Several Problems In Computerized Tomography With A Discontinuous Solution

In this thesis, we mainly discuss problems of solving Radon integral equations arisen in the field of Computerized Tomography. Because of the discontinuity of the original functions in CT, how to design an accurate and efficient algorithm to reconstruct the discontinuous solution is a difficult problem.We propose an algorithm named "piece-wise Tikhonov regularization" in Chapter 3. The form of the piece-wise Tikhonov regularization functional is like‖Rf-g‖2+α‖f‖Hm(∪Di)2. Compared to classical Tikhonov regularization, this kind of piece-wise method will function better in the reconstruction of a discontinuous solution, it has bet-ter resolution in the neighborhood of discontinuous boundaries. In the end of Chapter 3, we give a two-step adaptive method which can automatically detect the discontinuous boundaries, and reconstruct using our piece-wise Tikhonov regularization.In Chapter 4, we give an iterative algorithm using level set methods toward the problem of discontinuous solutions in CT. We use‖Rf -g‖2+α‖f‖BV+β‖f‖2 as a Tikhonov regularization functional. By applying the theory of shape sensitivity analysis, we easily get the Euler derivative of the functional, meanwhile, we deal with the change of the shape by level set methods. In the end of Chap.4, we give some numerical examples to demonstrate the algorithm.