Error analysis in the determination of the electron microscopical contrast transfer function parameters from experimental power Spectra

In this paper, we explore the effect of these estimation errors on the theoretical CTF. For the CTF model proposed in [1] we show which are the most sensitive CTF parameters as well as the most sensitive background parameters. Moreover, we provide a methodology to reveal the internal structure of the CTF model (which parameters influence in which parameters) and to estimate the accuracy of each model parameter. Finally, we explore the effect of the variability in the detection of the CTF for CTF phase and amplitude correction.We show that the estimation errors for the CTF detection methodology proposed in [1] does not show a significant deterioration of the CTF correction capabilities of subsequent algorithms. All together, the methodology described in this paper constitutes a powerful tool for the quantitative analysis of CTF models that can be applied to other models different from the one analyzed here.The transmission electron microscope distorts the structural information contained in the electron micrographs by changing the amplitude of the Fourier coefficients at all spatial frequencies and flipping their phase at certain annular regions [2]. This effect is usually modeled in Fourier space by the Contrast Transfer Function (CTF), which in turn has to be estimated from the electron micrographs. Normally, a theoretical model of the CTF is assumed and the parameters defining this model are optimized so that the experimentally observed PSD and the theoretically predicted PSD coincide as much as possible [1,3-10]. Therefore, the PSD has to be estimated first. This step is traditionally performed by the fast, although less accurate, periodogram averaging [10-13] or parametric methods, more accurate but much slower to compute [8,12]. The estimated periodogram can be further enhanced [14] to highlight the Thon rings and, therefore, facilitate the task of fitting the parameters of the theoretical model.Fully two-dimensional models multiply by three the number of param