It has been almost thirty years since Shannon introduced the sampling theorem to communications theory. In this review paper we will attempt to present the various contributions made for the sampling theorems with the necessary mathematical details to make it self-contained. We will begin by a clear statement of Shannon's sampling theorem followed by its applied interpretation for time-invariant systems. Then we will review its origin as Whittaker's interpolation series. The extensions will include sampling for functions of more than one variable, random processes, nonuniform sampling, nonband-limited functions, implicit sampling, generalized functions (distributions), sampling with the function and its derivatives as suggested by Shannon in his original paper, and sampling for general integral transforms. Also the conditions on the functions to be sampled will be summarized. The error analysis of the various sampling expansions, including specific error bounds for the truncation, aliasing, jitter and parts of various other errors will be discussed and summarized. This paper will be concluded by searching the different recent applications of the sampling theorems in other fields, besides communications theory. These include optics, crystallography, time-varying systems, boundary value problems, spline approximation, special functions, and the Fourier and other discrete transforms.

The Shannon sampling theorem&#8212;Its various extensions and applications: A tutorial review

Using cryo-electron microscopy (cryo-EM), we determined the structure of the Escherichia coli 70S ribosome with a global resolution of 2.0 A. The maps reveal unambiguous positioning of protein and RNA residues, their detailed chemical interactions, and chemical modifications. Notable features include the first examples of isopeptide and thioamide backbone substitutions in ribosomal proteins, the former likely conserved in all domains of life. The maps also reveal extensive solvation of the small (30S) ribosomal subunit, and interactions with A-site and P-site tRNAs, mRNA, and the antibiotic paromomycin. The maps and models of the bacterial ribosome presented here now allow a deeper phylogenetic analysis of ribosomal components including structural conservation to the level of solvation. The high quality of the maps should enable future structural analyses of the chemical basis for translation and aid the development of robust tools for cryo-EM structure modeling and refinement.

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Structures of the bacterial ribosome at 3.5 Å resolution

Fourier shell correlation threshold criteria

Three-dimensional reconstruction of single particles embedded in ice.

Introduction: Principles of Electron Tomography.- Sample Shrinkage and Radiation Damage of Plastic Sections.- Electron Tomography of Frozen-hydrated Sections of Cells and Tissues.- The Electron Microscope as a Structure Projector.- Cryotomography: Low-dose Automated Tomography of Frozen-hydrated Specimens.- Fiducial Marker and Hybrid Alignment Methods for Single- and Double-axis Tomography.- Markerless Alignment in Electron Tomography.- Algorithms for Three-dimensional Reconstruction From the Imperfect Projection Data Provided by Electron Microscopy.- Weighted Back-projection Methods.- Reconstruction With Orthogonal Functions.- Resolution in Electron Tomography.- Denoising of Electron Tomograms.- Segmentation of Three-dimensional Electron Tomographic Images.- Segmentation of Cell Components Using Prior Knowledge.- Motif Search in Electron Tomography.- Localization and Classification of Repetitive Structures in Electron Tomograms of Paracrystalline Assemblies.

Electron tomography : methods for three-dimensional visualization of structures in the cell

The signal-to-noise ratio of electron micrographs can be determined by two-dimensional digital cross correlation even though neither signal nor noise can be analysed separately. Such measurements suggest how best to make use of the electron microscope.

L. Al-Ali

Papers

Signal-to-noise ratio of electron micrographs obtained by cross correlation.