Medical Image Segmentation Based on Novel ... - Semantic Scholar

Medical Image Segmentation Based on Novel Local Order Energy ? LingFeng Wang1 , Zeyun Yu2 , and ChunHong Pan1 1

2

NLPR, Institute of Automation, Chinese Academy of Sciences [email protected], [email protected] Department of Computer Science University of Wisconsin-Milwaukee [email protected]

Abstract. Image segmentation plays an important role in many medical imaging systems, yet in complex circumstances it is still a challenging problem. Among many difficulties, problem caused by the image intensity inhomogeneity is the key aspect. In this work, we develop a novel localhomogeneous region-based level set segmentation method to tackle this problem. First, we propose a novel local order energy, which interprets the local intensity constraint. And then, we integrate this energy into the objective energy function. After that, we minimize the energy function via a level set evolution process. Extensive experiments are performed to evaluate the proposed approach, showing significant improvements in both accuracy and efficiency, as compared to the state-of-the-art.

1

Introduction

Level set model has been extensively applied to medical image segmentation, because it holds several desirable advantages over traditional image segmentation methods. First, it can achieve sub-pixel segmentation accuracy of object boundaries. Second, it can change the topology of the contour automatically. Existing level set methods can be mainly classified into two groups: i.e. the edge-based [1–3] methods and the region-based [4–14] methods. Edge-based methods utilize the local edge information to attract the active contour toward the object boundaries. These methods have two inevitable limitations [4], such as depending on the initial contour as well as sensitive to the noise. Region-based methods, on the other hand, are introduced to overcome the two limitations. These methods identify each interest region by a certain region descriptor, such as intensity, to guide the evolution of the active contour. The principle of region-based methods tend to rely on the assumption of homogeneity of the region descriptor. In this work, we improve the traditional region-based method to tackle the segmentation difficulties, such as the inhomogeneity of the image intensity. There are mainly two types of region-based segmentation methods: i.e. the global-homogeneous [4–6] and local-homogeneous [7–14]. One of the most important global-homogeneous methods is proposed by Chan et al. [4]. This method ?

This work was supported by the National Natural Science Foundation of China (Grant No. 60873161 and Grant No. 60975037)

2

LingFeng Wang, Zeyun Yu, and ChunHong Pan

first assumes that image intensities are statistically homogeneous in each segmented region. Then, it utilizes the global fitting energy to represent this homogeneity. Improved performance of the global-homogeneous methods has been reported in [4–6]. However, the details of the object always cannot be segmented out by these methods. As shown in Fig.1, parts of brain tissues are not segmented out. The local-homogeneous methods are proposed to solve intensity inhomogeneity. These methods assume that the intensities in a relatively small local region are separable. Li et al. [7, 10] introduced a local fitting energy due to a kernel function, which is used to extract the local image information. Motivated by the local fitting energy proposed in [10], the local Gaussian distribution fitting energy is presented in [14]. An et al. [8] offered a variational region-based algorithm, which combines the Γ -Convergence approximation with a new multi-scale piecewise smooth model. Unfortunately, the local-homogenous methods have two inevitable drawbacks. First, these methods are sensitive to the initial contour. Second, the segmentation results have obvious errors. As shown in Fig.1, although the two initial contours are very similar, the segmentation results of [10] are quite different. Furthermore, both the two results have obvious segmentation errors. We use the blue ellipses to indicate these errors. Motivated by the previous works [4, 7, 8, 10, 15], the local intensity constraint is adopted to realize our local-homogeneous region-based segmentation method. The purposes of using the local intensity constraint are from two aspects. First, the local characteristic can soften the global constraint of global-homogeneous methods. Second, using additional constraint can strengthen the instability of local-homogenous methods. In this work, we first develop a local order energy as local intensity constraint. The proposed local order energy assumes that intensities in a relatively small local region has order. For example, the intensities in the target are larger than those in the non-target. Generally, the local order assumption is easy to be satisfied in medical image. Then, we integrate the local order energy into the objective energy function, and minimize the energy function via a level set evolution process. Experimental results show the proposed method has many advantages and improvements. First, compared to the global-homogeneous methods, the detail information is well segmented. Second, the two main drawbacks of the traditional local-homogeneous methods are well solved. Third, the proposed method is also insensitive to noise. The remainder of the presented paper is organized as follows: Section 2 describes the proposed segmentation method in detail; Section 3 presents some experiments and results; the conclusive remark is given in Section 4.

