Figure 1
The mean and std of the \(d_1\) and \(d_2\) measures (illustrated as, respectively, a dot and an error bar) for different methods and different sampling schemes, in the presence of Gaussian noise (a) 488 (\(61\times 8\))21, and (b) 15221 (\(19\times 8\))31 directions (the y-axis in \(d_1\) is scaled logarithmically). Arithmetic sum: simple arithmetic averaging; Lebedev: weighted averaging by31; Knutsson: weighted averaging by43; SH: Spherical harmonic method for powder averaging by3 \(L = 2, \; 4, \; 6\), shows the order in spherical harmonic representation; trace(M)/3: powder average signal from Eq. (10); MAP: direction-averaged signal using MAP-MRI44 for \(N_{\max} = 6\) and 8; and MAPL: direction-averaged signal using MAP-MRI with Laplacian regularization51. The ‘s’ and ‘ns’ correspond to the shelled and non-shelled point sets, respectively. Different colors (blue, red, yellow, purple, green) show the results in different noise levels (\(\sigma _g = 0.1414,\, 0.0707,\, 0.0283,\, 0.0071,\, 0.0014\)).
