A direct Fourier method (DFM) for X-ray tomographic reconstructions and the accurate simulation of sinograms

This article illustrates the reconstruction of tomographic images by a direct Fourier method (DFM) and the results obtained from simulations and from experimental X-ray sinograms. The implementation of DFM, especially with regard to the resampling of the 2D Fourier transform, is based on a technique of Shannon reconstruction, devised by the authors, and on novel interpolating kernels. A short account is given on the principles and the implementation aspects of the interpolation technique. The DFM protocol developed by the authors has been tested, both for parallel and fan geometry, on simulated sinograms obtained from real images and from phantoms. The technique used to compute accurate projections is also described, since it might be useful in restoring missing parts of sinograms with processes based on ‘projections on convex sets’ (POCS) techniques. The results obtained from simulations and from the raw data of a third generation tomograph are presented and discussed. A comparison among reconstructions obtained from complete sinograms and from half of them suggests that adequate images could be obtained with a radiation dose lower than that used to obtain the experimental sinograms.     

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.     

Symmetry embedding in the reconstruction of macromolecular assemblies via the discrete Radon transform

In this paper we discuss the embedding of symmetry information in an algorithm for three-dimensional reconstruction, which is based on the discrete Radon transform. The original algorithm was designed for randomly oriented and in principal asymmetric particles. The expanded version presented here covers all symmetry point groups which can be exhibited by macromolecular protein assemblies. The orientations of all symmetry equivalent projections, based on the orientation of an experimental projection, are obtained using global group operators. Further, an improved interpolation scheme for the recovery of the three-dimensional discrete Radon transform has been designed for greater computational efficiency. The algorithm has been tested on phantom structures as well as on real data, a virus structure possessing icosahedral symmetry.     

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.     

Three-Dimensional Reconstruction from Reduced Sets of Very Noisy Images Acquired Following a Single-Axis Tilt Schema: Application of a New Three-Dimensional Reconstruction Algorithm and Objective Comparison with Weighted Backprojection

In this work we propose a reconstruction algorithm (ART with blobs) that has not been previously used in electron Tomography and we compare it with the standard method in the field (weighted back projection, WBP). We assume that only a limited set of very noisy images, collected around a single axis tilt, is available; which is a typical situation in Electron Tomography. In general, the reconstruction problem is underdetermined (due to the limited number of projections) and the data are inconsistent (due to the high level of noise). The evaluation of the results is performed in a rigorous way by a task-oriented approach which makes use of numerical observers. ART with blobs outperforms WBP for a number of key tasks. Results are presented both for simplified line integral data and for realistic simulations of macromolecular structures embedded in amorphous ice.     

Convolutional neural networks for reconstruction of undersampled optical projection tomography data applied to in vivo imaging of zebrafish

Optical projection tomography (OPT) is a 3D mesoscopic imaging modality that can utilize absorption or fluorescence contrast. 3D images can be rapidly reconstructed from tomographic data sets sampled with sufficient numbers of projection angles using the Radon transform, as is typically implemented with optically cleared samples of the mm‐to‐cm scale. For in vivo imaging, considerations of phototoxicity and the need to maintain animals under anesthesia typically preclude the acquisition of OPT data at a sufficient number of angles to avoid artifacts in the reconstructed images. For sparse samples, this can be addressed with iterative algorithms to reconstruct 3D images from undersampled OPT data, but the data processing times present a significant challenge for studies imaging multiple animals. We show here that convolutional neural networks (CNN) can be used in place of iterative algorithms to remove artifacts—reducing processing time for an undersampled in vivo zebrafish dataset from 77 to 15 minutes. We also show that using CNN produces reconstructions of equivalent quality to compressed sensing with 40% fewer projections. We further show that diverse training data classes, for example, ex vivo mouse tissue data, can be used for CNN‐based reconstructions of OPT data of other species including live zebrafish.      

