F1k2 F2k212V

Again, as in the definition of the DPR in the previous section, the notation under the sum refers to the terms that fall into a ring of certain radius and width. The FRC curve (see figure 3.21) starts with a value of 1 at low spatial frequencies, indicating perfect correlation, then falls off more or less gradually, toward a fluctuating flat region of the curve with values that originate from chance correlation.

Here, the resolution criterion is derived in two different ways: (i) by comparison of the FRC measured with the FRC expected for pure noise, FRCnoise = 1/(N[k, Ak])1/2, where %, Ak] denotes the number of samples in the Fourier ring zone with radius k and width Ak, or (ii) by comparison of the FRC measured with an empirical threshold value. The term FRCnoise is often denoted as "ct," even though it does not have the meaning of a standard deviation. Using this notation, the criteria based on the noise comparison that have been variably used are 2ct (van Heel and Stoffler-Meilicke, 1985), 3ct (e.g., Orlova et al., 1997), or 5ct (Radermacher et al., 1988, 2001). As empirical threshold, in the second group of FRC-based criteria, the value 0.5 (Bottcher et al., 1997) is most frequently used.

The FRC = 3ct criterion invariably gives much higher numerical resolution values than FRC = 0.5. A critical comparison of the 0.5 and the 3ct criteria, and a refutation of some of the comments in Orlova et al. (1997) are found in Penczek's appendix to Malhotra et al. (1998). A data set is considered that is split into halves, and the FRC is calculated by comparing the halves. It is shown here that FRC = 0.5 corresponds to SNR = 1. In other words, in the corresponding shell, the noise is already as strong as the signal. Therefore, the inclusion of data beyond this point appears risky. More about the application of resolution criteria to 3D

reconstructions, as opposed to 2D averages considered here, will be found in chapter 5.

DPR Versus FRC Regarding the relationship between FRC and DPR, experience has generally shown that the FRC = 3a gives consistently a more optimistic answer than DPR = 45°. In order to avoid confusion in comparisons of resolution figures, some authors have used both measures in their publications. Some light on the relationship between DPR and FRC has been shed by Unser et al. (1987), who introduced another resolution measure, the SSNR (see section 5.2.8). Their theoretical analysis confirmed the observation that always kFRCpa] > kDPR[45o, for the same model data set. Further illumination of the relative sensitivity of these two measures was provided by Radermacher (1988) who showed, by means of a numerical test, that an FRC = 2 x 1/N1/2 cutoff (the criterion initially proposed, before a factor of 3 was adopted) (Orlova et al., 1997) is equivalent to a SNR of 0.2, whereas the DPR 45° cutoff is equivalent to SNR = 1. Thus, the FRC cutoff, and even the DPR cutoff with its fivefold increased SNR seem quite optimistic; on the other hand, for well-behaved data the DPR curve is normally quite steep, so that even a small increase in the FRC cutoff will often lead to a rapid increase in SNR.

Another observation by Radermacher (1988), later confirmed by de la Fraga et al. (1995), was that the FRC cutoff of FRC = 2a corresponds to a DPR cutoff of 85°.

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