Elsevier

Signal Processing

Volume 147, June 2018, Pages 80-91
Signal Processing

Multilevel thresholding selection based on variational mode decomposition for image segmentation

https://doi.org/10.1016/j.sigpro.2018.01.022Get rights and content

Highlights

  • An equivalent objective function of VMD is proved.

  • The histogram is decomposed into several sub modes based on between-class variance function.

  • Two principles of threshold selection are proposed based on histogram decomposition.

  • The proposed method exhibits higher efficiency than others.

Abstract

Multilevel thresholding techniques based on gray histogram are usually computationally expensive for the image segmentation. In this paper, we propose a novel thresholding extraction method based on variational mode decomposition (VMD). The improved VMD is used to decompose the histogram non-recursively into several sub-modes for minimizing Otsu's objective function. Then, we can extract the thresholds easily by the minimum point search (MPS) method or the cross point search (CPS) method. The experimental results demonstrate that the proposed MPS scheme exhibits more excellent capability than CPS. Further, compared with other approaches namely particle swarm optimization algorithm (PSO), fuzzy c-means (FCM) algorithm and bacterial foraging (BF) algorithm, the proposed algorithm can get similar performance, but its computing speed is faster than others. Therefore, it could have some advantages in image preprocessing, such as fast target recognition and classification.

Introduction

The segmentation technique can effectively separate the different objects in an image via performing pixel categorization based on different gray levels. This method has a wide range of applications in machine vision, computer-aided diagnosis of the medical imaging, feature extraction and analysis. According to the number of classes, it involves bi-level thresholding and multilevel thresholding. Based on different principles, it can also be classified into connectivity-preserving relaxation methods, region-based techniques, edge-based methods, and threshold techniques. These strategies have their respective advantages and limitations. Thus, we only focus on the threshold extraction method based on the gray level histogram.

In general, the way to handle or understand the histogram is the key factor for the success of image segmentation. Some literatures propose a methodology to process image histogram curves directly. For example, we can smooth the histogram by the repetitive filtering method, including lowpass/highpass filter [1] or Gaussian filtering [2], and detect the valleys as thresholds by calculating the derivatives of the smoothed histogram [3]. The same effect can also be achieved by some new fitting techniques such as the convex hull theory for analyzing the concavities of the histogram [4], the support vector regression [5] and the Chan–Vese segmentation model which can approximate the gray-level histogram of the image by a weighted sum of Heaviside functions [6]. These methods for highlighting the valleys of the histogram are generally based on the assumption that there is a trough between two peaks. However, such assumption does not always hold in real images which could result in inaccurate thresholds.

Meanwhile, the histogram can be transformed based on the information entropy theory [7], [8]. The threshold selection becomes the problem of maximizing an objective function based on the conventional thermodynamic entropies [9], the Shannon entropy [10], Renyi entropy [11], Tsallis entropy [12] and Masi entropy [13]. By using the cross entropy, it can also be converted to the problem of minimizing objective functions [14], [15]. Furthermore, one can extend it to a 2-D maximum entropy thresholding [16], [17], [18]. Most of the related works are conducted using particle swarm optimization (PSO) [19], [20], ant colony optimization [21], bacterial foraging [22], honey bee mating optimization [23], the artificial bee colony algorithm [24], genetic algorithm [25], and fuzzy c-means (FCM) algorithm [26]. Besides, they can also be used to solve other forms of objective functions. Although the entropy-based thresholding methods have the rigorous theoretical basis of mathematics and good practical application or research significance, most of them are restricted to the bi-level thresholding and contain some adjustable parameters. Meanwhile, the results obtained by this way may be biased.

In most existing papers, the histogram is considered as a multimodal probability distribution. Based on the Gaussian probability distribution, it can be divided into parametric and nonparametric analysis methods. In the parametric approaches, the related parameters of probability distribution can be estimated using the expectation maximization algorithm [27] or the least-squares estimation method [28]. Besides, based on the Poisson probability distribution, the threshold selection problem can be solved by minimizing the cross entropy [29]. However, since the real image pixel distribution is generally more complex, the corresponding histogram may not conform to a particular model, hence the selected threshold is not accurate. Although Otsu's method [30], [31] can overcome these limitations, it is inefficient for multi-level thresholding. Therefore it appears to be more important to decompose probability of histogram efficiently.

Similar to the threshold selection problem for images, signal decomposition is also an adaptive target solution method. There are several kinds of signal decomposition algorithms, including empirical mode decomposition (EMD), local mean decomposition (LMD), Hilbert vibration decomposition (HVD) and variational mode decomposition (VMD). Compared to other methods, VMD algorithm has a more complete mathematical theory derivation process and the good resolution in spectrum analysis as it is a different equivalent filter bank structure essentially. Meanwhile, the algorithm is a non-recursively decomposition method [32], [33], [34]. But before we try to solve the image segmentation problem with the aid of VMD, the relationship between them should be established.

In this paper, we attempt to explain the meaning of histogram from the perspective of signal spectrum, establish the relationship between threshold segmentation and signal decomposition, and propose a histogram decomposition approach called Histogram Thresholding-VMD (HTVMD) algorithm. The remainder of the paper is organized as follows: Section 2 introduces the method of gray histogram to time-varying signal model conversion. Section 3 analyzes the essence of VMD, and presents an improved method for solving the between-class variance function. Furthermore, two principles of threshold selection, i.e. the minimum point search (MPS) method and the cross point search (CPS) method, are implemented in image segmentation. Subsequently, some experiments on performance evaluation are presented in Section 4. Finally, some conclusions are made in Section 5.

Section snippets

Histogram of image

Histogram is a statistical table used to express gray level distribution of an image. From the mathematical point, it calculates the number or the probability of each gray level in an image. From the perspective of graphics, it is a two-dimensional graph, where the abscissa represents the gray level of each pixel in the image and the ordinate is the number or frequency of the image pixels at each gray level. For an image I with L gray levels {0, 1, …, L − 1} and size M × N, its histogram can be

The proposed multilevel thresholding algorithm

In this section, we analyze the essence of the VMD decomposition process for the signal that represents the histogram of an image, and propose a histogram decomposition method using an improved VMD directly whose objective function is changed as the between-class variance function. Meanwhile, two threshold search strategies are applied to segment image. And the algorithm of the proposed method is narrated.

Experimental results and comparative performances

In this section, we will present some typical experiments and their analysis to show the performances of the two proposed algorithms, which are defined as MPS and CPS for simplicity, respectively. The parameter ε of HTVMD method is set to 1e−12, and the initial values of the center μk are uniformly distributed in the gray level range. Further, the algorithm is implemented in Matlab 7.0 on a personal computer with 2.4 GHz CPU, 1GB RAM running the Windows 7 operating system.

Meanwhile, the

Conclusion and outlook

In this paper, we analyze the feasibility of a threshold selection method, which is based on the decomposition of the histogram signal by VMD, and find that the threshold results are easily affected by phase. Through discussing the essence of VMD algorithm, we propose an improved VMD whose objective function is changed to meet the requirements of image segmentation. Then, MPS or CPS can be implemented to choose the thresholds.

Through a series of experiments, the proposed MPS scheme clearly

References (41)

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