基于变分偏微分方程的医学B超图像处理
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摘要
医用B超已越来越广泛地应用于临床诊断中,然而B超图像中存在大量的斑点噪声,不同于传统的加性噪声,斑点噪声是一种乘性噪声。乘性噪声广泛存在于合成孔雷达成像,超声成像,激光成像及显微镜图像中,相比较于加性噪声图像,乘性噪声对图像的损坏更为严重,且乘性噪声图像对比度往往更低。合理地去除乘性噪声,将极大地提高医生的分析效率及临床诊断的准确率。B超图像处理的另一项工作是图像分割。通过对组织或器官精确的分割和提取,可以定量地分析组织或器官的大小,形状等变化情况,从而判断组织的病理变化情况,协助医生进行诊断和手术等。B超图像精确分割的困难在于B超图像中存在的各种干扰信息,如大量斑点噪声、组织或器官的边缘缺失、阴影等。图像多相分割问题是图像处理及计算机视觉中的一项重要内容。过去,有许多针对加性高斯噪声图像的多相分割模型及算法。然而这些模型并不适合具有乘性噪声图像的多相分割。另外,以往的模型常考虑图像灰度为分片常数,然而,真实图像灰度常有起伏,因此,更实际的多相分割模型应考虑图像灰度的不均匀性。变分偏微分方程在对加性噪声图像的处理中已取得很大成功,已积累了许多经典模型,相应的理论分析工作也已开展得比较完善。然而乘性噪声与加性噪声是两种完全不同的噪声,因此,利用变分偏微分方程处理乘性噪声图像将面临新的挑战。一方面需要建立更适合斑点噪声图像特点的模型；另一方面,相应的理论分析工作需要近一步深入地探讨；最后,B超图像处理有很强的工程应用背景,对模型求解算法的精度和速度提出更高要求。
本文主要研究以下四方面内容：一是针对临床B超图像模型,提出去B超斑点噪声的凸模型,并针对所提模型设计行之有效的快速算法。二是利用B超图像的灰度分布特征,提出针对B超图像的分割模型。三是结合人体肾脏解剖结构的先验形状,考虑在边缘缺失的情形下,B超肾脏图像的分割问题。四是提出针对具有乘性Gamma噪声图像的多相分割模型及算法,并讨论解的存在性。
本文结构如下：
第一章我们简单介绍B超图像的成像原理及图像特点。并介绍基于变分偏微分方程的去斑点噪声模型,分割模型,以及基于TV模型的快速算法的研究背景、进展,以及本文的研究内容。
第二章我们研究B超图像的去斑点噪声问题。针对一种描述B超中斑点噪声的图像模型,我们提出一种去除B超图像斑点噪声的凸模型,并讨论所提模型解的存在惟一性及有界性问题。此外,我们提出利用分裂Bregman方法,将原模型分解为两个较简单的子问题,再分别交替求解两个子问题。数值试验表明,相比较于直接利用梯度下降法求解,所提方法速度提高近一倍。我们给出模拟图像和真实B超图像的数值试验,并同其它相关方法比较,试验结果表明所提方法对B超图像的斑点噪声处理效果更好。
第三章考虑B超图像的分割问题。通过对B超成像原理的学习,我们提出针对B超图像分割的活动轮廓线模型。在水平集方法的基础上,我们提出利用求解其相应的凸松弛模型来得到原模型的解。进一步,我们讨论了水平集模型与其对应的凸松弛模型之间的关系。针对凸松弛模型中TV正则项的不可微性问题,我们提出一种基于主对偶式的快速算法。数值试验表明,利用凸松弛方法及主对偶算法,可极大地提高分割速度,并且分割结果具有某种全局最优性。
第四章我们考虑B超肾脏图像的分割。结合肾脏解剖结构的先验形状,在第三章所提模型的基础上,我们提出基于广义超椭圆的B超肾脏图像分割模型。由于该方法利用肾脏解剖结构的全局先验形状,因此不需要建立肾脏先验形状的数据库,省时方便。试验结果表明,在具有边界缺失的B超肾脏图像中,该模型能有效分割出图像的肾脏器官。与人工手绘的器官形状相比,误差在6%以内。
第五章我们考虑具有乘性Gamma噪声和灰度不均匀的图像多相分割问题。该问题的求解涉及以下几方面的难题。一是图像的多相分割问题,相比较于二相分割,图像多相分割问题会增加更多的约束性条件,这使得算法的设计会困难很多。二是乘性噪声较加性噪声更强,对图像的损坏更为严重,乘性噪声图像的对比度往往更差,因此针对乘性噪声图像的多相分割更为复杂；三是考虑图像灰度分布的不均匀性。针对以上问题,我们提出一种处理具有乘性Gamma噪声和灰度分布不均匀的图像多相分割模型,并讨论解的存在性问题。结合分裂Bregman算法和交替极小算法,我们给出一种有效的求解算法。并给出模拟图像和真实SAR图像以及B超图像的数值试验来验证所提模型和算法的有效性。
Medical B ultrasound imaging has been largely used in clinical diagnosis. However, speckle noise widely exists in B ultrasound images. Unlike traditional additive white Gaussian noise, speckle noise is a type of multiplicative noise. Multiplicative noise widely exists in SAR im-ages, ultrasound images, laser images and microscope images,and multiplicative noise is much stronger than additive Gaussian noise. The contrast of images corrupted with multiplicative noise is quite low. Suitably removing speckle noise will largely improve the accuracy of clini-cal diagnosis and the efficiency of image analysis. Image segmentation is important for medical B-ultrasound images. In some clinical diagnosis, one needs to quantitatively measure the shape, area, volume and so on, therefore, accurate detection of tissue or organs from a ultrasound image is favorable to judge the pathology state of the tissue and is helpful for surgeons to perform an operation. The difficulties for accurate segmentation of ultrasound images are the appearance of speckle noise, boundary dropout, shadow, etc.. Multi-phase image segmentation is an important task in image processing and computer vision. In the past few years, many multi-phase segmen-tation models and numerical methods have been proposed for images corrupted with additive Gaussian noise. However, these models are not suitable for images corrupted with multiplica-tive noise. Besides, most of previous multi-phase segmentation models are based on piecewise constant image model. However, intensity inhomogeneity occurs in many real world images, for example, it can be caused by non-uniform illumination. Thus, more practical image models should consider the intensity inhomogeneity of images. Variational partial differential equations have been very successful in image processing and many classic models have been proposed to deal with images corrupted with additive Gaussian noise. Corresponding theoretical analysis of these models is well presented. However, multiplicative noise is totally different from additive noise, therefore, new mathematical models and the corresponding theoretical analysis should be developed.
