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.     

Electron tomography of amplified nanogold immunolabelling: Improvement of quality based on alignment of projections with sinograms and use of post-reconstruction deconvolution

Electron tomography of immunolabelled proteins identified with amplified nanogold particles imaged by Scanning and Transmission Electron Microscopy within thick sections is a powerful method to investigate the three-dimensional organization of complex cellular machineries. In order to increase the overall quality of the reconstructed cube, we have developed two methods that improve the tomographic reconstruction process. We first performed a very precise alignment of the projections before reconstruction with a technique using sinograms. After reconstruction, we propose to compute image restoration by calculating the Point Spread Function of the projection/back-projection system and to use it to deblur the reconstructed cubes. Improvement in the quality of the reconstructed cubes is demonstrated on images of nucleolar proteins tagged with EGFP and immunolabelled with nanogold particles.     

A multiresolution approach to orientation assignment in 3D electron microscopy of single particles

Three-dimensional (3D) electron microscopy (3DEM) aims at the determination of the spatial distribution of the Coulomb potential of macromolecular complexes. The 3D reconstruction of a macromolecule using single-particle techniques involves thousands of 2D projections. One of the key parameters required to perform such a 3D reconstruction is the orientation of each projection image as well as its in-plane orientation. This information is unknown experimentally and must be determined using image-processing techniques. We propose the use of wavelets to match the experimental projections with those obtained from a reference 3D model. The wavelet decomposition of the projection images provides a framework for a multiscale matching algorithm in which speed and robustness to noise are gained. Furthermore, this multiresolution approach is combined with a novel orientation selection strategy. Results obtained from computer simulations as well as experimental data encourage the use of this approach.     

Issues of resolution and polymorphism in single-particle reconstruction

Three-dimensional reconstruction from electron microscopic (EM) images of isolated macromolecular complexes is being employed by many laboratories. This approach is extremely powerful and continues to improve in resolution. In the absence of stereochemical constraints that can be used to assess the quality of a reconstruction, as exist in X-ray crystallography, several other measures have typically been used. A very useful assessment of quality can be made in the comparison between the projections of the three-dimensional reconstruction and averages generated from classes of images. The main quantitative measure has been that of resolution statistics, typically based upon Fourier shell correlations. We show, using only simulated noise for images, that impressive resolution statistics are generated that can even extend the apparent resolution of the starting model. When truly independent reconstructions are generated starting from different initial models, however, such artefacts are not possible. We also show, using real images of DnaB rings, that in the presence of polymorphism artefactual reconstructions can be generated whose projections match class averages. These averages, however, are themselves artefactual as they involve heterogeneous images. The issues presented here need to be considered when single-particle EM reconstructions are evaluated.     

Optimal calibration marker mesh for 2D X-ray sensors in 3D reconstruction

Image intensifiers suffer from distortions due to magnetic fields. In order to use this X-ray projections images for computer-assisted medical interventions, image intensifiers need to be calibrated. Opaque markers are often used for the correction of the image distortion and the estimation of the acquisition geometry parameters. Information under the markers is then lost. In this work, we consider the calibration of image intensifiers in the framework of 3D reconstruction from several 2D X-ray projections. In this context, new schemes of marker distributions are proposed for 2D X-ray sensor calibration. They are based on efficient sampling conditions of the parallel-beam X-ray transform when the detector and source trajectory is restricted to a circle around the measured object. Efficient sampling are essentially subset of standard sampling in this situation. The idea is simply to exploit the data redundancy of standard sampling and to replace some holes of efficient schemes by markers. Optimal location of markers in the sparse efficient sampling geometry can thus be found. In this case, the markers can stay on the sensor during the measurement with--theoretically--no loss of information (when the signal-to-noise ratio is large). Even if the theory is based on the parallel-beam X-ray transform, numerical experiments on both simulated and real data are shown in the case of weakly divergent beam geometry. We show that the 3D reconstruction from simulated data with interlaced markers is essentially the same as those obtained from data with no marker. We show that efficient Fourier interpolation formulas based on optimal sparse sampling schemes can be used to recover the information hidden by the markers.     

