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The VeseChan model without redundant parameter estimation for multiphase image segmentation
EURASIP Journal on Image and Video Processing volume 2020, Article number: 3 (2020)
Abstract
The VeseChan model for multiphase image segmentation uses m binary label functions to construct 2^{m} characteristic functions for different phases/regions systematically; the terms in this model have moderate degrees comparing with other schemes of multiphase segmentation. However, if the number of desired regions is less than 2^{m}, there exist some empty phases which need costly parameter estimation for segmentation purpose. In this paper, we propose an automatic construction method for characteristic functions via transformation between a natural number and its binary expression, and thus, the characteristic functions of empty phases can be written and recognized naturally. In order to avoid the redundant parameter estimations of these regions, we add area constraints in the original model to replace the corresponding region terms to preserve its systematic form and achieve high efficiency. Additionally, we design the alternating direction method of multipliers (ADMM) for the proposed modified model to decompose it into some simple subproblems of optimization, which can be solved using GaussSeidel iterative method or generalized soft thresholding formulas. Some numerical examples for gray images and color images are presented finally to demonstrate that the proposed model has the same or better segmentation effects as the original one, and it reduces the estimation of redundant parameters and improves the segmentation efficiency.
Introduction
Multiphase image segmentation under variational framework has found a lot of applications including multitarget detection and recognition, 3D segmentation and reconstruction in medical images, remote sensing images, etc. [1, 2], due to its property of multiple cue integration. The aim of multiphase image segmentation is to partition images into different regions without any overlaps and without any unlabeled region (called in the sequel vacuum) automatically. It is a natural extension of the twophase image segmentation based on the variational image analysis paradigm.
The MumfordShah model [3] is fundamental to variational image segmentation; it is a regionbased model which approximates an image to a piecewise smooth one and edges. To circumvent the difficulty of its implementation, Chan and Vese [4] proposed the classical VeseChan model under variational level set framework [5] based on reduced MumfordShah model with piecewise constant image assumption. The ChanVese model introduced onelevel set function to construct two characteristic functions for two regions. Using the same concept, [6] introduced nlevel set functions to partition n regions for Potts model [7], but a simplex constraint must be added to avoid vacuum and overlapping problem. In order to improve computational efficiency from the viewpoint of model, Pan et al. [8] proposed to use (n−1)level set functions to design n characteristic functions satisfying the simplex constraint naturally. Vese and Chan [9] proposed a strategy using mlevel set functions to design n=2^{m} characteristic functions for different regions with less evolution equations to be solved. Furthermore, Chung and Vese [10] proposed a more efficient scheme using only one smooth function to partition different regions.
Motivated by the relationship between the Heaviside function of a level set function and a binary label function, the piecewise constant level set function method was adopted to twophase image segmentation [11] combined with convex relaxation and thresholding techniques with high efficiency. The models for multiphase image segmentation using variational level set method as mentioned in the previous paragraph have been extended to the counterparts using piecewise constant level set function method successively, such as, [12] used n binary label functions to partition n regions, [13] used m binary label functions for n=2^{m} regions, and [14] used one piecewise constant level set function for all regions. Comparatively, VeseChan type scheme uses less label functions than [12], and its characteristic functions have lower degree than [14]. Obviously, lower degree characteristic functions are convenient to achieve minimum of the energy functional. To achieve higher computation efficiency for the optimization problems, Goldstein and Osher [15] proposed the split Bregman method, and Duan et al. [16] proposed some fast projection methods without reinitialization.
Our research in this paper starts from VeseChan model via binary label functions [11], and the goal is to improve the computational efficiency through modifying its original model with fewer parameters to be estimated. Using VeseChan model, the number n of regions is determined by binary label function number m (i.e., n=2^{m}). For instance, if we partition 9 regions, we need 4 binary functions to construct 16 characteristic functions, and thus 7 regions are empty; however, the parameters in these empty regions must be estimated when using the original model. To avoid the parameter calculations in these redundant regions, [17] proposed a modified scheme for characteristic functions after 2^{m−1} regions that include two parts to discard the last empty regions, which can overcome the problem of redundant parameter estimations.
In this paper, we use a unified characteristic function expression for all regions including redundant regions, but we add some simple area constraints of redundant regions to avoid estimations of redundant parameters, thus reducing costs. Although we transform the original model into a constrained optimization problem, we use ADMM [18–20] to solve it easily and systematically without additional constructions of characteristic functions for empty regions.
