Localised edge‐region‐based active contour for medical image segmentation

IET Image ProcessingVolume 15, Issue 7 p. 1567-1582 ORIGINAL RESEARCH PAPEROpen Access Localised edge-region-based active contour for medical image segmentation Hua-Xiang Liu, Corresponding Author Hua-Xiang Liu felicia_liu@126.com orcid.org/0000-0002-1243-7491 College of Mechanical and Electrical Engineering, Central South University, Changsha, China School of Mechanical and Electronic Engineering, East China University of Technology, Nanchang, China Correspondence Hua-Xiang Liu and Yong-Cheng Lin, College of Mechanical and Electrical Engineering, Central South University, Changsha, 410083, China. Email: felicia_liu@126.com and yclin@csu.edu.cnSearch for more papers by this authorJiang-Xiong Fang, Jiang-Xiong Fang School of Mechanical and Electronic Engineering, East China University of Technology, Nanchang, ChinaSearch for more papers by this authorZi-Jian Zhang, Zi-Jian Zhang Department of Radiation Oncology, Central South University, Changsha, ChinaSearch for more papers by this authorYong-Cheng Lin, Corresponding Author Yong-Cheng Lin yclin@csu.edu.cn College of Mechanical and Electrical Engineering, Central South University, Changsha, China Correspondence Hua-Xiang Liu and Yong-Cheng Lin, College of Mechanical and Electrical Engineering, Central South University, Changsha, 410083, China. Email: felicia_liu@126.com and yclin@csu.edu.cnSearch for more papers by this author Hua-Xiang Liu, Corresponding Author Hua-Xiang Liu felicia_liu@126.com orcid.org/0000-0002-1243-7491 College of Mechanical and Electrical Engineering, Central South University, Changsha, China School of Mechanical and Electronic Engineering, East China University of Technology, Nanchang, China Correspondence Hua-Xiang Liu and Yong-Cheng Lin, College of Mechanical and Electrical Engineering, Central South University, Changsha, 410083, China. Email: felicia_liu@126.com and yclin@csu.edu.cnSearch for more papers by this authorJiang-Xiong Fang, Jiang-Xiong Fang School of Mechanical and Electronic Engineering, East China University of Technology, Nanchang, ChinaSearch for more papers by this authorZi-Jian Zhang, Zi-Jian Zhang Department of Radiation Oncology, Central South University, Changsha, ChinaSearch for more papers by this authorYong-Cheng Lin, Corresponding Author Yong-Cheng Lin yclin@csu.edu.cn College of Mechanical and Electrical Engineering, Central South University, Changsha, China Correspondence Hua-Xiang Liu and Yong-Cheng Lin, College of Mechanical and Electrical Engineering, Central South University, Changsha, 410083, China. Email: felicia_liu@126.com and yclin@csu.edu.cnSearch for more papers by this author First published: 20 January 2021 https://doi.org/10.1049/ipr2.12126AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract Segmenting the region of interest (ROI) from medical images is a fundamental but challenging task due to the illumination change and imaging devices. Although many models based on the local region-based active contour model (LR-ACM) are proposed to deal with intensity inhomogeneity, it is still difficult for the global energy-based ACM with local image information to accurately extract the ROI from medical images. To solve this problem, this study proposes a novel localised ACM by constructing the gradient information based on the probability scores from the fuzzy k-nearest neighbour classifier. Different from the traditional LR-ACMs, our model is based on local rather than global image statistics, where a probability score-based edge detector is directly used for gradient information. The energy functional consists of localised region energy and an edge energy. By introducing a local characteristics function, the localised region energy with the probability-score-based edge information is formulated, which can make the evolution curve stop on the exact boundaries of ROI. The edge energy including the regularisation and the penalty terms is used to avoid the reinitialisation process and smooth the evolution curve during evolution. To maintain the signed distance property of the evolution curve, a novel potential function in the penalty term is designed, which can consistently control the diffusion direction of the evolution curve. Experiments on the medical images including the cardiac magnetic resonance imaging and the 3DIRCADb databases demonstrate that the proposed model is more robust and accurate to extract the ROI than the popular localised and region-based ACMs. The code is available at: https://github.com/HuaxiangLiu/Localized_ERAC/. 