Three-dimensional graphical reconstruction from HVEM stereoviews of biological specimens by means of a microcomputer

A microcomputer reconstruction technique has been developed in order to permit a larger exploitation of stereomicroscopy. The microcomputer facility consists of a digitizing tablet, a microcomputer, a graphics terminal, a graphics plotter and a printer. The technique has been applied to the study of HVEM stereopairs, performed by recording two images of the same area of a specimen (thick section of araldite-embedded leech ganglion neurons), tilted relative to the beam axis through an angle 0/20 degrees. Coordinates of N conjugate points of interest, expressed in a common reference system were obtained with the help of a digitizing tablet and the misorientation between the two images was determined by a method based on a least square technique. New projections of the object on different planes are provided by the microcomputer facility. Also the microcomputer method permits to obtain new stereopairs drawings, in various orientations and slices from a three-dimensional reconstruction of the object oriented in any direction in space. The method permits to obtain computed anaglyph drawings, printed here, which are stereoviews of the same object.     

Generation of clinical 177Lu SPECT/CT images based on Monte Carlo simulation with GATE

PurposePatient-specific dosimetry in MRT relies on quantitative imaging, pharmacokinetic assessment and absorbed dose calculation. The DosiTest project was initiated to evaluate the uncertainties associated with each step of the clinical dosimetry workflow through a virtual multicentric clinical trial. This work presents the generation of simulated clinical SPECT datasets based on GATE Monte Carlo modelling with its corresponding experimental CT image, which can subsequently be processed by commercial image workstations.MethodsThis study considers a therapy cycle of 6.85 GBq ¹⁷⁷Lu-labelled DOTATATE derived from an IAEA-Coordinated Research Project (E23005) on “Dosimetry in Radiopharmaceutical therapy for personalised patient treatment”. Patient images were acquired on a GE Infinia-Hawkeye 4 gamma camera using a medium energy (ME) collimator. Simulated SPECT projections were generated based on experimental time points and validated against experimental SPECT projections using flattened profiles and gamma index. The simulated projections were then incorporated into the patient SPECT/CT DICOM envelopes for processing and their reconstruction within a commercial image workstation.ResultsGamma index passing rate (2% − 1 pixel criteria) between 95 and 98% and average gamma between 0.28 and 0.35 among different time points revealed high similarity between simulated and experimental images. Image reconstruction of the simulated projections was successful on HERMES and Xeleris workstations, a major step forward for the initiation of a multicentric virtual clinical dosimetry trial based on simulated SPECT/CT images.ConclusionsRealistic ¹⁷⁷Lu patient SPECT projections were generated in GATE. These modelled datasets will be circulated to different clinical departments to perform dosimetry in order to assess the uncertainties in the entire dosimetric chain.     

A two-exposure technique for ice-embedded samples successfully reconstructs the chlorocruorin pigment of Sabella spallanzanii at 2. 1 Nm resolution.

A technique for reconstructing ice-embedded macromolecules from electron micrographs taken at two specimen tilts (+/-23 degrees ) has been used to determine the structure of chlorocruorin isolated from the Polychaete annelid Sabella spallanzanii. Images of individual molecules were extracted in couples from two micrographs of the same field of view so each couple consists of two projections of the same molecule. One couple was used as a fixed reference for alignment. Different references yielded reconstructions with different orientations. These were merged to give a model against which the orientation of 1624 first-exposure images was refined to give a final reconstruction at 2.1 nm resolution. The structure of this hematic pigment, essentially the same as that for Lumbricus terrestris, is a bilayer structure with overall symmetry D6, containing six hollow groups per layer. A hollow group is formed by six globular masses and has approximate threefold symmetry. Other structural elements connect the two layers and the hollow groups in a layer. This non-globin material occupies about 15% of the total molecular volume. The results show that the double-exposure strategy, previously described by some of the authors and tested in computer simulations, performs well in real experiments and could be used to obtain preliminary reconstructions in a semiautomatic way.     

Noise properties of reconstructed images in a kilo-voltage on-board imaging system with iterative reconstruction techniques: A phantom study

X-ray computed tomography (CT) images obtained with a kilo-voltage (kV) on-board imaging (OBI) system improve the accuracy of patient setup and treatment planning. The use of iterative reconstruction techniques (IRTs) for CT imaging can also reduce radiation dose compared to analytic reconstruction techniques. Despite these improvements, the image quality varies with IRTs, and the noise structure of reconstructed images can be distorted by IRTs. In this study, the noise properties and spatial resolution of the images reconstructed by IRTs were evaluated in terms of conventional noise metrics, high-order statistics, noise spectral density (NSD) and modulation transfer function (MTF) at different radiation doses. A kV OBI system mounted on a Varian Trilogy machine and a CATPHAN600 phantom were used to obtain projections, and the projections were reconstructed by Feldkamp (FDK), algebraic reconstruction technique (ART), maximum-likelihood expectation–maximization (MLEM) and total variation (TV) minimization algorithms. The reconstructed images were compared according to mean, standard deviation, skewness, kurtosis, NSD and MTF at different radiation doses. The results demonstrated that the noise properties and spatial resolution of reconstructed images depend on the type of IRT and the radiation dose. The noise structures are altered by IRTs and can be characterized by high-order statistics and NSD, as well as conventional noise metrics. In conclusion, high-order statistics and NSD should be considered in order to provide detailed information for the images reconstructed by IRTs. Also, trade-off among noise properties, spatial resolution and contrast is important to optimize image quality obtained using IRTs.     

