Precise and unbiased estimation of astigmatism and defocus in transmission electron microscopy

Defocus and twofold astigmatism are the key parameters governing the contrast transfer function (CTF) in transmission electron microscopy (TEM) of weak phase objects. We present a new algorithm to estimate these aberrations and the associated uncertainties. Tests show very good agreement between simulated and estimated defocus and astigmatism. We evaluate the reproducibility of the algorithm on experimental data by repeating measurements of an amorphous sample under identical imaging conditions and by analyzing the linearity of the stigmator response. By using a new Thon ring averaging method, the modulation depth of the rings in a 1D averaged power spectrum density (PSD) can be enhanced compared to elliptical averaging. This facilitates a better contrast transfer assessment in the presence of spherical aberration. Our algorithm for defocus and astigmatism estimation inverts the contrast of the Thon rings and suppresses the background in the PSD using an adaptive filtering strategy. Template matching with kernels of various ellipticities is applied to the filtered PSD after transformation into polar coordinates. Maxima in the resulting 3D parameter space provide multiple estimates of the long axis orientation, frequencies and apparent ellipticities of the rings. The frequencies of the detected rings, together with outlier rejection and assignment of an order to the CTF zeros, are used to estimate the defocus and its uncertainty. From estimations of defocus and ellipticity, we derive astigmatism and its uncertainty. A two-pass approach refines the astigmatism and defocus estimate by taking into account the influence of the known spherical aberration on the shape and frequencies of the rings. The implementation of the presented algorithm is freely available for non-commercial use.