Adaptively Active Contours Based on Variable Exponent L p (cid:2) |∇ I | (cid:3) Norm for Image Segmentation

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Introduction
Image segmentation has always been an essential problem in image analysis and computer vision.Segmentation is the process of partitioning an image into a set of distinct regions, each of which has a consistent trait intensity, etc. that is different from other regions in the image.Its goal is to change the representation of an image into something that is more meaningful and easier to analyze.So far, a large number of efficient algorithms and methodologies including active contour models have been proposed for image segmentation.In this study, we focus on implicit active contour models 1 , that is, active contour models in a level set formulation.
Implicit active contour models, which are implemented via curve evolution theory 2 and level set method 3 , have been proved to be an efficient framework for image segmentation.The basic idea is that an active contour is implicitly represented as the zero level set of a level set function defined in image domain, and then the level set function is deformed according to an evolution partial differential equation PDE .A remarkable merit of implicit models is that the level set function evolution allows for cusps, corners, and automatic topological change of active contour, which is generally impossible in traditional parametric active contours 4 when direct implementations are performed.Early implicit models 5, 6 first yield an evolution PDE of a parametrized contour, and then convert it to the evolution PDE for a level set function whose zero level set represents the parametrized contour.Alternatively, the evolution PDE for level set function can be directly derived from the minimization problem for an energy functional that takes a level set function rather than a curve as its argument by steepest descent 7-17 .
Existing implicit active contour models can be roughly categorized into two basic classes: edge-based models 5-10 , and region-based models 11-20 .Edge-based models utilize typically image gradient as an external force to attract the contour towards the boundaries of the desired objects.They have been successfully used to segment images with strong object boundaries defined by gradient, but they have difficulty in handling images with weak boundaries.Region-based models usually utilize region information rather than the gradients on the boundaries to perform segmentation; therefore, they generally have better performance than the former in the presence of weak object boundaries.In addition, region-based models are significantly less sensitive to the location of initial contours.One of the most popular region-based models is the two-phase piecewise constant segmentation model usually called C-V model proposed by Chan and Vese 11 .
The C-V model relies fully on the global information of image instead of its local gradient.It can realize the global optimization of image segmentation through the minimization of Mumford-Shah energy functional 21 , and thus works well on images with roughly constant foreground and background.But for more general images, such as images with low contrast and blurred boundaries, the C-V model shares the lower speed, and even fails to segment.
To address general images, some more sophisticated models than the C-V model have been proposed.For example, Li et al. 12 proposed recently the region-scalable fitting usually called RSF active contour model.However, the RSF model is presented to handle intensity inhomogeneity, and it is sensitive to the location of initial contours and needs timeconsuming convolution operations.To increase the segmentation speed, Bresson et al. 18 proposed a fast global minimization based on a dual formulation of the active contour/snake model; Chan et al. 19 proposed the algorithm for finding global minimizers of image segmentation; Goldstein et al. 20 proposed the Split Bregman Method for globally convex segmentation model.These works 18-20 represent the state-of-the-art of contour evolution methods which are efficient and fast in a global minimization framework, and they have been widely used in practical application.However, for some images with low contrast and blurred boundaries, these models also seem powerless.
In order to handle those images efficiently and quickly, our idea is to introduce a much simpler model which has many desirable properties as the model introduced in 11 and is efficient in numerical computation.In this study, we propose an improved active contours model based on L p |∇I| p |∇I| > 2 norm instead of the L 2 norm to define the external energy, and incorporate an extra internal energy into the overall energy.The proposed model has the following two advantages.Firstly, the proposed model shares the advantages of the C-V model, such as the robustness to noise and the location of initial contours, and it leads itself particularly well and very fast to the segmentation of those images that the C-V model can handle.Secondly, the proposed model based on L p |∇I| norm combines the gradient information with the region information, so it has the hope of segmenting those images with low contrast and blurred boundaries.
The remainder of this paper is organized as follows.Section 2 briefly reviews the C-V model.Section 3 introduces the proposed model.Section 4 presents numerical algorithm and experimental results for some synthetic and real images, followed by the discussion about the parameter used in the segmentation algorithm in Section 5.This paper is summarized in Section 6.