A measure for the angle between projections based on the extent of correlation between corresponding central sections

A pre-condition for the ab initio assignment of Euler angles to a set of projections from an asymmetric object is that at least three of the available projections correspond to rotations about different axes. For symmetric objects this condition may be relaxed. There are some applications of single-particle electron microscopy, such as the reconstruction of filamentous macromolecular assemblies, where all available projections more-or-less correspond to rotations about a common rotation axis making it difficult to satisfy this condition. Here, a method has been developed to overcome this problem, based on the fact that the correlation between two central sections of the Fourier transform of a compact object will not be limited to an infinitesimal central line but will have a finite extent, which is related to the angle between the corresponding projections. Projections from model filaments, with different degrees of rotational symmetry about the long axis, have been used to test the methodology. The results show that angle determination is robust down to signal-to-noise ratios as low as 2 and that, in general, the error decreases as the degree of symmetry increases. The method has been used to assign angles to a set of negatively stained muscle thick filament projections to obtain an initial 3D reconstruction. The main features of the projections are seen to be faithfully reproduced in the reprojections from the reconstruction. A real-space adaptation of this method is also discussed.     

Analysis of symmetry and three-dimensional reconstruction of thin gp32*I crystals.

Thin, multilayered crystals of gp32*I were analyzed by negative stain electron microscopy and image processing. Images of untilted crystals exhibited different projection symmetries and structural motifs. Systematic analysis of these images categorized the projections into four types. Areas producing the type 1 projection were reconstructed in three-dimensions from four tilt series containing 111 images. The three-dimensional data has excellent p121 plane group symmetry and reveals that the gp32*I molecule contains two large domains linked together by a small domain. Computer simulations utilizing projection data suggested that the type 2 and 3 projections arise from superposition of type 1 projections related by a 21 screw axis along the projection axis. The three-dimensional reconstruction was utilized in a final simulation that explained the occurrence of the fourth type of projection. This work provides a firm foundation for future high-resolution analysis of the crystal by electron cryomicroscopy.     

Iterative reconstruction of cryo-electron tomograms using nonuniform fast Fourier transforms

Algorithms for three-dimensional (3D) reconstruction of objects based on their projections are essential in various biological and medical imaging modalities. In cryo-electron tomography (CET) a major challenge for reconstruction is the limited range of projection angles, which manifests itself as a “missing wedge” of data in Fourier space making the reconstruction problem ill-posed. Here, we apply an iterative reconstruction method that makes use of nonuniform fast Fourier transform (NUFFT) to the reconstruction of cryo-electron tomograms. According to several measures the reconstructions are superior to those obtained using conventional methods, most notably weighted backprojection. Most importantly, we show that it is possible to fill in partially the unsampled region in Fourier space with meaningful information without making assumptions about the data or applying prior knowledge. As a consequence, particles of known structure can be localized with higher confidence in cryotomograms and subtomogram averaging yields higher resolution densities.     

An iterative Fourier-Bessel algorithm for reconstruction of helical structures with severe Bessel overlap

Classical Fourier-Bessel methodology fails when used to reconstruct helical structures with severe Bessel overlap on the layer lines. In the reconstruction of a peculiar type of double-layered helical tube of GDP-tubulin, we face the problem of Bessel overlap on all the layer lines due to the superposition of the Fourier components from the inner and outer layers of the tube. In order to decompose the Fourier terms of the inner and outer layers more than one image of the tubes must be combined and the orientations of their inner and outer layer helices must be determined. While there is no direct analytical method to determine these orientational parameters, we have devised an iterative Fourier-Bessel algorithm to calculate the correct orientations and thus allow us to obtain a reconstruction from multiple images of the double-layered tubes. The algorithm successfully works for the reconstruction of computer-modeled double-layered helical tubes as well as with real images obtained by cryo-electron microscopy. The algorithm has also been applied with very satisfactory results to the reconstruction of 13-protofilament microtubules, which is another helical structure that suffer Bessel overlap, suggesting its generality.     

