A hybrid regularization algorithm for high contrast tomographic image reconstruction

The common Level set based reconstruction method (LSRM) is applied to solve a piecewise constant inverse problem using one level set function and considers two different conductivity quantities for the background and the inclusion (two phases inclusion). The more the number of the piecewise constant conductivities in the medium, the higher the calculation effort of the LSRM, using multiple level set functions, will be. The assumption of piecewise constant conductivities (coefficients) is to discriminate between two regions with sharp conductivity interface; however, it may not be a realistic assumption when there are smooth conductivity gradients inside each region as well. In this paper, we propose a hybrid regularization method (HRM), which is a two steps solution, to solve ill-posed, nonlinear inverse problem with smooth conductivity transitions. The first step of this hybrid inversion framework plays the role of an initializing procedure for the second step, and acts in a similar way as a source type inversion method. In the first stage, the LSRM with one level set function is applied to determine the region of interest, which is defined as the region with sharpest interface. Then in the second stage, an inverse solver with penalty terms based on sum of absolute values (L1 norms), which are highly robust against measurement errors, is applied to reconstruct the conductivity changes inside the determined ROI. The generated forward solution in the final iteration of the level set is fed to the second stage where the L1 norms based penalty terms are minimized using primal-dual interior point method (PDIPM). The PDIPM has been shown to be effective in minimizing the L1 norms. The reconstructed images with the proposed HRM maintains the edge information as well as the smooth conductivity variations, a trait absent in all previously established level set based reconstruction method. The integration of the LSRM and the PDIPM can generate less noisy reconstructed images when comparing either with the reconstruction results of the PDIPM or with those of squared error based reconstruction methods, such as Gauss-Newton (GN) method. Our proposed HRM is tested on a circular 2D phantom with either sharp conductivity gradients (piecewise constant coefficients) or smooth conductivity transitions (smooth coefficients). We show that the proposed HRM maintains sharp edges and is robust against the measurement noise.