The outline of the paper is as follows. In Section 2, we present some classical models of variational image segmentation, covering the Potts model and VeseChan model for multiphase image segmentation using binary label functions. In Section 3, we propose the modified VeseChan model and design its ADMM. In Section 4, experimental results of gray images and color images are given to illustrate the efficiency of the proposed model in this paper. Concluding remarks are drawn finally.
Some previous works of variational image segmentation
Image segmentation under variational framework
The MumfordShah model and the Potts model are fundamental to variational image segmentation. The former one is to partition an image f(x):Ω→R into piecewise smooth parts u(x):Ω^{s}→R and edges Γ such that Ω=Ω^{s}∪Γ and Ω^{s}∩Γ=Ø, its energy functional minimization problem is stated as
where α,β,γ are penalty parameters. The second one is to partition the image domain Ω into
Its energy functional minimization problem is
where ∂Ω_{i} stands for the boundary length of each subregion.
In order to solve these problems in image domain, one approach is assigning a characteristic function for each region and making use of its total variation to replace length terms. If zero level set {x:ϕ(x)=0} of a level set function ϕ(x) is used to describe the curve for region partition implicitly, its Heaviside function H(ϕ(x)) can be used to define characteristic functions of regions. ϕ(x) and H(ϕ(x)) are defined respectively as
Then, the length of a closed curve Γ is
Using level set method, Chan and Vese [4] proposed the VeseChan model for twophase segmentation under variational level set framework based on reduced MumfordShah model with piecewise constant image assumption, and it is stated as
If ϕ(x) is defined as a signed distance function, it must fulfill the following Eikonal equation
For the case of multiphase segmentation, if each region is assigned a characteristic function χ_{i}(x)∈{0,1},(i=1,...,n), (i.e., χ_{i}(x)=1 for x∈Ω_{i},χ_{i}(x)=0 for x∉Ω_{i}), the Potts model (3) can be rewritten as
In order to fulfill the condition (2), χ_{i}(x) must satisfy the following simplex constraint
Using variational level set method, each region is described by onelevel set function ϕ_{i}(x), then, its characteristic function can be denoted by the Heaviside function of ϕ_{i}(x) (i.e., χ_{i}(x)=H(ϕ_{i}(x))). Replacing χ_{i}(x) with H(ϕ_{i}(x)) in Eqs. (9) and (10) leads to the Potts model under variational level set framework.
Motivated by the FourColor Theorem, Vese and Chan [9] proposed a strategy to partition n regions using mlevel set functions with n=2^{m}, which can avoid the problem of vacuum and overlaps, and where the model is expressed as
where χ_{i} is characteristic function with the form
For instance, if m=2,χ_{1}=H(ϕ_{1})H(ϕ_{2}),χ_{2}=H(ϕ_{1})(1−H(ϕ_{2})),χ_{3}=(1−H(ϕ_{1}))H(ϕ_{2}),χ_{4}=(1−H(ϕ_{1}))(1−H(ϕ_{2})).
Variational convex model for multiphase image segmentation
Taking into account the relationship of level set function and characteristic function, [11] rewrote the ChanVese model via a binary label function ϕ∈{0,1}
During automatic computation, ϕ is relaxed to a convex version ϕ∈[0,1], which is recovered to its binary label function finally. For the case of multiphase image segmentation, [13] introduced m binary label functions ϕ_{j}∈{0,1},(j=1,...,m) to define n=2^{m} characteristic functions for region partitioning, leading to the new VeseChan model in convex form
where the form of characteristic function is
Whenever the number of image regions is taken as N∈(2^{m−1},2^{m}), the problem of the automatic estimation of redundant parameters u_{i} arises, and this in turn increases the computation cost due to inherent redundancy. Assuming that the number of redundant regions start from N+1, Li et al. [17] proposed a scheme to avoid redundant parameter estimation using various characteristic function formulas. Their energy functional minimization problem is as follows
where \({u = \left \{ {{u_{i}}} \right \}_{i = 1}^{N}}, {\phi = \left \{ {{\phi _{j}}} \right \}_{j = 1}^{m}}, {\chi _{i}^{N}}\) is characteristic function of the ith region. If \(N = {2^{m}}, {\chi _{i}^{N}}\) is defined evenly
if 2^{m−1}<N<2^{m},
where \({m_{1}}=m  1, {{i_{0}}} = N  {2^{{m_{1}}}}, {{i_{1}} = i  {i_{0}}}, s_{i}^{m} = \sum \limits _{j = 1}^{m} {b_{j}^{m,i  1}}, b_{j}^{m,i  1} = \{ 0,1\} \). This scheme can handle different situations for different numbers of image regions. However, it destroys the original symmetric forms with complicated different characteristic functions. For example, there are different forms of characteristic function for N=5 and N=7, which are as follows
which will reduce the generality of characteristic function for different region numbers and increase the computation time. In the next section, we propose another modified VeseChan model to overcome these problems.