2

The proposed Method

In this section, we first briefly introduce two traditional segmentation methods, i.e. the global-homogenous methods and the local-homogenous methods. Then, we describe the proposed local order energy, which interprets the local intensity constraint, in detail. Finally, we use level set evolution process to minimize the energy function, and present our algorithm in Alg.1.

Medical Image Segmentation Based on Novel Local Order Energy

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’

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Fig. 1. A comparison of the proposed method to the traditional methods, i.e. the Chan et al. [4] (a global-homogeneous method) and the Li et al. [10] (a local-homogenous method). Note that, two different yet very similar initial contours are used to test the initialization sensitive of three algorithms.

2.1

Global-homogenous and Local-homogenous methods

Let Ω ⊂ R2 be the image domain, and I : Ω → R be the given gray-level image. Global-homogenous methods formulated the image segmentation problem as to find a contour C. The contour segments the image into two non-overlapping regions, i.e. the region of inside contour as Cin (the target region) and the region of outside contour as Cout (the non-target region). By specifying two constants cin and cout , the fitting energy (global fitting energy) is defined as follows, Z

I(x) − cin 2 dx E global (C, cin , cout ) = λin ZCin

I(x) − cout 2 dx + µ|C| + λout (1) Cout

where |C| is the length of the contour C, and λin , λout and µ are positive constants. As expressed in Eqn.1, the specified two constants cin and cout globally govern all the pixel intensities in Cin and Cout , respectively. Thus, Chan’s method is denoted as global-homogeneous method. The two constants cin and cout can further be interpreted as the two cluster centers, i.e. the target cluster cin and the non-target cluster cout . Motivated by Chan’s work, Li et al. [10] proposed a local fitting energy due to a kernel function, given by Z local f it E C, cin (x), cout (x) = E C, cin (x), cout (x) dx + µ|C| (2) yielding, Z

2 E f it C, cin (x), cout (x) = λin Kσ x − y I(y) − cin (x) dy ZCin

2 + λout Kσ x − y I(y) − cout (x) dy Cout

(3)

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where K(.) is the Gaussian kernel function Kσ x =

1 (2π)n/2 σ n

e−kxk

2

/2σ 2

pa-

rameterized with the scale parameter σ. More comprehensive interpretations of the local fitting energy are introduced in [10] in detail. The fitting values cin (x) and cout (x) are locally govern the pixel intensities centered at position x, whose size is controlled by σ. Thus, Eqn.2 is named as the local fitting energy, and Li’s method is denoted as local-homogeneous method. Similarly, cin (x) and cout (x) can be interpreted as two local cluster centers at position x. The main difference between the above two methods is how to choose the cluster centers. The global-homogenous method simply utilizes two constants cin and cout as the cluster centers, and formulates them as the global fitting energy. In contrary, the local-homogenous method adopts the local cluster centers, and formulates them as the local fitting energy. 2.2

Local Order Energy

Generally, medical image has inhomogeneity problem. Thus, local fitting energy interprets the segmentation problem better than global fitting energy. However, local fitting energy also has some problems. The most serious one is that local fitting energy function has many local optimal solutions. Thus, it is hard to reach the global optimal solution through the optimization method, such as the level set evolution process. Meanwhile, this problem brings two inevitable drawbacks. First, the solution is sensitive to the initialization. Second, the segmentation result has some obvious errors. Actually, the local optimal solutions of global fitting energy is less than those of local fitting energy. That is, the global-homogenous method has more stable solution than the local-homogenous method. The main reason is that global fitting energy utilizes a strong intensity constraint. However, the local fitting energy weakens this intensity constraint by using the local characteristic. Thus, it is reasonable to add other local intensity constraints. Here, we adopt the local order energy as the local intensity constraint. The proposes of using this energy are from two sides. First, using local constraints can solve the inhomogeneity problem better than using global constraints. Thus, the detail information can be well segmented out. Second, adding other constraints can obviously decrease local optimal solutions number. Thus, the segmentation result can be stable and with small errors. In the next, we first give a straightforward example to interpret the local optimal solution problem. Then, we present the local order energy, which is one of the important local intensity constraint. Fig.2 presents a straightforward example to interpret the local optimal solution problem. We use a simple synthetic image with half bright pixels and half dark pixels as the test image. As shown in Fig.2, the curves Cright (in red) and Cwrong (in blue) represent the right and wrong segmentation results, respectively. When specifying the local cluster centers cin (x) and cout (x), both two local fitting energies E local (Cright , cin (x), cout (x)) and E local (Cwrong , cin (x), cout (x)) calculated from Cright and Cwrong can reach to the local minimums. That is, both E local (Cright , cin (x), cout (x)) and E local (Cwrong , cin (x), cout (x)) are the local optimal solutions. As shown in the second row of Fig.2, we assign three boundary pix-


