Active Contours Model for Image Segmentation: A Review

As we all known, the start-of-the-art ACM methods can segment the objects actually. However, most of them are very time-consuming and inefficient. In this paper, we review all of these algorithms, discuss methods unifying energy minimization and curve evolution approaches and show the correspondence of level set methods to classical methods and recent advanced methods. Meanwhile, we proposed a novel fast active contours model for speed up the computational complexity.


Introduction
Image segmentation [1] is a fundamental process in image processing that has found application in many fields, ranging from neural image analysis to satellite imagery [2][3][4][5].The design of a segmentation method typically involves both modeling the data and spatial regularization of the solution.The active contour models (ACM) [6] are based on the theory of surface evolution and geometric flows, have been extensively studied and successfully utilized in image segmentation in recent years.
Generally speaking, the so called ACM can be categorized into two classes: edge-based models [7,8] and region-based models [9,10].Edge-based models have an edge stopping function and a balloon force function to control the motion of the contour.However, during some researches, we can found that appropriate choice of balloon force is difficult.If the balloon force is not appropriate, there will be getting a bad segment result.On the other hand, region-based active contour models have the following advantages: a) do not use the image gradient and therefore have a better performance for weak boundaries images.b) Insensitive to the location of initial contours.One of the most popular region-based active contour models is Chan-Vese model [11].This Chan-Vese model based on Mumford-Shah [12] segmentation techniques and successfully for images with two regions, each having a distinct mean of pixel intensity.In 2002, Vese and Chan extended their work by using a multiphase level set framework scheme, for piecewise constant (PC) [13] and piecewise smooth (PS) [14] optimal approximations.It has the advantages that the phases cannot overlap and minimizes as much as possible the computational cost, reducing the number of level set functions.These models effectively solve the fuzzy boundaries image or digital objects segmentation and detect interior contours in images.However, their main drawback is the additional computational complexity.Li et al. [15] proposed the local binary fitting (LBF) model by embedding the local information.LBF is able to segment images with intensity inhomogeneities and is much more efficient and accurate than the PS model.However, the computational complexity is still high.Therefore, improving the computational complexity of segmentation is a main motivation of this paper.In [24], we firstly review all of the above methods and then propose a novel ACM model, i.e.FACM, that improves the traditional level set methods by avoiding the calculation of signed distance function (SDF) and speed up the computational complexity of re-initialization.In the FACM, in order to speed up the computational complexity, we also provide a fast semi-implicit additive operator splitting (AOS) algorithm [14].
The remainder of this paper is organized as follows.In Section 2, we briefly review the stand-of-the-art models and point out the main drawbacks of these models.In Section 3, presents numerical results with synthetic and real-world images.Finally, Section 4 concludes the paper.

Curve evolution-based methods
Curve evolution based methods are often utilized in image segmentation.In these methods, partial differential equations are frequently used to evolve the curves.The equations are formed following a certain physical principle.And use the curve evolution problem as an energy minimization problem, to seek the solution of the corresponding Euler-Lagrange equation.In either case, partial differential equations are solved in an iterative manner to evolve the curve.Generally, the curve evolution based methods are divided into two categories: region based methods and edge based methods.In this section, we just review most epidemic models.

Mumford and shah model
The Mumford and shah (MS) model [12] the image segmentation problem as follows: assume the segmented image as piecewise constant, find an optimal piecewise smooth approximation function u of image u 0 , which varies smoothly within each subregion Ω i of image domain Ω⊂R 2 , and rapidly or discontinuously goes across the boundaries of Ω i .The MS energy functional is where |C| is the length of the contour C, μ,ν≥0 are fixed parameters.
For many applications, it is enough to assume that u is a piecewise constant function.In these cases, the second term disappears from the above minimization function.Among the ways of representing the unknown interfaces, the level set method is the most elegant due to its ability to deal with unknown topology.Its main drawback is expensive computation.

