Habib et al. [24] presented a hybrid surrogate assisted decomposition-based evolutionary algorithm. Yang et al. [25] proposed a multiple surrogate assisted multiobjective optimization algorithm using the reference vector-based final solution set generation strategy (MS-RV). MS-RV not only approximates fitness functions well, but also improves the reliability of surrogates. Comparing with K-RVEA using the Kriging model, MS-RV adopts the polynomial regression (PR) model as the coarse surrogate and the radial basis function (RBF) model as the fine surrogate. What is more important is that MS-RV does not execute the evaluations of real objective functions to update the surrogates during the optimization process. Amounts of experiments reveal that MS-RV behaves well in evolutionary performance and efficiency.

Taking into account the advantages of coarse–fine surrogate model, this paper introduces the coarse–fine surrogate model to guide the optimization search of multi-objective evolutionary fuzzy clustering algorithm for noisy image segmentation and proposes a coarse–fine surrogate model driven multi-objective evolutionary fuzzy clustering algorithm with dual membership functions (CFS-MOEFC). The attractive contributions of CFS-MOEFC can be summarized as follows: (1) To well deal with the uncertainty in the image, dual fuzzy membership degree functions are first designed by utilizing the local and non-local image spatial information. Then a novel dual memberships-based fuzzy compact fitness function with the local and non-local image spatial information is constructed to improve the robustness on images with multiple types of noise, and a fuzzy separation fitness function with joint membership derived from the dual membership functions is also designed in this paper. CFS-MOEFC aims to perform segmentation under these two complementary fitness functions. (2) Aiming to improve the evolutionary efficiency, the coarse–fine surrogate model is employed in CFS-MOEFC to guide the evolutionary optimization, including approximating the fitness functions and evolving satisfactory solutions. (3) The dual memberships-driven cluster validity index with the local and non-local spatial information is presented to obtain a single optimal solution from the final non-dominated solution set of CFS-MOEFC.

The rest of this paper is organized as follows. Section 2 introduces the relevant knowledge of the proposed algorithm. Section 3 describes CFS-MOEFC algorithm in detail. In Section 4, CFS-MOEFC is verified by experiments on different kinds of images. Section 5 gives a summary of current work and a discussion of future work.