Radial averages of astigmatic TEM images

The Contrast Transfer Function (CTF) of an image, which modulates images taken from a Transmission Electron Microscope (TEM), is usually determined from the radial average of the power spectrum of the image (Frank, J., Three-dimensional Electron Microscopy of Macromolecular Assemblies, Oxford University Press, Oxford, 2006). The CTF is primarily defined by the defocus. If the defocus estimate is accurate enough then it is possible to demodulate the image, which is popularly known as the CTF correction. However, it is known that the radial average is somewhat attenuated if the image is astigmatic (see Fernando, K.V., Fuller, S.D., 2007. Determination of astigmatism in TEM images. Journal of Structural Biology 157, 189-200) but this distortion due to astigmatism has not been fully studied or understood up to now. We have discovered the exact mathematical relationship between the radial averages of TEM images with and without astigmatism. This relationship is determined by a zeroth order Bessel function of the first kind and hence we can exactly quantify this distortion in the radial averages of signal and power spectra of astigmatic images. The argument to this Bessel function is similar to an aberration function (without the spherical aberration term) except that the defocus parameter is replaced by the differences of the defoci in the major and minor axes of astigmatism. The ill effects due this Bessel function are twofold. Since the zeroth order Bessel function is a decaying oscillatory function, it introduces additional zeros to the radial average and it also attenuates the CTF signal in the radial averages. Using our analysis, it is possible to simulate the effects of astigmatism in radial averages by imposing Bessel functions on idealized radial averages of images which are not astigmatic. We validate our theory using astigmatic TEM images.