Minimal Partition Problem
Given an observed image I : Ω ⊂ R 2 → R, the general Mumford-Shah functional 21 for image segmentation is given by 2.1 In the above, ν, μ > 0 are fixed scale parameters, u is an unknown piecewise smooth approximation of I, C is an unknown curve, |C| is the length of C, and Ω \ C are the domain excluding the curve C. Mumford and Shah 21 proposed that the segmentation of an image can be obtained by minimizing the functional 2.1 .
In order to obtain an optimal piecewise-constant approximation of the image I, Mumford and Shah 21 proposed the following minimal partition problem: given an image I : Ω → R, find a set of disjoint regions Ω i , such that u c i in each Ω i is a minimizer of the following functional, a special case of the functional 2.1 : where In practice, it is not a trivial task to minimize the functional 2.2 due to the unknown contour C of lower dimension and the nonconvexity of the functional.The existence and regularity for the piecewise constant model have been proved in the book 22 .

The C-V Model
In 11 , Chan and Vese solved a particular case of the minimal partition problem 2.2 for image segmentation using the curve evolution and the level set method, where the binary case of two regions was considered.Based on 2.2 and using the level set method, Chan and Vese proposed to minimize the following functional for a two-phase segmentation:

2.3
Here, H φ and δ φ are the one-dimensional Heaviside and Dirac function, respectively, and φ is the unknown level set function with the following properties: where in C and out C stand for the "inside" and "outside" regions divided by the curve C, respectively.Minimizing the functional 2.3 with respect to φ is done by introducing an artificial time variable, and moving φ in the steepest descent direction to steady state: with initial conditions φ 0, x, y φ 0 x, y , the signed distance function to the initial curve.In the above, c 1 and c 2 are, respectively, updated at each iteration by

2.6
The recovered image is a piecewise constant approximation to the image I.
Minimizing the functional 2.3 leads to a binary segmentation of the given image The C-V model has been generalized to the cases of more than two regions 13, 17, 23 .
In practice, the Heaviside and Dirac functions H φ and δ φ have to be approximated by smooth functions, which are typically defined by

2.7
The function δ ε φ is indeed an approximation to the Dirac function δ φ .To keep the notations simple, we still write H φ and δ φ instead of H ε φ and δ ε φ in what follows.

Description of Model
We use L p |∇I| -norm instead of L 2 -norm to define the external energy term of the C-V model, that is, the new external energy term is defined as 3.1 The exponent p s : 0, ∞ → R is a monotonically increasing function with 2 < p s ≤ K, where K is a constant.A simple example is where the constant K > 17/8 is tuned for a particular application.Clearly, 2 < p s ≤ K.
The external energy E ext c 1 , c 2 , φ combines the gradient information with the region information; its role is to force the level set function to move towards the boundaries of the objects.
In order to control the smoothness of the zero level set { x, y | φ x, y 0} and further avoid the occurrence of small isolated regions in the final segmentation, the regularization must be imposed on the zero level set:

3.3
In order to eliminate the need of the costly reinitialization procedure, we add an extra internal energy 7, 8 , to the energy E ext c 1 , c 2 , φ in 3.1 .This energy actually serves as a metric to characterize how close the level set function is to a signed distance function.Therefore, the total energy functional is given by

Mathematical Problems in Engineering
Keeping φ fixed and minimizing the energy E c 1 , c 2 , φ with respect to the constants c 1 and c 2 , we have

3.6
Since c 1 and c 2 do not have explicit expressions, we apply the fixed point algorithm to solve 3.6 , namely,

3.7
Keeping c 1 and c 2 fixed, and minimizing the functional 3.5 with respect to φ by the gradient descent, yields the associated Euler-Lagrange equation for φ as follows: with the initial and Neumann boundary conditions: where n denotes the exterior normal to the boundary ∂Ω and ∂φ/∂ n denotes the normal derivative of φ at the boundary.

