Abstract
We propose an -based adaptively active contours model for image segmentation which is derived from the well-known Chan-Vese (C-V) model. Unlike the C-V model, the proposed model uses the norm instead of the norm to define the external energy and incorporates an extra internal energy into the overall energy. Due to the variable exponent which could fit the image gradient information adaptively, the proposed -based model has the hope of segmenting those images with low contrast and blurred boundaries. Experimental results show that the proposed model with really can effectively and quickly segment those images with low contrast and blurred boundaries.
1. 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 z\ero 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 time-consuming 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 norm instead of the 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 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.
2. The C-V Model
2.1. Minimal Partition Problem
Given an observed image , the general Mumford-Shah functional [21] for image segmentation is given by
In the above, are fixed scale parameters, is an unknown piecewise smooth approximation of , is an unknown curve, is the length of , and are the domain excluding the curve . 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 , Mumford and Shah [21] proposed the following minimal partition problem: given an image , find a set of disjoint regions , such that in each 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 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].
2.2. 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: Here, and are the one-dimensional Heaviside and Dirac function, respectively, and is the unknown level set function with the following properties: where and stand for the “inside” and “outside” regions divided by the curve , respectively.
Minimizing the functional (2.3) with respect to is done by introducing an artificial time variable, and moving in the steepest descent direction to steady state: with initial conditions , the signed distance function to the initial curve. In the above, and are, respectively, updated at each iteration by The recovered image is a piecewise constant approximation to the image .
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 and have to be approximated by smooth functions, which are typically defined by
The function is indeed an approximation to the Dirac function . To keep the notations simple, we still write and instead of and in what follows.
3. The Proposed Model
3.1. Description of Model
We use -norm instead of -norm to define the external energy term of the C-V model, that is, the new external energy term is defined as The exponent : is a monotonically increasing function with , where is a constant. A simple example is where the constant is tuned for a particular application. Clearly, .
The external energy 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 and further avoid the occurrence of small isolated regions in the final segmentation, the regularization must be imposed on the zero level set:
In order to eliminate the need of the costly reinitialization procedure, we add an extra internal energy [7, 8], to the energy 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
Keeping fixed and minimizing the energy with respect to the constants and , we have Since and do not have explicit expressions, we apply the fixed point algorithm to solve (3.6), namely,
Keeping and fixed, and minimizing the functional (3.5) with respect to by the gradient descent, yields the associated Euler-Lagrange equation for as follows: with the initial and Neumann boundary conditions: where denotes the exterior normal to the boundary and denotes the normal derivative of at the boundary.
3.2. Why Use -Norm Instead of -Norm?
Since the image domain is bounded and , it is well known that with , where is a positive constant. In fact, by simple proof, we obtain , that is, , and it is independent of image intensity. That is to say, the norm is stronger than the norm .
The external energy term in (3.5) can be rewritten as thus minimizing this external energy term with respect to leads to the optimal binary segmentation of the given image : in the sense of the norm, where is a minimizer of the external energy. Because the is stronger than the norm, we have This implies that the binary image better approximates the original image than the binary image . When the original image approximates the , the speed of external energy term that approximates zero in the sense of norm is much faster than the speed in the sense of norm. Therefore, the segmentation speed of the proposed model is much faster than the C-V model.
On the other hand, the variable exponent 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 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 in -norm can achieve satisfactory segmentation results for those images with noise and low contrast. However, the use of fails to handle those images. At the same time, these results also show that the proposed model with can segment objects with blurred boundaries.
Figure 1 demonstrates the segmentation results using and , respectively, for three infrared images (left to right) with high noise (, ) and low contrast (). To make a fair comparison, we choose the best parameter (given in the caption) and contour initialization for . And we choose the same value of for both cases. The segmentation results of both cases are shown in the top row () and bottom row (), respectively. This experiment shows that the use of rather than 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 for these images.
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4. Implementation and Experimental Results
4.1. 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 , , and be the space step, the time step, and the grid points, respectively.
Let be an approximation of the level set function with , , where 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 , and compute the initial values and by (2.6) and variable exponent by (3.2), and then set .(2)Compute the values and using (3)Evolve the level set function by solving (4.2) to obtain .(4)If the zero level set of is exactly on the object boundary, then stop; otherwise, let , then return to step (2).
4.2. 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 is simply chosen as a binary step function as in [7, 8], which takes a positive constant value inside a region and a negative constant value outside . We choose for all experiments in this paper.
