Active Contour Model for Ultrasound Images with Rayleigh Distribution

Ultrasound images are often corrupted by multiplicative noises with Rayleigh distribution. The noises are strong and often called speckle noise, so segmentation is a hard work with this kind of noises. In this paper, we incorporate multiplicative noise removing model into active contour model for ultrasound images segmentation. To model gray level behavior of ultrasound images, the classic Rayleigh probability distribution is considered. Ourmodel can segment the noisy ultrasound images very well. Finally, a fast method called Split-Bregman method is used for the easy implementation of segmentation. Experiments on a variety of synthetic and real ultrasound images validate the performance of our method.


Introduction
Ultrasound imaging is a cheap and convenient technique in clinical diagnosis.However, the ultrasound image is often suffering from speckle noise.The noise is strong and often hinders diagnosis.Many authors model the speckle noise as multiplicative one with Rayleigh distribution.Ultrasound image segmentation is often considered a hard work because of the existence of the speckle noise.In this paper, we propose a new ultrasound image segmentation method using active contour model.Because the speckle noise is strong, we consider using an image restoration model for speckle noise removal as the fidelity term.
Most active contour models use edge detectors for assisting segmentation; however, clear boundaries between two areas with large noise are hard to be detected.These models need additional smoothing operation which will blur the edges.There are many traditional active contours based on an edge detection function for the stopping criteria [1][2][3][4][5][6][7][8].However, this kind of active contour models has failed in many applications such as in medical images where fuzzy edges can be encountered.Besides, edge-based active contours are very sensitive to noise and initial position.Then active contour models based on region statistics information have been proposed.The statistics information contains mean value, variance, probability density function (pdf), and so on.Chan and Vese [9] proposed famous CV model for two-phase segmentation based on the mean values of different areas.The contour will stop at the position when the difference of mean values inside and outside the contour gets minimized.From then on, active contour models based on region information are proposed.Segmentation method based on statistical moments [10] is proposed for segmentation.Zhu and Yuille [11], Paragios and Deriche [12], and Rousson and Deriche [13] proposed using pdf descriptor which can be modeled by using mixture of Gaussians for segmentation.Herbulot et al. [14], Jehan-Besson et al. [15], and Aubert et al. [16] proposed using shape gradient-based joint intensity distribution.In these models, the pdf of different areas is updated in every step of iteration until the optimal partition is reached.Liu et al. [17] use the difference of probability density among different areas for segmentation.Yuan [18] proposes an active 2 Mathematical Problems in Engineering contour model driven by local divergence energies for the segmentation.Pan et al. [19] proposed a geodesic active contour model based on global variables difference.The model can deal with medical images with strong contrast, complex topological structure, and low noise characteristics.
The segmentation methods mentioned above belong to the catalog of local minimization model which means that the quality of segmentation depends on the initial location of active contour.Bresson et al. [20] and Chan et al. [21] modified the classic CV model as a convex one which is called convex CV model.In these methods, the segmentation results are not affected by the initial condition.Besides, dual method and the splitting technique are adapted for the fast minimization with TV norm in [20].Recently, the fast Split-Bregman method [22] is also applied to solve the convex active contour model and proved faster than dual method.The Split-Bregman method has several advantages.It is superior over graph cut in efficiency, accuracy, and memory requirement.It is actually equivalent to the augmented Lagrangian method [23,24].In this paper, we also design the Split-Bregman method for the easy implementation of our proposed new segmentation model.
All the methods mentioned above almost use the statistical information of different areas.However, the speckle noise is very strong in ultrasound images; statistical methods do not always get good performance.So, in this paper, we consider incorporating multiplicative noise removal model into active contour model for ultrasound image segmentation.Because the multiplicative noise removal model and the active contour model are iterated in a relatively independent manner, our method can get the real results without the influence of noise.In our previous work [25], we have proposed a model for images with Rayleigh distribution but the smoothing term we used is total variation.Wang et al. [26] have proved that noise removal for images corrupted by noise of Rayleigh distribution by using Perona-Malik term is superior to by using total variation.So, in this paper we proposed a new active contour model coupling with multiplicative noise removal model by using Perona-Malik term.During the iteration process, the noise removal model can reduce the effect of noise and the active contour model can get the real position of the interesting area.The proposed active contour model does not use edge detection method and this property enables our model to segment medical images without clear gradient boundaries.It is an improvement model on the piecewiseconstant Mumford-Shah segmentation model [27].
The main contribution of the paper is as follows.First, we propose a new active contour model incorporated with ultrasound noise removing method.The novelty of the new model is that the proposed one does not do segmentation after noise removal; instead, these two processes are coupled together.Second, for easy implementation, the Split-Bregman method is designed for solving the model.
The outline of the paper is as follows.In the next section we introduce some multiplicative removing models.In Section 3 we propose our new active contour model for ultrasound images with multiplicative noise.Then we design the Split-Bregman methods for the easy implementation of the model.In Section 4, we test the proposed method on a variety of synthetic and real ultrasound images.The last section is conclusion.

