Fuzzy Active Contour Model Using Fractional-Order Diffusion Based Edge Indicator and Fuzzy Local Fitted Image

Local region-based active contour models (ACMs) can effectively segment images corrupted by intensity inhomogeneity, however, they always converge to local minimum and are sensitive to the initial position of contour. In this paper, a novel fuzzy ACM is proposed to tackle these problems. In order to deal with intensity inhomogeneity, the fuzzy local fitted image is first defined and utilized for constructing a local-region based fuzzy energy term, which is minimized in a variational level set framework to accurately segment inhomogeneous images. Second, the fractional-order diffusion based edge indicator is used to scale the local fuzzy energy term to reduce the effect of intensity inhomogeneity. Third, the fuzzy signed pressure force (FSPF) function defined by local image information is used for constructing the weighted area term to further improve the accuracy of the developed model. Finally, the global FSPF is formulated and used as an adaptive force, which can drive the level set function (LSF) to adaptively move up or down according to image intensity information. Therefore, the initial contour can be initialized as a constant function, which eliminates the problem caused by contour initialization. Moreover, the global FSPF makes the proposed model not easy to fall into local minimum. The results of experiments on synthetic and real images validate the accuracy of the proposed model for inhomogeneous image segmentation.


I. INTRODUCTION
Image segmentation is a primordial task in image analysis and computer vision. The main goal of image segmentation is to partition an image into non-intersected regions with approximately similar property, such as texture, color, intensity, etc [1]. Up to now, many methods have been proposed for image segmentation. Among these methods, ACMs initially proposed by Kass et al. [2] have received extensive attention and are widely used for image segmentation. The basic idea of the ACMs is to state the problem of image segmentation as the minimization of an energy functional. Moreover, level set based ACMs implicitly represent an evolving curve as the zero level set of a higher dimension function [3], [4], and the initialized curve is driven by a partial differential equation, which is obtained by minimizing a predefined energy The associate editor coordinating the review of this manuscript and approving it for publication was Giovanni Pau . functional. When the evolution of curve stops, the desired boundary of the target object is obtained.
In the past decades, a variety of level set based ACMs, embedding different image features, have been presented for image segmentation. However, the existing ACMs can be roughly dived into three categories: edge-based models [5]- [9], region-based models [10]- [19] and hybrid models [20]- [29]. In edge-based models, an edge indicator defined on image gradient information is used for guiding the evolving curve toward the interest boundary of the target object. These methods generally work well on images with strong boundaries. However, they have difficulty in extracting the weak edge of the target object and are more sensitive to noise, because the effective gradient information near weak edge and noise cannot be obtained.
Generally speaking, region-based models yield better performance on images with weak boundaries, because the curve evolution of these models is urged by a certain region VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ descriptor, which is independent of image gradient information. Specially, one of the typical global region-based models is the Chan-Vese (CV) model [10], which is based on the assumption that image intensities are piecewise constant in the foreground and background. The CV model has some advantages, such as being robust to noise and insensitive to the initial contour. Although global region-based models like the CV model can work well on bimodal images, they usually obtain inaccurate segmentation results in the presence of intensity inhomogeneity. The main reason is because only global image information is incorporated into the defined energy functional. To address the above problem of global region-based models, local statistical information is considered by researchers to construct local region-based models. By introducing a Gaussian kernel function into the energy functional, Li et al. [11], [12] propose the local binary fitting (LBF) model, which uses two local fitting functions to approximate the averages of intensities in a local window inside and outside the contour. In [14], the local entropy weighted region-scalable fitting energy (WRSF) model is proposed to enhance the robust of the LBF model. Zhang et al. [13] propose the local image fitting (LIF) model, which minimizes the differences between the original image and the local fitted image defined as the weighted average of two local intensity means. By minimizing the Kullback-Leibler divergence between original image, square image and their corresponding fitted images, Wang et al. [15] propose the local hybrid image fitting (LHIF) energy model, which is further improved in [16] by introducing a new inhomogeneity entropy, and they define an entropy weighted fitting (EWF) energy model. In [17], the bias field correction strategy is also used to tackle intensity inhomogeneity. In these models, image intensity is presented as the multiple of the true image and the bias component considered as the inhomogeneity of image. And then, object segmentation and bias component correction are obtained together in a minimization framework. By taking advantages of local image information, the above-mentioned local region-based models can accurately extract the boundary of the target object in the case of intensity inhomogeneity. However, they are heavily dependent on the initial contour. In other words, if the position of the initial contour is inappropriate, the segmentation results of these models may be unsatisfactory.
