A new solution to the curved Ewald sphere problem for 3D image reconstruction in electron microscopy

Ghostbuster: A phase retrieval diffraction tomography algorithm for cryo-EM

Ewald sphere curvature correction, which extends beyond the projection approximation, stretches the shallow depth of field in cryo-EM reconstructions of thick particles. Here we show that even for previously assumed thin particles, reconstruction artifacts which we refer to as ghosts can appear. By retrieving the lost phases of the electron exitwaves and accounting for the first Born approximation scattering within the particle, we show that these ghosts can be effectively eliminated. Our simulations demonstrate how such ghostbusting can improve reconstructions as compared to existing state-of-the-art software. Like ptychographic cryo-EM, our Ghostbuster algorithm uses phase retrieval to improve reconstructions, but unlike the former, we do not need to modify the existing data acquisition pipelines.

The pypadf package: computing the pair angle distribution function from fluctuation scattering data

The pair angle distribution function (PADF) is a three- and four-atom correlation function that characterizes the local angular structure of disordered materials, particles or nanocrystalline materials. The PADF can be measured using X-ray or electron fluctuation diffraction data, which can be collected by scanning or flowing a structurally disordered sample through a focused beam. It is a natural generalization of established pair distribution methods, which do not provide angular information. The software package pypadf provides tools to calculate the PADF from fluctuation diffraction data. The package includes tools for calculating the intensity correlation function, which is a necessary step in the PADF calculation and also the basis for other fluctuation scattering analysis techniques.

CryoEM single particle reconstruction with a complex-valued particle stack

Single particle reconstruction (SPR) in cryoEM is an image processing task with an elaborate hierarchy that starts with many very noisy multi-frame images. Efficient representation of the intermediary image structures is critical for keeping the calculations manageable. One such intermediary structure is called a particle stack and contains cut-out images of particles in square boxes of predefined size. The micrograph that is the source of the boxed images is usually corrected for motion between frames prior to particle stack creation. However, the contrast transfer function (CTF) or its Fourier Transform point spread function (PSF) are not considered at this step. Historically, the particle stack was intended for large particles and for a tighter PSF, which is characteristic of lower resolution data. The field now performs analyses of smaller particles and to higher resolution, and these conditions result in a broader PSF that requires larger padding and slower calculations to integrate information for each particle. Consequently, the approach to handling structures such as the particle stack should be reexamined to optimize data processing. Here we propose to use as a source image for the particle stack a complex-valued image, in which CTF correction is implicitly applied as a real component of the image. We can achieve it by applying an initial CTF correction to the entire micrograph first and perform box cutouts as a subsequent step. The final CTF correction that we refine and apply later has a very narrow PSF, and so cutting out particles from micrographs that were approximately corrected for CTF does not require extended buffering, i.e. the boxes during the analysis only have to be large enough to encompass the particle. The Fourier Transform of an exit-wave reconstruction creates an image that has complex values. This is a complex value image considered in real space, opposed to standard SPR data processing where complex numbers appear only in Fourier space. This extension of the micrograph concept provides multiple advantages because the particle box size can be small and calculations crucial for high resolution reconstruction such as Ewald sphere correction, aberration refinement, and particle-specific defocus refinement can be performed on the small box data.

Unified fast reconstruction algorithm for conventional, phase-contrast, and diffraction tomography

A unified method for three-dimensional reconstruction of objects from transmission images collected at multiple illumination directions is described. The method may be applicable to experimental conditions relevant to absorption-based, phase-contrast, or diffraction imaging using x rays, electrons, and other forms of penetrating radiation or matter waves. Both the phase retrieval (also known as contrast transfer function correction) and the effect of Ewald sphere curvature (in the cases with a shallow depth of field and significant in-object diffraction) are incorporated in the proposed algorithm and can be taken into account. Multiple scattering is not treated explicitly but can be mitigated as a result of angular averaging that constitutes an essential feature of the method. The corresponding numerical algorithm is based on three-dimensional gridding which allows for fast computational implementation, including a straightforward parallelization. The algorithm can be used with any scanning geometry involving plane-wave illumination. A software code implementing the proposed algorithm has been developed, tested on simulated and experimental image data, and made publicly available.