1 INTRODUCTION In the field of medical image analysis and machine vision, accurately extracting the region of interest (ROI) plays an essential role in clinical tasks [1], such as the diagnosis, the organ delineation, and the visualisation of disease treatment. A direct solution to solve this problem is to segment the ROI from the medical images with complex background, for example, magnetic resonance imaging (MRI) image, computed tomography (CT) image, and positron emission tomography image. Medical image segmentation has been a fundamental but important task in the field of medical image processing. However, there are at least two problems in the medical image processing. First, intensity inhomogeneity caused by the illumination change and imaging devices [2] leads to the information loss of the image boundaries. Second, the over- or under-segmentation easily occurs in the extraction of the complex ROI. Therefore, designing an efficient and robust segmentation method for extracting the ROI is still very challenging. In recent years, a number of segmentation methods for extracting the ROI have been proposed, and they can be roughly divided into four classes: Clustering-based method, graph-based method, deep learning-based method, active contour-based method. In the clustering-based method, each pixel is assigned to a class by a clustering method, such as fuzzy c-means (FCM) [3] and k-means [4], which easily generates discontinuous segmented regions. The graph-based method [5] is a graph partition problem, and its energy functional is built based on its correlations between pixels. But it is very difficult to accurately design the weights between pixels. The deep learning-based method [6, 7] can extract deep image features by using the convolutional neural network (CNN), and now shows a vast range of prospects. But the CNN-based method needs lots of training images with ground truth (GT) to train deep network, especially collecting the images, and masking the corresponding GT regions is a very tedious and time-consuming task. More importantly, the pooling layers in the encoder lose image details; and two common issues in the deep learning-based methods are still overfitting and computation time. In the active contour-based method [8, 9], the energy functional, by incorporating local image information, can reduce the effect of intensity inhomogeneity to some extent. Therefore, the active contour-based method is still one of the efficient and prevalent approaches in medical image segmentation. The active contour model (ACM), first proposed by Kass et al. [10], is widely applied to image segmentation by minimising the energy functional due to its capacity of handling complex topology structure. It aims to partition the given image into some non-overlapping regions with the level-set functions (LSFs), where each segmented region represented as a zero level set has certain similar characteristics [11], such as colour, intensity, and texture. According to the formulation of the energy functional, the existing ACMs can be sorted into two classes: Edge- and region-based ACMs. The edge-based ACMs utilise image gradient information to spur the movement of the evolution curve toward the object boundaries. The snake model [9] is the earliest edge-based ACM and consists of the inner and outer energies, where the inner energy uses the first- and second-order derivatives to smooth and regularise the contour curve while the outer energy takes the integral of the curve as the force to drive the movement of the evolution curve. The geodesic active contour [12] incorporates image gradient information and geometrically active contour into curve evolution theory to construct an edge-stopping function (ESF). Li et al. [13] designed a novel penalty term to maintain the signed distance characteristics and avoid the reinitialisation process. But the penalty term may cause an arbitrarily large diffusion rate. To solve this problem, the distance regularisation level set evolution (DRLSE) model [14] was proposed to avoid the reinitialisation process during evolution, which consists of a distance regularisation term and an external energy. The distance regularisation term is formulated based on a potential function, which has the forward-and-backward (FAB) diffusion effect and can maintain the signed distance characteristics of the LSF. The external energy with the ESF can make the evolution curve stop on the object boundaries. But the DRLSE model is sensitive to the objects with weak boundaries. To obtain more robust ESF in the DRLSE model, Pratondo et al. [15, 16] proposed a gradient function by incorporating probability