To deal with the shortcomings of global or local based ACMs and make full use of their advantages, many hybrid ACMs are proposed in the literature. Akram et al. [21] propose a local and global fitted image based model, and the Gaussian distribution is used to approximate the bias field. Similar to [29], the coefficient of the weighted area term need to be adjusted according to the position of the initial contour. In [22], global information learned by self-organising maps is used to construct a local energy term. However, if the number of layers is too much, a large computational cost is required. Otherwise, the model cannot effectively segment images with intensity inhomogeneity. In [25], the multiplicative and difference images are used to formulate a new hybrid ACM. However, finding an appropriate method to process the original images is a challenging task. Based on different weighting methods, global and local SPF are used to define hybrid ACMs [26]- [28]. In these hybrid ACMs, local information helps ACMs to accurately segment images with intensity inhomogeneity and global information improves the robustness of ACMs to the initial contour. Therefore, hybrid models always work well on inhomogeneous images.
From the perspective of clustering, region-based ACMs belong to the category of hard one [30]. In contrast, soft clustering methods can retain more information of the original image. The famous fuzzy energy based active contour (FEBAC) [31] is proposed by Stelios Krindis to tackle the problem of local minimum energy. Later on, many fuzzy set based ACMs are presented to enhance the segmentation accuracy of the model. References [32] and [33] use the fuzzy c-means method to initialize the initial contour and the clustering centers, respectively. In [34]- [36], the coefficient constructed by fuzzy method is invoked as the weight to adjust the evolution of level set function. In addition, many fuzzy ACMs [37]- [52] are constructed by using of pseudo LSF, which is identical to the LSF of traditional ACMs. In addition, the energy functional of fuzzy ACMs is often solved by two methods. The first is the Euler-Lagrange equation based gradient descent method [37]- [43]. The second is to directly minimize the difference between the new and old energies [43]- [52].
Specifically, Wu et al. [37] put forward an improved fuzzy ACM by introducing kernel metric (KFAC). To segment images having intensity inhomogeneity and slight texture, in [44], the coefficient of variation is used as a regional statistic. Similar to CV, these global fuzzy information based ACMs cannot obtain desirable segmentation results, because only global information of image is used to formulate the fuzzy energy functional. To overcome this problem, many fuzzy ACMs [38], [40], [41], [45], [46], [48], [52] make full use of local fuzzy statistics to cope with intensity inhomogeneity. To improve the robustness of model to initial contour, Sun et al. [38] fuse an adaptive contrast constraint into the presented model. By weighting global and local fitting energy, Fang et al. [48] present a novel fuzzy region-based ACM, called FRAGL. These models can segmentation inhomogeneous images to some extent. In addition, fuzzy information of the filtered image is also used to define FSPF [53], which guides the movement of the contour. And the optimal membership function is considered as the adjustment weight of each approximated local intensity. Moreover, many convex fuzzy ACMs [42], [44], [45], [48] is designed to prevent the evolution of curve from falling into local minimum.
As mentioned above, fusing local image information into the defined energy functional is an effective way to cope with intensity inhomogeneity. In this paper, we present a novel fuzzy energy-based ACM. First of all, the fuzzy local fitted image (FLFI) is defined as the weighted average of local fuzzy means on the two sides of the evolving curve, and the local-region based fuzzy energy term is defined as the Kullback-Leibler divergence between the original image and FLFI. Secondly, the fractional-order based diffusion method is utilized to smooth the original image, and the edge indicator is constructed on the filtered image. In order to improve the ability of the model to segment inhomogeneous images, the previously defined local fuzzy energy term is scaled by the edge indicator. And then, we respectively formulate local and global FSPF. The first one is considered as the weight of the area term, and the other one is taken as an adaptive force. Because their signs are opposite inside and outside the object, the weighted area term can automatically drive the evolving curve toward the desired object boundaries. Meanwhile, the adaptive force can drive the LSF to automatically increase or decrease according to image intensity information, which allows the initial contour of the proposed model to be initialized as a constant function. Therefore the problem caused by contour initialization can be eliminated. The experimental results demonstrate that the proposed method is able to segment the images with intensity inhomogeneity in terms of efficiency and accuracy.
In summary, the main contributions of the paper are listed as follows: 1) The fractional-order diffusion based edge indicator and FLFI are constructed to reduce the effect of intensity inhomogeneity in the given images.
2) The area term weighted by the defined local FSPF is proposed to enhance the ability of our model to segment inhomogeneous images. Moreover, local FSPF can control the direction of the evolving curve, which makes the initial contour automatically move toward the desired object boundaries.
3) The adaptive force based on the formulated global FSPF is used to cope with the position problem of the initial contour. And the evolution of the level set of the proposed model can start with a constant function without considering the position of the initial contour. The rest of the paper is organized as follows. Section 2 briefly introduces some related work and discusses their limitations to prepare the description of the proposed model, which will be described in detail in Section 3. Section 4 presents the experimental results. Finally, conclusions are drawn in Section 5.

