A robust level set method based on local statistical ...

Optik 125 (2014) 2199–2204

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A robust level set method based on local statistical information for noisy image segmentation Xiaomin Xie ∗ , Changming Wang, Aijun Zhang, Xiangfei Meng School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China

a r t i c l e

i n f o

Article history: Received 10 May 2013 Accepted 11 October 2013 Keywords: Active contour model Image segmentation Level set method Statistical information Noisy image

a b s t r a c t The paper presents an improved local region-based active contour model for image segmentation, which is robust to noise. A data fitting energy functional is defined in terms of curves and the energy terms which are based on the differences between the local average intensity and the global intensity means. Then the energy is incorporated into a level set variational formulation, from which a curve evolution equation is derived for energy minimization. And then the level set function is regularized by Gaussian filter to keep smooth and eliminate the re-initialization. By using the local statistical information, the proposed model can handle with noisy images. The proposed model is first presented as a two-phase level set formulation and then extended to a multi-phase one. Experimental results show desirable performances of the proposed model for both noisy synthetic and real images, especially with high level noise. © 2013 Elsevier GmbH. All rights reserved.

1. Introduction Image segmentation has always been a difficult task in image processing and computer vision, the goal of which is to divide images into some meaningful subsets. Noise often occurs in all kinds of images: laser images, microscope images, sonar images, photoelectric images and all that, owing to the influence of the equipment of image transmission and processing, etc. [1,2]. The noise will bring about challenges in image segmentation. In recent years, active contour models (ACM) implemented via level set methods have been successfully used in image segmentation [3–5]. The basic idea of ACM is to implicitly represent a contour as the zero level set of higher dimensional level set function, and formulate the evolution of the contour through the evolution of the level set function [6]. The pioneer work about ACM can be traced to Kass in 1988 [7], after which the approaches have developed in a variety of directions. Generally, the existing ACM can be broadly classified as either edged based models [7–10] or region based models [11–17]. Edged based models [1–4] utilize image gradient information to stop the curve evolution. For this kind of models, it is not necessary to place a global constraint on the level set. The Geodesic active contour (GAC) model is one of the famous models in this class [8]. However, for some type of images, which object boundaries are weak or corrupted by noise, the edge-based models are likely to pass through the object boundary or produce spurious boundaries.

∗ Corresponding author. Tel.: +86 13851754704; fax: +86 025 84315471. E-mail address: yu jian [email protected] (X. Xie). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.10.026

More recently, work has been focused on the region-based model. The region-based models [11–17] take advantage of a certain region descriptor (color, intensity, texture etc.). Therefore, they show better performance over the edge-based models in two aspects: First, the region-based models can obtain the promising results when handling with the weak boundaries. Second, the region-based models are less affected by the noise compared to the edge-based models, since they use the global region information. One of the most famous region-based models is the CV model [11], which is a simplified Mumford-Shah model [12]. The CV model assumes that each region of the image has a statistically homogeneous intensity. And its energy functional is based on the difference of each pixel and the region intensity means. In fact, the images suffer from various types of artifacts such as intensity non-uniformity and noise. In such cases, the CV model fails to detect the object boundaries accurately. However, the advent of the region based active contour models [13–17] has had a significant impact on the intensity inhomogeneity. The local models assume that the intensities in a relatively small local region are separable. For instance, the RSF model, which is proposed by Li et al. [13,14], draws upon the local region intensity information in spatially varying local region, and hence does well in segmenting images with intensity inhomogeneity. Since the region based models devote to the intensity inhomogeneity, they meet difficulties in segmenting images with high level noise. In this paper, we focus on the need for the segmentation of the noisy images. Given that the CV model is based on the difference of the pixel and region average intensities, it would become invalid when it is polluted by noise. In this paper, we describe

