Ewald sphere correction algorithm details

Hi @team,

Wondering if I might have some clarification on the Ewald sphere correction approach, mainly because it seems I have been scooped on a little side project I was considering but thankfully hadn’t started. From the Tutorial:

“As of v3.3, cryoSPARC supports the correction of Ewald sphere curvature during refinement and CTF refinement. While the standard reconstruction of a 3D density is based upon maximizing the likelihood of the data given the image poses, the algorithm used for Ewald sphere correction is an improvement on the “simple insertion” method developed by [3]. This itself is an approximation to the maximum likelihood method, while accounting for the geometry of the Ewald sphere.”

Where reference [3] is M. Wolf, D. DeRosier and N. Grigorieff, “Ewald sphere correction for single-particle electron microscopy”, Ultramicroscopy, vol. 106, no. 4-5, pp. 376-382, 2006. Available: 10.1016/j.ultramic.2005.11.001.

My reading of this is that, unlike in relion, there is no splitting the images into different “sectors” in the single-sideband approach suggested by Russo and Henderson and the Fourier coefficients are added back to the two different (Freidel related) Ewald spheres with the CTF (e^{iχ(k)}) removed, ie. the “simple insertion” method. At the moment relion doesn’t do iterative alignment of 2D images with the 3D model using an Ewald sphere corrected forward model, but it appears that cryosparc now does? I assume iterative refinement of the 3D volume solves the “two Ewald sphere” ambiguity that Wolf et al. spend much of their paper addressing and that Russo and Henderson solve with the “single-sideband” approach?

Appreciate that some of this might be the subject of an upcoming publication but would welcome any extra detail you can give.


1 Like

Hi @HamishGBrown,

Thanks for your post :slight_smile: You’re correct that the images are not split into sectors based on the Russo and Henderson approach. We essentially use the exact same method proposed by Wolf, DeRosier, and Grigorieff – in the paper this is equation (10). The details in this equation are all implicit, since all of the “corrected geometry” is in determining which 2D Fourier coefficients from the images contribute to which 3D Fourier coefficients in the volume. But the underlying principle is the same – the images are viewed as containing a noisy sum of the left and right Fourier coefficients, each phase shifted by e^{ +/- iχ(k)}. During reconstruction, each image is added onto the 3D volume grid twice (once for the left and once for the right side), with cCTF multiplication (different cCTF for each side) and opposite curvatures for each side (see Figure 3 in their paper). I apologize if this isn’t 100% clear – despite thinking about it for a long time I still struggle to explain it without just writing equations… regardless, yes it’s the same algorithm proposed by Wolf et al.

Our experimentation showed us that this was the best method when it came to the trade-off between computational cost, method complexity, and result quality, as reconstruction is only 2x as expensive than the standard method. As you mentioned, we also iteratively refine poses and CTF using the Ewald-sphere-informed image formation model, however, this didn’t end up being significant on most real datasets, only on synthetic data in various limits (e.g. very low voltage microscopes, for example).



Hi @mmclean,

Thanks for the explanation, that’s sufficiently clear to me and congrats on a successful implementation of this within cryosparc!


1 Like