II. RELATED WORK
ACMs have become one of the most popular methods for image segmentation due to easily handling the topological changes of evolving curve. In this section, we will give a brief review on related work and indicate their limitations.

A. CV MODEL
Let I : → R be an input gray level image in domain . A closed curve C is implicitly represented by the zero level set of the LSF φ(x). Variable x is a point in domain with intensity I (x). The purpose of the CV model [10] is to look for a optimal partition by minimizing the following energy functional: ∇ is gradient operator, and H ε (·) is the regularized Heaviside function formulated as: Dirac function δ ε (z) is the derivative of H ε (z), and ε is a small positive constant. λ 1 , λ 2 and µ are fixed positive parameters. c 1 and c 2 are two constants that denote the average intensities inside and outside the contour C, respectively. The LSF φ(x) is expressed by: By minimizing the energy functional (1) with regard to c i , i = 1, 2, the following equations can be obtained: Obviously, c 1 and c 2 respectively represent the global statistical information inside and outside the contour C. Therefore, if the intensity of image is inhomogeneous, they may be far different from the actual data. As a result, the CV model usually fails to segment images corrupted by intensity inhomogeneity. However, the CV model is less sensitive to initial contour and not easy to trap into local minimum.

B. LBF MODEL
To segment images with intensity inhomogeneity, the LBF model [11], [12] is proposed by introducing a Gaussian kernel function. The energy functional of the LBF model is expressed by: where λ 1 and λ 2 are positive constants. K σ is the Gaussian kernel function with standard deviation σ . Two smooth functions f 1 (x) and f 2 (x) are used to approximate the local averages of intensities inside and outside the contour in a local window. VOLUME 8, 2020 Keeping LSF φ(x) fixed and minimizing the energy functional (5), f 1 (x) and f 2 (x) can be obtained: Different from c 1 and c 2 in the CV model, f 1 (x) and f 2 (x) are local averages of the input image intensities in a local window. Therefore, the LBF model can effectively segment images with intensity inhomogeneity. However, the LBF model is easy to trap into local minimum and sensitive to the position of the initial contour.

C. LIF MODEL
The LIF model [13] assumes that the local region N x is piecewise smooth, and the center intensity of N x can be fitted by the local intensity averages. The energy functional of the LIF model aims to minimize the differences between the original image and the local fitted image, which is formulated as follows: where I LFI (x) is the local fitted image, which is expressed as: here f 1 (x) and f 2 (x) represent the local intensity averages inside and outside the contour in the local region N x . Segmentation results of the LIF model are similar to that of the LBF model. But compared with the LBF model, the LIF model needs less computational time, because the Gaussian filtering process is used to smooth the level set function. Moreover, the LIF model still suffers from the problem of the position of the initial contour, that is, whether the selected position of the initial contour is appropriate is an important factor affecting the accuracy of the segmentation results.

