Medical B-Ultrasound Image Processing Based On Variational PDEs

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.