Why Use
, where M is a positive constant.In fact, by simple proof, we obtain M, that is, M |Ω| K−2/2K , and it is independent of image intensity.That is to say, the The external energy term in 3.5 can be rewritten as thus minimizing this external energy term with respect to φ leads to the optimal binary segmentation of the given image I: On the other hand, the variable exponent p |∇I| reflects the local property of image, and so the proposed model adapts the exponent to fit the image information automatically.Therefore, the proposed model with p |∇I| > 2 can force the level set function to move towards the boundaries of the objects adaptively, and it has the hope of segmenting those images with low contrast and blurred boundaries.
Next, we use the experiment to show that the use of p |∇I| > 2 in L p |∇I| -norm can achieve satisfactory segmentation results for those images with noise and low contrast.However, the use of p |∇I| 2 fails to handle those images.At the same time, these results also show that the proposed model with p |∇I| > 2 can segment objects with blurred boundaries.

Numerical Algorithm
In this subsection, we briefly present the numerical scheme to solve the proposed model.We recall first the notations in the finite difference scheme.Let τ, Δt, and x i , y j iτ, jτ be the space step, the time step, and the grid points, respectively.
Let φ n i,j φ n i,j x i , y j , nΔt be an approximation of the level set function φ x, y, t with n ≥ 0, φ 0 φ 0 , where φ 0 is the initial level set function.The central differences of spatial partial derivatives are expressed in the following notations: Then, the numerical approximation to 3.8 can be simply written as where In summary, the main procedures of the improved algorithm are as follows.
1 Initialize the level set function φ 0 x, y φ 0 x, y , and compute the initial values c 0 4 If the zero level set of φ n 1 x, y is exactly on the object boundary, then stop; otherwise, let n n 1, then return to step 2 .

Experimental Results
We carry out several experiments on some synthetic and real images to demonstrate the performance of the proposed model.To make fair comparison, for the C-V model we added the internal energy 3.4 into the functional 2.3 to avoid the reinitialization step.For all experiments, the level set function for C-V model is initialized to the signed distance function to a circle in image domain; and for both RSF model and the proposed model, the initial level set function φ is simply chosen as a binary step function as in 7, 8 , which takes a positive constant value ρ inside a region R 0 ⊂ Ω and a negative constant value −ρ outside R 0 .We choose ρ 1 for all experiments in this paper.
Unless otherwise specified, we use the following default setting of the parameters for all the experiments: τ 1, Δt 0.1, μ 0.001 × 255 × 255, ν 0.04, and ε 1 for δ ε z .For the proposed model, in all experiments, the initial contours are chosen as circles with radius of fifteen excluding Figures 3-5 , located at the centre of image domain.In general, we choose the parameter K ranging from 2.2 to 7 in the proposed model.The full decision of value K depends on varied image scenes.We will give the exact value of K each time.
The first four experiments show the segmentation results of both the proposed model and C-V model for some synthetic and real images.
Figure 2 shows the segmentation results of both models for five synthetic images.As can be seen, the proposed model obtains the satisfactory segmentation results for these images left to right: K 2.7, 2.7, 2.8, 3, and 2.9 , which are almost the same as C-V model visually.Table 1 gives the CPU times and iteration numbers of both models.We can observe that both CPU times and iteration numbers of the proposed model are less than C-V model.Figures 3 and 4 demonstrate that the segmentation results of both models for a medical image 112 × 66 , and an aerial image 128 × 128 with five initial circles of different positions.For the proposed model, we choose only a parameter K for all different positions K 3.3 in Figure 3 and K 3.2 in Figure 4 , all those different initial curves give the same segmentation results in a single iteration.However, by inspecting these images carefully, we can see that only one initial curve gives accurate segmentation results by the C-V model see the last image in middle row of Figure 3 and the third image in middle row of Figure 4 ; and this shows that the C-V model is, to some extent, sensitive to the position of initial curve.
Figure 5 shows the segmentation results of C-V model, RSF model, and the proposed model for a noisy infrared image with four initial circles of different positions and sizes.For the proposed model, we simply choose a fixed K 3.6 for all those different initial curves and give the same segmentation results in a single iteration, as shown in the lower row of Figure 5.However, we observe from the second and third rows that only one initial curve gives correct segmentation results by both C-V model and RSF model.This shows that both C-V model and RSF model are, to some extent, sensitive to the initial circles of different positions and sizes.
The and third columns of Figure 6.In fact, no matter how many iterations, some part of the background/foreground is incorrectly identified as the foreground/background.Due to the introduction of L p |∇I| p |∇I| > 2 norm in the proposed model, by alternating K 6.6, 6.5, 6.4, and 5, the proposed model successfully detects the objects for all these images; see the last column of Figure 6.
Figure 7 shows the detective results of three models for three aerial images with low contrast and blurred boundaries 128 × 128 .For the proposed model, we simply choose a fixed K 4.4 for all these images.We clearly observe from  In Figure 8, we consider three range images of saddle and cone with step or roof edges top to bottom: 287 × 287, 290 × 290, and 285 × 282 , these of which are taken from that in 16 .The top row of Figure 8 is the segmentation of a saddle range image.This range image has complicated image intensity distribution, in which the end boundaries of the saddle are step edges, and the side boundaries are roof edges.The last two rows of Figure 8 are the segmentation results of two cones range images.The boundaries between the cones and the bottom planes are roof edges.As can be seen from the last two columns of Figure 8 6 .