D. FEBAC MODEL
The basic assumption of the FEBAC model [31] is similar to that of the CV model. That is, the image intensities in the foreground and background are piecewise constant. By introducing the membership value u(x), the FEBAC model first defines a pseudo LSF, which is given by: And then the fuzzy energy functionanl of the FEBAC model is expressed as: (10) where λ 1 and λ 2 are positive constants. u 1 (x) = u(x) ∈ [0, 1] and u 2 (x) = 1−u(x) indicate the degree of the membership of intensity I (x) belong to c 1 and c 2 , respectively. The weighting exponent m(m > 1) is usually set to 2.
Keeping u(x) fixed and minimizing the energy functional (10) with respect to c 1 and c 2 , the following equations used to update c 1 and c 2 can be obtained: Similar to the CV model, when the basic assumption that image intensities are piecewise constant in each region (foreground or background) is not met, FEBAC model usually cannot extract the desirable boundaries of the target objects.

III. THE PROPOSED MODEL
As mentioned in Section 2, global-based ACMs lack the ability to segment images with intensity inhomogeneity.
In contrast, the local-based ACMs can obtain more accurate segmentation results, however, these local-based ACMs are more sensitive to the position of the initial contour. To cope with these problems, in this section, we present an adaptive fuzzy energy based ACM and describe it in detail. We firstly present the fractional-order based diffusion filter, and define an edge indicator. Then, we formulate the global and local FSPF, and prove that their signs are opposite inside and outside the object. Finally, we define a novel fuzzy energy functional and give the algorithm steps.

A. FRACTIONAL-ORDER DIFFUSION BASED EDGE INDICATOR
Inspired by [54]- [56], in this subsection, we aim to extract the underlying object edge where the initial contour needs to stop. For this purpose, the fractional-order derivative based Perona-Malik diffusion is used to smooth the original image. This diffusion process can yield a piecewise constant results while preserving edges and suppressing staircase [55], [56]. Specially, the input image is first smoothed by the following fractional-order diffusion equation: where D α * x and D α * y are the conjugates of fractional-order derivatives D α x and D α y , α ∈ (1, 2). g(DC) = exp(−DC/100) is diffusivity function, and DC stands for difference curvature, which is expressed as: and where I x and I y represent the first-order derivatives, I xx and I yy denote the second-order derivatives. The partial differential equation defined in (12) can be solved iteratively in the frequency-domain, for details refer to [55], [56]. As pointed out in [57], the value of DC is large at the object edges, while it is small in flat regions. Therefore, the diffusion process conducted by (12) can efficiently smooth the flat regions while preserving important image edges, which are related to the boundaries of the target object. Let I k (s) be the result of the k iterations, the adaptive order-based edge indicator of the smoothed image I k (s) is defined as follows: where m(s) is the sample mean of the local rectangular window N s of size 5 × 5, and adaptive order function β(s) is defined by: here T is the maximum value of I 2 x (s) + I 2 y (s)(s ∈ ), and I is the normalized result of the smoothed image I k (s). It is worth noting that when the condition mean(|M k (s) − M k−1 (s)|) < 0.1 is met, iteration process conducted by (12) is terminated. As shown in Fig. 1, The proposed edge indicator can provide more reliable results compared to the traditional edge indicator (e.g. Laplacian of Gaussian (LoG) filter).

B. LOCAL FUZZY ENERGY TERM
Local image information plays an important role in dealing with intensity inhomogeneity. Therefore, we consider making full use of fuzzy local information to construct the proposed local energy term. We first defined the fuzzy local fitted image as follows: where m 1 (x) and m 2 (x) are local fuzzy clustering center functions and defined by Obviously, m 1 (x) and m 2 (x) calculate the local intensity means inside and outside the contour C in a neighbourhood of x. I FLFI can be considered as the fuzzy approximation of the original image I in a local window.
To take advantage of the defined FLFI, in this paper, Kullback-Leibler divergence is used to quantify the differences between the original image I and its approximation FLFI, and the edge indicator scaled local fuzzy energy term is defined by: where W (x) = 1/(1 + exp(−M (x)) is used to scale the Kullback-Leibler divergence between I and I FLFI . M (x) is the normalized form of the edge indicator defined in (16), which is obtained when the iteration process conducted by (12) is terminated.