Methods
Characteristic functions based on binary decomposition
According to the original scheme by [9], the intersection of m binary functions can generate n=2^{m} characteristic functions for region partition. Let the binary expression of characteristic functions χ_{i} be \(b_{1}^{i  1}b_{2}^{i  1}\cdots b_{m}^{i  1}\), where the value of \(b_{j}^{i  1}\) is related to the binary expression about the subtraction of index by one (i.e., i−1) for the ith region Ω_{i}, then the characteristic function of Ω_{i} is
where ϕ_{j}∈{0,1}, \(b_{j}^{i  1} = \left \{ {0,1} \right \}, i=1,\cdots,2^{m}, {j = 1,\cdots,m}\). Taking m=3 for example, 8 regions are partitioned as shown in Fig. 1, and the relevant characteristic functions are listed in Table 1.
The classic optimization scheme of the VeseChan model
Equation (14) stated a multiphase segmentation VeseChan model using m binary label functions to define 2^{m} characteristic functions, which is usually solved by alternative optimization method.
The idea of alternative optimization is to solve minimum problem of a variable by fixing others. Firstly fix ϕ to optimize u, we obtain
When u is fixed, we compute ϕ by gradient descent method, which is shown below
where \(\nabla \cdot \left (\frac {\nabla \phi }{\left \nabla \phi \right }\right)\) is the curvature term of ϕ. We use the upwind differential scheme [16] to compute this term
where
In Eq. (24), x and y represent coordinates in the image, h is spatial step, and ε is a small positive value.
For the optimization of variable ϕ, the curvature term of ϕ will be obtained if the alternative optimization method is used directly, causing complex difference scheme and low computational efficiency. However, the ADMM method can avoid computing the curvature term and simplify the solving process by introducing auxiliary variables and Lagrange multipliers.
The modified model for gray images and its ADMM
For a gray value image f(x):Ω→R with N<2^{m} regions, there exist 2^{m}−N redundant regions in the range [N+1,2^{m}]. To avoid the redundant parameter estimations, we add area constraints on these regions
We cast the VeseChan model into
with constraints (25), where \({u = \left \{ {{u_{i}}} \right \}_{i = 1}^{N}}\) consists of parameters need to be estimated, \({\phi = \left \{ {{\phi _{j}}} \right \}_{j = 1}^{m}}\) consists of binary label functions.
In order to improve the efficiency, ADMM is used in our method. We introduce auxiliary variable w=∇ϕ, Lagrange multipliers λ_{j},λ_{l} and positive penalty parameters θ,μ. We add constraints (25) into Eq. (26), which can be rewritten as the following form of alternative optimization
where k is the iteration step.
Using the idea of alternative optimization, Eq. (27) is transformed into the following subproblems of optimization in each loop
Using variational method to Eq. (28a), fix ϕ^{k} and w^{k} to optimize u^{k+1}, we obtain
For the problem (28b), since ϕ∈{0,1} is a nonconvex function, we first transform ϕ∈{0,1} to ϕ∈[0,1] with convex relaxation and then fix u^{k+1} and w^{k}, and the subproblem of optimization with respect to ϕ^{k+1} is as follows
we obtain the EulerLagrange equation of \({\phi ^{k + 1}_{j}}\)
where \(A=\sum \limits _{i = 1}^{N} {{\alpha }{{\left ({f  u_{i}^{k + 1}} \right)}^{2}}\frac {{\partial {\chi _{i}}\left (\phi \right)}}{{\partial \phi }}}, B=\sum \limits _{j = 1}^{m} \left (\nabla \cdot {\boldsymbol {\lambda }}_{j}^{k} + {\theta }\nabla \cdot ({\boldsymbol {w}}_{j}^{k}  \nabla {\phi _{j}}) \right), C = \sum \limits _{l = N + 1}^{{2^{m}}} {\left ({\lambda _{l}^{k} + {\mu }{\chi _{l}}\left (\phi \right)} \right)\frac {{\partial {\chi _{l}}\left (\phi \right)}}{{\partial \phi }}}\). \({\phi _{j}^{k + 1}}\) can be solved by GaussSeidel iterative method, then
For Eq. (28c), to fix u^{k+1} and ϕ^{k+1} to optimize w^{k+1}, we obtain the EulerLagrange equation
when w_{j}≠0. Equation (33) can be expressed as a following generalized soft thresholding formula in analytical form
The Lagrange multipliers \({\boldsymbol {\lambda }}_{j}^{k + 1}\) and \(\lambda _{l}^{k + 1}\) can be updated as the following
In order to represent the boundary of segmented images, it is necessary to transform ϕ_{j} after convex relaxation to binary label function as the following
where τ∈(0,1). The parameter τ is usually chosen as \(\tau = \frac {1}{2}\). The stopping criterion is based on the relative energy error formula E_{1}^{k+1}−E_{1}^{k}E_{1}^{k}≤ε, where ε is a small positive value.
