Electron tomography in conical tilt geometry. The accuracy of a direct Fourier method (DFM) and the suppression of non-tomographic noise

Cones and spirals: Multi-axis acquisition for scalar and vector electron tomography

Electron tomography (ET) has become an important tool for understanding the 3D nature of nanomaterials, with recent developments enabling not only scalar reconstructions of electron density, but also vector reconstructions of magnetic fields. However, whilst new signals have been incorporated into the ET toolkit, the acquisition schemes have largely kept to conventional single-axis tilt series for scalar ET, and dual-axis schemes for magnetic vector ET. In this work, we explore the potential of using multi-axis tilt schemes including conical and spiral tilt schemes to improve reconstruction fidelity in scalar and magnetic vector ET. Through a combination of systematic simulations and a proof-of-concept experiment, we show that spiral and conical tilt schemes have the potential to produce substantially improved reconstructions, laying the foundations of a new approach to electron tomography acquisition and reconstruction.

Feature Adaptive Sampling for Scanning Electron Microscopy

A new method for the image acquisition in scanning electron microscopy (SEM) was introduced. The method used adaptively increased pixel-dwell times to improve the signal-to-noise ratio (SNR) in areas of high detail. In areas of low detail, the electron dose was reduced on a per pixel basis, and a-posteriori image processing techniques were applied to remove the resulting noise. The technique was realized by scanning the sample twice. The first, quick scan used small pixel-dwell times to generate a first, noisy image using a low electron dose. This image was analyzed automatically, and a software algorithm generated a sparse pattern of regions of the image that require additional sampling. A second scan generated a sparse image of only these regions, but using a highly increased electron dose. By applying a selective low-pass filter and combining both datasets, a single image was generated. The resulting image exhibited a factor of ≈3 better SNR than an image acquired with uniform sampling on a Cartesian grid and the same total acquisition time. This result implies that the required electron dose (or acquisition time) for the adaptive scanning method is a factor of ten lower than for uniform scanning.

A simple Fourier filter for suppression of the missing wedge ray artefacts in single-axis electron tomographic reconstructions

The limited specimen tilting range that is typically available in electron tomography gives rise to a region in the Fourier space of the reconstructed object where experimental data are unavailable - the missing wedge. Since this region is sharply delimited from the area of available data, the reconstructed signal is typically hampered by convolution with its impulse response, which gives rise to the well-known missing wedge artefacts in 3D reconstructions. Despite the recent progress in the field of reconstruction and regularization techniques, the missing wedge artefacts remain untreated in most current reconstruction workflows in structural biology. Therefore we have designed a simple Fourier angular filter that effectively suppresses the ray artefacts in the single-axis tilting projection acquisition scheme, making single-axis tomographic reconstructions easier to interpret in particular at low signal-to-noise ratio in acquired projections. The proposed filter can be easily incorporated into current electron tomographic reconstruction schemes.

Fourier-based reconstruction for fully 3-D PET: optimization of interpolation parameters

Fourier-based approaches for three-dimensional (3-D) reconstruction are based on the relationship between the 3-D Fourier transform (FT) of the volume and the two-dimensional (2-D) FT of a parallel-ray projection of the volume. The critical step in the Fourier-based methods is the estimation of the samples of the 3-D transform of the image from the samples of the 2-D transforms of the projections on the planes through the origin of Fourier space, and vice versa for forward-projection (reprojection). The Fourier-based approaches have the potential for very fast reconstruction, but their straightforward implementation might lead to unsatisfactory results if careful attention is not paid to interpolation and weighting functions. In our previous work, we have investigated optimal interpolation parameters for the Fourier-based forward and back-projectors for iterative image reconstruction. The optimized interpolation kernels were shown to provide excellent quality comparable to the ideal sinc interpolator. This work presents an optimization of interpolation parameters of the 3-D direct Fourier method with Fourier reprojection (3D-FRP) for fully 3-D positron emission tomography (PET) data with incomplete oblique projections. The reprojection step is needed for the estimation (from an initial image) of the missing portions of the oblique data. In the 3D-FRP implementation, we use the gridding interpolation strategy, combined with proper weighting approaches in the transform and image domains. We have found that while the 3-D reprojection step requires similar optimal interpolation parameters as found in our previous studies on Fourier-based iterative approaches, the optimal interpolation parameters for the main 3D-FRP reconstruction stage are quite different. Our experimental results confirm that for the optimal interpolation parameters a very good image accuracy can be achieved even without any extra spectral oversampling, which is a common practice to decrease errors caused by interpolation in Fourier reconstruction.

Conical tomography of freeze-fracture replicas: a method for the study of integral membrane proteins inserted in phospholipid bilayers

We have used conical tomography to study the structure of integral proteins in their phospholipid bilayer environments. Complete conical series were collected from replicas of the water channel aquaporin-0 (AQP0), a 6.6 nm side tetramer with a molecular weight of approximately 120 kDa that was purified and reconstituted in liposomes. The replicas were tilted at 38 degrees , 50 degrees or 55 degrees and rotated by 2.5 degrees , 4 degrees , or 5 degrees increments until completing 360 degrees turns. The elliptical paths of between 6 and 12 freeze-fracture particles aligned the images to a common coordinate system. Using the weighted back projection algorithm, small volumes of the replicas were independently reconstructed to reconstitute the field. Using the Fourier Shell Correlation computed from reconstructions of even and odd projections of the series, we estimated a resolution of 2-3 nm, a value that was close to the thickness of the replica (approximately 1.5 nm). The 3D reconstructions exhibited isotropic resolution along the x-y plane, which simplified the analysis of particles oriented randomly in the membrane plane. In contrast to reconstructions from single particles imaged using random conical tilt [J. Mol. Biol. 325 (2003) 210], the reconstructions using conical tomography allowed the size and shape of individual particles representing the AQP0 channel to be identified without averaging or imposing symmetry. In conclusion, the reconstruction of freeze-fracture replicas with electron tomography has provided a novel experimental approach for the study of integral proteins inserted in phospholipid bilayers.

