A geometric image segmentation method based on a bi-convex,fuzzy,variational principle with teaching-learning optimization

Abstract

This paper proposes an efficient, bi-convex, fuzzy, variational (BFV) method with teaching and learning based optimization (TLBO) for geometric image segmentation. Firstly, we adopt a bi-convex, object function to process a geometric image. Then, we introduce TLBO to maximally optimize the length-penalty item, which will be changed under the teaching phase and the learner phase of the TLBO. This makes the length penalty item closer to the target boundary. Therefore, the length-penalty item can be automatically adjusted according to the fitness function, namely the evaluation standards of the image quality. At last, we combine the length-penalty item with the numerical remedy mechanism to achieve better results. Compared with existing methods, simulations show that our method is more effective.

Keywords

Geometric image segmentation CV BFV TLBO

1 Introduction

Image segmentation is a key step in image processing and image analysis. It is also an important direction of research for computer vision technology and has been highly appreciated by people for many years. On the one hand, it is the foundation of the target expression and has an important influence on the measurement of characteristics; on the other hand, because the image segmentation and expression of segmentation, feature extraction and target parameter measurement based on segmentation will convert the original image into a more abstract and more compact form. This makes it possible to have a higher level of image analysis and understanding.

Nowadays, there are already a lot of methods in the field of image segmentation. The important methods include the following: thresholding [22], edge detection [14], clustering [7, 16], regional active contour [6], and specific mathematical theory tools [20], etc.. Inaddition to the above methods, in recent years some new segmentation methods were also proposed, such as ACL [1], an improved firefly algorithm [3] and fuzzy c-means clustering [9], etc. They are relatively simple image segmentation methods and are most widely used, but there are still various deficiencies. The active contour model without edges, named the CV model, is one of the most successful models in image segmentation. However, the CV model also has drawbacks: (1) converging to local optima [5], (2) being sensitive to the selection of parameters [19] and (3) computational inefficiency [17]. In order to overcome these drawbacks, a novel bi-convex fuzzy variational (BFV) image can be proposed [10]. However, the bi-convex fuzzy method is only suitable for some special images and the length-penalty item is randomly initialized; there is no universality. This paper proposes an efficient, bi-convex, fuzzy method with teaching and learning based optimization (TLBO) for image segmentation. The TLBO approaches [12 –14] used to maximally optimize the length-penalty item will be changed under the teaching phase and the learner phase to achieve better results.

The article is organized as follows: In section 2, we simply introduce the CV model and the BFV model. Section 3 describes the TLBO algorithm. The ggeometric image segmentation aalgorithm based on BFV with TLBO is described in Section 4. In Section 5, experimental results are given to validate the validity and efficiency of the proposed model. The last section is the concluding remarks.

2 CV model and the BFV model

Chan and Vese simplify the Mumford-Shah model. They also propose a novel, active contour model based on region, namely the CV model [2 , 21]. The model assumes that the image is divided into two types of targets and backgrounds. The energy function is defined as follows: $\begin{matrix} E (C) = μ Length (C) + υ Area (C) + \\ λ_{1} \int_{inside (C)} {| I (x, y) - c_{1} |}^{2} dx + \\ λ_{2} \int_{outside (C)} {| I (x, y) - c_{2} |}^{2} dy \end{matrix}$ (1) where the parameter E presents the energy of the image, and I (x, y) is the original image, and the parameter C defines the contour of the original image. The function inside(C) indicates the region, which is inside the contour. The function outside(C) expresses the region, which is outside the contour. The latter two items are fitted to detect the target region and background region. The function Length(C) indicates the length of the contour, keeping the curve smooth. The function Area(C) shows the area inside the contour. The parameter c₁ indicates the average gray inside the contour, and c₂ indicates the average gray outside the contour. Here, the fixed parameters μ, v, λ₁, and λ₂ are all non-negative numbers, where λ₁, λ₂ are weights for the length-penalty item and the fitting item.

In order to minimize the energy function and achieve the best effect, we use a fuzzy energy functional and define our energy functional as: $\begin{matrix} E = λ_{1} \int_{Ω} μ^{m} {(I - c_{1})}^{2} dx + \\ λ_{2} \int_{Ω} {(1 - μ)}^{m} {(I - c_{2})}^{2} dy \end{matrix}$ (2) where u is the membership function and m is a constant positive integer. In the experiment λ₁ = λ₂ = 1.

We calculate the first order partial derivatives in Equation (2) with respect to c₁ and c₂, and then we set them to equal zero. c₁ and c₂ can be deduced as: $\begin{matrix} C_{1} = \frac{\int_{Ω} μ^{m} Idx}{\int_{Ω} μ^{m} dx} \\ C_{2} = \frac{\int_{Ω} (1 - μ)^{m} Idx}{\int_{Ω} (1 - μ)^{m} dx} \end{matrix}$ (3)

Another shortcoming of the CV model is that the length-penalty item restricts the choice of the initial value for the level set function. Also, the selection of the initial level set function depends on u. As a result, the proposed method is using TLBO to optimize the length item.