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an improved region based CV model in a variational level set formulation to segment noisy images. The basic idea of the proposed model is to exploit the difference of local statistical information and global image information inside and outside of the contour to conduct an energy functional. When the curve is exactly on the boundaries of the object, the energy functional is minimized. The energy functional is incorporated into a level set functional to achieve its minimization. In addition, the proposed model is extended to multi-phase one. Furthermore, the Gaussian filtering processing [15,17] is adopted to regularize the level set functional and reduce the affect of the noise to a certain degree. Experimental results on some noisy synthetic and real images show the advantages of our model in term of efficiency and robustness. Moreover, compared with the well-known CV model, our model shows better performance on various noises and high level noise. The remainder of this paper is organized as follows: A brief review of some well-known region based models for image segmentation is given in Section 2. Then, Section 3 devotes to the proposed model in this paper. In Section 4, implement and experiment results are presented and analyzed. This paper is summarized in Section 5. 2. Backgrounds Let ˝ be the image domain, and I: ˝ →  be a gray level image. x of the I(x) is the pixel in ˝. The goal of image segmentation is to divide the image into disjoint subregions ˝1 . . .˝N . On the basis of Mumford-Shah model, Chan and Vese proposed an active contour model (i.e. CV model [11]), energy of which in term of contour C can be written as:

The third term is the length term that regularizes the zero level contours, and the last term is the penalty term that penalizes the deviation of the level set function from a signed distance function. However, inspired by the thought of the RSF model, we propose an improved active contour model based on local intensity information for noisy image segmentation, which is an improved CV model. 3. The proposed model Let ˝ be the image domain, which can be partitioned to a set of disjoint regions noted as ∪N ˝, i=1 i

εCV (c 1 , c2 , C) = 1 ·

(I(x) − c1 ) dx



 

(1)

inside(C)

where outside (C) and inside (C) represent the regions outside and inside the curve C, respectively. c1 and c2 designate two constants that approximate the image intensities in outside (C) and inside (C), respectively. 1 , 2 , are fixed constants. The third term is the length of curve C for regularization. For parameter v ≥ 0, if we have to not detect smaller objects (like points, due to noise), v has to be larger [11]. In order to deal with intensity inhomogeneity, Li et al. brought forward a region based active contour model that draws upon intensity information in local regions at a controllable scale (RSF) [13,14]. Considering the two-phase model, the energy functional of the RSF model is expressed as:

 



  + 2

 +v



2

K (x − y) · I(y) − f1 (x) · H((y)) dy dx

εRSF (, f1 , f2 ) = 1



2



K (x − y) · I(y) − f2 (x) · H(−(y)) dy dx

  ∇ H((x)) dx + 



1 2 (∇ (x) − 1) dx 2

(2)

where 1 , 2 , are positive constants which govern the tradeoff between the first term and the second term. f1 (x), f2 (x) are two values that locally approximate image intensities on the two sides of C. The size of the local region can be controlled by the scale parameter␴ of kernel function K . It is chosen as a Gaussian kernel K(x−y) , which decreases drastically to zero as y goes away from x: 1 2 2 K (u) = e−|u| /2 2 2

i=1

. N is the number of the regions,

Let C be a closed contour in the image domain, which separates the domain into two parts: ˝1 outside (C) and ˝1 inside (C). inside (C) and outside (C) represent the regions inside and outside the contour C, respectively. Considering the two-phase model, we define the following two-phase model in term of contour C:



2

ε(ci , fi , C) = ε1 + ε2 =

(f1 (x) − c1 (x)) dx ˝1

 2

(f2 (x) − c2 (x)) dx

(4)

˝2

(I(x) − c2 )2 dx +  C 

+ 2 ·

N

3.1. Level set formulation: two-phase model

+

outside(C)

˝i

for i = / j, (˝i ∩ ˝j = ). and ˝ = For the CV model, the energy is based on the difference between each pixel and the average intensity of the region. However, when the image is polluted by noise, this model may result in accurate segmentation results. In our model, we utilize the local intensity average instead of the single pixel to establish the energy functional. In that case, the proposed model is less affected by the noise.