Emerging Themes in CryoEM─Single Particle Analysis Image Processing

Cryo-electron microscopy (CryoEM) has become a vital technique in structural biology. It is an interdisciplinary field that takes advantage of advances in biochemistry, physics, and image processing, among other disciplines. Innovations in these three basic pillars have contributed to the boosting of CryoEM in the past decade. This work reviews the main contributions in image processing to the current reconstruction workflow of single particle analysis (SPA) by CryoEM. Our review emphasizes the time evolution of the algorithms across the different steps of the workflow differentiating between two groups of approaches: analytical methods and deep learning algorithms. We present an analysis of the current state of the art. Finally, we discuss the emerging problems and challenges still to be addressed in the evolution of CryoEM image processing methods in SPA.

Method for virtual optical sectioning and tomography utilizing shallow depth of field

A method is proposed for high-resolution, three-dimensional reconstruction of internal structures of objects from planar transmission images. The described approach can be used with any form of radiation or matter waves, in principle, provided that the depth of field is smaller than the thickness of the sample. The physical optics basis for the method is elucidated, and the reconstruction algorithm is presented in detail. A simulated example demonstrates an application of the method to three-dimensional electron transmission imaging of a nanoparticle under realistic radiation dose and spatial resolution constraints. It is envisaged that the method can be applicable in high-resolution transmission electron microscopy, soft x-ray microscopy, ultrasound imaging, and other areas.

A Method for High-Resolution Three-Dimensional Reconstruction with Ewald Sphere Curvature Correction from Transmission Electron Images

A method for three-dimensional reconstruction of objects from defocused images collected at multiple illumination directions in high-resolution transmission electron microscopy is presented. The method effectively corrects for the Ewald sphere curvature by taking into account the in-particle propagation of the electron beam. Numerical simulations demonstrate that the proposed method is capable of accurately reconstructing biological molecules or nanoparticles from high-resolution defocused images under conditions achievable in single-particle electron cryo-microscopy or electron tomography with realistic radiation doses, non-trivial aberrations, multiple scattering, and other experimentally relevant factors. The physics of the method is based on the well-known Diffraction Tomography formalism, but with the phase-retrieval step modified to include a conjugation of the phase (i.e., multiplication of the phase by a negative constant). At each illumination direction, numerically backpropagating the beam with the conjugated phase produces maximum contrast at the location of individual atoms in the molecule or nanoparticle. The resultant algorithm, Conjugated Holographic Reconstruction, can potentially be incorporated into established software tools for single-particle analysis, such as, for example, RELION or FREALIGN, in place of the conventional contrast transfer function correction procedure, in order to account for the Ewald sphere curvature and improve the spatial resolution of the three-dimensional reconstruction.

Perspective: Emerging strategies for determining atomic-resolution structures of macromolecular complexes within cells

In principle, electron cryo-tomography (cryo-ET) of thin portions of cells provides high-resolution images of the three-dimensional spatial arrangement of all members of the proteome. In practice, however, radiation damage creates a tension between recording images at many different tilt angles, but at correspondingly reduced exposure levels, versus limiting the number of tilt angles in order to improve the signal-to-noise ratio (SNR). Either way, it is challenging to read the available information out at the level of atomic structure. Here, we first review work that explores the optimal strategy for data collection, which currently seems to favor the use of a limited angular range for tilting the sample or even the use of a single image to record the high-resolution information. Looking then to the future, we point to the alternative of so-called “deconvolution microscopy”, which may be applied to tilt-series or optically-sectioned, focal series data. Recording data as a focal series has the advantage that little or no translational alignment of frames might be needed, and a three-dimensional reconstruction might require only 2/3 the number of images as does standard tomography. We also point to the unexploited potential of phase plates to increase the contrast, and thus to reduce the electron exposure levels while retaining the ability align and merge the data. In turn, using much lower exposures per image could have the advantage that high-resolution information is retained throughout the full data-set, whether recorded as a tilt series or a focal series of images.

Relative roles of multiple scattering and Fresnel diffraction in the imaging of small molecules using electrons, Part II: Differential Holographic Tomography

It has been argued that in atomic-resolution transmission electron microscopy (TEM) of sparse weakly scattering structures, such as small biological molecules, multiple electron scattering usually has only a small effect, while the in-molecule Fresnel diffraction can be significant due to the intrinsically shallow depth of focus. These facts suggest that the three-dimensional reconstruction of such structures from defocus image series collected at multiple rotational orientations of a molecule can be effectively performed for each atom separately, using the incoherent first Born approximation. The corresponding reconstruction method, termed here Differential Holographic Tomography, is developed theoretically and demonstrated computationally on several numerical models of biological molecules. It is shown that the method is capable of accurate reconstruction of the locations of atoms in a molecule from TEM data collected at a small number of random orientations of the molecule, with one or more defocus images per orientation. Possible applications to cryogenic electron microscopy and other areas are briefly discussed.