C. GLOBAL AND LOCAL FSPF
In our works [41], a global FSPF is defined as: where c 1 and c 2 are defined in (11). It has been proved that spf (I (x)) has values in the range [−1, 1]. Although it can efficiently modulate the signs of the pressure forces inside and outside the region of interest, the constant weights (e.g., 0.5) cannot carry more information of the images.
To overcome this drawback, in this subsection, we propose two novel FSPF. Let spf G (I (x)) and spf L (I (x)) be global FSPF and local FSPF, respectively, then they are defined by: where c 1 , c 2 are defined in (11) and m x (x), m 2 (x) are defined in (19). From the definition, we can see that spf G (I (x)) is a global measure which contains the global statistical information, and spf L (I (x)) is a local measure which contains the local statistical information. The FSPF functions defined in (22) and (23) also have values in range [−1, 1]. For simplicity, we prove this property only for a binary image. Let \ω and ω be the background and the object, respectively, then a binary image can be expressed as: where a, b > 0 with a = b. VOLUME 8, 2020 Theorem 1: Let I be a binary image defined by (24). Then one has Proof: see Appendix A. Theorem 2: Let I be a binary image defined by (24). Then one has Proof: see Appendix B. Compared with the FSPF proposed in [53], the proposed FSPF has the following differences: 1) On one hand, c 1 and c 2 of the proposed spf G (I (x)) denote the fuzzy average intensities inside and outside the contour C, respectively. On the other hand, the weights of c 1 and c 2 are not constant (eg. 0.5), but membership values. Therefore, the designed spf G (I (x)) can retain more information of the original image.
2) The spf G (I (x)) is designed to reduce the dependence of ACMs on initial contour, not to control the direction of the evolving curve, which is why the level set of the proposed model can start with a constant function.
3) The purpose of spf L (I (x)) defined in (23) is the same as that of the FSPF proposed in [53], which is to control the direction of the evolving curve. However, the spf L (I (x)) is defined by the membership value weighted local mean, which enables the direction of the evolving curve make adjustments automatically based on local information.

D. EVOLUTION EQUATION OF THE PROPOSED MODEL
The fuzzy energy functional of the proposed model is expressed as: where λ, µ, ν and γ are positive constants. u(x) is pseudo LSF defined in (9). L(u(x)) and A(u(x)) are defined by: and where δ ε (z) is the derivative of the regularized Heaviside function H ε (z) defined in (2). The energy functional L(u(x)) is length term, which is used to smooth the pseudo zero level contour. A(u(x)) computes the weighted area of the region {x ∈ : u(x) > 0.5}. Since signs of spf L (I (x)) are opposite inside and outside the object, the pseudo zero level contour can automatically shrink or expand. And the global FSPF spf G (I (x)) can be seen as an adaptive force, which drives the pseudo LSF to automatically increase or decrease according to image intensity information. Keep m i (x)(i = 1, 2) and c i (i = 1, 2) fixed, we minimize the fuzzy energy functional (27) with respect to u(x). The following formulation is obtained to update the pseudo level set u(x): and where div(·) is the divergence operator. Next, we analyse the characteristic of the global FSPF spf G (I (x)) based on the following equation: From the Theorem 1, It can be seen that the spf G (u(x)) has opposite signs inside and outside the object. If spf G (u(x)) > 0, then ∂u/∂t > 0, and (32) makes u increase; Conversely, if spf G (u(x)) < 0, then ∂u/∂t < 0, and (32) makes u decrease. This characteristic can drive the pseudo LSF u to move up or down according to the image intensity information. Therefore, a constant function can be used as the initialization of u(x). Taking a binary image (Fig. 2a) as an example, the initial contour is set to be a constant function (e.g., u(x) = 0.5, x ∈ ) shown in Fig. 2c. The segmentation result of the proposed model is shown in Fig. 2b, and Fig. 2d shows the 3D plot of the final pseudo LSF, and the contour C = {x ∈ : u(x) = 0.5} is marked by red line. This example demonstrates that the proposed model is less troubled by the problem of the initial contour.
It is worth noting that the proposed model is different from the hybrid ACMs proposed in [27], [48]. In one hand, the gradient information of LSF used in [27] is sensitive to the edge of object, which cannot effectively deal with weak boundaries. On the other hand, the global SPF proposed in [27] and the proposed model are used to enhance the robustness of the models to initial contour. However, the proposed global SPF constructed by fuzzy image information can makes the level set evolution start with a constant function. In addition, the hybrid averages in FRAGL [48] based on global and local intensity ones may not fit the local information, because the global averages will be far different from the actual data the intensity of image is inhomogeneous. Therefore, FRAGL cannot deal with images with serious intensity inhomogeneity. However, in the proposed model, Kullback-Leibler divergence based local term is used to effectively capture the local information, and local FSPF can automatically adjust the direction of the evolving curve in a local manner. They make the proposed more robustness to intensity inhomogeneity.

