A Primal Dual - Interior Point Framework for Using the L1-Norm or the L2-Norm on the Data and Regularization Terms of Inverse Problems

Maximum A Posteriori (MAP) estimates in inverse problems are often based on quadratic formulations, corresponding to a Least Squares fitting of the data and to the use of the L2 norm on the regularization term. While the implementation of this estimation is straightforward and usually based on the Gauss Newton method, resulting estimates are sensitive to outliers, and spatial distributions of the estimates that are smooth. As an alternative, use of the L1 norm on the data term renders the estimation robust to outliers, and use of the L1 norm on the regularization term allows reconstructing sharp spatial profiles. The ability therefore of using the L1 norm either on the data term, on the regularization term, or on both is desirable. Use of this norm results though in non-smooth objective functions which require more sophisticated implementations compared to quadratic algorithms. Methods for L1 norm minimization have been studied in a number of contexts, including in the recently popular Total Variation regularization. Different approaches has been used and methods based on Primal Dual Interior Point Methods (PD-IPM) have been shown to be particularly efficient. In the present manuscript we derive a PDIPM framework for using the L1 norm indifferently on the two terms of an inverse problem. We use Electrical Impedance Tomography as an example inverse problem to demonstrate the implementation of the algorithms we derive, and the effect of choosing the L2 or the L1 norm on the two terms of the inverse problem. Pseudo codes for the algorithms and a public domain implementation are provided.