The ADMM of modified model for gray images can be described as Algorithm 1.
The modified model for color images and its ADMM
Different from a scalar image, a color image consists of three layers, so parameter estimation is needed for each layer in different regions. Let f=(f_{1},f_{2},f_{3})∈R^{3} be a color image, u_{i}=(u_{i1},u_{i2},u_{i3}) be the piecewise constant parameters to be estimated in the ith region, ϕ consists of m binary functions as the previous subsection, and the variational model for multiphase color image segmentation is
it is also subjected to the constraints (25).
To design its ADMM method, we introduce auxiliary variable w=∇ϕ, Lagrange multipliers λ_{j},λ_{l} and positive penalty parameters θ,μ. Then we add constraints (25) into Eq. (37) and transform it into the following form of alternative optimization
Using the same procedure as the last subsection in each loop of optimization, we get successively
Fixing \({u_{ip}^{k + 1}}\) and w^{k}, we obtain the EulerLagrange equation of ϕ^{k+1} as the following
where \(A_{i}={\alpha }\sum \limits _{i = 1}^{N} {\sum \limits _{p = 1}^{3} {{\left ({{f_{p}}  u_{ip}^{k + 1}} \right)^{2}}\frac {{\partial {\chi _{i}}\left (\phi \right)}}{{\partial \phi }}} }\). We design GaussSeidel iterative method from Eq. (40) to compute \({\phi _{j}^{k + 1}}\).
\({\boldsymbol {w}}_{j}^{k + 1}\) can be solved from Eq. (34). The Lagrange multipliers \({\boldsymbol {\lambda }}_{j}^{k + 1}\) and \(\lambda _{l}^{k + 1}\) can be updated as Eq. (35).
The ADMM of the modified model for color images can be described as Algorithm 2.
Results and discussion
In this section, we present two groups of numerical experiments for segmented images including gray images and color images to study the effectiveness and efficiency of our method. For Eqs. (27) and (38), if the number of regions N=2^{m}, this method represents traditional 2^{m}phase segmentation with ADMM. If N<2^{m}, it represents the proposed method in this paper. The initial binary label function ϕ is initialized as m circles, there are two cases, in the circle ϕ^{0}=1 otherwise ϕ^{0}=0. In our experiments, We set α=1,μ=1, h=1,ε=10^{−6}, ε=10^{−5}. All experiments are performed using MATLAB R2016a on a Window 7 platform with an Intel(R) Core(TM) i54590 CPU at 3.30GHz and 4.00GB memory.
Numerical experiments for gray image segmentation
Firstly, we compare the VeseChan model without redundant regions by using classic optimization method shown in Section 3.2 and ADMM method in Section 3.3 to study advantages of the latter method. Besides, we compare Algorithm 1 with the original VeseChan model in [9] and TV regularization method in [17] for images with redundant regions.
Comparisons between the classic optimization method and ADMM
Figure 2 a shows a synthetic image with four phases, so we use two binary label functions to design characteristic functions. Figure 2b and c are segmentation results for classic optimization method and ADMM method, respectively, which contains four meaningful phases (Fig. 2d–g, h–k). We conclude that the above two methods obtain the same segmentation effectiveness.
In Fig. 3, we compare the classic optimization method and ADMM method for a brain MR image. The four phases of two methods are displayed in the second row and the third row, respectively. The ADMM method can give the better segmentation result from the comparison of Fig. 3f and g.