E. ALGORITHM
In this subsection, a finite difference scheme is used to numerically solve the proposed model. The iterative formula of (30) can be simply written as: where t is the time step, u k is the result of the kth iteration, and A(u k ) is the approximation of the right hand side of (30).
In order to obtain a stable updating of u, the input image is first normalized by the following equation: (34) where I min and I max represent the minimum and maximum of image intensity. Note that u may break the constraint (0 ≤ u ≤ 1) during the evolution. Therefore, u is reset at the end of each iteration according to the following approach: And the result of each iteration is smoothed by the Gaussian filtering method, which is expressed as: where K ρ is the Gaussian kernel function with standard deviation ρ. For simplicity, in this paper, the parameter ρ is set to 0.55 for all experiments. In summary, the main procedures of the proposed algorithm are given as follows: 1: Normalize the input image by (34). 2: Initialize the pseudo LSF u to be a constant function or a step function defined in (9). 3: Compute c i and m i (i = 1, 2) according to (11) and (19). 4: Update u according to (33), and then normalize it by (35).  (36). 6: Check whether the evolution is stationary. If not, return to step 2.

IV. EXPERIMENTAL RESULTS
In this section, the proposed model will be tested on synthetic and real images. All experiments are conducted in Matlab (R2019b) programming environment on Intel(R) Core(TM) i7-4720HQ CPU 2.6GHz and RAM 16.0GB. Unless otherwise specified, the parameters are set by default as follows: λ = ν = 1, γ = 0.05, µ = 0.5, t = 0.05 and α = 1.5. To validate the effectiveness of the developed model, we compare the developed model with CV [10], LBF [12], LIF [13], EWF [16], FEBAC [21], FRAGL [48] and GLSEPF [28] to display their performance differences. The parameters of the compared methods are set according to the original papers.

A. EFFECTIVENESS OF THE PROPOSED MODEL
In this subsection, we first validate the effectiveness of the proposed model on four images, and the initial contour is set to be a constant function (e.g., u(x) = 0.5, x ∈ ).  Fig. 3(e), and the 3D plot of the final pseudo LSF are shown in Fig. 3(f). It can be observed that the developed model can obtain desirable segmentation results for these images, whether it is noisy images or images with intensity inhomogeneity. This experiment demonstrates that the defined global FSPF spf G (I (x)) can drive the pseudo level set function u(x) to automatically move up or down, which allows u(x) to be initialized as a constant function. Therefore, the problem caused by the position of the initial contour can be alleviated to some extent. In addition, the corresponding fuzzy local fitted images are shown in row 1 of Fig. 4, which demonstrate that a large number of the undesired background information of the original images is greatly suppressed. Moreover, these fitted images I FLFI highlight the importance of the desirable objects and reduce the adverse effects of the complex VOLUME 8, 2020  background in image segmentation of the developed model. Row 2 of Fig. 4 shows the corresponding edge indicators, which provide the underlying boundaries of the target objects and efficiently suppress the complex texture components. Therefore, the proposed edge indicator weighting method can hance the model's ability to segment images with intensity inhomogeneity.
To demonstrate the contribution of the proposed local FSPF spf L (I (x)), we conduct the following experiment shown in Fig. 5, and the initial contours are set as a step function according to the (9). Fig. 5(a) shows the original images with initial contour. Figs. 5(b)-(f) show evolution results after 50, 100, 150, 200, 250 iterations, respectively. Rows 1 and 2 are the evolution process of the curve by setting spf L (I (x)) to 1. We can discover that the contours are gradually contracted, if fact, the contours will disappear after 300 iterations. When spf L (I (x)) is reserved, the proposed model can obtain desirable segmentation results. The main reason is that the proposed local FSPF can efficiently modulate the direction of the evolution of the curve.

