由線性彈性模型引起的多維系數反問題在很多範疇都有其應用，如斷層探測、油田檢測、鹽石檢測、礦石檢測及醫療成像等。瞬時成像技術是其中最有用的應用。它提供了一個快速及安全的醫療成像技術，可以用來檢測在身體內快速移動的器官的一些異常組織，如肝腫瘤。在這篇論文中，我們會重點討論兩個解決瞬時弹性成像反問題的數值方法，即水平集反演方法和近似全局收斂方法。我們會研究這兩種方法的推導和數值結果。 / 特別地，近似全局收斂方法是一種由Klibanov 新提出用來解決由雙曲偏微分引起的多維系數反問題的方法。因為這佪方法沒有使用求泛函極小值的步驟，因此能避免了一些眾所周知的問題，所以它特別穩定。數值結果顯示近似全局收斂方法對噪聲有很高的穩定性。這表明近似全局收斂方法是一個解決由線性彈性模型引起的多維系數反問題的其中一個有效方法。 / Multi-dimensional coefficient inverse problem (MCIP) in linear elasticity has found many applications, such as crack detection, oil/salt/ore detection, medical imaging. Transient elastography is among one of the most useful applications, providing a fast and safe medical imaging technique which can be used to detect tumors or abnormal tissue in “fast-moving“ organs such as the liver. In this thesis focus is casted on two of the numerical algorithms to solve inverse problems related to transient elastography, namely the level-set inversion method and the approximate globally convergent method. The derivations of both methods and numerical results are presented. In particular, the approximate globally convergent method is a newly developed stable method to solve coefficient determination inverse problem for hyperbolic partial differential equation proposed by Beilina and Klibanov in [6]. It achieves pproximately a global convergence by avoiding construction of a least squares functional, thus averting some of the well-known problems of trapping in the neighborhoods of local minima when one minimizes such a nonlinear functional. The results of the approximate globally convergent method have shown its strong stability and robustness. This suggests a good way for the reconstruction of the distribution of the shear modulus in the coefficient inverse problem of linear elasticity. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Chow, Yat Tin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 98-102). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Linear Elasticity Model --- p.9 / Chapter 2.1 --- Introduction to Linear Elasticity Model --- p.9 / Chapter 2.2 --- Physical Meanings of Elasticity Equation --- p.12 / Chapter 2.3 --- Derivations of Linear Elasticity Equation --- p.14 / Chapter 2.4 --- Discussion of Christoffel’s Equation --- p.17 / Chapter 3 --- Formulations of the Forward and Inverse Problem --- p.27 / Chapter 3.1 --- The Forward Problem --- p.29 / Chapter 3.2 --- The Inverse Problem --- p.29 / Chapter 3.3 --- A Uniqueness Result --- p.30 / Chapter 4 --- Algorithms for Inverse Problems in Elasticity --- p.33 / Chapter 5 --- Level Set Inversion Method --- p.37 / Chapter 5.1 --- Arrival Time Acquisition: Cross-Correlation --- p.37 / Chapter 5.2 --- The Distance Inversion Method --- p.41 / Chapter 5.3 --- Solving the Forward Eikonal Equation --- p.43 / Chapter 5.3.1 --- Discretizing the eikonal equation --- p.43 / Chapter 5.3.2 --- A forward eikonal solver: fast marching algorithm --- p.47 / Chapter 5.3.3 --- A forward eikonal solver: fast sweeping algorithm --- p.50 / Chapter 5.4 --- Level Set Inversion Scheme --- p.54 / Chapter 5.5 --- Numerical Implementation --- p.57 / Chapter 5.6 --- Results of Reconstructions --- p.58 / Chapter 6 --- Approximate Globally Convergent Method --- p.63 / Chapter 6.1 --- The Forward Problem --- p.65 / Chapter 6.1.1 --- Forward problem in time domain --- p.65 / Chapter 6.1.2 --- Forward problem in Laplace domain --- p.67 / Chapter 6.2 --- The Inverse Problem --- p.67 / Chapter 6.2.1 --- Inverse problem in time domain --- p.67 / Chapter 6.2.2 --- Inverse problem in Laplace domain --- p.68 / Chapter 6.3 --- A Nonlinear Integral Differential Equation --- p.69 / Chapter 6.4 --- Approximation of the First Tail --- p.71 / Chapter 6.5 --- The Algorithm --- p.72 / Chapter 6.6 --- Notes About the Convergence Analysis --- p.76 / Chapter 6.6.1 --- Approximate global convergence --- p.76 / Chapter 6.6.2 --- Basic formulation of Theorem 2.9.4 of [6] --- p.78 / Chapter 6.6.3 --- Some ideas of the convergence analysis for the algorithm in section 6.5 --- p.80 / Chapter 6.7 --- Numerical Implementation --- p.81 / Chapter 6.8 --- Results of Reconstructions --- p.88 / Chapter 7 --- Conclusions --- p.96 / Bibliography --- p.98