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els (in green dashed curve) with cin (x) = 1 and other pixels with cin (x) = 1 and cout (x) = 0. Here, the local fitting energy E local (Cright , cin (x), cout (x)) reaches to the local minimum. As shown in the third row of Fig.2, we assign three boundary pixels with cout (x) = 1 and other pixels with cin (x) = 1 and cout (x) = 0. Here, the local fitting energy E local (Cwrong , cin (x), cout (x)) reaches to the local minimum. Note that, the curve Cwrong is the wrong segmentation result. We can also enumerate other wrong results, which are similar with Cwrong .

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x x x x 1 0 0 0 0 0 x x x x 1 0 0 0 0 0 x x x x 1 0 0 0 0 0

Fig. 2. Overview of the comparison of two segmentation results. 1 and 0 are the intensity values of the image. The first row gives two segmentation curves Cright (in red) and Cwrong (in blue). The second row presents the cin (x) and cout (x) of each pixel when curve is Cright , while the third row presents them when curve is Cwrong . Note that, the symbol x means the random value.

There are two types of values. The first one is the value of non-target cluster centers cout (x) inside the curve C. The second one is the value of target cluster centers cin (x) outside the curve C. From the Eqn.2 and Eqn.3, we find that these two types of values are none sense when calculating the local fitting energy. That is, we can set them as the random values. As shown in Fig.2, we use symbol x

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to represent the random value. However, these values have useful meanings in practice. Such as, cin (x) means the local target cluster center, and the cout (x) means the local non-target cluster center. Moreover, these values are recursive used when perform level set evolution process. The evolution process will be described in Subsection 2.3. Or you can find the similar process in [10] in detail. In this work, these values are used to formulate the local order energy, which is a local intensity priori constraint. Actually, in a small local region of a medical image, the intensity values always have order. Such as the target intensity values are larger than the non-target ones. Thus, the target cluster center value cin (x) is larger than the non-target cluster center value cout (x) at position x. The local order information is represented by the local order energy, given by, Z (4) E order C, cin , cout = sgn cout (x) − cin (x) dx Ω

where sgn(.) is the sign function, ( sgn(x) =

1 x>0 0 Else

(5)

As shown in Eqn.4, the local order energy uses all cluster centers, which include the above two types of values. Furthermore, the local order energy, is one of the important local intensity constraint, because it restricts the local target intensity should larger than the non-target intensity statistically. Combining the local fitting energy defined in Eqn.2, we can get the new local energy E E = E local + E order

(6)

where E local and E order are defined by Eqn.2 and Eqn.4, respectively. Since the local fitting energy and the local order energy are both defined from the contour C, the final energy E is represented by the contour C too. 2.3

Level Set Formulation and Energy Minimization

Level set formulation is performed to solve the above energy E. In level set methods, a contour C is represented by the zero level set of a Lipschitz function φ : Ω → R, which is called a level set function. Accordingly, the energy of our method can be defined as ZZ

2 F φ, cin , cout = λin Kσ x − y I(y) − cin (x) H φ(y) dydx ZZ

2

+ λout Kσ x − y I(y) − cout (x) 1 − H φ(y) dydx Z +γ sgn cout (x) − cin (x) dx + µL(φ) + νP(φ) (7) Ω

where λin , λout , γ,R µ, and ν are the weighting constants, H(.) is the Heaviside function, L(φ) = δ(φ(x))|∇φ(x)|dx is the length term, δ(.) is the derivative


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R function of Heaviside function, and P(φ) = 12 (|∇φ(x)| − 1)2 dx. The term P(φ) is the regularization term [3], which serves to maintain the regularity of the level set function. In implement, Heaviside function is approximated by a smooth function H (x) = 12 1 + π2 arctan( x ) , and the corresponding derivative function is defined by δ (x) = H0 (x) = π(2+x2 ) . When implementation, the coefficient is set as = 1.0. Standard gradient descent method is performed to minimize the energy function F(φ, cin , cout ) based on two sub-steps. For the first step, we fix the level set functional φ, and calculate two cluster centers cin (x) and cout (x). For the second step, we fix the two cluster centers cin (x) and cout (x), and calculate the level set functional φ. Variational method is first applied to calculate cin and cout for a fixed level set functional φ, given by, cin = max b cin , b cout cin = min b cin , b cout (8) yielding, Kσ (x) ⊗ H φ(x) I x b cin = Kσ (x) ⊗ H φ(x)

b cout

Kσ (x) ⊗ 1 − H φ(x) I x = Kσ (x) ⊗ 1 − H φ(x)