B. COMPARISONS WITH THE EXISTING SEGMENTATION ALGORITHMS 1) ROBUSTNESS TO INITIAL CONTOUR
To demonstrate the robustness of the developed method to initial contour, the following experiments are conducted on  two vessel images with intensity inhomogeneity. Given that CV, FEBAC and FRAGL cannot efficiently segment images with intensity inhomogeneity, which will be shown in the subsequent experiments, we just compare the developed model with LBF, LIF, EWF and GLSEPF. The segmentation results of the related models are shown in Figs. 6 and 7. In Fig. 6, LBF, LIF and EWF only get the desirable segmentation results in the row 1. GLSEPF falls into local minimum in rows 2 and 5. Fig. 7 shows that LBF, LIF and EWF can correctly segment the input image in rows 1 and 2, and their performance in other situation are worse. It is also can be seen that the developed model and GLSEPF work well on the second image with all initial contours. From experiments in Figs. 6 and 7, we can get the following conclusions to a certain extent. On one hand, the traditional local information  based models can deal with images with intensity inhomogeneity, but they are sensitive to the position of the initial contour. On the other hand, introducing global image information into ACM is necessary to improve the robustness of the method to initial contour. Although, both GLSEPF and our model are a hybrid model based on global and local information, our model performs better. The main reason is that the proposed adaptive fuzzy force greatly avoids the proposed model trapping into local minimum.

2) SEGMENTATION OF BIMODAL AND INHOMOGENEOUS IMAGES
In Fig. 8, two bimodal images and four inhomogeneous images are used to visually evaluate the performance of the proposed model and the compared methods. Fig. 8(a) shows the original images with the initial contours. Figs. 8(b)-(i) are the segmentation results of the CV, EWF, FEBAC, FRAGL, LBF, LIF, GLSEPF and the propose model, respectively. From rows 1 and 2, we can observe that all methods can work  well on bimodal images. But CV and FEBAC cannot efficiently segment inhomogeneous images as shown in columns 2 and 3 due to only using the global mean of intensities. Even though FRAGL is the local-region based model and has better performance in row 5, it cannot get desirable segmentation results in rows 3, 4 and 6, the reason may be that the weight used to compute the local means cannot efficiently capture the local information of images. On the contrary, EWF, LIF, LBF, GLSEPF and the proposed model obtain satisfactory segmentation results for these inhomogeneous images. This demonstrates that efficiently extracting local information of images plays a vital role in segmenting images with intensity inhomogeneity.

3) SEGMENTATION OF MEDICAL IMAGES
In computer-aid medical image analysis, correctly extracting the required boundaries of the target objects is very important for the doctor's diagnosis. For example, the boundaries of skin lesion can be utilized to asses the risk of melanoma. In order to further test the ability to segment images with complex background, we conduct the following experiments shown in Figs. 9 and 10. Fig. 9 shows the vessel segmentation of different modalities: pulmonary vessel (ultrasonic imaging), abdominal vessel (computed tomography) and carotid vessel (magnetic resonance angiography). Segmentation results of three skin lesion images corrupted by noise are shown in Fig. 10. From these segmentation results, it can be seen that the proposed model works well on these images. GLSEPF obtains desirable results of skin lesion images, but fails to extract the boundary of the carotid vessel image as shown in row 3 of Fig. 9. However, the other compared models cannot get satisfactory segmentation results. However, CV, FEBAC and FRAGL are better than EWF, LBF and LIF. This just verifies what we mentioned earlier that global-based models are less likely to fall into local minimum than local-based models. But the proposed model avoids this drawback by integrating local fuzzy statistical information and global based adaptive force into the energy functional, which makes the proposed more effective to deal with intensity inhomogeneity and more robust to images with complex background. Fig. 11 shows the fuzzy local fitted images corresponding to images in Figs. 9 and 10. We can find that the proposed I FLFI is more capable of effectively highlighting the desirable objects than I LFI proposed in [13], which makes the proposed model less susceptible to complex backgrounds in image segmentation.

