Fourier shell correlation

The quality of 3-D reconstructions should be assessed at various steps of the data processing to ensure data convergence and to measure the effective resolution of the final 3-D map. An obvious measurement that should be performed is to calculate the cross-correlation coefficient of the computed 3-D reconstruction and the true structure of the object. However, because the true structure is the very entity that is being reconstructed and, thus, is unknown in most situations, it is impossible to perform such a comparison. Instead, a widely used approach is to estimate how well two independent reconstructions agree with one another by calculating the differential phase residues (DPR) (Fig. 6a) and the Fourier shell correlation (FSC) coefficients (Fig. 6b) between the two maps (Bottcher et al, 1997a Zhou et al, 1994 van Heel, 1987). The spatial frequency, where the FSC reaches 0.5 or the DPR increases to 45°, is commonly considered to be the effective resolution of the reconstructions therefore, the term resolution does not really mean resolvability so much as reproducibility. 

Fig. 6. Resolution assessments, (a) Differential phase residual method based on the 45 phase difference criterion used in the 8.5-A resolution (indicated by arrow) structure of the herpesvirus capsid (Zhou et al., 2000). (b) Fourier shell correlation method based on the 0.5 Fourier shell correlation coefficient criterion used in the assessment of effective resolution of the rice dwarf virus (RDV) structure at 6.8 A (Zhou et al, 2001). The imperfect FSC value (<1.0) is partly due to the presence of nonicosahedrally ordered dsRNA genomes within the RDV virions. [Adapted from Zhou et al. (2000, 2001), with permissions from the publishers.]...

The reliability and resolution of the final reconstruction can be measured by use of a variety of indices. For example, the differential phase residual (DPR) (133), the Fourier shell correlation (FSC) (134), and the Q-factor (135) are three such measures. DPR is the mean phase difference, as a function of resolution, between the structure factors from two independent reconstructions, often calculated by splitting the image data into two halves. FSC is a similar calculation of the mean correlation coefficient between the complex structure factors of the two halves of the data as a function of resolution. The Q-factor is the mean ratio of the vector sum of the individual structure factors from each image divided by the sum of their moduli, again calculated as a function of resolution. Perfectly accurate measurements would have values of DPR, FSC, and Q-factor of O ", 1.0, and 1.0 respectively, whereas random data containing no informa-... 

From equation 5, it is apparent that each shell of scatterers will contribute a different frequency of oscillation to the overall EXAFS spectrum. A common method used to visualize these contributions is to calculate the Fourier transform (FT) of the EXAFS spectrum. The FT is a pseudoradial-distribution function of electron density around the absorber. Because of the phase shift [< ( )], all of the peaks in the FT are shifted, typically by ca. —0.4 A, from their true distances. The back-scattering amplitude, Debye-Waller factor, and mean free-path terms make it impossible to correlate the FT amplitude directly with coordination number. Finally, the limited k range of the data gives rise to so-called truncation ripples, which are spurious peaks appearing on the wings of the true peaks. For these reasons, FTs are never used for quantitative analysis of EXAFS spectra. They are useful, however, for visualizing the major components of an EXAFS spectrum. 

Alternatively, the fluctuating velocity field can be characterized by the energy spectrum defined as the Fourier transform of the two-point velocity correlation integrated over a spherical shell of wavenumbers with magnitude k ... 

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