Gridding-based direct Fourier inversion of the three-dimensional ray transform

We describe a fast and accurate direct Fourier method for reconstructing a function f of three variables from a number of its parallel beam projections. The main application of our method is in single particle analysis, where the goal is to reconstruct the mass density of a biological macromolecule. Typically, the number of projections is extremely large, and each projection is extremely noisy. The projection directions are random and initially unknown. However, it is possible to determine both the directions and f by an iterative procedure; during each stage of the iteration, one has to solve a reconstruction problem of the type considered here. Our reconstruction algorithm is distinguished from other direct Fourier methods by the use of gridding techniques that provide an efficient means to compute a uniformly sampled version of a function g from a nonuniformly sampled version of Fg, the Fourier transform of g, or vice versa. We apply the two-dimensional reverse gridding method to each available projection of f, the function to be reconstructed, in order to obtain Ff on a special spherical grid. Then we use the three-dimensional gridding method to reconstruct f from this sampled version of Ff. This stage requires a proper weighting of the samples of Ff to compensate for their nonuniform distribution. We use a fast method for computing appropriate weights that exploits the special properties of the spherical sampling grid for Ff and involves the computation of a Voronoi diagram on the unit sphere. We demonstrate the excellent speed and accuracy of our method by using simulated data.

Fast and accurate three-dimensional reconstruction from projections with random orientations via radon transforms

A new algorithm for three-dimensional reconstruction from randomly oriented projections has been developed. The algorithm recovers the 3D Radon transform from the 2D Radon transforms (sinograms) of the projections. The structure in direct space is obtained by an inversion of the 3D Radon transform. The mathematical properties of the Radon transform are exploited to design a special filter that can be used to correct inconsistencies in a data set and to fill the gaps in the Radon transform that originate from missing projections. Several versions of the algorithm have been implemented, with and without a filter and with different interpolation methods for merging the sinograms into the 3D Radon transform. The algorithms have been tested on analytical phantoms and experimental data and have been compared with a weighted back projection algorithm (WBP). A quantitative analysis of phantoms reconstructed from noise-free and noise-corrupted projections shows that the new algorithms are more accurate than WBP when the number of projections is small. Experimental structures obtained by the new methods are strictly comparable to those obtained by WBP. Moreover, the algorithm is more than 10 times faster than WPB when applied to a data set of 1000-5000 projections. Copyright 1999 Academic Press.

Three-dimensional reconstruction of metal replicas of the Helicobacter pylori vacuolating cytotoxin

The Helicobacter pylori vacuolating cytotoxin (VacA) forms high molecular weight homooligomers which contain either six or seven copies of a 95-kDa polypeptide. Electron microscope visualization of carbon platinum replicas of quick-freeze, deepetched, preparations of VacA has revealed that the oligomers are arranged in flower-like structures with six- or sevenfold radial symmetry, depending on the number of 95-kDa oligomers that they contain. Each monomer is structured in two subunits of 37 and 58 kDa connected by an exposed loop which is a site for proteolytic cleavage. In preparations of VacA which had undergone extensive cleavage at the exposed loop, oligomers of both six- and seven-fold symmetry which appeared flatter were observed; these latter were interpreted as molecules which had lost a complete set of one of the subunits. We exploited a 3D reconstruction of metal replicas of quick-freeze, deep-etched, oligomers, representing the four types of molecules described. All the molecules appear to adhere with the same face toward the mica. Images of rotary shadowed oligomers were processed by multivariate statistical analysis to evidence clusters of equivalent and homogeneous oligomers. 3D reconstructions of the replicas so classified were performed by random conical tilt tomography. In the case of intact molecules (not cleaved) the reconstructions represent both the outer and the inner surfaces of the mold; the latter gives a reasonably accurate sense of the upper surface of the VacA oligomers. These data support the hypothesis that VacA is an AB type toxin and suggest a model in which the smaller of the two subunits is arranged in a uniform ring on the surface of the molecule in such a way as to contribute to the overall stability of the molecule.

Fast direct Fourier methods, based on one- and two-pass coordinate transformations, yield accurate reconstructions of x-ray CT clinical images

The conversion from polar to Cartesian coordinates can be carried out with two-pass algorithms. The paper describes two different methods based on concentric square frames and octagonal frames and their results, obtained with accurate interpolations based on the "moving window Shannon reconstruction' (MWSR). The embedding of these algorithms in direct Fourier methods (DFMs) of tomographic reconstruction is discussed. With respect to one-pass methods and to the use of octagonal frames, the square frame method makes it possible to carry out the first pass, a radial resampling, in the direct space, before computing 1D Fourier transforms (FTs) of projections. Reconstructions of clinical images from the raw data of a third-generation x-ray tomograph are presented and compared with those obtained with one-pass DFMs and with the convolution back-projection method (CBPM) performed by the instrument. The simple algorithm using square frames yields results in complete agreement with other DFM protocols and the CBPM. On a general-purpose computer, the execution of DFM protocols based on one-pass and two-pass coordinate transformations is 35 to 55 times faster than the CBPM and make the algorithms attractive for modern instrumentation.