在各种临床应用广泛使用的诊断和治疗工具中，超声成像是其中的一个。与其他成像模式相比，比如计算机断层照相法和磁共振成像，超声波检查法有许多优点:没有辐射风险，设备价格低以及能够实时获取图像。很多超声应用的第一步通常是对感兴趣组织和结构的检测和定位。然而，超声图像存在一些特有的伪影，比如高噪声，低信噪比和灰度不均，这些伪影使得检测任务变得困难。此外，感兴趣区域之间的低对比度也使得这一任务变得更加复杂。在这篇论文里，我们深入研究这些图像伪影并提出新的方法来促进临床中的超声应用。 / 首先，我们提出一个多尺度的框架来进行超声图像的分割，这个框架是基于各向异性去噪扩散和测地主动轮廓线的。各向异性去噪扩散是对边缘敏感且专门用于斑点噪声图像的扩散过程，这里它被用来去除超声图像的斑点噪声，我们对每幅输入图像构造一个多尺度的表示方法，随着尺度的增加，噪声被逐渐地消除。之后，多尺度测地主动轮廓线从粗到细渐进地应用到这些尺度来提取物体的边界线。为了避免在低对比度区域出现边界泄漏的情况，我们把不同尺度之间的边界形状相似性结合到传统的测地主动轮廓线模型里作为一个外部约束来指导轮廓线的演化。在合成和临床图像的实验结果证明了我们的方法的优越性。 / 其次，我们提出一个基于相位的方法来检查和测量超声图像里的胎儿腹部轮廓线。我们定义了一个基于局部相位的度量来检测胎儿腹部的边界线，这个度量称为多尺度特征非对称性，它与图像的亮度无关，并且能为图像里特征的重要程度提供一个绝对的测量。为了估计与腹部轮廓线相吻合的椭圆，我们使用一个迭代随机霍夫变换来排除内腹部边界线的影响，从而使得估计的椭圆逐渐收敛到外边界线。在临床超声图像里进行腹部周长测量的实验结果验证了我们的方法与手工的方法有很高的一致性，这也表明我们的方法可以作为一个可靠的工具来进行产科的护理和诊断。 / Ultrasound imaging is one of the most widely used diagnostic and therapeutic tools for a variety of clinical applications. Compared with other imaging modalities, such as computed tomography and magnetic resonance imaging, ultrasonography has a lot of advantages: free of radiation risk, low cost of acquisition and images are available in real-time. The first step in many ultrasonic applications is usually the detection and localization of interested tissues and structures. However, there are a number of characteristic artifacts in ultrasound images that make the task difficult such as high speckle noise, low signal-to-noise ratio and intensity inhomogeneity. Besides, the low contrast between regions of interest further complicates the processing. In this thesis, we deeply investigate these image artifacts and propose new techniques to facilitate ultrasonic applications in clinic. / First, we propose a multiscale framework for ultrasound image segmentation based on speckle reducing anisotropic diffusion(SRAD) and geodesic active contours (GAC). SRAD is an edge-sensitive diffusion tailored for speckled images, and it is adopted here to reduce speckle noise by constructing a multiscale representation for each input image, where the noise is gradually removed as the scale increases. Multiscale geodesic active contours are then applied along the scales in a coarse-to-fine manner to capture the object boundaries progressively. To avoid boundary leakages in low contrast regions, traditional GAC model is modified by incorporating the boundary shape similarity between different scales as an external constraint to guide the contour evolution. Experimental results in both synthetic and clinical images demonstrate the superiority of the proposed approach. / Second, we propose a phase-based approach for fetal abdominal contour detection and measurement in ultrasound images.We define a local phase-based measure, called multiscale feature asymmetry (MSFA), from the monogenic signal to detect the boundaries of fetal abdomen. The MSFA measure is intensity invariant and provides an absolute measurement for the significance of features in the image. In order to estimate the ellipse that fits to the abdominal contour, we employ an iterative randomized Hough transform to exclude the interferences of the inner boundaries of the abdomen, after which the estimated ellipse gradually converges to the outer boundaries. Experimental results in clinical ultrasound images validate the high agreement between our approach and manual approach in the measurement of abdominal circumference, indicating that the proposed approach can be used as a reliable tool for obstetric care and diagnosis. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Wang, Weiming . / Thesis (Ph.D.) Chinese University of Hong Kong, 2014. / Includes bibliographical references (leaves 68-84). / Abstracts also in Chinese. / Wang, Weiming .

