Figure 1.

Flowchart for the regularized optimization (RO) algorithm. Input parameters are: experimental projections D⃗ ; angles Θ⃗ at which projections were measured; goals of regularization terms: ${\mathrm{\Gamma }}_{\mathrm{R}}^{\mathrm{g}},{\mathrm{\Gamma }}_{\mathrm{O}}^{\mathrm{g}}$ with error limits: δ_R, δ_O; expected lower and upper limits for linewidths dH_min and dH_max; positions in the sample with significant radical concentration, Ω; and the lineshape description. The goal of RO is to find the profiles ρ⃗_O and ρ⃗_R that minimize discrepancy with the data provided that $\left|\Vert {\mathrm{L}}_{\mathrm{O}}{\overrightarrow{\mathrm{\rho }}}_{\mathrm{O}}\Vert -{\mathrm{\Gamma }}_{\mathrm{O}}^{\mathrm{g}}\right|\le {\mathrm{\delta }}_{\mathrm{O}}\text{ and }\left|\Vert {\mathrm{L}}_{\mathrm{R}}{\overrightarrow{\mathrm{\rho }}}_{\mathrm{R}}\Vert -{\mathrm{\Gamma }}_{\mathrm{R}}^{\mathrm{g}}\right|\le {\mathrm{\delta }}_{\mathrm{R}}$. The search strategy can be divided into two steps. In the first step the discrepancy error ∑ is minimized without keeping track of changes in smoothness of the profiles. When the lowest error is reached, the algorithm searches (step 2) for solutions that are smooth enough to match the criteria ${\mathrm{\Gamma }}_{\mathrm{R}}^{\mathrm{g}}\text{ and }{\mathrm{\Gamma }}_{\mathrm{O}}^{\mathrm{g}}$. At this step the discrepancy error may increase at the expense of smoothing of the profiles and improving the accuracy of ρ⃗_O and ρ⃗_R.