Computing and fitting SSNMR powder patterns with the arithmetic-geometric mean and edge detection

This article presents a highly efficient technique for approximating the ideal SSNMR powder pattern using the arithmetic-geometric mean and demonstrates finding an initial fitting of the ideal powder pattern to an experimental spectrum via Marr-Hildreth edge detection. In particular, the edge detection approach is used to identify possible values for the principal values of the chemical shielding tensor. These possibilities are then evaluated using a heuristic approach for choosing the best estimates of the principal values based on a measure of edge strength and the sign of the third derivative of the broadened experimental spectrum. We present a detailed mathematical development of the ideal SSNMR powder spectrum and of the arithmetic geometric mean and summarize the fundamental ideas of line broadening and edge detection. The algorithms in this article are demonstrated in a program supplied in the appendix and are applied to experimental data from |^l3C ₁|-leucine.