Exploratory Consensus of Hierarchical Clusterings for Melanoma and Breast Cancer

Finding subtypes of heterogeneous diseases is the biggest challenge in the area of biology. Often, clustering is used to provide a hypothesis for the subtypes of a heterogeneous disease. However, there are usually discrepancies between the clusterings produced by different algorithms. This work introduces a simple method which provides the most consistent clusters across three different clustering algorithms for a melanoma and a breast cancer data set. The method is validated by showing that the Silhouette, Dunne's and Davies-Bouldin's cluster validation indices are better for the proposed algorithm than those obtained by k-means and another consensus clustering algorithm. The hypotheses of the consensus clusters on both the data sets are corroborated by clear genetic markers and 100 percent classification accuracy. In Bittner et al.'s melanoma data set, a previously hypothesized primary cluster is recognized as the largest consensus cluster and a new partition of this cluster into two subclusters is proposed. In van't Veer et al.'s breast cancer data set, previously proposed "basal” and "luminal A” subtypes are clearly recognized as the two predominant clusters. Furthermore, a new hypothesis is provided about the existence of two subgroups within the "basal” subtype in this data set. The clusters of van't Veer's data set is also validated by high classification accuracy obtained in the data set of van de Vijver et al.     

New method for yeast identification using Burrows-Wheeler transform

The explosive growth in biological data in recent years has led to the development of new methods to identify DNA sequences. Many algorithms have recently been developed that search DNA sequences looking for unique DNA sequences. This paper considers the application of the Burrows-Wheeler transform (BWT) to the problem of unique DNA sequence identification. The BWT transforms a block of data into a format that is extremely well suited for compression. This paper presents a time-efficient algorithm to search for unique DNA sequences in a set of genes. This algorithm is applicable to the identification of yeast species and other DNA sequence sets.     

Efficient implementation of a filtered back-projection algorithm using a voxel-by-voxel approach

The large amount of image data necessary for high-resolution 3D reconstruction of macromolecular assemblies leads to significant increases in the computational time. One of the most time consuming operations is 3D density map reconstruction, and software optimization can greatly reduce the time required for any given structural study. The majority of algorithms proposed for improving the computational effectiveness of a 3D reconstruction are based on a ray-by-ray projection of each image into the reconstructed volume. In this paper, we propose a novel fast implementation of the "filtered back-projection" algorithm based on a voxel-by-voxel principle. Our version of this implementation has been exhaustively tested using both model and real data. We compared 3D reconstructions obtained by the new approach with results obtained by the filtered Back-Projections algorithm and the Fourier-Bessel algorithm commonly used for reconstructing icosahedral viruses. These computational experiments demonstrate the robustness, reliability, and efficiency of this approach.     

一种优化的生物数据多层关联规则挖掘算法

关联规则挖掘技术是寻找基因间关系的有效手段,但现有算法未针对高通量生物数据的特点进行优化,而存在着效率低下等缺点。提出的MAGO-FP算法,使用Gene Ontology(GO)的概念分层结构,通过对FP-Growth算法的扩展,具有一定的性能优势。在此基础上,应用该算法分析了一组由S.cerevisiae酵母菌cDNA微阵列芯片产生的实验数据,发现了一些候选关联规则。并针对其中一些重要的关联规则,通过相关文献证实了其真实性,表明该算法在基因表达分析等研究中具有应用价值。     

Elastic phantoms generated by microfluidics technology: validation of an imaged-based approach for accurate measurement of the growth rate of lung nodules

This paper focuses on validating our approach for monitoring the development of lung nodules detected in successive chest low-dose computed tomography (LDCT) scans of a patient. Our methodology for monitoring detected lung nodules includes 3D LDCT data registration, which is a non-rigid technique and involves two steps: (i) global target-to-prototype alignment of one scan to another using the experience gained from a prior appearance model, followed by (ii) local alignment to correct for intricate relative deformations. We propose a new approach for validating the accuracy of our algorithm for elastic lung phantoms constructed with state-of-the-art microfluidics technology and in vivo data. Fabricated from a flexible transparent polymer, i.e. polydimethylsiloxane (PDMS), the phantoms mimic the contractions and expansions of the lung and nodules as seen during normal breathing. The in vivo data in our study had been collected from a small control group of four subjects and a larger test group of 27 subjects with known ground truth (biopsy diagnosis. The growth rate and diagnostic results for both phantoms and in vivo data confirm the high accuracy of our algorithm.     

