It is important to know how large n must be made for the variance estimate to be reliable. Penczek (2002b) found that in a typical case, about 30 reconstructions will suffice. Despite the high speed of today's computers, the computation of this number of reconstructions is still substantial. Penczek (2002b) therefore suggested that the Fourier interpolation method of reconstruction be used instead of the weighted back-projection, for increased speed. Fourier interpolation, even though it results in a much cruder reconstruction quality, has the advantage that it allows the different reconstructions from permutations of the projection set to be obtained in a very simple way, by subtracting or adding Fourier transforms of projections on the appropriate central planes according to the realization of the random draw. [Note, though, that the recent addition of a fast gridding method (Penczek et al., 2004b) promises a way to obtain reconstructions both fast and accurately, which means that the accuracy of the variance computation does not need to be compromised.]
It must be noted that the variance obtained by the bootstrapping method is different from, and normally larger than, the variance of the original volumes. The reason is that in addition to the desired structural component, there are two terms, one stemming from the reconstruction method itself and its numerical errors, the other from the unevenness of the angular distribution of projections.
An application of the bootstrapping method to a realistic problem is found in Spahn et al. (2004a): given was a heterogeneous data set of ribosome images. The source of the heterogeneity was the partial occupancy of a ligand, EF-G: a certain percentage of the ribosomes were bound with EF-G, the rest were empty. As expected, the variance estimate showed high variance in the region of EF-G binding.
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