Restoration of weak phase-contrast images recorded with a high degree of defocus: the "twin image" problem associated with CTF correction

Abstract

Relatively large values of objective-lens defocus must normally be used to produce detectable levels of image contrast for unstained biological specimens, which are generally weak phase objects. As a result, a subsequent restoration operation must be used to correct for oscillations in the contrast transfer function (CTF) at higher resolution. Currently used methods of CTF correction assume the ideal case in which Friedel mates in the scattered wave have contributed pairs of Fourier components that overlap with one another in the image plane. This "ideal" situation may be only poorly satisfied, or not satisfied at all, as the particle size gets smaller, the defocus value gets larger, and the resolution gets higher. We have therefore investigated whether currently used methods of CTF correction are also effective in restoring the single-sideband image information that becomes displaced (delocalized) by half (or more) the diameter of a particle of finite size. Computer simulations are used to show that restoration either by "phase flipping" or by multiplying by the CTF recovers only about half of the delocalized information. The other half of the delocalized information goes into a doubly defocused "twin" image of the type produced during optical reconstruction of an in-line hologram. Restoration with a Wiener filter is effective in recovering the delocalized information only when the signal-to-noise ratio (S/N) is orders of magnitude higher than that which exists in low-dose images of biological specimens, in which case the Wiener filter approaches division by the CTF (i.e. the formal inverse). For realistic values of the S/N, however, the "twin image" problem seen with a Wiener filter is very similar to that seen when either phase flipping or multiplying by the CTF is used for restoration. The results of these simulations suggest that CTF correction is a poor alternative to using a Zernike-type phase plate when imaging biological specimens, in which case the images can be recorded in a close-to-focus condition, and delocalization of high-resolution information is thus minimized.

Figure 1

The wedge-shaped pattern of radial spokes shown here provides an object in which each spatial frequency is localized at one particular position along the horizontal axis. The height of the panel represents 30 nm in these simulations. (A) The original pattern. (B) The pattern when imaged with a defocus of 2 μm and an electron energy of 300 keV. There are clearly resolved zeros and contrast reversals in certain zones of spatial frequency, while at the higher spatial frequencies it is seen that the Fourier components are shifted so much that two sets of spokes separate from each other. (C, D) single sideband images of (A) computed with the same effective defocus as in (B). Here the effects of the wave aberration can be seen as producing vertical shifts of the Fourier components by amounts that are proportional to the frequency of that component. The image in (B) is the sum of those shown in (C) and (D).

Figure 2

Comparison of the restoration of delocalized information that is achieved by phase flipping and by multiplication by the CTF. (A) A spatially bounded cross-grating pattern is formed as the product of two perpendicular sine waves. With the size-scale set to 0.1 nm per pixel, the period is 1.3 nm. (B) The image of the object in (A) that is computed with an effective defocus of 2 µm. (C) Restoration of (B) obtained by “phase flipping” – i.e. inverting the sign of the Fourier transform of (B) in alternate zones of the CTF. (D) Restoration of (B) computed by multiplying the Fourier transform by the original CTF. Insets show a section of the Fourier transform, with the origin near the lower left corner.

Figure 3

Simulation of the delocalization effect in an image of the large ribosomal subunit, and examples of the restoration achieved with a Wiener filter for different levels of the SNR. Images are not shown on the same relative scale of contrast, since the contrast after Wiener filtration depends upon the value of SNR that is used. (A) The initial image that is obtained when phase contrast is produced by using a defocus of 2 μm. (B) Restoration of the original object from the image in (A) is almost perfect when the Wiener filter assumes that SNR(s) = 900. Low-frequencies are still not well-represented in the restoration, however, since the CTF asymptotically goes to zero at low resolution. (C) Simulation of image restoration when the Wiener filter assumes that SNR(s) = 0.09, a value that is more realistic for cryo-EM images. (D) Restoration already fails to recover all of the delocalized information when the Wiener filter assumes that SNR(s) = 9, a value that is still unrealistically high for cryo-EM images.

Figure 4

The phase-contrast CTF for a defocus of 2 μm and electron energy of 300 keV, and the weighting (resultant “transfer function”) that is provided when a Weiner filter is used for image restoration. The CTF is shown by the black curve, while the product of the CTF and the Wiener filter is shown as differently colored curves for which the value of the SNR is identified in the insert. See the text for further explanation.