Automatic particle pickup method using a neural network has high accuracy by applying an initial weight derived from eigenimages: a new reference free method for single-particle analysis

The single-particle analysis is a structure-determining method for electron microscope (EM) images which does not require crystal. In this method, the projections are picked up and averaged by the images of similar Euler angles to improve the signal to noise ratio, and then create a 3-D reconstruction. The selection of a large number of particles from the cryo-EM micrographs is a pre-requisite for obtaining a high resolution. To pickup a low-contrast cryo-EM protein image, we have recently found that a three-layer pyramidal-type neural network is successful in detecting such a faint image, which had been difficult to detect by other methods. The connection weights between the input and hidden layers, which work as a matching filter, have revealed that they reflect characters of the particle projections in the training data. The images stored in terms of the connection weights were complex, more similar to the eigenimages which are created by the principal component analysis of the learning images rather than to the averages of the particle projections. When we set the initial learning weights according to the eigenimages in advance, the learning period was able to be shortened to less than half the time of the NN whose initial weights had been set randomly. Further, the pickup accuracy increased from 90 to 98%, and a combination of the matching filters were found to work as an integrated matching filter there. The integrated filters were amazingly similar to averaged projections and can be used directly as references for further two-dimensional averaging. Therefore, this research also presents a brand-new reference-free method for single-particle analysis.     

High-resolution pyrimidine- and ribose-specific 4D HCCH-COSY spectra of RNA using the filter diagonalization method

The NMR spectra of nucleic acids suffer from severe peak overlap, which complicates resonance assignments. 4D NMR experiments can overcome much of the degeneracy in 2D and 3D spectra; however, the linear increase in acquisition time with each new dimension makes it impractical to acquire high-resolution 4D spectra using standard Fourier transform (FT) techniques. The filter diagonalization method (FDM) is a numerically efficient algorithm that fits the entire multi-dimensional time-domain data to a set of multi-dimensional oscillators. Selective 4D constant-time HCCH-COSY experiments that correlate the H5-C5-C6-H6 base spin systems of pyrimidines or the H1'-C1'-C2'-H2' spin systems of ribose sugars were acquired on the (13)C-labeled iron responsive element (IRE) RNA. FDM-processing of these 4D experiments recorded with only 8 complex points in the indirect dimensions showed superior spectral resolution than FT-processed spectra. Practical aspects of obtaining optimal FDM-processed spectra are discussed. The results here demonstrate that FDM-processing can be used to obtain high-resolution 4D spectra on a medium sized RNA in a fraction of the acquisition time normally required for high-resolution, high-dimensional spectra.     

Brain PET imaging optimization with time of flight and point spread function modelling

PurposeTo assess the influence of reconstruction algorithms and parameters on the PET image quality of brain phantoms in order to optimize reconstruction for clinical PET brain studies in a new generation PET/CT.MethodsThe 3D Hoffman phantom that simulates ¹⁸F-fluorodeoxyglucose (FDG) distribution was imaged in a Siemens Biograph mCT TrueV PET/CT with Time of Flight (TOF) and Point Spread Function (PSF) modelling. Contrast-to-Noise Ratio (CNR), contrast and noise were studied for different reconstruction models: OSEM, OSEM + TOF, OSEM + PSF and OSEM + PSF + TOF.The 2D multi-compartment Hoffman phantom was filled to simulate 4 different tracers' spatial distribution: FDG, ¹¹C-flumazenil (FMZ), ¹¹C-Methionine (MET) and 6-¹⁸F-fluoro-l-dopa (FDOPA). The best algorithm for each tracer was selected by visual inspection. The maximization of CNR determined the optimal parameters for each reconstruction.ResultsIn the 3D Hoffman phantom, both noise and contrast increased with increasing number of iterations and decreased with increasing FWHM. OSEM + PSF + TOF reconstruction was generally superior to other reconstruction models. Visual analysis of the 2D Hoffman brain phantom suggested that OSEM + PSF + TOF is the optimum algorithm for tracers with focal uptake, such as MET or FDOPA, and OSEM + TOF for tracers with diffuse cortical uptake (i.e. FDG and FMZ). Optimization of CNR demonstrated that OSEM + TOF reconstruction must be performed with 2 iterations and a filter FWHM of 3 mm, and OSEM + PSF + TOF reconstruction with 4 iterations and 1 mm FWHM filter.ConclusionsOptimization of reconstruction algorithm and parameters has been performed to take particular advantage of the last generation PET scanner, recommending specific settings for different brain PET radiotracers.     