We calculated the Gateaux derivative for the energy functional according to Equation (1) and the derivative function on variables δ. The level set function is expressed as φ, which shows the initial surface of the contour. It should be noted that the contour is the zero level set. The purpose of the evolution process is to get the derivative of the function φ on variable t, as shown in formula (4). $\frac{\partial_{φ}}{\partial_{τ}} = δ (φ) [\begin{matrix} μ div (\frac{\nabla_{φ}}{| \nabla_{φ} |}) - v \\ - λ_{1} (1 - c_{1})^{2} + \\ λ_{2} (1 - c_{2})^{2} \end{matrix}]$ (4) where div is the divergence operator, and δ (φ) is the approximating solution of the Dirac function, and ∇ is the gradient operator. In the experiment, v = 0 and λ₁ = λ₂ = 1. In the same way, the evolution function for the level set of Equation (1) is as follows: $\begin{matrix} \frac{\partial_{μ}}{\partial_{τ}} = m (- λ_{1} μ^{m - 1} {(I - c_{1})}^{2} \\ + λ_{2} (1 - μ)^{m - 1} (I - c_{2})^{2}) \end{matrix}$ (5)

We apply standard Von Neumann analysis [20] to study the time stability. The following formula is used to compute: $\begin{matrix} μ_{n} = μ_{n + 1} + Δ tg (p (| \nabla I, s |), s) Δ μ_{n + 1 / 2} \\ p (| \nabla I, s |) = ɛ (s - s_{0}) | \nabla I | \\ + (1 - ɛ (s - s_{0})) | G * \nabla I | \\ ɛ (s) = \frac{1}{1 + e^{- s}} \\ g_{\max} = \max {g (p (| \nabla I |))} \end{matrix}$ (6) where s is the SNR of the test image, and G is a Gaussian kernel, and * is the operation of convolution.

3 Teaching-learning-based optimization

The TLBO put forward by Rao et al. [14, 15] is a novel, heuristic algorithm. The model can be described as randomly generating a series of solutions in the constraints space. These solutions can be regarded as a group of “students,” and one of the best is recognized as a “teacher”. The teacher imparts knowledge and answers to students’ questions. Students enrich their knowledge from the teacher. This is the first process of TLBO algorithm, which is called the teaching phase. The learning phase, which is the second process, can be described as communicating with others and exchanging experiences to encourage each other. After a period of time, the students’ knowledge becomes higher and higher. In other words, it is more and more tending towardthe optimal solution in the constraint space. The whole process is shown in Fig. 1.

The optimal model is

min f (X) or max f (X)

subject to X ∈ S

g_i (X) ≤0, (i = 1, 2, …, m)

where f (X) is the optimized objective function, searching any point Xj = (x₁, x₂, …, x_d), j = 1, 2, …, Sp, S_P is the number of species and d is the dimensions of X. Continuous variables $S = {X | x_{q}^{L} \leq x_{q} \leq x_{q}^{U}, q = 1, 2, \dots, d}$ , and $x_{q}^{L}$ and $x_{q}^{U}$ are the lower and upper bound of each dimension weight of X, respectively. Discrete variables x_q ∈ S = {X₁, X₂, …, X_P}, and p are a number of a discrete set. In the TLBO algorithm, the relations of class, students and teacher are as follows:

Class: set {X^j, j = 1, 2, …, Sp}.

Students: set X^j = (x₁, x₂, …, x_d), where d is the subjects of teaching.

Teacher: the best fitness value f (X) of set {X^j, j = 1, 2, …, Sp}. The Class matrix can be expressed as: $\begin{matrix} (\begin{matrix} f (X^{1}) \\ f (X^{2}) \\ ⋮ \\ f (X^{Sp}) \end{matrix} | \begin{matrix} X^{1} \\ X^{2} \\ ⋮ \\ X^{Sp} \end{matrix}) = (\begin{matrix} x_{1}^{1} & x_{2}^{1} & \dots & x_{d}^{1} \\ x_{1}^{2} & x_{2}^{2} & \dots & x_{d}^{2} \\ ⋮ & ⋮ & \dots & ⋮ \\ x_{1}^{Sp} & x_{2}^{Sp} & \dots & x_{d}^{Sp} \end{matrix}) \\ X_{teacher} = min f (X^{j}), j = 1, 2 \dots, Sp \end{matrix}$

We define the marks distribution curve of the class as a normal distribution, as formulation (6). In spite of having certain deviation from the actual, it can still be helpful for analysis. We can describe the above definition as: $f (X) = \frac{1}{σ \sqrt{2 π}} e^{\frac{- (x - μ)^{2}}{2 σ^{2}}}$ (7) where σ is the variance and x is any value in a certain range, and μ is the mean.