Table 2 demonstrates the iterations and computational time for Figs. 2a and 3a. ADMM method consumes less computational time and iterations than the classic optimization method. We can conclude that using ADMM method can improve computational efficiency for image segmentation.
Comparisons between Algorithm 1 and other methods
Figure 4 a shows a picture consists of a little bottle, a cup, and background, which occupy three regions. Using VeseChan scheme for division, we need two binary label functions to design characteristic functions. The original VeseChan method uses N=4 with one empty region, TV regularization method and our method use N=3 without empty regions. Three methods get the same segmentation results as shown in Fig. 4b–d, which contain three meaningful regions (Fig. 4e–g, i–k, and m–o) and one empty region (Fig. 4h, l, and p). The differences of their computation efficiency are listed lately together with other experiments.
Figure 5 presents a threephase segmentation problem also, but the image is a synthetic one with noises as shown in Fig. 5a. Figure 5 b–d show the segmentation results via the original VeseChan method, TV regularization method, and our method, respectively. Comparison between four characteristic functions of every method (Fig. 5e–p) shows that three methods obtain the same segmentation result.
In Fig. 6, we compare the original VeseChan method, TV regularization method, and our method for a threephase CT image slice of brain, and the segmentation results using these three methods are given in Fig. 6b–d, respectively. Different regions of three methods are shown in the last three rows. From experiment results, we conclude that three methods can obtain the similar segmentation effectiveness.
Figure 7 a shows a synthetic image consists of six regions, and we need three binary label functions to design characteristic functions. The original VeseChan method uses N=8 with two empty regions, TV regularization method, and our model use N=6 without empty regions. Three methods get the same segmentation results as shown in Fig. 7b–d, which can obtain the same meaningful phases. Therefore, we only show six different regions obtained via our method in Fig. 7e–j.
We record iterations and computational time of three methods for Figs. 4a, 5a, 6a, and 7a in Table 3. From Table 3, we reach that TV regularization method and our method both consume less computational time than original VeseChan method, and the reason of which is the first two methods reduce estimation of redundant parameters. Our method can obtain the highest computational efficiency. Next, we will study the reasons for this result.
TV regularization method uses an acceleration method, and our method also uses ADMM acceleration method. In order to further explore the efficiency of our method, we compare four methods, including TV regularization without acceleration method (TVRWAM), our method without ADMM, TV regularization method, and our method. The first two methods use upwind differential scheme as shown in Section 3.2 to compute label function ϕ. Figure 8 presents a sixphase segmentation problem, the image is a remote sensing one of coastline as shown in Fig. 8a. Figure 8 b–e show the segmentation results via the above four methods, respectively. Figure 8f–i are gray images according to four final segmentations. The iterations and computational time of four methods are showed in Table 4.
From Table 4, our method without ADMM consumes less iterations and computational time than TV regularization without acceleration method, because we add area constraints of redundant regions and simplify the expression of characteristic functions. Besides, our method consumes less computational time than TV regularization method, and the reason of which is we use ADMM method to compute energy optimization, improving the computational efficiency.
Numerical experiments for color image segmentation
Color image segmentation is an extension of gray image segmentation. We compare the original VeseChan method and TV regularization method with our method presented in Section 3.4 for color images.
Figure 9 presents a threephase segmentation problem, the color image is a part of flower as shown in Fig. 9a. Figure 9b–d show the segmentation results via the above three methods respectively. Figure 9e–h show different regions via the original VeseChan method. Because the TV regularization method and our method obtain the same result, we only show different regions about our method (Fig. 9i–l). Comparison between empty regions (Fig. 9h and l) shows that our method and TV regularization method give better segmentation result.
Figure 10a presents a sixphase synthetic color image, the segmentation results using the original VeseChan method, TV regularization method and our method are given in Fig. 10b–d, respectively, which obtain the same meaningful regions. Therefore we only show six meaningful regions via our method in Fig. 10e–j.
Figures 11a, 12a, and 13a are synthetic color images with nine, ten and eleven different regions respectively, so we need four binary label functions to design characteristic functions. We use three methods: the original VeseChan method, TV regularization method and the proposed method. The first one uses N=16 for three images, causing seven, six and five empty regions separately. Our method uses N=9,N=10 and N=11 without empty regions respectively. For Fig. 11a, results are shown in Fig. 11b–d. Three methods obtain the same segmentation results. As a representative, Fig. 11e–m show nine different regions obtained via our method. Figure 12b–d show segmentation results for a tenphase image. The first eight different regions for Fig. 12 are the same as Fig. 11, and two other regions as shown in Fig. 12e–f. The segmentation results for Fig. 13a are shown in Fig. 13b–d, and its 11 different regions include Figs. 11e–l, 12e, and 13e–f.
