Weighted Backprojection Research Articles

Entropy-regularized deconvolution of cellular cryotransmission electron tomograms

Cryo-electron tomography (cryo-ET) allows for the high-resolution visualization of biological macromolecules. However, the technique is limited by a low signal-to-noise ratio (SNR) and variance in contrast at different frequencies, as well as reduced Z resolution. Here, we applied entropy-regularized deconvolution (ER-DC) to cryo-ET data generated from transmission electron microscopy (TEM) and reconstructed using weighted back projection (WBP). We applied deconvolution to several in situ cryo-ET datasets and assessed the results by Fourier analysis and subtomogram analysis (STA).

Modified weighted back-projection algorithm (MWBP) for 3D electrical impedance mammography systems with the planar electrode array

Electrical impedance mammography (EIM), as one of the EIT medical applications, is applied for diagnostics and imaging of the breast. Quality of reconstructed images in this method depends on precision of the EIM hardware and efficiency of the image reconstruction algorithm. One of the linearized methods in EIT image reconstruction is Weighted Back-Projection (WBP). The purpose of this paper is to present a novel method, which is a modified version of the current 3D WBP algorithm (MWBP), to improve the quality of reconstructed images in electrical impedance mammography. In order to calibrate the 3D WBP algorithm with regards to real equipotential surfaces and output results, some attempts have been made to model the real EIM conditions in terms of body shape and electrode positions by means of a saline-filled phantom. We have proposed a few modifications which improve the quality of images greatly, in terms of sensitivity, resolution, contrast, and accuracy in size and position of objects under imaging. The mentioned modifications contain: changing the problem assumptions, adding a new weighting function to the current 3D WBP equation, and defining an exciting radius for the pixel being reconstructed. The final results (containing phantom and in vivo images) prove the efficiency of the proposed modifications.

A joint deep learning model to recover information and reduce artifacts in missing-wedge sinograms for electron tomography and beyond

We present a joint model based on deep learning that is designed to inpaint the missing-wedge sinogram of electron tomography and reduce the residual artifacts in the reconstructed tomograms. Traditional methods, such as weighted back projection (WBP) and simultaneous algebraic reconstruction technique (SART), lack the ability to recover the unacquired project information as a result of the limited tilt range; consequently, the tomograms reconstructed using these methods are distorted and contaminated with the elongation, streaking, and ghost tail artifacts. To tackle this problem, we first design a sinogram filling model based on the use of Residual-in-Residual Dense Blocks in a Generative Adversarial Network (GAN). Then, we use a U-net structured Generative Adversarial Network to reduce the residual artifacts. We build a two-step model to perform information recovery and artifacts removal in their respective suitable domain. Compared with the traditional methods, our method offers superior Peak Signal to Noise Ratio (PSNR) and the Structural Similarity Index (SSIM) to WBP and SART; even with a missing wedge of 45°, our method offers reconstructed images that closely resemble the ground truth with nearly no artifacts. In addition, our model has the advantage of not needing inputs from human operators or setting hyperparameters such as iteration steps and relaxation coefficient used in TV-based methods, which highly relies on human experience and parameter fine turning.

Characterisation of flame-generated soot and soot-in-oil using electron tomography volume reconstructions and comparison with traditional 2D-TEM measurements

This work characterises soot nanoparticles by electron tomography using Weighted Back Projection algorithm and appraises the uncertainties in two-dimensional calculations by comparison with 3D parameters for flame-generated soot and diesel soot-in-oil. Bright field TEM was used to capture 2D images of soot. Large uncertainties exist in 2D-measured morphological parameters. The flame-generated particle showed an extensive 3D structure while the soot-in-oil was notably two-dimensional. Morphological parameters of flame-generated soot and diesel soot-in-oil were different; primary particles, volume, and surface area varied significantly over the range of viewing angle, with differences as large as 60%. 2D flame-generated soot volume underestimated 3D measurements by 38%; soot-in-oil 2D- and 3D-derived volumes were within 4%. 2D calculations of fractal dimension generally underestimate the 3D value.