In this paper, we are concentrated on following four problems:first, for a clinical ultra-sound image model, we propose a convex speckle reduction model, and design a fast and effec-tive algorithm for the proposed model; secondly, we propose a ultrasound image segmentation model by combining the gray level statistics of clinical ultrasound images, then, we propose a fast primal-dual algorithm to solve the proposed model; thirdly, to deal with the boundary dropout of ultrasound kidney images, we propose a ultrasound kidney segmentation model by incorporation the prior shape information of kidney; at last, we propose a multi-phase image segmentation model and algorithm for images with multiplicative Gamma noise and intensity inhomogeneity, and give an existence theorem of solutions.
This paper is organized as follows:
In the first chapter, we briefly introduce the procedure of ultrasound imaging and the char-acteristics of ultrasound images. Then, we introduce the backgrounds and development of vari-ational partial differential equations in speckle reduction and image segmentation. At last, we show the goal of our research work.
In the second chapter, we work on removing speckle noise in ultrasound images. For a clinical ultrasound image model, we propose a convex variational model and existence, u-niqueness and boundedness of the solution to the variational model is discussed. Besides, we introduce an auxiliary variable, and solve an equivalent constrained problem by using a Breg-man iteration method. By doing this, we get two simple subproblems, and each subproblem is solved alternatively until a stopping rule satisfied. The numerical experiments show that the proposed algorithm is about two times faster than the traditional gradient descent method. We also compared our method with other popular methods on synthetic images and real ultrasound images, and the experiments show the advantage of the proposed method.
In the third chapter, we consider fast segmentation of ultrasound images. By considering the ultrasound imaging procedure and the gray level statistics of log-compressed ultrasound images, we propose a ultrasound image segmentation model based on a maximum likelihood estimation method. On the basis of level set method, we propose to solve the corresponding relaxed model related to a membership function. The relaxed model is convex with respect to the membership function. We discuss the relation between the proposed active contour model and the corresponding relaxed model. To overcome the non-differentiable TV term, we pro-pose a primal dual algorithm to solve the relaxed model. Numerical experiments show that the proposed method is quite fast and efficient, and the solution is in some sense of a global optima.
In the fourth chapter, we consider the segmentation of ultrasound kidney images. Based on the segmentation model in the third chapter, we propose a ultrasound kidney segmentation mod-el by combining the prior shape information of kidney anatomical structure. The prior shape information relies on the anatomical structure of the kidney, therefore, it is not necessary to con-struct a prior shape data base. As we know, the construction of a prior shape data base is very time consuming and quite complicated. Numerical experiments show that the proposed method can effectively segment a ultrasound kidney images when the ultrasound kidney images suffer from boundary dropout. We compare the segmentation results of our method with the hand drawing kidney boundaries by radiologists, and the errors are below6%in all our experiments.