Dissussion
In this section, we simply discuss about the parameter K.For the proposed model, K for the function p |∇I| is very important to the segmentation results.Theoretically, if K is larger, then the p |∇I| norm is stronger.That is to say, when larger K is served as external energy term for the proposed model, there have been better approximation effects for those images with low contrast and blurred boundaries.Here, we discuss the setting of K when it is chosen by hand as some fixed value.Experimentally, our observations are as follows: K is typically chosen as a constant between 2.2 and 7.For synthetic images and range images, we choose K in the range between 2.2 and 3; for aerial images and some medical images excluding Figure 10 e , we choose K in the range between 3.1 and 4.9; for infrared images and those medical images with low contrast and blurred boundaries, we choose K in the range between 5 and 7.
It is worth emphasizing that the proposed model is robust to K.For this purpose, we apply the proposed model to two synthetic images 102 × 103, 128 × 128 and two real images 128×128, 128×128 , top to bottom as shown in the first column of Figure 12.For each image, we choose default initial contour, and all those different K give the almost same segmentation results in a single iteration, as shown in Figures 12 b , 12 c , 12 d , 12 e , 12 f , 12 h , 12 i , 12 j , 12 k , 12 l , 12 n , 12 o , 12 p , 12 q , 12 r , 12 t , 12 u , 12 v , 12 w and 12 x .

Conclusion
In this paper, following the well-known C-V model we propose a region-based implicit active contour model based on L p |∇I| p |∇I| > 2 norm for images segmentation.First, the proposed model shares the advantages of the C-V model, such as the robustness to noise and the location of initial contours, and allows us to segment those images that the C-V model can handle but the segmentation speed of the proposed model is much faster than C-V model .Second, due to the function p |∇I| fits the gradient information, the proposed model can effectively and quickly segment those images with low contrast and blurred boundaries.It should be pointed out that the segmentation speed of the proposed model is faster than Bresson et al.'s model 18 , but it is lower than Goldstein et al.'s model 20 .In the future, we will introduce Split Bregman method to the proposed model.
demonstrates the segmentation results using p |∇I| 2 and p |∇I| > 2, respectively, for three infrared images left to right with high noise 105 × 144, 159 × 163 and low contrast 159 × 165 .To make a fair comparison, we choose the best parameter μ given in the caption and contour initialization for p |∇I| 2. And we choose the same value of μ for both cases.The segmentation results of both cases are shown in the top row p |∇I| 2 and bottom row p |∇I| > 2 , respectively.This experiment shows that the use of p |∇I| > 2 rather than p |∇I| 2 can really segment the infrared images with noise and low contrast accurately.It should be pointed out that all other different initial curves give the same segmentation results only in a single iteration in the case of p |∇I| > 2 for these images.