The Multiplicative Noise Removing Models
In recent years, much attention was attracted to multiplicative noise removal.Many methods were proposed for multiplicative noise removal by using variational method [28][29][30][31].RLO method [28] is the first variational method for multiplicative noise removal by using TV (total variation) method.AA model [29] is established by using maximum a posteriori (MAP) estimator.The energy function has proved that the minimum corresponds to the denoised image.Huang et al. [30] incorporate the modified total variation regularization in the objective function to recover image edges.The proposed model is a strictly convex objective function for multiplicative noise removal problems.Durand et al. [31] enhance the multiplicative noise removal ability by using L1 fidelity.Setzer et al. [32] proposed a variational method for multiplicative noise with Poissonian distribution by MAP method.Shi and Osher [33] proposed a convex model of multiplicative noise removal by using nonlinear inverse scale space.Shen [34] modified the TV model by dividing the gray value and called the modified one a Weberized TV model for multiplicative noise removal.Denis et al. [35] proposed a model for ultrasound noise removal with Rayleigh distribution using MAP.The model is very effective and we will incorporate it in our model.
The relative model function of [35] is shown in the following: where  is the original noisy image,  is the denoised image we want, and  is the balance parameter.We refer the reader to [35] for the derivation of this fidelity term using Bayesian statistics.In the next section, we will couple the above model with active contour model for ultrasound image segmentation.

Active Contour Model Coupling Multiplicative Noise Removing
The models for images segmentation corrupted by multiplicative noise with Rayleigh are not well developed.In [36], the authors proposed a model for multiplicative noise images segmentation, but from the model we find that it is a model fit for images with Gaussian distribution.In [37], the authors use variance information for segmentation introduced by MAP of Rayleigh distribution.The method does not consider the trait of multiplicative noise.So it is not fit for multiplicative noise very well.
To overcome the drawback of segmentation for images corrupted by multiplicative noise with Rayleigh distribution, where Ω is the image domain,   (b) The segmentation result using [36].(c) The segmentation result using [37].(d) The segmentation result using our proposed model.
Equation ( 2) can be divided into the following three subproblems of minimization: arg min where In the following, we will solve (3), (4), and (5) by using the Split-Bregman method [23].
For solving (3), we introduce auxiliary vector variable  ⇀  1 and Bregman iterative parameter ⃗  1 and transform (3) into the following equivalent minimization problems:
Fix ⃗ V to solve   = arg min By variational method, the Euler-Lagrange equation of ( 15) is Using gradient descent method, we can get : and  is the time step and in the whole paper we set  = 1.
Then we construct  by projecting it on [0, 1]; that is, Fix  to solve ⃗ V The Euler-Lagrange equation of ( 19) is ⃗ V +1 is also got using a generalized shrinkage formula [39] The initialization is also important; we set The segmentation result using [36].(d) The segmentation result using [37].(e) The segmentation result using proposed model.
In the end of calculation, we set Thus the segmentation gets the real result. = 1 corresponds to the segmentation target and  = 0 corresponds to the background.Our proposed new model is also globally convex due to  ∈ [0, 1].That is to say, the position of  does not need to be initialized.

