Fig. 1. (a) Averaged power spectra from 10,029 tiles of a cryo-EM image of a bacteriorhodopsin 2D crystal. Pixel size was 1.15 Å and tile size 100 pixels. Tiles were padded into 400 pixel wide squares, Fourier transformed, and the intensities were averaged. The diffraction spots of the 2D crystal appear blurred due to the small tile size. (b) Average power along the red line in (a).

Fig. 2. (a) Average power spectrum as in Fig. 1, but calculated from the noise-whitened image. (b) Average power along the red line, as in (a).

Fig. 3. Histogram of the grey-values in the noise-whitened image (red line) that was used for Fig. 2. For comparison, the histogram of the same image before whitening is shown (black dashed line). After appropriate scaling, the two histograms are virtually identical.

Fig. 4. Flow-chart of the implemented ML algorithm.

Fig. 5. ML processing of the image of a negatively stained 2D crystal of QmpF. (a–c) Intermediate maps obtained after noise whitening and 5, 15 and 20 iterations, respectively. (d–f) ML processing of the same data when using the modified whitening procedure that excludes peaks in the Fourier spectrum produced by periodic image features.

Fig. 6. Application of the ML (a) and CC (b) algorithms to a frozen hydrated 2D crystal image of bacteriorhodopsin. Threefold symmetry was applied during processing. (c) The Fourier ring correlation between two half-data sets of the final maps. (d) Image processing result from one image, after unbending and Fourier extraction using the MRC crystallographic programs, with p3 symmetrization.

Fig. 7. Processing of a negatively stained, partially disordered 2D crystal image of the ammonium transporter AmtB. Final maps with 2-fold averaging using the ML (a) and the CC (b) alignment approach. (c) The Fourier ring correlation between structures calculated from two halves of the data. (d) Structure obtained using the crystallographic approach, with p2 symmetrization.

Fig. 8. Image processing of a negatively stained, badly ordered 2D crystal image of a tetrameric potassium channel MloK1. (a and b) Final maps with 4-fold averaging using the ML (a), and CC (b) alignment approach. (c) The Fourier ring correlation between structures calculated from two halves of the data. (d) Image processing result from one image, after unbending and Fourier extraction using the MRC programs, with p4 symmetrization.

Fig. 9. Computer simulation to detect over-fitting of noise. (a) Test pattern used to generate 2000 noisy images with a SNR of 0.005 (b). Average calculated from the noise component of the test images after one iteration of CC alignment (c), and ML estimation (d). Averages similar to (c) and (d) are shown after 20 iterations for CC alignment (e) and ML estimation (f). (g) FRC curves between the noise-free test pattern (a) and the noise averages in (e) and (f).