(9) where ⊗ is the convolution operation. Then, keeping cin and cout fixed, the energy functional is minimized with respect to the level set functional φ by using the standard gradient descent, given by, ∂φ = − δ (φ) λin Tin − λout Tout ∂t ∇φ ∇φ + ν ∇2 φ − div (10) + µδ div |∇φ| |∇φ| where Tin and Tout are defined as follows, Z

2

Tin (x) = Kσ (y − x) I(y) − cin (x) dy Z

2 Tin (x) = Kσ (y − x) I(y) − cin (x) dy

(11)

The more information of the implementation of the proposed local prior intensity level set method is illustrated in Alg.1 in detail.

3

Experiment Results

A number of real medical images are used to evaluate the performance of the proposed scheme by comparing the two traditional methods, i.e. Chan et al. [4] and Li et al. [10], which are the deputations of global-homogeneous and localhomogeneous methods, respectively. When implementation, the coefficients are

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Algorithm 1: Local Intensity Priori Level Set Segmentation

1 2 3 4 5 6 7 8 9

Data: initial level set φinitial , input image I, max iteration maxiter, and weighting constants Result: level set φ and segmentation result. Initial the output level set function φ1 = φinitial ; for i ← 1 to maxiter do Calculate cin and cout based on φi by Eqn.8 and Eqn.9; Calculate level set φi+1 based on cin and cout by Eqn.10 and Eqn.11; if φi equals to φi+1 then Break the iteration; end end Saving level set φ, and segmenting the input image based on φ;

empirically set as λin = 1, λout = 1, ν = 1, µ = 0.004 · h · w, where h and w are the height and width of the test image. All the medical images are downloaded from the web-site 3 and 4 . The visual comparison is first shown in Fig.1, and the more comparisons are illustrated in Fig.3, Fig.4, and Fig.5, respectively. We use three different image acquisition techniques, i.e. CT, MR, and Ultrasound, to evaluate our approach. As illustrated in these figures, although the images are rather noisy and part of the boundaries are weak, the detail information are still segmented very well comparing to the Chan’s [4] and Li’s [10] methods. Furthermore, the proposed approach is not sensitive to the initialization compared to the Li’s [10] method. The numeric comparisons are presented in Fig.6 and Fig.7. The segmentation results are compared to the ground truth by using the following two error measures: i.e. false negative (FN), which interprets number of target pixels that are missed, and false positive (FP), which interprets number of non-target pixels are marked as target. Denoting Rour target are the result of target region of by gt li , our method. Similarly, Rchar , R target Rtarget are the Chan’s, Li’s, and ground in truth. Thus, the FN and FP are defined as follows FP =

R∗target − R∗target ∩ Rgt target Rgt target

FN =

Rt argetgt − R∗target ∩ Rgt target Rgt target

(12) where {∗|∗ ∈ chan, li, our}. In Fig.6, six different scenes are choosed, and for each scene, 100 experiments are performed with the different initial contour. As shown in the figure, both the means and variances of the above two error measures are smaller than the traditional methods. That is to say, the proposed method is of better accuracy (smaller error mean) and stability (smaller error variance) than the traditional ones. Furthermore, the computation cost of our approach is similar with the Li’s approach. Fig.7 shows the comparisons when 3 4

http://www.bic.mni.mcgill.ca/brainweb/ http://www.ece.ncsu.edu/imaging/Archives/ImageDataBase/Medical/index.html


Initializations

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Li’ s Method

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Proposed Method

Fig. 3. The comparison of our method to Chan [4] and Li [10] on CT image.

adding noise to the images. As illustrated in the figure, the error of our method increase slower than the traditional methods when the power of noise enlarging. That is, the proposed method is less sensitive to the noise. We also use Chan’s, Li’s , and our methods to medical image reconstruction. The visual result of our method is shown in Fig.8. As shown in this figure, the brain tissues are preserved well. The main reason is our segmentation method gives excellent the segmentation result. The numerical comparison of reconstruction error is shown in Tab.1. The error is defined as the mean of all slice’s FN and FP, given by N

X d= 1 FN FNi N i=1

N

X d= 1 FP FPi N i=1

(13)

where N is the slice number, the FNi , FPi are the ith slice’s FN and FP (defined in Eqn.12). As shown in this table, the our two errors are obviously smaller than the Chan’s and Li’s.

