We present a new reconstruction method that takes advantage of the fact that many biological macromolecular assemblies show a preferred orientation with respect to the plane of the specimen grid in the electron microscopic preparation. From one micrograph taken of such a specimen tilted by a large angle, a conical tilt series with random azimuthal angles can be extracted and used for a three-dimensional reconstruction. Our technique allows the determination of the molecular structure under low-dose conditions, which are not achievable with reconstruction methods that use conventional tilt series. The reconstruction method combines a number of existing image processing techniques with a newly developed weighted back-projection algorithm designed for three-dimensional reconstruction from projections taken with arbitrary projecting directions. The method is described as it was applied to the three-dimensional reconstruction of the structure of the 50S ribosomal subunit of Escherichia coli (E. coli).

https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1365-2818.1987.tb01333.x

Three-dimensional reconstruction from a single-exposure, random conical tilt series applied to the 50S ribosomal subunit of Escherichia coli

Three-dimensional structure determination by electron microscopy of two-dimensional crystals

A method is given for the evaluation, in transmission electron microscopy, of the amplitude and phase from the intensity distribution of an electron micrograph. The method requires a minimum of two micrographs taken under different defocus conditions. The iterative scheme requires only the relative defocus between micrographs, and the procedure is valid both in bright-field and dark-field microscopy for any specified coherence of the electron source. Assumptions on the scattering properties of the specimen, such as the weak-phase-weak-amplitude object, are not required. For a complete determination of the amplitude-phase distribution for electron transmission through the specimen, the electron micrograph must be corrected for the effect of lens aberrations and defocusing to give the electron wavefunction immediately after transmission; only in the case of a weak-phase object can this wavefunction be directly related to the projected potential distribution in the object. Inelastic electron scattering is explicitly omitted from the analysis presented.

https://iopscience.iop.org/article/10.1088/0022-3727/6/1/102/pdf

A method for the solution of the phase problem in electron microscopy

Comment onA method for the solution of the phase problem in electron microscopy

An evaluation is made of an iterative method for determining the amplitude and phase from the image intensity recorded in optical systems. The method, which requires two images recorded at different lens defocus values, is tested with simulated data subject to error arising from the photographic recording of the image. In the case of error-free data, the solution for the phase distribution appears to be indeterminate to within a constant. The results for photographic noise levels of up to 20% of the maximum image intensity reflect the small effect of error on the calculated phase distribution. The calculation of phase distributions for both symmetric and asymmetric amplitude-phase distributions shows that the use of two images, taken at defocus values differing by about 100 nm in electron optics and about 1 mm in optics (depending on the numerical aperture of the objective lens), may be used to determine the complex object wave-function in both dark-field and bright-field optics.

http://iopscience.iop.org/article/10.1088/0022-3727/6/18/305/pdf

An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations

Electron microscopy using microscopes with conventional magnetic lenses is restricted in resolution by spherical aberration. The 'zonal correction principle' allows this limit to be overcome. Its use in image reconstruction schemes allows separation of the influence of defocusing, astigmatism, etc., from the physically significant structure. The use of three-dimensional methods of image differencing is necessary in order to get interpretable results in the study of the not 'infinitely thin' objects in electron microscopy at atomic level. The use of redundancy principles (similar to X-ray structure determination) is discussed and demonstrated. The most severe difficulty, especially in electron microscopy of organic specimens, is radiation damage.

Use of zone correction plates and other techniques for structure determination of aperiodic objects at atomic resolution using a conventional electron microscope

W. Hoppe

Papers

Use of zone correction plates and other techniques for structure determination of aperiodic objects at atomic resolution using a conventional electron microscope