Fully three-dimensional defocus-gradient corrected backprojection in cryoelectron microscopy

Abstract

Recognizing that the microscope depth of field is a significant resolution-limiting factor in threedimensional cryoelectron microscopy, Jensen and Kornberg proposed a concept they called defocus-gradient corrected backprojection (DGCBP) and illustrated by computer simulations that DGCBP can effectively eliminate the depth of field limitation. They did not provide a mathematical justification for their concept. Our paper provides this, by showing (in the idealized case of noiseless data being available for all projection directions) that the reconstructions obtained based on DGCBP from data produced with distance-dependent blurring are essentially the same as what is obtained by a classical method of reconstruction of a 3D object from its line integrals. The approach is general enough to be applicable for correcting for any distancedependent blurring during projection data collection. We present a new implementation of the DGCBP concept, one that closely follows the mathematics of its justifications, and illustrate it using mathematically-described phantoms and their reconstructions from finitely-many distancedependently blurred projections.

Characterization of transfer function, resolution and depth of field of a soft X-ray microscope applied to tomography enhancement by Wiener deconvolution

“…Consequently, reconstruction will improve if standard deconvolution is applied to experimental data. The image formation process within this assumption, known as the X-ray transform, is described as [35]:…”

“…The inversion of Eq. (4) has already been proved to recover the information of the standard projection [35]:…”

Three-dimensional CTF correction improves the resolution of electron tomograms

“…Importantly, the coordinate system used here is the same as that of the microscope (Kazantsev et al, 2010). After determining a feature e.g.…”

“…Conceptually, the most complete approach was proposed by Jensen and Kornberg (Jensen and Kornberg, 2000), which performed a truly 3D CTF correction for the tilt-induced and sample-thickness-induced focus gradient in the tilt-series. While the mathematical justification has been provided later (Kazantsev et al, 2010), a feasible implementation was still missing. A reliable 3D CTF correction with a proper reconstruction technique could hold the promise of reaching a level of quality that would allow for the detection of individual macromolecular complexes in the cellular context without the need for sub-tomogram averaging.…”

“…Previously, it was shown mathematically that this can lead to a accurately CTF-corrected reconstruction (Kazantsev et al, 2010). Current computational solutions in Fourier space , even based on algorithmic simplifications (Voortman et al, 2011), are computationally so expensive, that render a practicable reconstruction difficult.…”