Wavelets filtering for classification of very noisy electron microscopic single particles images- application on structure determination of VP5-VP19C recombinant

Background

Images of frozen hydrated [vitrified] virus particles were taken close-to-focus in an electron microscope containing structural signals at high spatial frequencies. These images had very low contrast due to the high levels of noise present in the image. The low contrast made particle selection, classification and orientation determination very difficult. The final purpose of the classification is to improve the signal-to-noise ratio of the particle representing the class, which is usually the average. In this paper, the proposed method is based on wavelet filtering and multi-resolution processing for the classification and reconstruction of this very noisy data. A multivariate statistical analysis (MSA) is used for this classification.

Results

The MSA classification method is noise dependant. A set of 2600 projections from a 3D map of a herpes simplex virus -to which noise was added- was classified by MSA. The classification shows the power of wavelet filtering in enhancing the quality of class averages (used in 3D reconstruction) compared to Fourier band pass filtering. A 3D reconstruction of a recombinant virus (VP5-VP19C) is presented as an application of multi-resolution processing for classification and reconstruction.

Conclusion

The wavelet filtering and multi-resolution processing method proposed in this paper offers a new way for processing very noisy images obtained from electron cryo-microscopes. The multi-resolution and filtering improves the speed and accuracy of classification, which is vital for the 3D reconstruction of biological objects. The VP5-VP19C recombinant virus reconstruction presented here is an example, which demonstrates the power of this method. Without this processing, it is not possible to get the correct 3D map of this virus.

     

The effect of overabundant projection directions on 3D reconstruction algorithms

The experimental process of collecting images from macromolecules in an electron microscope is such that it does not allow for prior specification of the angular distribution of the projection images. As a consequence, an uneven distribution of projection directions may occur. Concerns have been raised recently about the behavior of 3D reconstruction algorithms for the case of unevenly distributed projections. It has been illustrated on experimental data that in the case of a heavily uneven distribution of projection directions some algorithms tend to elongate the reconstructed volumes along the overloaded direction so much as to make a quantitative biological analysis impossible. In answer to these concerns we have developed a strategy for quantitative comparison and optimization of 3D reconstruction algorithms. We apply this strategy to quantitatively analyze algebraic reconstruction techniques (ART) with blobs, simultaneous iterative reconstruction techniques (SIRT) with voxels, and weighted backprojection (WBP). We show that the elongation artifacts that had been previously reported can be strongly reduced. With our specific choices for the free parameters of the three algorithms, WBP reconstructions tend to be inferior to those obtained with either SIRT or ART and the results obtained with ART are comparable to those with SIRT, but at a very small fraction of the computational cost of SIRT.     

Three-dimensional spectral signal-to-noise ratio for a class of reconstruction algorithms

A three-dimensional (3D) version of the spectral signal-to-noise ratio (SSNR)-based resolution measure is introduced. The measure is defined for a class of 3D reconstruction algorithms that use interpolation in Fourier space. The statistical properties of the SSNR are discussed and related to the properties of another resolution measure, the Fourier shell correlation (FSC). The new measure was tested on 3D structures calculated from a simulated set of quasi-evenly spaced 2D projections using a nearest-neighbor interpolation and a gridding algorithm. In the latter case, the results agree very well with the FSC-based estimate, with the exception of very high SSNR values. The main applicability of the 3D SSNR is tomography, where due to the small number of projections collected, FSC cannot be used. The new measure was applied to three sets of tomographic data. It was demonstrated that the measure is sufficiently sensitive to yield theoretically expected results. Therefore, the 3D SSNR opens up the possibility of evaluating the quality of tomographic reconstructions in an objective manner. The 3D distribution of SSNR is of major interest in single-particle analysis. It is shown that the new measure can be used to evaluate the anisotropy of 3D reconstructions. The distribution of SSNR is characterized by three anisotropy indices derived from principal axes of the 3D inertia covariance matrix of the SSNR. These indices are used to construct a 3D Fourier filter which, when applied to a 3D reconstruction of a macromolecule, maximizes the SNR in real space and minimizes real-space artifacts caused by uneven distribution of 2D projections.     