Numerical Experiments
We test our method on a variety of synthetic images with multiplicative noise of Rayleigh distribution and real ultrasound images.We also compare the segmentation results of our method with the state-of-art methods in [36,37].For further comparison, we also give the quantitative analysis which includes the comparison of the results of our segmentation with the results derived from manual tracing.The segmentation result using [36].(d) The segmentation result using [37].(e) The segmentation result using proposed model.
In all the experiments, we set  = 30 in nonlinear function  because this value is a stable one for noise removal.Figure 1 is a simple synthetic image corrupted by Rayleigh multiplicative noise and the variance is 0.08. Figure 1(a Figure 2 is the same image corrupted by larger noise with the same distribution as Figure 1 and the variance is 1. Figure 2 The parameters of our model referred to in the experiments are the same as in Figure 1,  = 200,  1 = 1,  2 = 1,  1 = 0.5,  2 = 0.5,  1 = 0.1,  2 = 0.1, and th = 0.006, and iteration number is 20.From this experiment we can find that although there are some outliers and the edges are not well preserved, the method in [36] can still get proper segmentation result.But the method in [37] cannot get the segment result because of the large noise.
From Figure 2, we can find that although the image is very noisy, our proposed model still gets proper result and also captures the real edges.The last three experiments are tested on real ultrasound images.Figure 3 is an experiment on a real ultrasound image and the noise model is often considered as multiplicative noise with Rayleigh distribution.From the results on real ultrasound images in Figures 3-5, we find that results using method in [36] cannot get good result because the model does not fit for multiplicative noise of Rayleigh distribution.The results using model in [37] get proper results but there are still some outliers because the model is proposed for Rayleigh distribution but it does not consider that the noise is also a multiplicative one.
For further explanation of the reliability of our proposed model, we compare our segmentation result with the gold standard which is represented by manual tracing by an expert observer.The operator manually traced the contour of the area of interest which is time consuming.A set of 28 ultrasound images were manually traced by an expert.The manually traced contours were compared point-to-point to the corresponding active contour by the calculation of the Hausdorff distance [40].From comparison between the manually and automatically traced contour, we find that they are in good agreement with error percentage of 2.1% ± 1.8%.The mean value of the Hausdorff distance between the manually and automatically traced contour is 1.2 pixels (range: 0.7-1.7 pixels).
Three examples of the automated segmentation versus the contour delineated by a radiologist are shown in Figures 6,  7, and 8 where the good correspondence can be appreciated.
From the experiments, we also find that our results get better results than the other methods.All the experiments mentioned above show that our proposed method is very effective in ultrasound image segmentation.

Conclusion
In this paper, a new model active contour for ultrasound images is proposed.The model couples with multiplicative noise removal method for ultrasound image segmentation.For the easy implementation, the Split-Bregman method of the new model is also designed.The experiment results on various images show that the proposed method is good.

Figure 1 :
Figure 1: Synthetic image corrupted by multiplicative noise with Rayleigh distribution for segmentation.(a) Original noised image.(b)The segmentation result using[36].(c) The segmentation result using[37].(d) The segmentation result using our proposed model.

1 ,
1 ,  2 ,  2 are the positive parameters,  is a standard level set function and (||) =  2 ln(1 + (|| 2 / 2 )), and  is the nonlinear diffusion function which can deal with noise very well.This is a global segmentation minimization problem due to convex set  ∈ [0, 1].Chan et al. [21] transformed the original active contour model to a convex minimization problem by relaxing  ∈ {0, 1} to  ∈ [0, 1] and showed that the characteristic function is the global minimizer.Authors can refer to [21] for the details of algorithm implementation and the globally convex effects.We modified the original active contour model as the form of (2) mainly because the gray values in different areas are not always the same.We replace the fidelity term | − | 2 of VC model with ∫ Ω ((1/2)(/) 2 + log ), so our model can segment images with multiplicative noise of Rayleigh distribution.This is a natural extension of CV[9] and VC[38] model for piecewise constant or smooth model.Because of the change of fidelity term, our model also encompasses pre-Rayleigh and post-Rayleigh distributions which occur in real ultrasound images.

Figure 2 :
Figure 2: Synthetic image corrupted by multiplicative noise with Rayleigh distribution for segmentation.(a) Original real ultrasound image.(b)The segmentation result using[36].(c) The segmentation result using[37].(d) The segmentation result using our proposed model.

Figure 6 :
Figure 6: Image for segmentation.(a) Original real ultrasound image.(b) The ground truth of the segmentation.(c) The segmentation result using[36].(d) The segmentation result using[37].(e) The segmentation result using proposed model.

Figure 7 :
Figure 7: Image for segmentation.(a) Original real ultrasound image.(b) The ground truth of the segmentation.(c) The segmentation result using[36].(d) The segmentation result using[37].(e) The segmentation result using proposed model.
) is the origin synthetic noised image; Figures 1(b), 1(c), and 1(d) are the segmentation results using different methods.The parameters of our model referred to in the experiment are  = 200,  1 = 1,  2 = 1,  1 = 0.5,  2 = 0.5,  1 = 0.1,  2 = 0.1, and th = 0.006 and iteration number is 20.Figures 1(c) and 1(d) are almost the same, but the segmentation of Figure 1(b) is not quite accurate and the corners and edges are not well segmented.
(a) is the noisy image; Figures 2(b), 2(c), and 2(d) are the segmentation results using different methods.

Figure 3 (Figure 8 :
Figure 8: Image for segmentation.(a) Original real ultrasound image.(b) The ground truth of the segmentation.(c) The segmentation result using[36].(d) The segmentation result using[37].(e) The segmentation result using proposed model.