Automated determination of parameters describing power spectra of micrograph images in electron microscopy

The current theory of image formation in electron microscopy has been semi-quantitatively successful in describing data. The theory involves parameters due to the transfer function of the microscope (defocus, spherical aberration constant, and amplitude constant ratio) as well as parameters used to describe the background and attenuation of the signal. We present empirical evidence that at least one of the features of this model has not been well characterized. Namely the spectrum of the noise background is not accurately described by a Gaussian and associated “ B-factor;” this becomes apparent when one studies high-quality far-from focus data. In order to have both our analysis and conclusions free from any innate bias, we have approached the questions by developing an automated fitting algorithm. The most important features of this routine, not currently found in the literature, are (i) a process for determining the cutoff for those frequencies below which observations and the currently adopted model are not in accord, (ii) a method for determining the resolution at which no more signal is expected to exist, and (iii) a parameter—with units of spatial frequency—that characterizes which frequencies mainly contribute to the signal. Whereas no general relation is seen to exist between either of these two quantities and the defocus, a simple empirical relationship approximately relates all three.