3.1 Teaching phase

In the beginning of the teaching phase, students’ values are relatively scattered and the average grades are S_A. Meanwhile, the teacher’s is T_A. After a period of teaching, students’ results gradually improve, and the average grade increases to S_B. The teacher’s also update to T_B. Students learn knowledge from the teacher, as shown by the difference between the average of the teacher and the student in the teaching phase. In short, we update the solution with Equation (7). If the new is better than the existing, replace the existing with the new solution. $\begin{matrix} Difference_M {ean}_{new, i} \\ = r_{i} (M_{new, i} - T_{F} M_{i}) \end{matrix}$ (8) where r_i is a random number, and r_i ∈ [0, 1], and T_F is a teaching factor which decides how the mean value is to be changed. Because TF is either 1 or 2, it can be designed as follows: $T_{F} = round [1 + rand (0, 1)]$ (9)

3.2 Learner phase

Solutions are updated according to the following procedure. We obtain the value of the objective function first. If the new solution is better than the existing, we replace it with the new one.

ii ← random(pop_rize) (ii≠i)

if Xi is better than Xii_then

X_i,new = X_i + r (X_i - X_ii)

else

X_i,new = X_i + r (X_ii - X_i)

end if

evaluate (X_i,new)

if X_i,new is better than X_i, then

X_i ← X_i,new

end if

The whole process is shown in Fig. 1.

4 Geometric Image segmentation algorithm based on BFV with TLBO

In this paper, the fitness function applies the Jaccard Similarity (JS) [11] value and the Standard Deviation (Std) as the quantitative indexes in order to evaluate the results. The JS value isexplained as: $JS (S_{1}, S_{2}) = \frac{| S_{1} \cap S_{2} |}{| S_{1} \cup S_{2} |}$ (10) where S₂ is the ground truth data and S₁ is the segmentation results.

The Std is defined as: $S_{td} = 100 \times \frac{1}{{(\frac{1}{n - 1} \sum_{i = 1}^{n} (x_{i} - \bar{x})^{2})}^{\frac{1}{2}}}$ (11) where $\bar{x}$ is the mean of image pixels and x_i is the value of image pixels.

From the above analysis, we can see that it is not necessary to record experimental results each time. We can partially take a reaction to T with the following step Δt (T = nΔt, where n is a positive integer).

In summary, the flowchart of the proposed algorithm (TLBO-BFV) can be summed up as follows.

Step 1. Set n = 0. It uses the TLBO method to optimize u and calculate g using Equation (6).

Step 2. Calculate c₁ and c₂ using Equation (3).

Step 3. Set t = 0. Compute μ^n+1/2 using Equation (5). If t + Δt ≤ T, then set t = t + Δt and μ_n = μ_n+1/2 and it goes to step 2; otherwise, it continues.

Step 4. It Computes μ_n+1 using Equation (6).

Step 5. If μ_n+1 satisfy the termination condition, it stops; otherwise, set n = n + 1 and it returns tostep 2.

5 Experiment results

We employ the CV model segmentation method and the BFV segmentation method to compare with the TLBO-BFV method proposed in this paper. In the experiments, the same iteration number is set.

According to the numerical results of three Figs, we draw the chart with JS and Std of the three methods, respectively.

We can notice that as show in Figs. 2(b), 3(b), 4(b) the loss of information about spectral characteristics is obvious, which using the fusion method of CV, As show in Figs. 2(c), 3(c), 4(c) the fusion method of BFV only retained a part of spectral characteristics information. The last images in each group (d) is the method based on TLBO-BFV and compared with the image fusion method above this method has more clear fusion boundary, richer local contrastinformation.

Tables 1–3 provide objective evaluations of the three methods. From the statistics on three tables we can see that the value of JS and Std in proposed method is higher than others and Figs. 5–6 more obvious show this trend. All of this above show that the information of image in proposed method is richer, the edge is more clear and our algorithm is better for analysis and study in the practicalapplication.

6 Conclusion

Aimed toward the limitation of the bi-convex, fuzzy, variational principle, this paper presents a novel segmentation method, named TLBO-BFV. The TLBO approach is used to maximally optimize the length-penalty item, so as to improve the accuracy of segmentation. Simulations indicate that the proposed method improves the robustness and the recognition rate, and also indicates that the algorithm has advantages in stability and effectiveness.

Footnotes

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under grant Nos. 61472204, 61272283, 61201416, the open funding project of state key Laboratory of Virtual Reality Technology and systems, Beihang University under No. BUAA-VR-16KF-10, and the Science Research Program Project of Educational Committee of Shaanxi Province under grant No. 14JS071.

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