Inverse Laplace Transform of Multidimensional Relaxation Data Without Non-Negativity Constraint

An algorithm based on Tikhonov regularization in generalized form is described to perform an inverse Laplace transform of multidimensional data without a non-negativity (NN) constraint for spectrum conditioning. Uniform penalty (UP) regularization is used to reduce the requirement for NN, and a further penalty is introduced for zero-crossing (ZC) of the spectrum. This ZC term is weighted with the slope of the curve, which does not prevent negative modes in the spectrum but makes nonphysical undershooting in the vicinity of narrow peaks more expensive. The performance of this algorithm is demonstrated using synthetic data, and the optimization of the free parameters for calculating the regularization matrix is discussed.