A study on regularization parameter choice for interior tomography based on truncated Hilbert transform

Abstract

ABSTRACT For interior tomography based on truncated Hilbert transform (THT), the recently proposed truncated singular value decomposition (TSVD) reconstruction method uses a regularizati on parameter given directly. In this paper, a method of choosing the regularization parameter is presented based on L-curve to get an optimal regularization parameter in theoretical sense. Furthermore, we develop a Tikhonov regula rization method in comparison to TSVD. Simulation results indicate that both of the two regularization methods with the optimal regularization parameters have good performances on the image quality for both cases of noise-free and noisy projections. Keywords: Interior Tomography, Truncated Hilbert Transform, L-curve, Truncated Singular Value Decomposition, Tikhonov Regularization 1. INTRODUCTION It had been believed that the solution to the interior tomography is not unique fo r a long time. Recently, inspired by the concept of the differential backprojection (DBP) proposed by Noo et al [1] and Pan et al [2], a novel solution for the interior problem was proposed with numerical results demonstrat ed that the interior problem can be uniquely and stably solved if a sub-region inside a region of interest (ROI) is known [3]-[5]. Similar results were also independently proposed by other researchers [6]-[8]. In these papers, image reconstruction was reduced to the inversion of THT via the projection onto convex set (POCS) method, which is iterativ e and computationally expensive. To over above limitations, TSVD as a regularization method has been proposed for inversing the THT, which runs two orders of magnitude faster than the iterative method [9]. However the regularization parame ter in this method is given directly without theoretical analysis. The selection of the regularization parameter that controls the trade-off between fidelity to the data and constraint on the reconstruction result is a primary problem in the regularization method. In this paper, we present a method of selecting the optimal regularization based on the L-curve for TSVD method for solving the interior tomography based on truncated Hilbert transform. In this method, the optimal regularization parameter is computed theoretically using the maximum curvature of the L-curve. Moreover, we develop a Tikhonov regularization method in comparison to TSVD. Root Mean Sq uare Error (RMSE) of reconstr ucted image with respect to different regularization parameters validates our selecting regularization parameter method. The paper is organized as follows: section 2 gives Mathematical formulation, TSVD, Tikhonov regularization, the L-curve. Section 3 exhibits the simulation results. Section 4 is conclusion.