scores from the classifier, which can improve the gradient magnitudes of the ESF. However, Pratondo's model easily fails to stop on the object boundaries when the boundaries of the ESF is blurred especially for images with intensity inhomogeneity. However, the region-based ACMs utilise image region characteristics to drive the evolution curve to move towards the ROI. Chan–Vese model (C-V model) [17] is a well-known region-based ACM and driven by the average intensities inside and outside the evolution curve, which is a special case of the Mumford–Shah model [18]. Generally speaking, the region-based ACMs can be divided into the global region-based ACMs (GR_ACMs) and the local region-based ACMs (LR_ACMs). Since the C-V model has been proposed, many improved GR_ACMs have been improved, such as convex variational model [19] and the piecewise smooth model [20]. Different from the GR_ACMs, the LR_ACMs are mainly used to segment the ROI from the images with intensity inhomogeneity by incorporating local intensity information. The local binary fitting (LBF) model [21] uses a convolutional kernel function to extract local image information, where the spatially varying weight can control a scalable parameter to extract image features. But the model needs to perform four convolution operations in each iteration, which has a relatively high computational cost. Moreover, the LBF model is highly dependent on initialisation. By introducing a fitting image with a Gaussian kernel function, a local image fitting (LIF) model [22] was proposed based on the intensity difference between the input image and the fitted image, where the fitted image is from the average intensity of the inner and outer regions of the evolution curve. Most of the LR_ACMs are constructed by designing different types of energy functions, for example, the local hybrid image fitting energy [23], the local likelihood image fitting energy [24], the local cosine fitting energy [25], and the local Gaussian distribution fitting (LGDF) energy [26]. To segment different types of images, the GR_ACM and LR_ACM were proposed by combining the GR_ACM and LR_ACM with weighted coefficients. The weighted hybrid models [27-33] consist of a global and a local region-based energy by incorporating global and local image information. To adaptively adjust the local window size, the region-scalable fitting models [34, 35] are proposed. But these mentioned LR_ACMs cannot obtain the specified ROI from inhomogeneous images since they are concentrated on a local energy rather than a local method. Later on, Lankton et al. [36] proposed a localising region-based framework to localise region-based energy. Its improvement is that the framework can segment the ROI from the images with a heterogeneous feature. But Lankton's model cannot segment the ROI when the initial curve does not include the object region since it gradually shrinks towards the object boundaries. More importantly, Lankton's model with the localised region-based energy is sensitive to intensity inhomogeneity. From the above analysis, the LR_ACMs consider global rather than local image statistics. It difficult for these models to extract a specified ROI from medical images since they use a standard global method. Inspired by Pratondo's model [15, 16], by integrating the edge detector based on the probability scores from the fuzzy k-nearest neighbour (k-NN) [37], we proposed a novel localised edge region-based ACM to segment the ROI from the inhomogeneous medical images. Our model consists of the localised region energy and the edge energy. The localised region energy can make the evolution curve stop on the exact boundaries where the global energy fails. The edge energy includes the regularisation term and the penalty term, where the regularisation term can smooth the evolution curve while the penalty term can avoid the reinitialisation process during evolution. Experiments and results on the cardiac MRI and the 3DIRCADb databases show that the proposed method is more accurate and robust than the localised and region-based ACM. In our present study, we make three main contributions: First, by designing a multi-dimension vector for each point in the training and testing sets, the probability scores instead of the class labels are generated from the fuzzy k-NN classifier, which is used to construct a non-linear ESF. Different from the traditional ESF which is sensitive to the objects with blurred boundaries, the novel ESF can help the proposed model to distinguish between the foreground and the background. Second, a novel localised ACM with the edge information is proposed