Computer Visualization of Three-Dimensional Image Data Using IMOD

Dual-axis tomography: an approach with alignment methods that preserve resolution.

3D electron microscopy in the physical sciences: the development of Z-contrast and EFTEM tomography

A method for Bayesian reconstruction which relies on updates of single pixel values, rather than the entire image, at each iteration is presented. The technique is similar to Gauss-Seidel (GS) iteration for the solution of differential equations on finite grids. The computational cost per iteration of the GS approach is found to be approximately equal to that of gradient methods. For continuously valued images, GS is found to have significantly better convergence at modes representing high spatial frequencies. In addition, GS is well suited to segmentation when the image is constrained to be discretely valued. It is shown that Bayesian segmentation using GS iteration produces useful estimates at much lower signal-to-noise ratios than required for continuously valued reconstruction. The convergence properties of gradient ascent and GS for reconstruction from integral projections are analyzed, and simulations of both maximum-likelihood and maximum a posteriori cases are included. >

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A local update strategy for iterative reconstruction from projections

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

Under certain conditions it is possible to obtain the approximate projected density of a specimen from an electron micrograph, and by taking electron micrographs in different directions a number of different projections of a specimen can be determined. Several methods have been proposed for reconstructing the three-dimensional density distribution of an object from a series of projections whose planes are equally spaced in angle about an axis. One of these methods uses the Fourier transforms of the projections, while other methods operate directly on the projection data. The limitations of the direct method of 'back-projection' are discussed, and it is shown how a valid reconstruction may be achieved by back-projection from modified projected densities. This direct method is shown to be equivalent to the Fourier method and also to certain methods of reconstruction developed for the solution of analogous problems in other fields. An expression is given for the resolution of the reconstruction obtainable from a finite number of projections. Because of its computational simplicity this method is very suitable for reconstructing the density of objects with rotational or helical symmetry.

P. F. C. Gilbert

Papers

The Reconstruction of a Three-Dimensional Structure from Projections and Its Application to Electron Microscopy. II. Direct Methods