On cross-correlations, averages and noise in electron microscopy

Abstract

Biological samples are radiation-sensitive and require imaging under low-dose conditions to minimize damage. As a result, images contain a high level of noise and exhibit signal-to-noise ratios that are typically significantly smaller than 1. Averaging techniques, either implicit or explicit, are used to overcome the limitations imposed by the high level of noise. Averaging of 2D images showing the same molecule in the same orientation results in highly significant projections. A high-resolution structure can be obtained by combining the information from many single-particle images to determine a 3D structure. Similarly, averaging of multiple copies of macromolecular assembly subvolumes extracted from tomographic reconstructions can lead to a virtually noise-free high-resolution structure. Cross-correlation methods are often used in the alignment and classification steps of averaging processes for both 2D images and 3D volumes. However, the high noise level can bias alignment and certain classification results. While other approaches may be implicitly affected, sensitivity to noise is most apparent in multireference alignments, 3D reference-based projection alignments and projection-based volume alignments. Here, the influence of the image signal-to-noise ratio on the value of the cross-correlation coefficient is analyzed and a method for compensating for this effect is provided.

Keywords: 3D reference-based projection alignment; cross-correlation; image processing; multireference alignment; signal-to-noise ratio.

Figure 1

Test data. (a) Motif, (b) one of the 400 images with added noise with SNR = 0.5.

Figure 2

Cryo-electron microscopy data for complex I from Y. lipolytica. (a) Area of micrograph. (b) Selection of boxed-out images and average image. (c) The same images as in (b), low-pass filtered. The scale bar is 100 Å in length.

Figure 3

Compensation for the negative effect of the SNR on the cross-correlation. (a) The ten motifs used in model calculation. (b) Example images of the ten motifs with added noise, SNR = 0.5. (c) Averages used as references calculated from 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100 images (from left to right). (d) Result of a classification using conventional multireference correlation without correction. (e) Result of a classification using multireference correlation with the correction described in (10) implemented.

Figure 4

Value of the cross-correlation coefficient between an image with SNR = α and an average image, depending on the number of images averaged in the reference image. Curves are shown for four different SNR values. Abscissa, number of images used to calculate the reference average image. Ordinate, cross-correlation coefficient C.

Figure 5

Asymptotic value of the cross-correlation coefficient in cross-correlations of an image with SNR = α with a virtually noise-free average. Abscissa, SNR of a single image. Ordinate, cross-correlation coefficient C.