by introducing a characteristics function to define a local region, which avoids the difficulty of the parameter settings and improves the ability to handle intensity inhomogeneity. Third, to ensure and maintain the distance property and stability of the evolution curve, a novel double-well potential in penalty term is designed to reduce the effect of intensity inhomogeneity in given medical images. The rest of the study is arranged as follows. In Section 2, the previous work including the LBF model and the DRLSE model is reviewed. Section 3 describes the proposed model, including the edge detector based on the probability scores, the formulation of the localised region energy and the region energy, the level-set formulation, and the implementation. Experiments and results, including the experimental settings, the probability score edge detector and segmentation results, and comparison with the popular localised ACMs and the popular region-based ACMs are discussed in Section 4. Section 5 concludes the whole study. 2 RELATED WORK 2.1 LBF model The LBF model [21] is constructed by incorporating local image information to deal with intensity inhomogeneity, which is implemented by a Gaussian kernel function. Suppose I ( x ) : Ω → R 2 represents a segmented image in the image domain Ω, a contour curve C is represented by a zero level set of a Lipschitz function, and the LSF ϕ ( x ) is defined as follows: ϕ ( x ) > 0 if x ∈ In ( C ) ϕ ( x ) = 0 if x ∈ On ( C ) ϕ ( x ) < 0 if x ∈ Out ( C ) (1) With the defined LSF ϕ ( x ) in Equation (1), the energy functional in the LBF model is written as E L B F ( ϕ , f 1 , f 2 ) = ∑ i = 1 2 λ i ∫ Ω ∫ Ω K ( x − y ) I ( y ) − f i ( x ) 2 M i ( ϕ ( x ) ) d x d y + β 2 ∫ ∇ ϕ ( x ) − 1 2 d x + v ∫ δ ( ϕ ( x ) ) ∇ ϕ ( x ) d x (2)where λ i , β, and ν are positive constants, point y is a spatial pixel independent of point x, K ( · ) is a Gaussian kernel function, f1 and f2 are two smooth functions to estimate the local intensities inside and outside the contour C, respectively, ∇ denotes the Hamilton operator and ( ∇ ϕ ) x , y = ( ∂ ϕ ∂ x , ∂ ϕ ∂ y ) is the gradient of ϕ. M 1 ( φ ( x ) ) = H ε ( φ ( x ) ) and M 2 ( φ ( x ) ) = 1 − H ε ( φ ( x ) ) are two membership functions, where H ε ( φ ( x ) ) is a Heaviside function with a small constant ε, and the Dirac function δ ε ( φ ( x ) ) is the derivative of H ε ( φ ( x ) ) . Two functions H ε ( φ ) and δ ε ( φ ) are written as H ε ( x ) = 1 2 1 + x ε + 1 π sin ( π x π x ε ε ) | x | ≤ ε 1 x > ε 0 x < ε (3) δ ε ( x ) = 1 2 ε 1 + cos π x ε | x | < ε 0 | x | > ε (4) The last two terms in Equation (2) are called as the penalising and the regularisation terms, respectively. Compared with the C-V model, the LBF model assumes the intensity as inhomogeneous. But the model still needs four convolution operators at each iteration, which causes a high computational cost. In additions, the segmentation results are highly dependent on the initial localisation and sensitive to illumination inconsistency [2]. 2.2 Distance regularized level set evolution The aim of the DRLSE model is to avoid reinitialisation during evolution, which often causes many serious problems to solved, such as numerical accuracy and how to perform in an uncertain way. To maintain the signed distance characteristics, the DRLSE model is defined as E D R L S E ( φ ) = μ ℜ p ( φ ) + λ ¯ g ( φ ) = μ 2 ∫ Ω p ( | ∇ φ ( x ) | ) d x + λ ¯ g ( φ ) = μ 2 ∫ Ω ( | ∇ φ ( x ) | − 1 ) 2 d x + l 1 ∫ Ω g δ ( φ ( x ) ) × | ∇ φ ( x ) | d x + l 2 ∫ Ω g H ( − φ ( x ) ) d x (5)where μ, l 1 , and l 2 are three positive constants, and g is the edge detector. In Equation (5), the term ℜ p ( φ ) is the distance regularisation term, and the term λ ¯ g ( φ ) is the external energy. The potential function p ( s ) = 1 2 ( s − 1 ) 2 = 1 2 ( | ∇ φ ( x ) | − 1 ) 2 forces the signed distance function (SDF) to maintain the property | ∇ φ ( x ) | = 1 when the distance regularisation term ℜ p ( φ ) is minimised. The edge detector g, also called the ESF, is defined as g = 1 1 + | ∇ ( G σ ∗ I ( x ) ) | q , q = 1 , 2 , … , I ( x ) ∈ Ω (6)where G σ is a Gaussian kernel with standard deviation σ. The standard gradient flow is applied to obtain the evolving equation by minimising the energy functional: ∂ φ ∂ t = − ∂ E D R L S E ∂ φ = μ div ( d p ( | ∇ φ ( x ) | ) ∇ φ ( x ) ) + λ δ ε ( φ ( x ) ) div g ∇ φ ( x ) | ∇ φ ( x ) | + α g δ ε ( φ ( x ) ) (7)where div ( · ) is the divergence operator and d p is the potential function d p ( s ) = p ′ ( s ) / s . The effect of the distance regularisation term is represented in the following form of the diffusion equation: ∂ φ ∂ t = μ div d p ( ∇ φ ( x ) ) ∇ φ ( x ) = div D ∇ φ ( x ) (8)where D = μ d p ( | ∇ φ ( x ) | ) is the diffusion rate. The potential function d p ( | ∇ φ ( x ) | ) = 1 − 1 / | ∇ φ ( x ) | controls the FAB diffusion. If the diffusion rate d p ( | ∇ φ ( x ) | ) is positive, the diffusion is forward to decrease | ∇ φ ( x ) | . Conversely, if the diffusion rate d p ( | ∇ φ ( x ) | ) is negative, the diffusion is backward to increase | ∇ φ ( x ) | . Therefore, the reinitialisation process is avoided during iteration since the distance regularisation term can maintain the signed distance property | ∇ φ ( x ) | = 1 . 