Figure 2 :
Figure 2: Segmentation results of both models for five synthetic images.Upper row: initial contours.Middle row: C-V model.Lower row: proposed model.

Figure 3 :Figure 4 :Figure 5 :
Figures3 and 4demonstrate that the segmentation results of both models for a medical image 112 × 66 , and an aerial image 128 × 128 with five initial circles of different positions.For the proposed model, we choose only a parameter K for all different positions K 3.3 in Figure3and K 3.2 in Figure4, all those different initial curves give the same segmentation results in a single iteration.However, by inspecting these images carefully, we can see that only one initial curve gives accurate segmentation results by the C-V model see the last image in middle row of Figure3and the third image in middle row of Figure4; and this shows that the C-V model is, to some extent, sensitive to the position of initial curve.Figure5shows the segmentation results of C-V model, RSF model, and the proposed model for a noisy infrared image with four initial circles of different positions and sizes.For the proposed model, we simply choose a fixed K 3.6 for all those different initial curves and give the same segmentation results in a single iteration, as shown in the lower row of Figure5.However, we observe from the second and third rows that only one initial curve gives correct segmentation results by both C-V model and RSF model.This shows that both C-V model and RSF model are, to some extent, sensitive to the initial circles of different positions and sizes.The next five experiments show the segmentation results of Bresson et al.'s model 18 , Goldstein et al.'s model 20 , and the proposed model for some real images.To make a fair

Figure 7 Figure 6 :
Figure 6: Detective results of three models for four infrared images with low contrast and blurred boundaries.The first column: original images.The second column: Bresson et al.'s model.The third column: Goldstein et al.'s model.The last column: proposed model, at the 1st iteration.

Figure 7 :
Figure 7: Detective results of three models for aerial images with low contrast and blurred boundaries.The first column: original images.The second column: Bresson et al.'s model.The third column: Goldstein et al.'s model.The last column: proposed model, at the 1st iteration.

Figure 8 :
Figure 8: Segmentation results of three models for three range images.The first column: original images.The second column: Bresson et al.'s model.The third column: Goldstein et al.'s model.The last column: proposed model, at the 1st iteration.

Figure 9 :Figure 10 :Figure 11 :
Figure 9: Applications of three models to four real images.Top row: original images.The second row: Bresson et al.'s model.The third row: Goldstein et al.'s model.Bottom row: proposed model, at the 1st iteration.

Table 1 :
CPU times in seconds and iteration numbers in Figure2

Table 2 :
CPU times in seconds in Figure7.
that the objects are accurately extracted by the Goldstein et al.'s model and the proposed model top to bottom: K 2.9, 2.4, and 2.2 .However, by carefully inspecting these images in the second column, Bresson et al.'s model cannot extract the real object boundaries, which leads

Table 3 :
CPU times in seconds in Figure8.

Table 4 :
CPU times in seconds in Figure9.

Table 5 :
CPU times in seconds in Figure10.For the heart image, the desired result is to capture the bright ventricle.For all images, segmentation of objects of interests is difficult because of low contrast and blurred boundaries.To make a fair comparison, we choose the best parameters for the two models Bresson et al.'s model and Goldstein et al.'s model and contour initialization for C-V model, and K 5, 5.4, 5.6, and 5.8 for the proposed model.We observe from the second row that the C-V model cannot segment them correctly, in which appears some noise points around objects.The last three rows show that Bresson et al.'s model successfully extracts the objects only for the first two images and Goldstein et al.'s model only segments the first image

Table 6 :
CPU times in seconds in Figure11.