8.4.4 Denoising
While denoising is typically done after reconstruction, removing noise in the source
images could improve alignment and reconstructions. Maiorca et al. [41] implemented a non-linear anisotropic diffusion (NAD) algorithm to denoise the tilt series
micrographs. They showed an improvement in segmentation results.
8.5 Contrast Transfer Function (CTF) Correction
To obtain good contrast in micrographs, the electron microscope is typically
defocused. This has the consequence of imposing an oscillating CTF on the image.
In single particle analysis, the CTF is routinely corrected to obtain reconstructions
at high resolution. In tomography, the tilting of the specimen means that there is a
focus gradient across the image, complicating both parameter determination and
correction. Nevertheless, to recover more detail, the micrographs need to be corrected for the CTF prior to reconstruction. Many reconstructions schemes incorporate this as part of the algorithm [24, 42].
8.5.1 CTF Theory
The CTF equation most commonly used is:
CTFðsÞ ¼ a cosðcðsÞÞ À
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À a 2
p
sinðcðsÞÞ
where s is the spatial frequency (1/Å), a is the amplitude contrast fraction (typically
about 0.07 for frozen specimens) and the phase shift due to defocus is:
cðsÞ ¼
p
2
C s k
3 s
4
À pkDfs
2
where C s is the spherical aberration (typically about 2 mm), k is the electron
wavelength (a function of the acceleration voltage), and Δf is the defocus. The latter
is the complicated part of the CTF for tilted specimens, and is given by:
Df ðn; r; fÞ ¼ Df a þ Df d cosð2ðn À vÞÞ þ r sinðb À fÞ tanðaÞ
The three terms are the average defocus, Δf a , the astigmatism and the tilt
compensation. The astigmatism term relates to frequency space, where Δf d is the
defocus deviation due to astigmatism, and n − v is the difference angle between the
point in frequency space and the astigmatism angle. The tilt compensation term
relates to real space, where r is the distance of the point from the tilt axis origin,
b − f is the difference angle between the point and the tilt axis angle and a is the tilt
8 Tomographic Reconstruction from Electron Micrographs
221
While denoising is typically done after reconstruction, removing noise in the source
images could improve alignment and reconstructions. Maiorca et al. [41] implemented a non-linear anisotropic diffusion (NAD) algorithm to denoise the tilt series
micrographs. They showed an improvement in segmentation results.
8.5 Contrast Transfer Function (CTF) Correction
To obtain good contrast in micrographs, the electron microscope is typically
defocused. This has the consequence of imposing an oscillating CTF on the image.
In single particle analysis, the CTF is routinely corrected to obtain reconstructions
at high resolution. In tomography, the tilting of the specimen means that there is a
focus gradient across the image, complicating both parameter determination and
correction. Nevertheless, to recover more detail, the micrographs need to be corrected for the CTF prior to reconstruction. Many reconstructions schemes incorporate this as part of the algorithm [24, 42].
8.5.1 CTF Theory
The CTF equation most commonly used is:
CTFðsÞ ¼ a cosðcðsÞÞ À
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À a 2
p
sinðcðsÞÞ
where s is the spatial frequency (1/Å), a is the amplitude contrast fraction (typically
about 0.07 for frozen specimens) and the phase shift due to defocus is:
cðsÞ ¼
p
2
C s k
3 s
4
À pkDfs
2
where C s is the spherical aberration (typically about 2 mm), k is the electron
wavelength (a function of the acceleration voltage), and Δf is the defocus. The latter
is the complicated part of the CTF for tilted specimens, and is given by:
Df ðn; r; fÞ ¼ Df a þ Df d cosð2ðn À vÞÞ þ r sinðb À fÞ tanðaÞ
The three terms are the average defocus, Δf a , the astigmatism and the tilt
compensation. The astigmatism term relates to frequency space, where Δf d is the
defocus deviation due to astigmatism, and n − v is the difference angle between the
point in frequency space and the astigmatism angle. The tilt compensation term
relates to real space, where r is the distance of the point from the tilt axis origin,
b − f is the difference angle between the point and the tilt axis angle and a is the tilt
8 Tomographic Reconstruction from Electron Micrographs
221