3 PROPOSED MODEL To accurately segment the ROI from the complex medical images with severe intensity inhomogeneity, a robust edge region-based ACM with a localised region energy and an edge energy is developed by integrating the PSED, and the framework of the proposed model is described in Figure 1. In this model, the probability scores are first generated by the fuzzy k-NN classifier, which are used for the formulation of the non-linear PSED. Then, by introducing a characteristics function, a localized region energy with the PSED-based edge information is constructed, also called the data term. Next, the regularisation term is constructed by incorporating the PSED-based ESF. Finally, the penalty term with a novel double-well potential is designed, which can continuously control the moving direction of the diffusion rate. The energy functional is defined E ( u , g , I ) = E r e g ( I , u , g ) + E e d g e ( I , u ) (9)where E r e g and E e d g e are the localised region and the edge energies, respectively, wherein, the edge energy includes the regularisation and the penalty terms. FIGURE 1Open in figure viewerPowerPoint The framework of the proposed model 3.1 Edge detector based on the probability scores Let X R = { x i } i = 1 L be a training set of multi-dimensional valued points, L R = { l i } i = 1 L be its corresponding set of class label, and Y R = { y i } i = 1 M be a set of testing points, where L and M are the number of elements in the training and testing sets, respectively. Each of the data points of X R can belong to one of the class labels from the label set l i ∈ { 0 , 1 } since we only consider the two-phase image segmentation. The fuzzy k-NN classifier [34] can generate a matrix of probability scores from the input image and includes the training and prediction processes. For the training process, the class label of a valued point x i ∈ X R is known. In the prediction process, for each point y i ∈ Y R to be classified, the fuzzy k-NN rule assigns a decision vector to each classified point, where the decision vector includes two values: A class label of the k-NN and a probability score. In this study, we use the probability scores W R = { w i } i = 1 M to design the edge detector, where i is the ith point in the segmented image. The probability score w i means the point y i and belongs to the probability of the object, and the set of probability score W R can be written as ∑ i = 1 M w i = 1 , 0 ≤ w i ≤ 1 (10)where w = ( w 1 , w 2 , … , w M ) denotes the probability scores of the testing set. To present the formulation of the vectors in the training and testing sets, as shown in Figure 2, for point x (red rectangle region) in the image domain Ω, its valued point is constructed with a set of the image intensities within the neighbourhood region, and four rectangle regions with 9-, 16-, 16-, and 25-dimensional vectors are shown from left to right corresponding to Figures 2(a)–(d), respectively. FIGURE 2Open in figure viewerPowerPoint The formulation of the current point and its corresponding valued point in the training and testing sets. (a)–(d) The valued point include 9-,16-, 16-, and 25-dimensional data, respectively In fact, the binary classifier provides a classification algorithm to generate the results of all the pixels. The binary results are obtained by setting the fixed threshold at 0.5 based on the probability scores. In the vicinity of the object boundary, the values of the probability scores are close to 0.5, which easily lead to an ambiguous classification when the segmented image includes severe intensity inhomogeneity. To overcome the problem, in this study, the probability scores instead of the binary results are further used to formulate the edge detector. Therefore, we should find the optimal solution where the function g ( w ) is written as g ( w ) = w w ∈ [ 0 , 1 ] (11) It is worth noticing that the membership function g ( w ) means each pixel belongs to the probability of the object. Figure 3(a) shows the binary function while