Determination of astigmatism in TEM images

We have developed a new two-step algorithm to determine the astigmatism of images from transmission electron microscopes (TEMs). Instead of computing the radial average of the power spectrum, we divide the power spectrum of a TEM image 1 to m (typically 32) sectors. We use a technique based on perturbation analysis of the contrast transfer function (CTF) to assimilate sector averages of the power spectrum of an image, which are incoherent in the presence of astigmatism, to a coherent radial average corresponding to a nominal defocus value. This is based on the fact that small defocus change from a nominal value can be considered to be equivalent to a perturbation on the spatial frequency spectra. Thus, instead of measuring the angular defocus variations, we optimise the frequency change required to obtain a coherent radial average. Numerically, this is achieved by minimizing sigma(2)/sigma(1) of a matrix formed from the sector averages, where sigma(i) denotes the ith singular value of the matrix. After the minimisation procedure, the second singular value should be very small compared with the first singular value, indicating that the matrix is nearly rank unity. In the second step, the nominal defocus can be obtained from the coherent radial average using any good defocus estimation program, which assumes zero astigmatism. The defocus value at a sector can be obtained from this nominal defocus value and one of the parameters from the unconstrained optima. Our algorithm is tested on astigmatic images of carbon film, 2D crystals of bacteriorhodopsin and cryo-images of HIV cores.