MOTIVATION: A number of macromolecular assemblies are being reconstructed in 3D from electron micrographs. The analysis yields a 3D matrix representing the protein density map. In reconstruction processes and in comparing the results of different experiments, it is often necessary to obtain all models oriented the same way in three dimensions. The problem is not trivial since there exist no 3D counterpart of correlation analysis used for 2D images. It is usually solved by time consuming trial and error algorithms. RESULTS: 3D density distributions can be brought to a 'canonical' orientation. The tensor of inertia of the distribution is determined and its eigenvectors are oriented along the coordinate axes. The method is fast and essentially free of reference. It is suitable for structures whose inertial axes do not completely degenerate as they do in icosahedral viruses or if symmetry is cubic. Applications are presented for asymmetric objects and for molecules possessing symmetry axes higher than twofold. IMPLEMENTATION: The implementation simply requires the accumulation of the inertial tensor and its diagonalisation. Volume data rotation has been already illustrated in this journal by the authors.

MOTIVATION: Arrays of three-dimensional (3D) data are ubiquitous in structural biology, biomedicine and clinical imaging. The Radon transform can be implied in their manipulation mainly for the solution of the inverse tomographic problem, since experimental data are often collected as projections or as samples of the Radon space. In electron tomography, new applications of the transform may become convenient if a fast and accurate transformation algorithm is adopted. RESULTS: A direct Fourier method (DFM) is proposed to compute the 3D Radon transform from a sampled function with compact support. This paper describes an already known two-step algorithm and illustrates its DFM implementation by coordinate transformations in 2D Fourier space. The algorithm is easily inverted to obtain a density distribution from the Radon transform. The main applications are in the field of electron tomography, especially in processes of angular refinement, since whatever projection of a structure can be retrieved from its Radon transform in a fast and accurate way. The times required to compute a number of projections with use of the Radon transform are compared with those required by other algorithms. Further uses of the Radon transform can be foreseen in applications based on 'projection onto convex sets' (POCS). AVAILABILITY: Software is available free of charge upon request to the authors. CONTACT: [email protected]

A direct Fourier method (DFM) to compute sinograms is described. Since the DFM is much faster than the customary rotation and projection process, novel uses of sinograms can be devised. As an example and an exercise on the properties of sinograms, a fast sinogram-based method to align large sets of images is described. The method is based on shift-invariant functions to detect rotations, and on tomographic reconstructions of cross-correlation functions to detect relative shifts. It has been implemented in a novel library, SIGNAL, used to align images of macromolecular assemblies observed in the electron microscope. A comparative analysis of the complexity of sinogram- and imagebased methods, as well as quite a few tests with EM images, show that SIGNAL runs faster, the accuracy being the same.

Three different implementations of the 3-pass algorithm of image and volume data rotation are illustrated and discussed. The three protocols use interpolation in real domain, with a peculiar implementation of the Shannon reconstruction, or phase shifts in Fourier domain. Accuracy and speed of the three methods are compared with corresponding values obtained with a 1-pass method. The results indicate that for low or moderate accuracy, 1-pass is more convenient than 3-pass rotation for both accuracy and speed. Very accurate rotations can be obtained in reasonable time if all steps of 3-pass rotation are performed in the Fourier domain.

A single projection of a helical distribution of matter allows one to obtain the complete three-dimensional reconstruction of the structure. This task is usually performed by a Fourier-Bessel algorithm, which is more efficient than a customary fast Fourier transform inversion. This article describes how to achieve such a result by a direct Fourier method in a reasonable time. Once the two-dimensional transform of the projection is obtained from the source image, it is possible to build up the three-dimensional transform array, in Cartesian coordinates, that yields the reconstruction by a straightforward Fourier inversion. Images of projected helices should be studied with high sampling rates to enhance the resolution, and the segments of helix should be long enough to give a satisfactory signal-to-noise ratio. These conditions result in three-dimensional transform arrays that would require one or more gigabytes of storage. The strategy proposed here requires much less storage and is fast enough to allow the reconstruction to be performed with different parameters and filters in a very short time without any sacrifice in resolution.

FT3D is a self-contained package of tools for three-dimensional Fourier analysis, written in the C language for Unix workstations. It can evaluate direct transforms of three-dimensional real functions, inverse transforms, auto- and cross-correlations and spectra. The library has been developed to support three-dimensional reconstructions of biological structures from projections obtained in the electron microscope. This paper discusses some features of the library, which has been implemented in such a way as to profit from the resources of modern workstations. A table of elapsed times for jobs of different dimensions with different RAM buffers is reported for the particular hardware used in the authors' laboratory.

The library POLCA implements the averaging of biological structures whose images are recorded in digital form from electron micrographs. The averaging protocol is based upon a method developed about ten years ago, which allows one to operate on a sequence of objects oriented and displaced at random within their frame; the relative rotations and the displacements of the structures are detected with the use of correlation algorithms and modified to make all objects appear the same, apart from their noisy components. The average image is then obtained by a simple addition and the signal-to-noise ratio is improved by a factor equal to the square root of the number of objects used to calculate the average. With respect to the original implementation of the method, two novel features characterize the library: the first one deals with the functions that are cross-correlated to determine the relative rotations of the structures; the functions used here are the inverse transforms of the amplitude spectra (IAS functions), which give rise to sharp maxima when they are cross-correlated. The second peculiarity is the systematic adoption, in the transformations of coordinates and in other circumstances, of an interpolation technique based upon the Fourier series kernel. POLCA is written in C and runs on a VME machine under the UNIX V/68 operating system. A programming style has been adopted to exploit fully the machine resources.