FjkF2k cos[2kxAx 01 k 02k

The third term, the Young's fringes term proper, is modulated by a cosine pattern whose wavelength is inversely proportional to the size of the image shift, and whose direction is in the direction of the shift. Since a fixed phase relationship holds only within the domain where the Fourier transform is dominated by the signal common to both superimposed images, while the relationship is random outside of that domain, the cosine fringe pattern induced by the shift can be used to visualize the extent of the resolution domain.

Figure 3.22 Computed diffraction patterns showing Young's fringes. The patterns are obtained by adding two different electron micrographs of the same specimen area and computing the power spectrum of the resulting image. (a-c) The micrographs are added with different horizontal displacements (corresponding to 10, 30, and 50 A). The intensity of the resulting pattern follows a cosine function. (d) A pattern with approximately rectangular profile is obtained by linear superposition of the patterns (a-c) using appropriate weights. From Zemlin and Weiss (1993), reproduced with permission of Elsevier.

Figure 3.22 Computed diffraction patterns showing Young's fringes. The patterns are obtained by adding two different electron micrographs of the same specimen area and computing the power spectrum of the resulting image. (a-c) The micrographs are added with different horizontal displacements (corresponding to 10, 30, and 50 A). The intensity of the resulting pattern follows a cosine function. (d) A pattern with approximately rectangular profile is obtained by linear superposition of the patterns (a-c) using appropriate weights. From Zemlin and Weiss (1993), reproduced with permission of Elsevier.

Moreover, the position of the Young's fringes is sensitive to the phase difference between the two transforms. When two images of an object are obtained with the same defocus setting, the phases are the same. Consistent shifts affecting an entire region of Fourier space show up as shifts of the fringe system. When two images of an object are obtained with different defocus settings, then the fringe system shifts by 180° wherever the contrast transfer functions differ in polarity (Frank, 1972a).

Using digital Fourier processing, the waveform of the fringes can be freely designed, by superposing normal cosine-modulated patterns with different frequencies (Zemlin and Weiss, 1993). The most sensitive detection of the band limit is achieved when the waveform is rectangular. Zemlin and Weiss (1993) obtained such a pattern of modulation experimentally (figure 3.22d).

5.2.6. Statistical Limitations of Halfset Criteria

Criteria based on splitting the data set in half in a single partition (as opposed to doing it repeatedly for a large number of permutations) are inferior because of large statistical fluctuations. The results of an evaluation by de la Fraga et al.

(1995) are interesting in this context. By numerical trial computations, these authors established confidence limits for DPR and FRC resolution tests applied to a data set of 300 experimental images of DnaB helicase. The data set was divided into halfsets using many permutations, and corresponding averages were formed in each case. The resulting DPR and FRC curves were statistically evaluated. The confidence limits for both the DPR determination of 10 Fourier units and the FRC determination of 13 units were found to be ±1 unit. This means that even with 300 images, the resolution estimates obtained by DPR45 or FRC0.5 may be as much as 10% off their asymptotic value. Fortunately, though, the number of particles in typical projects nowadays go into the thousands and tens of thousands, so that the error will be much smaller.

Another obvious flaw has not been mentioned yet: the splitting of the data set in half worsens the statistics, and will inevitably underreport resolution. Since the dependence of the measured resolution on the number of particles N entering the reconstruction is unknown, there is no easy way to correct the reported figure. Morgan et al. (2000) chose to determine this relationship empirically, by extrapolating from the trend of a curve of FSC0.5 as a function of log(N), which the authors obtained by making multiple reconstructions with increasing N.

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