2



(3)

where c1 (x) and c2 (x) are constants which stand for the average intensities of ˝1 and ˝2 , respectively. f1 (x) and f2 (x) indicate the weighted intensity means of the neighborhood region partitioned by the contours. When the energy is minimized, the curve is exactly on the object boundaries. In the level set formulation, active contours    are represented by the zero level set C(t) = (x, y) (x, y, t) = 0 of a level set function . Then ˝1 and ˝2 can be represented as the two regions outside and inside the zero level set of , i.e., ˝1 = { > 0} and ˝2 = { < 0}. Using Heaviside function H, the energy in Eq. (4) can be expressed as an energy functional in terms of , ci and fi as below: ε(c1 , c2 , f1 , f2 , ) =

2  

2

(fi (x) − ci (x)) Mi ((x)) dx

(5)

i=1

where M1 ((x)) = H((x)), M2 ((x)) = 1 − H((x)), in practice, Heaviside function H((x)) and its derivative Dirac function ı((x)) are approximated respectively as smooth functions Hε ((x)) and ıε ((x)), defined by:

 x  ⎧ 1 2 ⎪ ⎨ Hε (x) = 2 1 +  arctan ε ⎪ 1 ε ⎩ ıε (x) = H (x) =



·

(6)

ε2 + x2

where the parameter ε can be set to 1.0 as in [11]. And to keep the notions simple, we still write H() and ı() instead of H␧ () and ı␧ (). ci , fi are the region average intensities and local intensity means, respectively, which are defined by the following functions:



ci (x) =

I(x)Mi ((x)) dx



Mi ((x)) dx

(7)

X. Xie et al. / Optik 125 (2014) 2199–2204

fi (x) =

K (x) ∗ (I(x)Mi ((x))) K (x) ∗ Mi ((x))

(8)

where K is a Gaussian kernel function which is defined as Eq. (3). The parameter  is a vital parameter in our model, which determines the size of the local region. Smaller  indicates that it will increase the noise sensitivity. When  is closed to 0, the proposed model is approximately taken as the CV model. *is convolution operator. Fixing ci and fi , we minimize the energy ␧(c1 , c2 , f1 , f2 , ) with respect to . And the minimization can be obtained by solving the following gradient descent flow equation: ∂ 2 2 = −ı()[(f1 − c1 ) − (f2 − c2 ) ] ∂t

(9)

Note that the above gradient descent flow equation looks like the CV model. However, in CV model, the energy takes each pixel into consideration, while the proposed model uses the intensity means of the neighborhood region centered at x (including x itself) to obtain the differences with the region average intensities, which shows better performance in the presence of high level noise. 3.2. Level set formulation: multi-phase model The two-phase model is not able to segment multiple regions which are adjacent to each other. Then, similar to the multi-phase CV models [18], we extend the model to multi-phase model. In multi-phase formulation, n level set functions 1 ,. . .n , which are denoted as ˚ = (1 ,. . .n ), can represent N = 2n regions. The energy of the multi-phase formulation can be written as: ε(ci , fi , ˚) =

N  

2

(fi − ci ) Mi (˚(x)) dx

(10)

i=1

where Mi (˚) are functions of ˚ such that:



M(1 (x), . . ., n (x)) =

1 x ∈ ˝i

(11)

0 else

We focus on the three-phase case in this paper, which uses two level set functions to define the partitions of the image domain. We can define M1 (1 , 2 ) = H(1 )H(2 ), M2 (1 , 2 ) = H(1 ) (1 − H(2 )), and M3 (1 , 2 ) = (1 − H(1 )). Then the energy functional of threephase model can be written as: ε(ci , fi , ˚) =

