A method for choosing the regularization parameter in generalized Tikhonov regularized linear inverse problems

This paper presents a systematic and computable method for choosing the regularization parameter appearing in Tikhonov-type regularization based on non-quadratic regularizers. First, we extend the notion of the L-curve, originally defined for quadratically regularized problems, to the case of non-quadratic functions. We then associate the optimal value of the regularization parameter for these non-quadratic problems with the corner of the resulting generalized L-curve. We identify the corner of this L-curve as the point of tangency between a straight line of arbitrary slope and the L-curve. This definition results in a corresponding algebraic equation which the optimal regularization parameter must satisfy. This algebraic equation naturally leads to an iterative algorithm for the optimal value of the regularization parameter. The convergence of this iterative algorithm is established. Simulation results confirm that the proposed method yields values of the regularization parameters that result in good reconstructions for non-quadratic problems.