In the last chapter, we consider multi-phase segmentation of images in presence of inten-sity inhomogeneity and multiplicative noise. This problem involves following three difficult subproblems. The first problem is the multi-phase image segmentation problem. Comparing with binary segmentation, multi-phase segmentation models usually bring more constraints, which is much more difficult to be numerically solved. The second problem is the existence of multiplicative noise, which is much stronger than additive noise. The contrast of a multiplica-tive noise image is quite low, therefore, the multi-phase segmentation of a multiplicative noise image is more difficult. The third problem is the combination of intensity inhomogeneity in multi-phase segmentation. We first present a multi-phase segmentation model for images with intensity inhomogeneity and multiplicative Gamma noise, and show the existence of a mini-mizer of the proposed model. Then, we propose an effective numerical algorithm by combining split Bregman method and alternating minimization method. Numerical experiments are shown by comparison with other two methods on both synthetic images and real images to illustrate the efficiency of the proposed method.
本文主要研究以下四方面内容：一是针对临床B超图像模型,提出去B超斑点噪声的凸模型,并针对所提模型设计行之有效的快速算法。二是利用B超图像的灰度分布特征,提出针对B超图像的分割模型。三是结合人体肾脏解剖结构的先验形状,考虑在边缘缺失的情形下,B超肾脏图像的分割问题。四是提出针对具有乘性Gamma噪声图像的多相分割模型及算法,并讨论解的存在性。
本文结构如下：
第一章我们简单介绍B超图像的成像原理及图像特点。并介绍基于变分偏微分方程的去斑点噪声模型,分割模型,以及基于TV模型的快速算法的研究背景、进展,以及本文的研究内容。
第二章我们研究B超图像的去斑点噪声问题。针对一种描述B超中斑点噪声的图像模型,我们提出一种去除B超图像斑点噪声的凸模型,并讨论所提模型解的存在惟一性及有界性问题。此外,我们提出利用分裂Bregman方法,将原模型分解为两个较简单的子问题,再分别交替求解两个子问题。数值试验表明,相比较于直接利用梯度下降法求解,所提方法速度提高近一倍。我们给出模拟图像和真实B超图像的数值试验,并同其它相关方法比较,试验结果表明所提方法对B超图像的斑点噪声处理效果更好。
第三章考虑B超图像的分割问题。通过对B超成像原理的学习,我们提出针对B超图像分割的活动轮廓线模型。在水平集方法的基础上,我们提出利用求解其相应的凸松弛模型来得到原模型的解。进一步,我们讨论了水平集模型与其对应的凸松弛模型之间的关系。针对凸松弛模型中TV正则项的不可微性问题,我们提出一种基于主对偶式的快速算法。数值试验表明,利用凸松弛方法及主对偶算法,可极大地提高分割速度,并且分割结果具有某种全局最优性。
第四章我们考虑B超肾脏图像的分割。结合肾脏解剖结构的先验形状,在第三章所提模型的基础上,我们提出基于广义超椭圆的B超肾脏图像分割模型。由于该方法利用肾脏解剖结构的全局先验形状,因此不需要建立肾脏先验形状的数据库,省时方便。试验结果表明,在具有边界缺失的B超肾脏图像中,该模型能有效分割出图像的肾脏器官。与人工手绘的器官形状相比,误差在6%以内。
第五章我们考虑具有乘性Gamma噪声和灰度不均匀的图像多相分割问题。该问题的求解涉及以下几方面的难题。一是图像的多相分割问题,相比较于二相分割,图像多相分割问题会增加更多的约束性条件,这使得算法的设计会困难很多。二是乘性噪声较加性噪声更强,对图像的损坏更为严重,乘性噪声图像的对比度往往更差,因此针对乘性噪声图像的多相分割更为复杂；三是考虑图像灰度分布的不均匀性。针对以上问题,我们提出一种处理具有乘性Gamma噪声和灰度分布不均匀的图像多相分割模型,并讨论解的存在性问题。结合分裂Bregman算法和交替极小算法,我们给出一种有效的求解算法。并给出模拟图像和真实SAR图像以及B超图像的数值试验来验证所提模型和算法的有效性。
Medical B ultrasound imaging has been largely used in clinical diagnosis. However, speckle noise widely exists in B ultrasound images. Unlike traditional additive white Gaussian noise, speckle noise is a type of multiplicative noise. Multiplicative noise widely exists in SAR im-ages, ultrasound images, laser images and microscope images,and multiplicative noise is much stronger than additive Gaussian noise. The contrast of images corrupted with multiplicative noise is quite low. Suitably removing speckle noise will largely improve the accuracy of clini-cal diagnosis and the efficiency of image analysis. Image segmentation is important for medical B-ultrasound images. In some clinical diagnosis, one needs to quantitatively measure the shape, area, volume and so on, therefore, accurate detection of tissue or organs from a ultrasound image is favorable to judge the pathology state of the tissue and is helpful for surgeons to perform an operation. The difficulties for accurate segmentation of ultrasound images are the appearance of speckle noise, boundary dropout, shadow, etc.. Multi-phase image segmentation is an important task in image processing and computer vision. In the past few years, many multi-phase segmen-tation models and numerical methods have been proposed for images corrupted with additive Gaussian noise. However, these models are not suitable for images corrupted with multiplica-tive noise. Besides, most of previous multi-phase segmentation models are based on piecewise constant image model. However, intensity inhomogeneity occurs in many real world images, for example, it can be caused by non-uniform illumination. Thus, more practical image models should consider the intensity inhomogeneity of images. Variational partial differential equations have been very successful in image processing and many classic models have been proposed to deal with images corrupted with additive Gaussian noise. Corresponding theoretical analysis of these models is well presented. However, multiplicative noise is totally different from additive noise, therefore, new mathematical models and the corresponding theoretical analysis should be developed.