3  

2

(fi − ci ) Mi (˚(x)) dx

(12)

i=1

Minimization of the energy functional in Eq. (12) with respect to 1 , we obtain the following gradient descent flow equation: ∂1 2 2 2 = −ı(1 )[(f1 − c1 ) H(2 ) + (f2 − c2 ) (1 − H(2 )) − (f3 − c3 ) ] ∂t (13) In the same manner, minimization of the energy functional with respect to 2 can be obtained by solving the gradient descent flow equation: ∂2 2 2 = −ı(2 )[(f1 − c1 ) H(1 ) − (f2 − c2 ) H(1 ) ∂t

(14)

2201

is proposed by Li et al. [9,13,14]. It has been verified that the Gaussian filter is a good substitute for the penalty term as described in [15,17]. The filtering process can be expressed as: ik+1 = G ∗ ik

(15)

k is the iteration number and is the variance which should be set according to the experiment. If the noise is high, a larger

should be chosen [15]. The Gaussian kernel is truncated as a (4 + 1) × (4 + 1) mask. For the proposed model, the Gaussian filter process not only keeps smooth and eliminates the re-initialization, but also handles with noisy images to some extent. In this paper, in order to show the localization property of our model, we use the Gaussian filter to smooth the CV model instead of the regularization term, noted as GCV model. The evolution function of GCV model is: ∂ = −ı()[(I − c1 )2 − (I − c2 )2 ] ∂t

(16)

And then regularize the level set function by Gaussian filter. We compare the model with the GCV model to demonstrate the desirable performances of our model. In this paper, i,0 is simply initialized as a binary function (take the three-phase case for example):



1,0 =

 2,0 =

−c0

x ∈ R1

c0

x∈ / R1

−c0

x ∈ R2

c0

x∈ / R2

(17)

(18)

where c0 = 2 in this paper. R1 and R2 are arbitrarily given subsets in the image domain. The steps of our model are as follows: Step 1: Initialize the level set functions 1 , 2 according to Eqs. (17) and (18). Step 2: Calculate fi , ci , according to their expressions, respectively. Step 3: Update the level set function 1 according to Eq. (13). Step 4: Smooth the level set function 1 by Eq. (15). Step 5: Update the level set function 2 according to Eq. (14). Step 6: Smooth the level set function 2 by Eq. (15). Step 7: Check whether the evolution is stationary. If not, return to step 2.

4. Experimental results and analysis In this section, numerical examples are shown to validate the effectiveness of the proposed models for images segmentation. All the experiments are conducted in Matlab R2010a, on a personal computer with an Intel (R) Core (TM) Duo CPU and 2.00GB memory. In this paper, four region overlap metrics are adopted to compare the performances of the models quantitatively. They are the jaccard similarity (JS) [19,21], the dice similarity coefficient (DSC) [20,21], and the false positive ratio (RFP), the false negative ratio (RFN). S1 represents the foreground of the ground truth image while S2 stands for the foreground obtained by the models. O is the common part of S1 and S1 . N() indicates the pixel numbers of the region. These metrics are defined as: N(S1 ∩ S2 ) , N(S1 ∪ S2 )

3.3. Implementation

JS =

In order to regularize the level set, we use a Gaussian filtering process to smooth the level set instead of the penalty term which

RFN =

N(S2 \O) N(S2 )

DSC =

2N(S1 ∩ S2 ) , N(S1 ) + N(S2 )

RFP =

N(S1 \O) , N(S1 )

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Fig. 1. Experiments with a synthetic image. (a) Original image and initial contour, (b) result of the CV model, (c) result of the proposed model.

Fig. 2. Experiments with real noisy images. (a) Original images and initial contours, (b) results of the CV model, (c) results of the proposed model.