Snakes model
A snake is a continuous curve that, form an initial state, tries to position itself dynamically on image features.It is deformed as a result of the influence of local forces derived from edge points, while this deformation remains smooth due to the effect of internal forces [6].The energy of the snake is typically given by where C(s) is the evolving curve parameterized by s, and s is the curve arc-length.The first term of right hand side of this equation penalizes stretching and bending of the curve.The second term attracts the evolving curve towards image features such as bright or dark line, edges et al.
The snake model has two main drawbacks.One is that it is highly sensitive to the prarmeterisation of the curve.The other is the explicit representation of the curves prohibits any topological changes during evolution.

Level set methods
Level set methods (LSM) [17] evolve the curve C along the normal direction.Different to the others, the level set methods embed the curve into a higher dimensional where, Ω is image domain.Noting that the curve C is always given byφ(C(t),t)=0, and then by taking the total derivative of the left hand side of this equation we can get where V is a speed function.The speed function V is used to define the energy minimization problem.The level set evolution problem can also be solved as energy minimization problem [16].Let us define energy functional as where ϕ s is the derivative of ϕ with respect to s.Then, use the Euler-Lagrange, we get Then we can formulate the level set function by equation (4), as where e is used to define the energy minimization problem.One drawback of the level set method is its high computational expense because it expands the domain of computation from the interface to a grid in one higher dimension.

Geodesic active contours
Geodesic active contours (GAC) [7] are an extension of the classical snakes described above.The GAC energy functional can be given by, where u 0 is the image, g is a strictly decreasing function such that g(w)→0 as w→∞.The curve evolution equation is obtained via Euler-Lagrange formulation as where the curvature κ is now computed on the level sets of ϕ, referred to as the mean curvature.The GAC model only considers the local information.Moreover, it needs to be initialized in a way that is either completely outside or inside the actual boundary.

Active contours without edges
Chan and Vese proposed a method (CV) [11] based on the Mumford-Shah model and the level set method.Set an image u 0 (x,y).Let us give the curve an open subset and two unknown constants c 1 and , the energy function is (see also [15]).

Geodesic active regions
The geodesic active regions (GAR) model [19] utilizes boundary and region based information to find a set of minimal length curves.The GAR energy function is where α is a positive constant that balance the contributions of the two terms, 0≦α≦1.g is a Gaussian function with zero mean and variance σ 2 .The first term on the right hand side is region based and represents the negative log-likelihood of a pixel belonging to a particular region.u is the image, and boundary probabilities P C (・) are used.P X (・) is the conditional intensity density function with respect to the hypothesis R 1 and R 2 .The level set update equation can be given as follows, ( ) This model overcomes many drawbacks of the other models.However, the Gaussian function used to describe intensities may be inadequate in many cases.

Piecewise smooth model
In order to overcome the general problem of MS model in image processing, Chan and Vese proposed piecewise smooth (PS) model [14] which aims at expressing the intensities inside and outside the contour as piecewise smooth functions instead of constants.Assume that u + , u -, are curve C functions on ϕ≧0 and on ϕ≦0 respectively.The unknowns' u and ϕ can be expressed by introducing two functions u + and u -as, Then, the 2-Dimensional PS model energy function was defined, where ν, μ≧0 are fixed parameters.u 0 is the original image.Minimizing the above equation, we get the following Euler-Lagrange equations (20) where n  ∂ ∂ / denotes the partial derivative in the normal direction ń at the corresponding boundary.
We can see that the computational cost is very expensive.Furthermore, the u + and u -must be extended to the whole image domain, which is difficult to implement and also increases the computational cost.[15,20] by embedding the local image information.The LBF model introduces a kernel function to define the LBF energy function as

Li et al. proposed the local binary fitting (LBF) model
where λ 1 , λ 2 >0 are fixed parameters.K σ is a kernel function with standard deviation σ. f 1 (x) and f 2 (x) are two numbers that fit image intensities near the point x.Rewritten the equation ( 21) with level set representation.And keeping f 1 and f 2 fixed, and minimizing the energy function, we derive the gradient descent flow as (22) where, , and e 1 and e 2 are the functions as below, . The parameters are difficult to choose in this model.The computational complexity is still very high in LBF model.

