Non-negative Least Squares using Multi-regularization and a Gaussian Basis, with Application to Magnetic Resonance Relaxometry

The discretized Fredholm equation of the first kind (FEFK) is encountered frequently in physics. Major applications include magnetic resonance relaxometry (MRR) and fluorescence decay analysis. Tikhonov regularization is often used to stabilize the ill-posed problem of solving the FEFK from data.  However, the choice of regularization parameter, is problematic. Current approaches identify an optimal value of, with the corresponding solution taken as the recovered distribution function (DF).  However, we observe that DF’s obtained from nearly indistinguishable signals respond differently to regularization over a range of values, indicating the substantial information content of regularized solutions corresponding to non-optimal. Accordingly, we have developed a multiple regularization (MultiReg) approach to solving the FEFK by incorporating a linear combination of solutions corresponding to a set of values. We demonstrate an on average >30% improved accuracy and precision across a wide range of synthetic target DF, and find a markedly reduced reliance on the selection of an optimal.  We show the application of MultiReg to MRR T2 relaxometry data.  MultiReg represents a potentially significant advance in regularization of ill-posed inverse problems arising from the FEFK in physics.