The closer the JS and DSC values to 1, and the RFP and RFN values to 0, the better the segmentation results. Fig. 1(a) is a synthetic image containing three separate objects, which is added Gaussian noise with zero mean and standard deviation 0.04. The initial contour is chosen as a circle. The segmentation results of the CV model and the proposed model are shown in Fig. 1(b) and (c), respectively. In order to improve the robustness to noise, we choose a lager parameter v = 0.01 × 2552 for the length term [11]. It can be seen that the CV model fails to detect the objects boundaries, taking some noise pots as the objects wrongly. By contrast, our model, which utilizes the local intensity information, possesses capability of handling noisy images. Fig. 2 shows experimental results on two real noisy images: a CT image (210 × 210) and a noisy Printed Circuit Board (PCB) photoelectric image (246 × 200). The images are quite noisy and the

object boundaries of the PCB are weak, which render it a challenging task for segmentation. Fig. 2(a) shows the initial contours that are chosen as circles. The segmentation results of the CV model and the proposed model are shown in Fig. 2(b) and (c), respectively. As we can see, by using the proposed model, the initial contours successfully evolve to the object boundaries. Experiments in Figs. 1 and 2 demonstrate better performances of the proposed model than the CV model in segmenting both synthetic and real noisy images clearly. The experiments in Fig. 3 demonstrate the robustness to two types of noises with different levels using our model. The test images are shown in Fig. 3(a), in which (left to right) the first three images are generated by adding the Gaussian noise of mean 0 and variance 0.01, 0.05 and 0.1, respectively, and the last three images are added speckle noise of variance 0.01, 0.05 and 0.01, respectively. The corresponding results of the CV model and our model are shown in Fig. 3(b) and (c), respectively. In this experiment, we set v = 0.03 × 2552 for the CV model. It can be seen that, when the noise is less (variance is small), the CV model can detect the object boundary. This is due to the fact that the CV model has a global dependence and the curve is automatically attracted toward the objects [11]. However, for the images added with Gaussian noise or speckle noise, with the increasing of variance, the segmentation results get worse. On the contrary, satisfactory segmentation results have been obtained for the challenging Gaussian noisy images and Speckle noisy images by using our model. Moreover, compared with the CV model, the proposed model is less sensitive with the noise level. The experiments in Fig. 4 are to validate the effect of the localization property of our model. The test image is a simple synthetic leaf image shown as the left of the upper row, for which the correct segmentation is immediately obvious and could be obtained easily by a thresholding model shown in the lower row. Note that, the synthetic image has been used in Fig. 3, by which we show that the proposed model succeed to segment images of various noises. The original image is contaminated by varying levels of Gaussian noise with zero mean and variance 0.03, 0.06, 0.09, 0.12 and 0.2, respectively, shown in Fig. 4(b)–(f). The related segmentation results of the GCV model and our model are shown in first row and second row of Fig. 4(b)–(f), respectively. The results of the GCV model for the noisy images are similar to that of our model by visual comparison, showing certain ability of the Gaussian filtering process in handling noise. However, by

Fig. 3. Segmentation results of The CV model and the proposed model with various noises. (a) Original images and initial contour, (b) results of the CV model, (c) results of the proposed model. (The first column to the third column is added by Gaussian noise with zero mean and variance 0.01, 0.05 and 0.1, respectively. The fourth column to the sixth column is added by speckle noise with variance 0.01, 0.05 and 0.1, respectively).

X. Xie et al. / Optik 125 (2014) 2199–2204

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Fig. 4. Comparisons with the GCV model conducted on a synthetic leaf image with different level noises. (a) Upper row: Original image; Lower row: ground truth (the object region is labeled by white color). ((b)–(f)) Results of the images with Gaussian noise of zero mean and variance 0.03, 0.06, 0.09, 0.12 and 0.2, respectively. Upper row: results of the GCV model. Lower row: results of the proposed model. Table 1 JS, DSC, RFP and RFN values for the results in Fig.4. Fig. 4(b)

JS DSC RFP RFN

Fig. 4(c)

Fig. 4(d)

Fig. 4(e)

Fig. 4(f)