In this paper, we are concentrated on following four problems:first, for a clinical ultra-sound image model, we propose a convex speckle reduction model, and design a fast and effec-tive algorithm for the proposed model; secondly, we propose a ultrasound image segmentation model by combining the gray level statistics of clinical ultrasound images, then, we propose a fast primal-dual algorithm to solve the proposed model; thirdly, to deal with the boundary dropout of ultrasound kidney images, we propose a ultrasound kidney segmentation model by incorporation the prior shape information of kidney; at last, we propose a multi-phase image segmentation model and algorithm for images with multiplicative Gamma noise and intensity inhomogeneity, and give an existence theorem of solutions.
This paper is organized as follows:
In the first chapter, we briefly introduce the procedure of ultrasound imaging and the char-acteristics of ultrasound images. Then, we introduce the backgrounds and development of vari-ational partial differential equations in speckle reduction and image segmentation. At last, we show the goal of our research work.
In the second chapter, we work on removing speckle noise in ultrasound images. For a clinical ultrasound image model, we propose a convex variational model and existence, u-niqueness and boundedness of the solution to the variational model is discussed. Besides, we introduce an auxiliary variable, and solve an equivalent constrained problem by using a Breg-man iteration method. By doing this, we get two simple subproblems, and each subproblem is solved alternatively until a stopping rule satisfied. The numerical experiments show that the proposed algorithm is about two times faster than the traditional gradient descent method. We also compared our method with other popular methods on synthetic images and real ultrasound images, and the experiments show the advantage of the proposed method.
In the third chapter, we consider fast segmentation of ultrasound images. By considering the ultrasound imaging procedure and the gray level statistics of log-compressed ultrasound images, we propose a ultrasound image segmentation model based on a maximum likelihood estimation method. On the basis of level set method, we propose to solve the corresponding relaxed model related to a membership function. The relaxed model is convex with respect to the membership function. We discuss the relation between the proposed active contour model and the corresponding relaxed model. To overcome the non-differentiable TV term, we pro-pose a primal dual algorithm to solve the relaxed model. Numerical experiments show that the proposed method is quite fast and efficient, and the solution is in some sense of a global optima.
In the fourth chapter, we consider the segmentation of ultrasound kidney images. Based on the segmentation model in the third chapter, we propose a ultrasound kidney segmentation mod-el by combining the prior shape information of kidney anatomical structure. The prior shape information relies on the anatomical structure of the kidney, therefore, it is not necessary to con-struct a prior shape data base. As we know, the construction of a prior shape data base is very time consuming and quite complicated. Numerical experiments show that the proposed method can effectively segment a ultrasound kidney images when the ultrasound kidney images suffer from boundary dropout. We compare the segmentation results of our method with the hand drawing kidney boundaries by radiologists, and the errors are below6%in all our experiments.
In the last chapter, we consider multi-phase segmentation of images in presence of inten-sity inhomogeneity and multiplicative noise. This problem involves following three difficult subproblems. The first problem is the multi-phase image segmentation problem. Comparing with binary segmentation, multi-phase segmentation models usually bring more constraints, which is much more difficult to be numerically solved. The second problem is the existence of multiplicative noise, which is much stronger than additive noise. The contrast of a multiplica-tive noise image is quite low, therefore, the multi-phase segmentation of a multiplicative noise image is more difficult. The third problem is the combination of intensity inhomogeneity in multi-phase segmentation. We first present a multi-phase segmentation model for images with intensity inhomogeneity and multiplicative Gamma noise, and show the existence of a mini-mizer of the proposed model. Then, we propose an effective numerical algorithm by combining split Bregman method and alternating minimization method. Numerical experiments are shown by comparison with other two methods on both synthetic images and real images to illustrate the efficiency of the proposed method.
引文
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