Adaptive multiresolution method for MAP reconstruction in electron tomography

3D image reconstruction with electron tomography holds problems due to the severely limited range of projection angles and low signal to noise ratio of the acquired projection images. The maximum a posteriori (MAP) reconstruction methods have been successful in compensating for the missing information and suppressing noise with their intrinsic regularization techniques. There are two major problems in MAP reconstruction methods: (1) selection of the regularization parameter that controls the balance between the data fidelity and the prior information, and (2) long computation time. One aim of this study is to provide an adaptive solution to the regularization parameter selection problem without having additional knowledge about the imaging environment and the sample. The other aim is to realize the reconstruction using sequences of resolution levels to shorten the computation time. The reconstructions were analyzed in terms of accuracy and computational efficiency using a simulated biological phantom and publically available experimental datasets of electron tomography. The numerical and visual evaluations of the experiments show that the adaptive multiresolution method can provide more accurate results than the weighted back projection (WBP), simultaneous iterative reconstruction technique (SIRT), and sequential MAP expectation maximization (sMAPEM) method. The method is superior to sMAPEM also in terms of computation time and usability since it can reconstruct 3D images significantly faster without requiring any parameter to be set by the user.

Progressive Stochastic Reconstruction Technique (PSRT) for cryo electron tomography.

Cryo Electron Tomography (cryoET) plays an essential role in Structural Biology, as it is the only technique that allows to study the structure of large macromolecular complexes in their close to native environment in situ. The reconstruction methods currently in use, such as Weighted Back Projection (WBP) or Simultaneous Iterative Reconstruction Technique (SIRT), deliver noisy and low-contrast reconstructions, which complicates the application of high-resolution protocols, such as Subtomogram Averaging (SA). We propose a Progressive Stochastic Reconstruction Technique (PSRT) - a novel iterative approach to tomographic reconstruction in cryoET based on Monte Carlo random walks guided by Metropolis-Hastings sampling strategy. We design a progressive reconstruction scheme to suit the conditions present in cryoET and apply it successfully to reconstructions of macromolecular complexes from both synthetic and experimental datasets. We show how to integrate PSRT into SA, where it provides an elegant solution to the region-of-interest problem and delivers high-contrast reconstructions that significantly improve template-based localization without any loss of high-resolution structural information. Furthermore, the locality of SA is exploited to design an importance sampling scheme which significantly speeds up the otherwise slow Monte Carlo approach. Finally, we design a new memory efficient solution for the specimen-level interior problem of cryoET, removing all associated artifacts.

Super-sampling SART with ordered subsets.

In tomography, the quality of the reconstruction is essential because the complete cascade of the subsequent analysis is based on it. To date, weighted back-projection (WBP) has been the most commonly used technique due to its versatility and performance in sub-tomogram averaging. Here we present super-sampling SART that is based on the simultaneous algebraic reconstruction technique. While algebraic reconstruction techniques typically produce better contrast and lately showed a significant improvement in terms of processing speed, sub-tomogram averages derived from those reconstructions were inferior in resolution compared to those derived from WBP data. Super-sampling SART, however, outperforms both in term of contrast and the resolution achieved in sub-tomogram averaging several other tested methods and in particular WBP. The main feature of super-sampling SART, as the name implies, is the super-sampling option - by which parameter-based up-sampling and down-sampling are used to reduce artifacts. In particular, the aliasing that is omnipresent in the reconstruction can be practically eliminated without a significant increase in the computational time. Furthermore, super-sampling SART reaches convergence within a single iteration, making the processing time comparable to WBP, and eliminating the ambiguity of parameter-controlled convergence times. We find that grouping of projections increases the contrast, while when projections are used individually the resolution can be maximized. Using sub-tomogram averaging of ribosomes as a test case, we show that super-sampling SART achieves equal or better sub-tomogram averaging results than WBP, which is of particular importance in cryo-electron tomography.