GCV

Ours

GCV

Ours

GCV

Ours

GCV

Ours

GCV

Ours

0.9748 0.9881 0.0095 0.0113

0.9814 0.9906 0.0092 0.0096

0.9674 0.9834 0.0146 0.0185

0.9739 0.9868 0.0101 0.0162

0.9579 0.9785 0.0187 0.0242

0.9699 0.9847 0.0118 0.0187

0.9471 0.9728 0.0292 0.0252

0.9614 0.9803 0.0170 0.0222

0.9265 0.9618 0.0320 0.0439

0.9552 0.9769 0.0159 0.0301

the quantitative comparison using the metrics stated above (see Table 1), the proposed model which draws upon local intensity information achieves more accurate segmentation results. The following experiments are conducted to show the results of our three-phase model for multiple tissue extraction. Two zero level sets of 1 , 2 with two circles are initiated. In this experiment, the segmentation results are visualized by displaying

3 

ci Mi , as

i=1

shown in Figs. 5(c) and 6(d). Fig. 5 shows a synthetic brain image which contains three distinct intensities. We add Gaussian noise with zero mean and variance 0.001, 0.005, 0.01 and 0.02 to the synthetic image,

Fig. 5. Experiments with a synthetic image using our multi-phase model. (a) Original image and initial contour, (b) final results of our multi-phase model, (c) visual results of the proposed model (the first column to the fourth column is added by Gaussian noise with zero mean and variance 0.001, 0.005, 0.01 and 0.02, respectively).

respectively, shown in Fig. 5(a) (from left to right). As the variance increases to 0.02, the objects boundaries become blurred. As depicted in Fig. 5(b), excellent contours are still obtained with the proposed three-phase model, despite of the heavy noise. The last experiments are carried out to show the segmentation process of a group of real noisy images using the multi-phase model. The first row shows the process and the final result of tumor segmentation in a brain image, which size is 109 × 119. The initial contours are chosen as two circles (shown in the left). The segmentation result in given in the right, which shows the satisfactory results of the tumor and the brain edges. The second row shows a lungs image (which size is 256 × 256), in which the noise problem is obvious. Our model achieves successful results for the image. The third row is a plane image with shadows, the size of which is 319 × 127. The boundaries of the shadows are badly blurred. By

Fig. 6. Experiments with real images using our multi-phase model. (a) Original images and initial contours, (b) intermediate segmentation results, (c) final results, (d) visual results of the proposed model.

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using our model, the plane and the shadows are detected successfully, shown in the bottom row of Fig. 6(d). 5. Conclusions In this paper, we propose a novel active contour model in a variational level set formulation for noisy image segmentation. The proposed model efficiently utilizes the local intensity mean of the image, which is responsible for the robustness to noise. Hence, our model can handle with images with various types noises and high level noise. In addition, we use the Gaussian filter process to regularize the level set functional and alleviate the sensitivity to noises to a certain extent. We further extend the two-phase model into a multi-phase one. Experimental results show the desirable performance both of the two-phase model and multi-phase model with noise. Acknowledgements The research is supported by the Open Project Program of Key Laboratory of Intelligent Perception and systems for HighDimensional Information (Nanjing University of Science and Technology), Ministry of Education (Grant No. 30920130122005). References [1] N. Qiao, B. Zou, A segmentation method for noisy photoelectric image, Optik Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2012.12.046. [2] J.W. Deng, H.T. Tsui, A fast level set method for segmentation of low contrast noisy biomedical images, Pattern Recognit. Lett. 23 (2002) 161–169. [3] S. Lankton, A. Tannenbaum, Localizing region-based active contours, IEEE Trans. Image Process. 17 (11) (2008) 2029–2039. [4] O. Michailovich, Y. Rathi, A. Tannenbaum, Image segmentation using active contours driven by the Bhattacharyya gradient flow, IEEE Trans. Image Process. 15 (11) (2007) 2787–2801.

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