Fast Active Contour Model
In [25], we also use a semi-implicit additive operator splitting (AOS) [16] method rather than explicit schemes to implement the discrete level set processing.The basic idea of the AOS scheme is to split the m-dimensional spatial operator into a set of one-dimensional space discretizations that can be efficiently solved with Gaussian elimination algorithm named Thomas Algorithm.
To simplified form of the gradient descent flow function, can be written by the semi-implicit AOS scheme [16] as Note that, by evaluating only image positions with , the denominator in this scheme cannot vanish.τ is the time step.In matrix-vector notation this becomes where A l describes the interaction in l direction.x,y respectively the x-direction and y-direction (2D).With the definition of A l (ϕ n ) = a ij (ϕ n ), and a ij (ϕ n ) can be expressed as According to above equations, the formulation ( 23) can be re-expressed as So, the gradient descent flow function finally can be written as [25],

Experimental Results
The proposed improved fast active contour model has been applied to a variety of synthetic and real images.The results calculated on a standard personal computer with Windows XP, Intel Core 2.0 GHz, and 1G RAM.During the experiments, we compare the results with iterations, CPU time and distance of different models.The parameters of the experiments are depended on the images.
Form the following Figure 1, the experiment validates that our method can achieve sub-pixel segmentation accuracy.The CV model can segment the inner details.However, the CV model need to initialize, it costs a lot of CPU time.The GAC model cannot segment the inner details.Moreover, the GAC model is difficult to control the number of iterations and sometimes may produce over-segmentation results.Compared to LBF model, the computational time is also longer than our proposed model.LIF model [23] achieves satisfying segmentation result.However, the number of iterations by the LIF model is higher than the proposed model.In real-time system, the computing time and calculation accuracy are very important.Compared with abundant methods, the proposed FACM model performs the best result.2, we found that the GAC model cannot segment the interior boundaries.We also can found that, the boundary of our model is smoother than the other models.The FACM can be better approaching the edges of the graphics in Figure 2. Furthermore, our model can well handle images with weak edges.Meanwhile, our model has fewest CPU time during processing.

Conclusions
We presented a survey of the class of region-based active contour model segmentation methods and detailed how they can be derived from a common statistical framework.The common goal of these approaches is to identify boundaries such that the texture, dynamic texture or motion in each of the separated regions is optimally approximated by simple statistical models.
In numerous experimental results, we demonstrate that this class of active contour models allows to partition images into domains of texture, dynamic texture or motion.In particular, we show that in contrast to the traditional edge-based segmentation schemes, these PDE-or ODE-based approaches are quite robust to noise and to varying initialization, making them well-suited for local optimization methods such as the level set method.In this survey, we reviewed some recent advances regarding the introduction of statistical shape knowledge into active contour based segmentation schemes.

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If and only if the curve C on the boundary of homogeneity area, the above function obtain minimum value.Set ϕ as the level set function.Then, the energy respect to the constants c 1 and c 2 , we can get the following formulas with }

Fig. 1 .
Segmenting of two cells medical image using different methods: (a) initial image; (b) segmentation result by the C-V model; (c) segmentation result by the GAC model; (d) segmentation result by the LBF model; (e) segmentation result by the LIF model; (f) FACM.

Fig. 2 .
Segmenting of synthetic square image using different methods: (a) initial image; (b) segmentation result by the CV model; (c) segmentation result by the GAC model; (d) segmentation result by the LBF model; (e) segmentation result by the LIF model; (f) FACM.

Table 1 .
Performance comparison of different models in computing two cells medical image.

Table 2 .
Performance comparison of different models in computing square image.