Methods of least squares and SIRT in reconstruction.

In this paper we show that a particular version of the Simultaneous Iterative Reconstruction Technique (SIRT) proposed by Gilbert in 1972 strongly resembles the Richardson least-squares algorithm.By adopting the adjustable parameters of the general Richardson algorithm, we have been able to produce generalized SIRT algorithms with improved convergence.A particular generalization of the SIRT algorithm, GSIRT, has an adjustable parameter σ and the starting picture ρ⁰ as input. A value $\text{1}\text{2}$ for σ and a weighted back-projection for ρ⁰ produce a stable algorithm.We call the SIRT-like algorithms for the solution of the weighted leastsquares problems LSIRT and present two such algorithms, LSIRT1 and LSIRT2, which have definite computational advantages over SIRT and GSIRT.We have tested these methods on mathematically simulated phantoms and find that the new SIRT methods converge faster than Gilbert's SIRT but are more sensitive to noise present in the data. However, the faster convergence rates allow termination before the noise contribution degrades the reconstructed image excessively.     

An iterative deconvolution algorithm for image recovery in clinical CT: A phantom study

PurposeTo study the feasibility of using an iterative reconstruction algorithm to improve previously reconstructed CT images which are judged to be non-diagnostic on clinical review. A novel rapidly converging, iterative algorithm (RSEMD) to reduce noise as compared with standard filtered back-projection algorithm has been developed.Materials and methodsThe RSEMD method was tested on in-silico, Catphan^®500, and anthropomorphic 4D XCAT phantoms. The method was applied to noisy CT images previously reconstructed with FBP to determine improvements in SNR and CNR. To test the potential improvement in clinically relevant CT images, 4D XCAT phantom images were used to simulate a small, low contrast lesion placed in the liver.ResultsIn all of the phantom studies the images proved to have higher resolution and lower noise as compared with images reconstructed by conventional FBP. In general, the values of SNR and CNR reached a plateau at around 20 iterations with an improvement factor of about 1.5 for in noisy CT images. Improvements in lesion conspicuity after the application of RSEMD have also been demonstrated. The results obtained with the RSEMD method are in agreement with other iterative algorithms employed either in image space or with hybrid reconstruction algorithms.ConclusionsIn this proof of concept work, a rapidly converging, iterative deconvolution algorithm with a novel resolution subsets-based approach that operates on DICOM CT images has been demonstrated. The RSEMD method can be applied to sub-optimal routine-dose clinical CT images to improve image quality to potentially diagnostically acceptable levels.     

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.     

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.     

A neural network model for the prediction of membrane-spanning amino acid sequences

The architecture and weights of an artificial neural network model that predicts putative transmembrane sequences have been developed and optimized by the algorithm of structure evolution. The resulting filter is able to classify membrane/nonmembrane transition regions in sequences of integral human membrane proteins with high accuracy. Similar results have been obtained for both training and test set data, indicating that the network has focused on general features of transmembrane sequences rather than specializing on the training data. Seven physicochemical amino acid properties have been used for sequence encoding. The predictions are compared to hydrophobicity plots.