Second order total generalized variation for speckle reduction in ultrasound images

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Abstract

Image denoising is one of the most important issues in image processing. For removing the speckle noise in ultrasound images, researchers have proposed the minimization models based on the total variation (TV), which effectively preserve the sharp edges. But they simultaneously suffer form the undesired artifacts, such as the staircase effect. To overcome this shortcoming, we propose a convex model by combining with the total generalized variation (TGV) regularization for retaining the fine detail and reducing the staircase effect. Furthermore, we develop an alternating direction method of multiplier (ADMM) to solve the proposed model. Experimental results demonstrate that our model outperforms some state-of-the-art methods in terms of visual and quantitative measures.

Introduction

Ultrasound imaging is widely employed in diagnosis of the clinical disease due to its low cost, safety, noninvasive measuring and real-time imaging [1], [2]. Unfortunately, the speckle noise in ultrasound images influences the diagnosis accuracy of the pathological and normal tissues. Speckle noise is a granular pattern based on the scattered ultrasound beam from the body’s tissues, which degrades the image contrast and resolution. Thus, it is a daunting challenge to eliminate the speckle noise in ultrasound images and simultaneously retain the important features.

As far as we know, the speckle noise in coherent imaging system follows Gamma distribution [3], [4]. However, the speckle noise in ultrasound images does not satisfy Gamma distribution due to the different imaging theory, see [5], [6], [7] and references therein. According to [5], [8], the degradation model of ultrasound images is formulated as the following form: u0=u+uη,where u > 0 is the underlying image on a rectangle ΩR2, u0 is the degraded image, and η denotes the Gaussian noise with mean 0 and standard deviation σ. We aim at recovering u from its noisy observation u0.

In recent decades, numerous denoising methods have been applied to recover medical ultrasound images corrupted by the speckle noise. The major methods are divided in three categories, including locally adaptive statistic filters [9], [10], [11], [12], wavelet transform based thresholding methods [13], [14], [15] and variational methods [16], [17], [18], [19], [20], [21], [22], [23]. In [17], motivated by the classical ROF model [24], the authors proposed a convex variational model for removing the speckle noise in ultrasound images: minu{Ω|u|dx+λΩ(u0u)2udx},where Ω|u|dx=sup{Ωudivφdx|φCc1(Ω,Rn),|φ|1} is referred as the TV regularization term, λ > 0 represents the regularization parameter which compromises between the data fidelity term and the TV regularization term. They utilized the gradient projection method to solve the model (2) and gave the relevant theoretical analysis. In [17], the regularization parameter λ is adaptively updated based on the estimated standard variance of the noise σ. But it was hard to accurately estimate σ. In [18], Huang et al. applied a generalized Kullback-Leibler (KL) distance [25], [26] to measure the discrepancy between u and u0. Their variational model was given as follows: minu{Ω|u|dx+λΩ(u0ulogu0uu0u+u)dx}.Since the above model is nonconvex, they took z=logu and regarded z as an image in the logarithm domain. Thus, the minimization model for removing the speckle noise in ultrasound images was rewritten as minz{Ω|z|dx+λΩ(u0ez/2logu0ezu0ez/2+ez/2)dx}.Note that as u0 > 0, the objective function of Eq. (3) is strictly convex. They solved this model by the split Bregman iterative method. Finally, the denoising result is attained by u*=ez*.

The TV regularization effectively reduce the noise and preserve the sharp edges. But there still exists some undesired staircase effect in smooth regions of the restored images. However, the TGV regularization is a useful tool to remove the staircase effect as well as preserve the sharp edge. In [27], [28], [29], [30], the researches show that TGV regularization involves the higher-order derivatives of images, which restores fairly accurate pixels in homogeneous regions and wipe off the staircase effect. Fig. 1 shows the denoising performance of the TV and TGV regularization for the noisy image corrupted by the additive Gaussian noise. From Fig. 1, we find that the TGV regularization accurately describes the intensity variations in homogeneous regions while preserves the edges.

Recently, most literatures have studied the TGV regularization to remove the additive Gaussian noise [31], [32] or the multiplicative noise [33], [34]. The TGV regularization substantially reduce the staircase effect, while effectively remove the noise and preserve the sharp edges. Motivated by the property of the TGV regularization, we propose a new convex model which consists of the second-order TGV regularization and the data fidelity term Ω(u0u)2udx. Furthermore, the ADMM algorithm is developed to solve the proposed model. The numerical experiments demonstrate the efficiency and effectiveness of the proposed model.

The remainder of this paper is organized as follows. In Section 2, the TGV regularization and the ADMM algorithm are reviewed briefly. Then we propose a novel variational model to reduce the speckle noise in ultrasound images and prove the existence of the solutions in Section 3. Meanwhile, the ADMM algorithm is developed for solving the proposed model. In Section 4, numerical results show that the performance of our proposed model is better than some existing approaches. Finally, the conclusions are given in Section 5.

Section snippets

A review of the TGV regularization and ADMM

In this section, we give a short review about the TGV regularization and the ADMM algorithm for the sake of completeness.

The proposed model based on TGVα2

According to [8], [29], we propose a new variational model which combines the data fidelity term Ω(u0u)2udx with the second-order TGV regularization for the speckle reduction. Consequently, the proposed denoising model is given by minu{TGVα2(u)+λΩ(u0u)2udx},where λ is the positive regularization parameter.

In order to illustrate the existence of solutions for Eq. (7), we firstly consider the general minimization problem: minuLp(Ω,R){TGVα2(u)+F(u)},where p ∈ (1, ∞), pd/(d1) and F:Lp(Ω,R)(

Experiments

In this section, we present several experimental results on both synthetic and real ultrasound images, respectively. All of numerical experiments are performed under Windows 7 and Matlab Version 7.10 (R2012a) running on a desktop with 3.4 GHz Intel Core i3-2130 CPU and 4 G RAM memory. In order to demonstrate the efficiency and effectiveness of the proposed TGV-based model, we give the sufficient experiments by compared with the TVGD model [17], the TVADM model, the TVKL model [18] and the TGVPD

Conclusion

In this paper, we propose a new variational model based on the TGV regularization. The new model effectively eliminates the speckle noise in ultrasound image as well as alleviates the staircase effect when utilizing the TV regularization. We develop the ADMM algorithm to solve the proposed model. The numerical experiments illustrate the efficiency and effectiveness of the proposed model. Compared with some state-of-the-art models, the proposed model achieves the visual and quantitative

Acknowledgments

We would like to thank the authors of [17] for providing the codes and the authors of [53] for offering the real ultrasound images. This research is supported by NSFC (61772003, 61402082, 11401081) and the Fundamental Research Funds for the Central Universities (ZYGX2016J129).

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