4) QUANTITATIVE EVALUATION
In this section, Dice Similarity Coefficient (DSC) [18] is used to quantitatively evaluate the accuracy of the proposed model, and the DSC is given by: where |G|, |T | and |G T | denote the pixel number of the regions G, T and their union area, respectively. G is the VOLUME 8, 2020 ground-truth segmentation. T is the segmentation results of the corresponding methods. Obviously, the value of DSC varies between 0 and 1, and the closer DSC value is to 1, the better the segmentation is. The DSC of images in Figs. 8-10 are listed in Table 1, in which the proposed model obtains comparable results. In addition, we can find that the accuracy of global and local based models (GLSEPF and our model) is better than that of global or local based models.
In addition, natural images from MSRA-B are used to further evaluate the proposed model. It is worth noting that the used color images are first transformed into a gray image before segmentation according to the formula: I = 0.299 * I R + 0.587 * I G + 0.114 * I B , here I R , I G and I B present the pixel values corresponding to the R, G and B channels, respectively. From Figs. 12 and 13, we can find that the proposed model and GLSEPF can exactly extract the desirable boundaries of the objects in the tested images, which include  Fig. 10. severe intensity inhomogeneity. By contrast, other compared models fail to segment these images. To clearly compare the performance of all methods, the segmentation accuracy of images shown in Fig. 12 in terms of DSC are listed in Table 2. Compared with other models, the proposed model and GLSEPF obtain more satisfactory results. But the segmentation accuracy of the proposed model is slightly higher than that of GLSEPF.

A. INITIALIZATION OF PSEUDO LEVEL SET FUNCTION
As we mentioned earlier that the pseudo LSF of the proposed model can be initialized as a constant function (e.g. u(x) = 0.5, x ∈ ), and the effectiveness of this strategy has been confirmed in some experiments. But this does not mean it can work well on all types of images. An example is shown in Fig. 14. It can be observed that the developed method can exactly extract the boundaries of the target objects with the position of the initial contour shown in row 1 of Fig. 12. However, when the pseudo LSF is initialized as a constant function, the proposed model cannot successfully segment all images as shown in Fig. 14. From the corresponding gray images, we can get the following inference. If the target object is darker or brighter than the background region, the proposed model can obtain desirable segmentation result. On contrary, if a large part of the target object is similar to the background region, the proposed model will not work well and even gets wrong segmentation results.

B. INFLUENCE OF THE PARAMETER γ
The proposed adaptive force, which is constructed by using global information of image, plays a key role in rejecting the shortcomings suffered by local region-based models. In the following experiment, we show the effect of parameter γ on the proposed model. Fig. 15 shows the segmentation results under the parameter γ , which is set to be 0.05, 0.1, 0.2, 0.3 and 0.4, respectively. It can be seen that two images are corrupted by intensity inhomogeneity and the proposed model obtain the desirable results by setting γ = 0.05 and γ = 0.01. However, the objects cannot be segmented successfully in other cases. In fact, the parameter γ should be set according to the image. In general, If the image contains severe intensity inhomogeneity, a small value γ should be chosen, and vice versa. Fortunately, γ = 0.05 works well on all tested images in this paper.

VI. CONCLUSION
In this work, a fuzzy local-region based ACM is presented to segment images with intensity inhomogeneity. The proposed model has some advantages over several famous models due to the following reason. Firstly, the defined fuzzy local fitted images effectively reduces a large number of undesired background information and fits the target objects well. Secondly, the proposed weighting method based on the edge indicator, which is defined on fractional-order diffusion, provides underlying boundaries of the target objects and helps the proposed model to segmentation images with weak boundaries. Thirdly, the local FSPF makes the evolving curve automatically move toward the boundaries of the target objects. Finally, the global FSPF prevents the proposed model from falling into local minimum and allows our model to have more flexible initialization method. Moreover, the level set evolution can start with a constant function. Experimental results demonstrate that the desirable performance of our model for segmenting both synthetic and real images with intensity inhomogeneity in terms of accuracy. But the proposed model cannot work well on the images when a large part of the target object is very similar to the background region. VOLUME 8, 2020