A Multi-Scale Weighted Back Projection Imaging Technique for Ground Penetrating Radar Applications

In this paper, we propose a new ground penetrating radar (GPR) imaging technique based on multi-scale weighted back projection (BP) processing. Firstly, the whole imaging region is discretized by large scale and low-resolution imaging result is obtained by using traditional BP imaging technique. Secondly, the potential targets regions (PTR) are delineated from low-resolution imaging result by using intensity detection method. In the PTR, small scale discretization is implemented and higher resolution imaging result is obtained by using weighted BP imaging technique. A weight factor is designed by analyzing the statistical characteristics of scattering data on the time-delay curve. The above “discretization-imaging-PTR delineation” processing continues until the imaging resolution reaches the specified requirement. In the multi-scale imaging result, the resolution in other regions is not as high as that in PTR. This algorithm can get higher resolution imaging results with much lower computation compared with traditional BP imaging algorithm. The simulation of this algorithm is processed and experimental results validate the feasibility of this method.

Weighted simultaneous iterative reconstruction technique for single-axis tomography

Tomographic techniques play a crucial role in imaging methods such as transmission electron microscopy (TEM) due to their unique capabilities to reconstruct three-dimensional object information. However, the accuracy of the two standard tomographic reconstruction techniques, the weighted back-projection (W-BP) and the simultaneous iterative reconstruction technique (SIRT) is reduced under common experimental restrictions, such as limited tilt range or noise. We demonstrate that the combination of W-BP and SIRT leads to an improved tomographic reconstruction technique: the weighted SIRT. Convergence, resolution and reconstruction error of the W-SIRT are analyzed by a detailed analytical, numerical, and experimental comparison with established methods. Our reconstruction technique is not restricted to TEM tomography but can be applied to all problems sharing single axis imaging geometry.

Electron tomography based on a total variation minimization reconstruction technique

The 3D reconstruction of a tilt series for electron tomography is mostly carried out using the weighted backprojection (WBP) algorithm or using one of the iterative algorithms such as the simultaneous iterative reconstruction technique (SIRT). However, it is known that these reconstruction algorithms cannot compensate for the missing wedge. Here, we apply a new reconstruction algorithm for electron tomography, which is based on compressive sensing. This is a field in image processing specialized in finding a sparse solution or a solution with a sparse gradient to a set of ill-posed linear equations. Therefore, it can be applied to electron tomography where the reconstructed objects often have a sparse gradient at the nanoscale. Using a combination of different simulated and experimental datasets, it is shown that missing wedge artefacts are reduced in the final reconstruction. Moreover, it seems that the reconstructed datasets have a higher fidelity and are easier to segment in comparison to reconstructions obtained by more conventional iterative algorithms.

Matrix Implementation of Simultaneous Iterative Reconstruction Technique (SIRT) on GPUs

Electron tomography (ET) is an important technique in biosciences that is providing new insights into the cellular ultrastructure. Iterative reconstruction methods have been shown to be robust against the noise and limited-tilt range conditions present in ET. Nevertheless, these methods are not extensively used due to their computational demands. Instead, the simpler method weighted backprojection (WBP) remains prevalent. Recently, we have demonstrated that a matrix approach to WBP allows a significant reduction in processing time both on central processing units and on graphics processing units (GPUs). In this work, we extend that matrix approach to one of the most common iterative methods in ET, simultaneous iterative reconstruction technique (SIRT). We show that it is possible to implement this method targeted at GPU directly, using sparse algebra. We also analyse this approach on different GPU platforms and confirm that these implementations exhibit high performance. This may thus help to the widespread use of SIRT.

Fast tomographic reconstruction on multicore computers

Tomo3D implements a multithreaded vectorized approach to tomographic reconstruction that takes full advantage of the computer power in modern multicore computers. Full resolution tomograms are generated at high speed on standard computers with no special system requirements. Tomo3D has the most common reconstruction methods implemented, namely weighted Back-projection (WBP) and simultaneous iterative reconstruction technique (SIRT). It proves to be competitive with current graphic processor unit solutions in terms of processing time, in the order of a few seconds with WBP or minutes with SIRT. The program is compatible with standard packages, which easily allows integration in the electron tomography workflow.