[Truncated abstract] Transrectal ultrasound (TRUS) has become the urologist’s primary tool for diagnosing and staging prostate cancer due to its real-time and non-invasive nature, low cost, and minimal discomfort. However, the interpretation of a prostate ultrasound image depends critically on the experience and expertise of a urologist and is still difficult and subjective. To overcome the subjective interpretation and facilitate objective diagnosis, computer aided analysis of ultrasound images of the prostate would be very helpful. Computer aided analysis of images may improve diagnostic accuracy by providing a more reproducible interpretation of the images. This thesis is an attempt to address several key elements of computer aided analysis of ultrasound images of the prostate. Specifically, it addresses the following tasks: 1. modelling B-mode ultrasound image formation and statistical properties; 2. reducing ultrasound speckle; and 3. extracting prostate contour. Speckle refers to the granular appearance that compromises the image quality and resolution in optics, synthetic aperture radar (SAR), and ultrasound. Due to the existence of speckle the appearance of a B-mode ultrasound image does not necessarily relate to the internal structure of the object being scanned. A computer simulation of B-mode ultrasound imaging is presented, which not only provides an insight into the nature of speckle, but also a viable test-bed for any ultrasound speckle reduction methods. Motivated by analysis of the statistical properties of the simulated images, the generalised Fisher-Tippett distribution is empirically proposed to analyse statistical properties of ultrasound images of the prostate. A speckle reduction scheme is then presented, which is based on Mallat and Zhong’s dyadic wavelet transform (MZDWT) and modelling statistical properties of the wavelet coefficients and exploiting their inter-scale correlation. Specifically, the squared modulus of the component wavelet coefficients are modelled as a two-state Gamma mixture. Interscale correlation is exploited by taking the harmonic mean of the posterior probability functions, which are derived from the Gamma mixture. This noise reduction scheme is applied to both simulated and real ultrasound images, and its performance is quite satisfactory in that the important features of the original noise corrupted image are preserved while most of the speckle noise is removed successfully. It is also evaluated both qualitatively and quantitatively by comparing it with median, Wiener, and Lee filters, and the results revealed that it surpasses all these filters. A novel contour extraction scheme (CES), which fuses MZDWT and snakes, is proposed on the basis of multiresolution analysis (MRA). Extraction of the prostate contour is placed in a multi-scale framework provided by MZDWT. Specifically, the external potential functions of the snake are designated as the modulus of the wavelet coefficients at different scales, and thus are “switchable”. Such a multi-scale snake, which deforms and migrates from coarse to fine scales, eventually extracts the contour of the prostate