Dipolar coupling between nitroxide spin labels: the development and application of a tether-in-a-cone model

A tether-in-a-cone model is developed for the simulation of electron paramagnetic resonance spectra of dipolar coupled nitroxide spin labels attached to tethers statically disordered within cones of variable halfwidth. In this model, the nitroxides adopt a range of interprobe distances and orientations. The aim is to develop tools for determining both the distance distribution and the relative orientation of the labels from experimental spectra. Simulations demonstrate the sensitivity of electron paramagnetic resonance spectra to the orientation of the cones as a function of cone halfwidth and other parameters. For small cone halfwidths (< approximately 40 degrees ), simulated spectra are strongly dependent on the relative orientation of the cones. For larger cone halfwidths, spectra become independent of cone orientation. Tether-in-a-cone model simulations are analyzed using a convolution approach based on Fourier transforms. Spectra obtained by the Fourier convolution method more closely fit the tether-in-a-cone simulations as the halfwidth of the cone increases. The Fourier convolution method gives a reasonable estimate of the correct average distance, though the distance distribution obtained can be significantly distorted. Finally, the tether-in-a-cone model is successfully used to analyze experimental spectra from T4 lysozyme. These results demonstrate the utility of the model and highlight directions for further development.     

A Novel Weighted Total Difference Based Image Reconstruction Algorithm for Few-View Computed Tomography

In practical applications of computed tomography (CT) imaging, due to the risk of high radiation dose imposed on the patients, it is desired that high quality CT images can be accurately reconstructed from limited projection data. While with limited projections, the images reconstructed often suffer severe artifacts and the edges of the objects are blurred. In recent years, the compressed sensing based reconstruction algorithm has attracted major attention for CT reconstruction from a limited number of projections. In this paper, to eliminate the streak artifacts and preserve the edge structure information of the object, we present a novel iterative reconstruction algorithm based on weighted total difference (WTD) minimization, and demonstrate the superior performance of this algorithm. The WTD measure enforces both the sparsity and the directional continuity in the gradient domain, while the conventional total difference (TD) measure simply enforces the gradient sparsity horizontally and vertically. To solve our WTD-based few-view CT reconstruction model, we use the soft-threshold filtering approach. Numerical experiments are performed to validate the efficiency and the feasibility of our algorithm. For a typical slice of FORBILD head phantom, using 40 projections in the experiments, our algorithm outperforms the TD-based algorithm with more than 60% gains in terms of the root-mean-square error (RMSE), normalized root mean square distance (NRMSD) and normalized mean absolute distance (NMAD) measures and with more than 10% gains in terms of the peak signal-to-noise ratio (PSNR) measure. While for the experiments of noisy projections, our algorithm outperforms the TD-based algorithm with more than 15% gains in terms of the RMSE, NRMSD and NMAD measures and with more than 4% gains in terms of the PSNR measure. The experimental results indicate that our algorithm achieves better performance in terms of suppressing streak artifacts and preserving the edge structure information of the object.     

Conformational heterogeneity and probability distributions from single-particle cryo-electron microscopy

Single-particle cryo-electron microscopy (cryo-EM) is a technique that takes projection images of biomolecules frozen at cryogenic temperatures. A major advantage of this technique is its ability to image single biomolecules in heterogeneous conformations. While this poses a challenge for data analysis, recent algorithmic advances have enabled the recovery of heterogeneous conformations from the noisy imaging data. Here, we review methods for the reconstruction and heterogeneity analysis of cryo-EM images, ranging from linear-transformation-based methods to nonlinear deep generative models. We overview the dimensionality-reduction techniques used in heterogeneous 3D reconstruction methods and specify what information each method can infer from the data. Then, we review the methods that use cryo-EM images to estimate probability distributions over conformations in reduced subspaces or predefined by atomistic simulations. We conclude with the ongoing challenges for the cryo-EM community.     

Towards a biomechanics-based technique for assessing myocardial contractility: an inverse problem approach

This work presents the initial development and implementation of a novel 3D biomechanics-based approach to measure the mechanical activity of myocardial tissue, as a potential non-invasive tool to assess myocardial function. This technique quantifies the myocardial contraction forces developed within the ventricular myofibers in response to electro-physiological stimuli. We provide a 3D finite element formulation of a contraction force reconstruction algorithm, along with its implementation using magnetic resonance (MR) data. Our algorithm is based on an inverse problem solution governed by the fundamental continuum mechanics principle of conservation of linear momentum, under a first-order approximation of elastic and isotropic material conditions. We implemented our technique using a subject-specific ventricle model obtained by extracting the left ventricular anatomical features from a set of high-resolution cardiac MR images acquired throughout the cardiac cycle using prospective electrocardiographic gating. Cardiac motion information was extracted by non-rigid registration of the mid-diastole reference image to the remaining images of a 4D dataset. Using our technique, we reconstructed dynamic maps that show the contraction force distribution superimposed onto the deformed ventricle model at each acquired frame in the cardiac cycle. Our next objective will consist of validating this technique by showing the correlation between the presence of low contraction force patterns and poor myocardial functionality.