Weighted L1 and L2 Norms for Image Reconstruction: First Clinical Results of Electrical Impedance Tomography Lung Data
Abstract
Image reconstruction is an inverse problem which can be formulated using quadratic objective functionals (Least Square fittings or L2 norms) and absolute values sum- mations (L1 norms). The L1 and L2 norms can be inde- pendently applied over the data mismatch and the regular- ization terms (image term) of an inverse problem. In this manuscript, we investigate weighted L1 and L2 norms in constituting a general inverse problem and reconstruct im- age using Primal-Dual Interior Point Method (PDIPM). We propose a generalized inverse problem to independently mix the smooth properties of the L2 norm based objective func- tionals with the blocky effect of the L1 norm based objec- tive functionals on a element by element basis through a weighting strategy. In our implementation, we use Electrical Impedance Tomography (EIT) as an instance of ill-posed, non-linear inverse problem. We investigate the effectiveness of different combinations of weighted L2 and L1 norms in dealing with measurement uncertainties, such as measure- ment noise and data outliers, using both EIT simulated data, and EIT human lung data. The simulated data is produced for a 2D circular phantom and EIT conductivity images are reconstructed. The first clinical results of applying weighted L1 and L2 norms to reconstruct image of EIT lung data us- ing a 2D thorax-shape mesh are reported.