Theory Of Ultrasonic Tomography

Ultrasound tomography technique is to determine from scattered ultrasonic wave the internal material properties of object. Due to ultrasound nonionizing radiation, harmless to human, and cheap to produce, ultrasound tomography technique has broadly applied to various fields such as biomedical engineering, geophysics, nondestructive evaluation (NDT) and pattern recognition etc.. When ultrasonic wave transmit in inhomogeneous media, the interaction between ultrasonic wave and the inhomogeneous media are complex, thus, make the relation between scattered fields and unknown function, which described the object's properties of ultrasound, nonlinear. In order to solve the unknown function from the nonlinear scattering equations, one must appeal to iterative method. Various iterative methods were proposed by pioneers who devoted to ultrasound tomography technique, but all these methods will meet an ill-posed equation and how to solve the ill-posed equation is the key for these iterative methods. For ill-posed problem, little change in data will cause a great variation in solution, so the convergence of these iterative methods will depend strongly on the regularization method for solving it. This thesis dedicate to how to deal with the ill-posed problem in iterative procedure. We first analyze various models by Picard theory, and give a simple tool for analyzing the degree of the model contaminated by noise. Then, we proposed four methods which fall into two classes for handling the ill-posed problems. By revised the strategies for choosing regular parameter, all these methods convergence to the true solution of the problems for objects with high contrast.The first class of regular method is named stationary method which include two methods, truncated singular value decomposition (TSVD) method and truncated total least squares (TTLS) method. For TSVD method, by choosing an appropriate truncate parameter k, we truncated the small singular value after k which in the expression of least squares solution, because small singular values will affect the solution greatly when there are noises in data of the ill-posed problem. For TTLS method, we consider both data and coefficient matrix of equation system are contaminated by noise. By using TSVD and TTLS regular method and revised the strategies for choosing truncated parameters k, the outer iteration convergence to the true solution of the problem for high contrast object with no extra information.We call the second class of regular method for solving ill-posed problem toiterative method, which include conjugate gradient least squares (cgls) method and LSQR method are belong to projection methods on Krylov. Cgls is a method which applied CG method on normal equation while LSQR method is the method use the Lancozs bidiagongal procedure on normal equation. For these two methods, the regular parameter is the iterative number and how to choose the parameter is the key for solving inverse scattering problem. By studying the spectrum properties of coefficient matrix and the regular parameters at the iterative procedure of the first class regular method (TSVD and TTLS), we assign a uniform parameter for the inner iteration (cgls and LSQR). Simulation results show that this method for choosing regular parameter is correct and the outer iterate convergence for high contrast object. By analyzing in theory and simulating in practice, we prove that all these methods we proposed equipped with an appropriate strategy for choosing regular parameter can solve the nonlinear ultrasound inverse scattering problem well for object with high contrast.