Figure 3(b) shows the linear function in Equation (11). To better obtain the edge detector, a non-linear monotonically increasing function g ( w ) ∈ [ 0 , 1 ] satisfies that its value should be more closer to 0 when w is in the vicinity of 0, and be more close to 1 when w is in the vicinity of 1. The following function meets g ( w ) = sin ( w ) w ∈ [ 0 , 1 ] (12) FIGURE 3Open in figure viewerPowerPoint The function g(w) based on the probability score w. (a) The binary function. (b)The linear function in Equation (11). (c) The non-linear function in Equation (12) The function in Equation (12) g ( w ) can be replaced as the edge detector in Equation (6) to deal with the blurred boundaries, and Figure 3(c) shows the non-linear function. 3.2 Localised region energy To accurately extract the ROI in given images, a local region in the vicinity of the evolution curve is used to construct the energy functional. Let x be a point on the evolution curve C, Ω y be a local region around the evolution curve, and the local region be divided into two regions: In ( Ω y ) and Out ( Ω y ) . In Figure 4, the blue curve denotes the evolution contour, the red region denotes the local region, and the evolution contour divides the local region into the inside local region In ( Ω y ) and the outside local region Out ( Ω y ) . By introducing a characteristics function, a mask region D ( x ) (the red region in Figure 4) is defined as D ( x , y ) = 1 y ∈ { | ϕ ( x ) | ≤ A } 0 otherwise , (13)where A is a positive constant. The function D ( x ) is 1 when the absolute value of the LSF is smaller than A and 0 otherwise. FIGURE 4Open in figure viewerPowerPoint The formulation of the localised region energy. The red region denotes the local region Ωx, and the blue curve denotes the evolution curve, which divides the local region into two local regions inside and outside the evolution curve, respectively By incorporating the probability-score-based edge information, the localised region energy is defined as E r e g ( I , u , g ) = ∫ Ω g δ ε ( ϕ ( x ) ) ∫ I n ( Ω y ) D ( x , y ) λ 1 ( I ( x ) − m 1 ) 2 d y + λ 2 ∫ O u t ( Ω y ) ( I ( x ) − m 2 ) 2 D ( x , y ) d y d x + α ∫ I n ( Ω ) g d x (14)where λ 1 , λ 2 , and α are three positive constants, I ( x ) is the input image, and m 1 and m 2 are two average intensities corresponding to the inside (object) and outside (background) regions in the local region Ω y , respectively. Two functions m 1 and m 2 are defined as m 1 = mean ( I ( y ) | y ∈ Ω x ∩ u ( y ) > 0 ) m 2 = mean ( I ( y ) | y ∈ Ω x ∩ u ( y ) < 0 ) , (15) It is worth noting that two average intensities are equal when the evolution curve is inside the object region, which causes the evolution curve to disappear. To solve this problem, the area term with the edge detector in the last term in Equation (14) is added to force the evolution curve move outward the object boundaries. 3.3 Edge energy In order to maintain the smoothness of the evolution curve, the edge energy E e d g e is comprised of regularisation and penalty terms. The edge energy is written as E e d g e ( g , ϕ ) = E p e n a l t y ( ϕ ) + E l e n ( g , ϕ ) , (16) E p e n a l t y ( ϕ ) = β 1 ∫ Ω d p ( ∇ ϕ ( x ) ) d x , (17) E l e n ( g , ϕ ) = β 2 ∫ Ω g δ ( ϕ ( x ) ) ∇ ϕ ( x ) d x , (18)where β 1 and β 2 are two positive constants. The regularisation term E l e n denotes the length of the evolution curve, which is used to keep the smoothness of the LSF. The penalty term E p e n a l t y is used to the distance consistency of the SDF and the LSF. The standard gradient descent flow of the edge energy E e d g e ( g , ϕ ) is obtained as ∂ φ ∂ t = − ∂ E e d g e ∂ φ = β 1 div d p ( ∇ φ ( x ) ) ∇ φ ( x ) + β 2 δ ε ( φ ( x ) ) div g ∇ φ ( x ) ∇ φ ( x ) (19)where d p ( | ∇ φ | ) is a diffusion rate, which is used to control the diffusion direction of the evolution curve [14]. If the function d p ( | ∇ φ | ) is positive, the diffusion will be expanded forward to decrease | ∇ φ | . On the contrary, the diffusion will shrink backwards to increase | ∇ φ | when the function d p ( | ∇ φ | ) is negative. The diffusion function d p ( | ∇ φ | ) can successfully maintain the signed distance feature and avoid the reinitialisation process. Li et al. [13] used a potential function p ( s ) = ( s − 1 ) 2 / 2 for the penalty term, and the diffusion rate is written as d p ( s ) = p ′ ( s ) / p ′ ( s ) s s = 1 − 1 / s . The diffusion rate d p ( s ) is close to ∞ when the variable s approaches 0. The arbitrarily large value of the diffusion rate leads to a skip evolution. To solve this problem, a double-well potential function p ( s ) [14] is defined as