A matrix approach to tomographic reconstruction and its implementation on GPUs

Electron tomography allows elucidation of the molecular architecture of complex biological specimens. Weighted backprojection (WBP) is the standard reconstruction method in the field. In this work, three-dimensional reconstruction with WBP is addressed from a matrix perspective by formulating the problem as a set of sparse matrix-vector products, with the matrix being constant and shared by all the products. This matrix approach allows efficient implementations of reconstruction algorithms. Although WBP is computationally simple, the resolution requirements may turn the tomographic reconstruction into a computationally intensive problem. Parallel systems have traditionally been used to cope with such demands. Recently, graphics processor units (GPUs) have emerged as powerful platforms for scientific computing and they are getting increasing interest. In combination with GPU computing, the matrix approach for WBP exhibits a significant acceleration factor compared to the standard implementation.

Three-dimensional shapes and distribution of FePd nanoparticles observed by electron tomography using high-angle annular dark-field scanning transmission electron microscopy

We have studied three-dimensional shapes and distribution of FePd nanoparticles, prepared by electron beam deposition and postdeposition annealing, by means of single-axis tilt tomography using atomic number contrasts obtained by high-angle annular dark-field scanning transmission electron microscopy. Particle size, shape, and locations were reconstructed by weighted backprojection (WBP), as well as by simultaneous iterative reconstruction technique (SIRT). We have also estimated the particle size by simple extrapolation of tilt-series original data sets, which proved to be quite powerful. The results of the two algorithms for reconstruction have been compared quantitatively with those obtained by the extrapolation method and those independently reported by electron holography. It was found that the reconstructed intensity map by WBP contains a small amount of dotlike artifacts, which do not exist in the results by SIRT, and that the particle surface obtained by WBP is rougher than that by SIRT. We demonstrate, on the other hand, that WBP yields a better estimation of the particle size in the z direction than SIRT does, most likely due to the presence of a “missing wedge” in the original data set.

Radiation dose reduction and image enhancement in biological imaging through equally-sloped tomography

Electron tomography is currently the highest resolution imaging modality available to study the 3D structures of pleomorphic macromolecular assemblies, viruses, organelles and cells. Unfortunately, the resolution is currently limited to 3-5nm by several factors including the dose tolerance of biological specimens and the inaccessibility of certain tilt angles. Here we report the first experimental demonstration of equally-sloped tomography (EST) to alleviate these problems. As a proof of principle, we applied EST to reconstructing frozen-hydrated keyhole limpet hemocyanin molecules from a tilt-series taken with constant slope increments. In comparison with weighted back-projection (WBP), the algebraic reconstruction technique (ART) and the simultaneous algebraic reconstruction technique (SART), EST reconstructions exhibited higher contrast, less peripheral noise, more easily detectable molecular boundaries and reduced missing wedge effects. More importantly, EST reconstructions including only two-thirds the original images appeared to have the same resolution as full WBP reconstructions, suggesting that EST can either reduce the dose required to reach a given resolution or allow higher resolutions to be achieved with a given dose. EST was also applied to reconstructing a frozen-hydrated bacterial cell from a tilt-series taken with constant angular increments. The results confirmed similar benefits when standard tilts are utilized.

Efficient parallel implementation of iterative reconstruction algorithms for electron tomography

Electron tomography (ET) combines electron microscopy and the principles of tomographic imaging in order to reconstruct the three-dimensional structure of complex biological specimens at molecular resolution. Weighted back-projection (WBP) has long been the method of choice since the reconstructions are very fast. It is well known that iterative methods produce better images, but at a very costly time penalty. In this work, it is shown that efficient parallel implementations of iterative methods, based primarily on data decomposition, can speed up such methods to an extent that they become viable alternatives to WBP. Precomputation of the coefficient matrix has also turned out to be important to substantially improve the performance regardless of the number of processors used. Matrix precomputation has made it possible to speed up the block-iterative component averaging (BICAV) algorithm, which has been studied before in the context of computerized tomography (CT) and ET, by a factor of more than 3.7. Component-averaged row projections (CARP) is a recently introduced block-parallel algorithm, which was shown to be a robust method for solving sparse systems arising from partial differential equations. It is shown that this algorithm is also suitable for single-axis ET, and is advantageous over BICAV both in terms of runtime and image quality. The experiments were carried out on several datasets of ET of various sizes, using the blob model for representing the reconstructed object.

