Multilevel thresholding selection based on variational mode decomposition for image segmentation
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)
- et al.
A fast multilevel thresholding method based on lowpass and highpass filtering
Pattern Recognit. Lett.
(1997) - et al.
Multi-modal gray-level histogram modeling and decomposition
Image Vision Comput.
(2002) - et al.
Seeking multi-thresholds directly from support vectors for image segmentation
Neurocomputing
(2005) - et al.
Fast multilevel thresholding for image segmentation through a multiphase level set method
Signal Process.
(2013) A step beyond Tsallis and Rényi entropies
Phys. Lett. A
(2005)- et al.
A novel fuzzy classification entropy approach to image thresholding
Pattern Recognit. Lett.
(2006) - et al.
Threshold selection using Rényi's entropy
Pattern Recognit.
(1997) - et al.
Image thresholding using Tsallis entropy
Pattern Recognit. Lett.
(2004) - et al.
A novel generalized entropy and its application in image thresholding
Signal Process.
(2017) - et al.
A multilevel color image thresholding scheme based on minimum cross entropy and differential evolution
Pattern Recognit. Lett.
(2015)
Automatic thresholding of gray-level pictures using two-dimensional entropy
Comput. Vision Graph. Image Process.
Multi-threshold image segmentation using maximum fuzzy entropy based on a new 2D histogram
Optik
Multilevel minimum cross entropy threshold selection based on particle swarm optimization
Appl. Math. Comput.
A study on particle swarm optimization and artificial bee colony algorithms for multilevel thresholding
Appl. Soft Comput.
Optimal multilevel thresholding using bacterial foraging algorithm
Expert Syst. Appl.
A multilevel image thresholding using the honey bee mating optimization
Appl. Math. Comput.
Multilevel thresholding selection based on the artificial bee colony algorithm for image segmentation
Expert Syst. Appl.
A multilevel automatic thresholding method based on a genetic algorithm for a fast image segmentation
Comput. Vision Image Understand.
A fast and robust image segmentation using FCM with spatial information
Digital Signal Process.
A new image thresholding method based on Gaussian mixture model
Appl. Math. Comput.
Cited by (51)
Detection of the pipeline elbow erosion by percussion and deep learning
2023, Mechanical Systems and Signal ProcessingALVLS: Adaptive local variances-Based levelset framework for medical images segmentation
2023, Pattern RecognitionCitation Excerpt :Image segmentation is an important part of computer vision, in addition to pattern recognition [1,2], target tracking [3,4] and so on. The threshold method [5–7], the neural network method [8,9] and the active contour method [10–13] have been gradually used to segment synthetic images, real images, and medical images, but there are still some limitations. At present, some achievements have been made in semantic segmentation, and the accuracy has been greatly improved.
Smart Metro Station Systems: Data Science and Engineering
2022, Smart Metro Station Systems: Data Science and EngineeringColor image segmentation using Kapur, Otsu and Minimum Cross Entropy functions based on Exchange Market Algorithm
2021, Expert Systems with ApplicationsWind speed forecasting based on variational mode decomposition and improved echo state network
2021, Renewable EnergyCitation Excerpt :It turns out that the VMD-based models perform better than WT-based and EMD-based models [21,23]. In recent years, VMD method has been applied in signal processing [44], image segmentation [45], and diagnosis of centrifugal pump bearing defects [46]. The original wind speed series has large volatility and randomness, so it is difficult to directly model it, which means that it is not easy to directly forecast the original series.