Unless otherwise specified, we use the following default setting of the parameters for all the experiments: , and (for ). 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 ranging from 2.2 to 7 in the proposed model. The full decision of value depends on varied image scenes. We will give the exact value of 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: ), 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.
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Figures 3 and 4 demonstrate that the segmentation results of both models for a medical image (), and an aerial image () with five initial circles of different positions. For the proposed model, we choose only a parameter for all different positions ( in Figure 3 and 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 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 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 comparison, we choose the best parameters for both Bresson et al.'s model and Goldstein et al.'s model.
In Figure 6, we apply the three models to four infrared images with low contrast and blurred boundaries. The test images are of size , , , and , which are shown in the first column of Figure 6. For such images, both Bresson et al.'s model and Goldstein et al.'s model show erroneous results, as shown in the second 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 norm in the proposed model, by alternating , the proposed model successfully detects the objects for all these images; see the last column of Figure 6.
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Figure 7 shows the detective results of three models for three aerial images with low contrast and blurred boundaries (). For the proposed model, we simply choose a fixed for all these images. We clearly observe from Figure 7 that both the Bresson et al.'s model and the proposed model can achieve the satisfactory detective results for these images, while Goldstein et al.'s model fails to handle them. Besides, from Table 2, we can see that CPU times of the proposed model are very small compared to Bresson et al.'s model.
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In Figure 8, we consider three range images of saddle and cone with step or roof edges (top to bottom: , , and ), 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 that the objects are accurately extracted by the Goldstein et al.'s model and the proposed model (top to bottom: ). However, by carefully inspecting these images in the second column, Bresson et al.'s model cannot extract the real object boundaries, which leads to the boundary leakage. Here, we notice that the proposed model needs more CPU the times than Goldstein et al.'s model (see Table 3).
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Figures 9 and 10 show the segmentation results of the three models for six real images. The test images are six images, in which four images are shown in the first row of Figure 9 (left to right: , , , and ) and two images are shown in the first column of Figure 10 (, ). We choose for these images in Figure 9, and for the two images in Figure 10. As can be seen from Figures 9 and 10 that the three models can obtain satisfactory segmentation results for all these images. Besides, we observe from Tables 4 and 5 that the CPU times of the proposed model are less than the Bresson et al.'s model [18], but more than the Goldstein et al.'s model [20].
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The last experiment shows the segmentation results for four medical images with low contrast and blurred boundaries using C-V model, Bresson et al.'s model, Goldstein et al.'s model, and the proposed model. The test images, shown in the top row of Figure 11, (left to right) are two real images of digital mammography (, ), a real corpus callosum MR image () and a heart image (). For the third image, the object is to segment the blurry corpus callosum region which is surrounded by the dim brain tissues. 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 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 correctly, while the proposed model effectively and quickly extracts objects for all images in a single iteration. Besides, the proposed model needs less CPU times than both Bresson et al.'s model and Goldstein et al.'s model (see Table 6).
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5. Dissussion
In this section, we simply discuss about the parameter . For the proposed model, for the function is very important to the segmentation results. Theoretically, if is larger, then the norm is stronger. That is to say, when larger 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 when it is chosen by hand as some fixed value. Experimentally, our observations are as follows: is typically chosen as a constant between 2.2 and 7. For synthetic images and range images, we choose in the range between 2.2 and 3; for aerial images and some medical images (excluding Figure 10(e)), we choose in the range between 3.1 and 4.9; for infrared images and those medical images with low contrast and blurred boundaries, we choose in the range between 5 and 7.
It is worth emphasizing that the proposed model is robust to . For this purpose, we apply the proposed model to two synthetic images (, ) and two real images (, ), (top to bottom) as shown in the first column of Figure 12. For each image, we choose default initial contour, and all those different 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).
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6. Conclusion
In this paper, following the well-known C-V model we propose a region-based implicit active contour model based on 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 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.
Acknowledgments
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve this paper. The authors are very grateful to Professor Xavier Bresson (the first author of [18], and the second author of [20]) for sharing the papers along with the evaluation codes. Besides, this work was supported by the Fundamental Research Funds for the Central Universities Grant No. (CDJXS11100021), and the Education Committee Project Research Foundation of Chongqing no. (KJ120709) and the National Natural Science Foundation of China under Grant no. 61202349 and 61271452.