Parameter optimization in 3D reconstruction on a large scale grid

Reaching as high structural resolution as possible in 3D electron microscopy of biological specimens is crucial to understanding their function and interactions. Technical and biological limitations make electron microscopy projections of such specimens quite noisy. Under those circumstances, the broadly used Weighted Back-Projection algorithm presents some limitations for 3D reconstruction. Iterative tomographic reconstruction algorithms are well suited to provide high resolution 3D structures under such noisy conditions. Nevertheless, these iterative algorithms present two major challenges: computational expensiveness and some free parameters which need to be correctly tuned to obtain the best possible resolution. This work applies global optimization techniques to search for the optimal set of parameters and makes use of the high-throughput capabilities of grid computing to perform the required computations. Fault tolerance techniques have been included in our application to deal with the dynamic nature and complexity of large scale computational grids. The approach for parameter optimization presented here has been successfully evaluated in the European EGEE grid, obtaining good levels of speedup, throughput and transfer rates.

A novel dual-axis reconstruction algorithm for electron tomography

A new algorithm for computing electron microscopy tomograms which combines iterative methods with dual-axis geometry is presented. Initial modelling using test data shows several improvements over both the weighted back-projection (WBP) and Simultaneous Iterative Reconstruction Technique (SIRT) method, and, with increased stability and tomogram fidelity under high-noise conditions.

Comparison of intensity distributions in tomograms from BF TEM, ADF STEM, HAADF STEM, and calculated tilt series

The three-dimensional (3D) morphology of a nanometer-sized object can be obtained using electron tomography. Variations in composition or density of the object cause variations in the reconstructed intensity. When imaging homogeneous objects, variations in reconstructed intensity are caused by the imaging technique, imaging conditions, and reconstruction. In this paper, we describe data acquisition, image processing, and 3D reconstruction to obtain and compare tomograms of magnetite crystals from bright field (BF) transmission electron microscopy (TEM), annular dark-field (ADF) scanning transmission electron microscopy (STEM), and high-angle annular dark field (HAADF) STEM tilt series. We use histograms, which plot the number of volume elements (voxels) at a given intensity vs. the intensity, to measure and quantitatively compare intensity distributions among different tomograms. In combination with numerical simulations, we determine the influence of maximum tilt angle, tilt increment, contrast changes with tilt (diffraction contrast), and the signal-to-noise ratio (SNR) as well as the choice of the reconstruction approach (weighted backprojection (WB) and sequential iterative reconstruction technique (SIRT)) on the histogram. We conclude that because ADF and HAADF STEM techniques are less affected by diffraction, and because they have a higher SNR than BF TEM, they are better suited for tomography of nanometer-sized crystals.

High-performance electron tomography of complex biological specimens

We have evaluated reconstruction methods using smooth basis functions in the electron tomography of complex biological specimens. In particular, we have investigated series expansion methods, with special emphasis on parallel computation. Among the methods investigated, the component averaging techniques have proven to be most efficient and have generally shown fast convergence rates. The use of smooth basis functions provides the reconstruction algorithms with an implicit regularization mechanism, very appropriate for noisy conditions. Furthermore, we have applied high-performance computing (HPC) techniques to address the computational requirements demanded by the reconstruction of large volumes. One of the standard techniques in parallel computing, domain decomposition, has yielded an effective computational algorithm which hides the latencies due to interprocessor communication. We present comparisons with weighted back-projection (WBP), one of the standard reconstruction methods in the areas of computational demand and reconstruction quality under noisy conditions. These techniques yield better results, according to objective measures of quality, than the weighted backprojection techniques after a very few iterations. As a consequence, the combination of efficient iterative algorithms and HPC techniques has proven to be well suited to the reconstruction of large biological specimens in electron tomography, yielding solutions in reasonable computation times.