Table 1. The comparison of reconstruction errors. Algorithm Chan’s [4] Li’s [10] Ours FN 0.161 0.143 0.019 FP 0.016 0.126 0.017

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Fig. 4. The comparison of our method to Chan [4] and Li [10] on Ultrasound image.

4

Conclusions and Future Works

In this work, we propose a novel region-based level set segmentation method. The main contribution of this work is proposing the local order energy, which is used to represent the local intensity priori constraint. The proposed algorithm has been shown on a number of experiments to address the medical image segmentation problem both efficiently and effectively. Meanwhile, compared to the traditional methods, the proposed method has many advantages and improvements: 1. the detail information is well segmented; 2. the main drawbacks of the traditional local-homogeneous methods are well solved; 3. the proposed method is insensitive to noise. Future works will be adding the other local intensity constraints to improve the segmentation algorithm, or adding the shape information to enhance the segmentation results. Moreover, the proposed algorithm can not only be used in medical image segmentation, but also be applied on other types of images.

References 1. Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. IJCV 22 (1997) 61–79 2. Yezzi, A., Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A.: A geometric snake model for segmentation of medical imagery. IEEE Trans. on MI 16 (1997) 199–209 3. Li, C., Xu, C., Gui, C., Fox, M.D.: Level set evolution without re-initialization: A new variational formulation. In: CVPR. (2005) 430–436 4. Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. on IP 10 (2001) 266–277 5. Tsai, A., Anthony Yezzi, J., Willsky, A.S.: Curve evolution implementation of the mumfordcshah functional for image segmentation, denoising, interpolation, and magnification. IEEE Trans. on IP 10 (2001) 1169–1186


Initializations

Chan’ s Method

Li’ s Method

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Fig. 5. The comparison of our method to Chan [4] and Li [10] on MR image.

6. Vese, L.A., Chan, T.F.: A multiphase level set framework for image segmentation using the mumford and shah model. IJCV 50 (2002) 271–293 7. Li, C., Kao, C.Y., Gore, J.C., Ding, Z.: Implicit active contours driven by local binary fitting energy. In: CVPR. (2007) 1–7 8. An, J., Rousson, M., Xu, C.: γ-convergence approximation to piecewise smooth medical image segmentation. In: MICCAI. (2007) 495–502 9. Piovano, J., Rousson, M., Papadopoulo, T.: Efficient segmentation of piecewise smooth images. In: SSVMCV. (2007) 709–720 10. Li, C., Kao, C.Y., Gore, J.C., Ding, Z.: Minimization of region-scalable fitting energy for image segmentation. IEEE Trans. on IP 17 (2008) 1940–1949 11. Li, C., Huang, R., Ding, Z., Gatenby, C., Metaxas, D., Gore, J.: A variational level set approach to segmentation and bias correction of images with intensity inhomogeneity. In: MICCAI. (2008) 1083–1091 12. Li, C., Gatenby, C., Wang, L., Gore, J.C.: A robust parametric method for bias field estimation and segmentation of mr images. In: CVPR. (2009) 13. Li, C., Li, F., Kao, C.Y., , Xu, C.: Image segmentation with simultaneous illumination and reflectance estimation: An energy minimization approach. In: ICCV. (2009) 14. Wang, L., Macione, J., Sun, Q., Xia, D., , Li, C.: Level set segmentation based on local gaussian distribution fitting. In: ACCV. (2009) 15. Xiang, S., Nie, F., Zhang, C., Zhang, C.: Interactive natural image segmentation via spline regression. IEEE Trans. on IP 18 (2009) 1623–1632

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Fig. 6. An overview of the comparisons to Chan’s [4] and Li’s [10] by using two error measures, i.e. the (FN) and (FP).

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Fig. 7. An comparisons to Chan’s [4] and Li’s [10] when adding noise to the images. The horizontal axis shows the power of noise, while the vertical axis shows two error measures, i.e. the false negative (FN) and false positive (FP).

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Fig. 8. Medical image reconstruction result by our method.