We record total iterations and computational time of the above three methods for Figs. 9a, 10a, 11a, 12a, and 13a in Table 5, from which we can conclude that compared to the original VeseChan method and TV regularization method, and our method can improve computational efficiency for color images. With increase of the empty regions of images, it is more obvious that the computational efficiency is improved. Besides, the iterations and computational time are positively correlated with the image sizes.
From the above experiments, we can conclude that the computational cost of our method includes two parts: parameter computations for ϕ,w,λ and parameter estimations for u, both of which are relative to the image sizes. Besides, the former one is also relative to the number of binary label function m. When there exist redundant regions in the images, (i.e., N<2^{m}), parameter estimations are relevant to N.
The number of regions N can be detected according to the number of target objects in the image. In the paper, the segmented images are simple and easy to determine the value of N. However, if the image is complex, the value of N is difficult to determine, which is a limit that needs to be solved in the future.
Conclusions
In order to improve the computation efficiency of the VeseChan model for multiphase image segmentation in a systematic form, we have designed a modified VeseChan model by introducing some simple area constraints. The forms of characteristic functions are unified, and the redundant parameter estimations are not needed, therefore, the cost of computation is reduced. Obviously, the computational efficiency is higher with the increase of the empty regions. Additionally, we formulate the ADMM for the proposed model to further improve efficiency. Some numerical examples for gray image and color image multiphase segmentation are presented to demonstrate that the proposed model has the same or better segmentation effects and higher efficiency. Our method can be applied into motion segmentation, surface segmentation, and 3D reconstruction in the future work.
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Abbreviations
 ADMM:

Alternating direction method of multipliers
 TVRWAM:

TV regularization without acceleration method
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Acknowledgements
The authors thank the editor and anonymous reviewers for their helpful comments and valuable suggestions.
Funding
This work was supported by the National Natural Science Foundation of China (Grant No.61772294).
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Authors’ contributions
All authors took part in the discussion of the work described in this paper. The author JW carried out the experiments of the paper and wrote the paper. The author ZX participated in experiments. The author ZP designed the algorithms of this work and revised the paper. The author WW and GW helped conduct the experiments. All authors read and approved the final manuscript.
Authors’ information
Jie Wang was born in 1994. She is a postgraduate student of College of Computer Science and Technology, Qingdao University, Qingdao, China. Her current research interests include variational image restoration and segmentation. Email: wangjqdy@163.com.
Zisen Xu was born in 1965. He received Master degree in College of Information Engineering, Qingdao University, Qingdao, China in 2007. He is a Senior Engineer in The Affiliated Hospital of Qingdao University, Qingdao, China. His current research interests include medical engineering and postprocessing of medical image.
Zhenkuan Pan was born in 1966. He received his Ph.D. degree in Engineering Mechanics from Shanghai Jiao Tong University, Shanghai, China in 1992. He is a Professor in College of Computer Science and Technology, Qingdao University, Qingdao, China. His current research interests include dynamics and optimization of multibody systems, medical simulation and variational image processing.
Weibo Wei was born in 1981. He received his Ph.D. degree from Nanjing University of Science and Technology, Nanjing, China in 2006. He is an Associate Professor in College of Computer Science and Technology, Qingdao University, Qingdao, China. His current research interests include image processing, biometric identification, automatic target recognition and tracking. Email: njustwwb@163.com.
Guodong Wang was born in 1980. He received his Ph.D. degree in Pattern Recognition and Intelligent System in Huazhong University of Science and Technology, Wuhan, China in 2008. Now he is an Associate Professor in College of Computer Science and Technology, Qingdao University, Qingdao, China. His research interests include variational image science, face recognition, intelligent video surveillance, 3D reconstruction and medical image processing and analysis. Email: doctorwgd@gmail.com.
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Correspondence to Zhenkuan Pan.
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Wang, J., Xu, Z., Pan, Z. et al. The VeseChan model without redundant parameter estimation for multiphase image segmentation. J Image Video Proc. 2020, 3 (2020). https://doi.org/10.1186/s1364001904886
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Keywords
 Multiphase image segmentation
 VeseChan model
 Parameter estimation
 Binary label function
 Alternating direction method of multipliers