Missing Wedge Research Articles

A deep learning method for simultaneous denoising and missing wedge reconstruction in cryogenic electron tomography

Cryogenic electron tomography is a technique for imaging biological samples in 3D. A microscope collects a series of 2D projections of the sample, and the goal is to reconstruct the 3D density of the sample called the tomogram. Reconstruction is difficult as the 2D projections are noisy and can not be recorded from all directions, resulting in a missing wedge of information. Tomograms conventionally reconstructed with filtered back-projection suffer from noise and strong artefacts due to the missing wedge. Here, we propose a deep-learning approach for simultaneous denoising and missing wedge reconstruction called DeepDeWedge. The algorithm requires no ground truth data and is based on fitting a neural network to the 2D projections using a self-supervised loss. DeepDeWedge is simpler than current state-of-the-art approaches for denoising and missing wedge reconstruction, performs competitively and produces more denoised tomograms with higher overall contrast.

Investigating the missing-wedge problem in small-angle X-ray scattering tensor tomography across real and reciprocal space.

Small-angle-scattering tensor tomography is a technique for studying anisotropic nanostructures of millimetre-sized samples in a volume-resolved manner. It requires the acquisition of data through repeated tomographic rotations about an axis which is subjected to a series of tilts. The tilt that can be achieved with a typical setup is geometrically constrained, which leads to limits in the set of directions from which the different parts of the reciprocal space map can be probed. Here, we characterize the impact of this limitation on reconstructions in terms of the missing wedge problem of tomography, by treating the problem of tensor tomography as the reconstruction of a three-dimensional field of functions on the unit sphere, represented by a grid of Gaussian radial basis functions. We then devise an acquisition scheme to obtain complete data by remounting the sample, which we apply to a sample of human trabecular bone. Performing tensor tomographic reconstructions of limited data sets as well as the complete data set, we further investigate and validate the missing wedge problem by investigating reconstruction errors due to data incompleteness across both real and reciprocal space. Finally, we carry out an analysis of orientations and derived scalar quantities, to quantify the impact of this missing wedge problem on a typical tensor tomographic analysis. We conclude that the effects of data incompleteness are consistent with the predicted impact of the missing wedge problem, and that the impact on tensor tomographic analysis is appreciable but limited, especially if precautions are taken. In particular, there is only limited impact on the means and relative anisotropies of the reconstructed reciprocal space maps.

Pillar data-acquisition strategies for cryo-electron tomography of beam-sensitive biological samples.

For cryo-electron tomography (cryo-ET) of beam-sensitive biological specimens, a planar sample geometry is typically used. As the sample is tilted, the effective thickness of the sample along the direction of the electron beam increases and the signal-to-noise ratio concomitantly decreases, limiting the transfer of information at high tilt angles. In addition, the tilt range where data can be collected is limited by a combination of various sample-environment constraints, including the limited space in the objective lens pole piece and the possible use of fixed conductive braids to cool the specimen. Consequently, most tilt series are limited to a maximum of ±70°, leading to the presence of a missing wedge in Fourier space. The acquisition of cryo-ET data without a missing wedge, for example using a cylindrical sample geometry, is hence attractive for volumetric analysis of low-symmetry structures such as organelles or vesicles, lysis events, pore formation or filaments for which the missing information cannot be compensated by averaging techniques. Irrespective of the geometry, electron-beam damage to the specimen is an issue and the first images acquired will transfer more high-resolution information than those acquired last. There isalso an inherent trade-off between higher sampling in Fourier space and avoiding beam damage to the sample. Finally, the necessity of using a sufficient electron fluence to align the tilt images means that this fluence needs to be fractionated across a small number of images; therefore, the order of data acquisition is also a factor to consider. Here, an n-helix tilt scheme is described and simulated which uses overlapping and interleaved tilt series to maximize the use of a pillar geometry, allowing the entire pillar volume to be reconstructed asa single unit. Three related tilt schemes are also evaluated that extend the continuous and classic dose-symmetric tilt schemes for cryo-ET to pillar samples to enable the collection of isotropic information across all spatial frequencies. Afourfold dose-symmetric scheme is proposed which provides a practical compromise between uniform information transfer and complexity of data acquisition.

Real space iterative reconstruction for vector tomography (RESIRE-V)

Tomography has had an important impact on the physical, biological, and medical sciences. To date, most tomographic applications have been focused on 3D scalar reconstructions. However, in some crucial applications, vector tomography is required to reconstruct 3D vector fields such as the electric and magnetic fields. Over the years, several vector tomography methods have been developed. Here, we present the mathematical foundation and algorithmic implementation of REal Space Iterative REconstruction for Vector tomography, termed RESIRE-V. RESIRE-V uses multiple tilt series of projections and iterates between the projections and a 3D reconstruction. Each iteration consists of a forward step using the Radon transform and a backward step using its transpose, then updates the object via gradient descent. Incorporating with a 3D support constraint, the algorithm iteratively minimizes an error metric, defined as the difference between the measured and calculated projections. The algorithm can also be used to refine the tilt angles and further improve the 3D reconstruction. To validate RESIRE-V, we first apply it to a simulated data set of the 3D magnetization vector field, consisting of two orthogonal tilt series, each with a missing wedge. Our quantitative analysis shows that the three components of the reconstructed magnetization vector field agree well with the ground-truth counterparts. We then use RESIRE-V to reconstruct the 3D magnetization vector field of a ferromagnetic meta-lattice consisting of three tilt series. Our 3D vector reconstruction reveals the existence of topological magnetic defects with positive and negative charges. We expect that RESIRE-V can be incorporated into different imaging modalities as a general vector tomography method. To make the algorithm accessible to a broad user community, we have made our RESIRE-V MATLAB source codes and the data freely available at https://github.com/minhpham0309/RESIRE-V.

No ground truth needed: unsupervised sinogram inpainting for nanoparticle electron tomography (UsiNet) to correct missing wedges

Complex natural and synthetic materials, such as subcellular organelles, device architectures in integrated circuits, and alloys with microstructural domains, require characterization methods that can investigate the morphology and physical properties of these materials in three dimensions (3D). Electron tomography has unparalleled (sub-)nm resolution in imaging 3D morphology of a material, critical for charting a relationship among synthesis, morphology, and performance. However, electron tomography has long suffered from an experimentally unavoidable missing wedge effect, which leads to undesirable and sometimes extensive distortion in the final reconstruction. Here we develop and demonstrate Unsupervised Sinogram Inpainting for Nanoparticle Electron Tomography (UsiNet) to correct missing wedges. UsiNet is the first sinogram inpainting method that can be realistically used for experimental electron tomography by circumventing the need for ground truth. We quantify its high performance using simulated electron tomography of nanoparticles (NPs). We then apply UsiNet to experimental tomographs, where >100 decahedral NPs and vastly different byproduct NPs are simultaneously reconstructed without missing wedge distortion. The reconstructed NPs are sorted based on their 3D shapes to understand the growth mechanism. Our work presents UsiNet as a potent tool to advance electron tomography, especially for heterogeneous samples and tomography datasets with large missing wedges, e.g. collected for beam sensitive materials or during temporally-resolved in-situ imaging.

Simulation-Assisted Augmentation of Missing Wedge and Region-of-Interest Computed Tomography Data.

This study reports a strategy to use sophisticated, realistic X-ray Computed Tomography (CT) simulations to reduce Missing Wedge (MW) and Region-of-Interest (RoI) artifacts in FBP (Filtered Back-Projection) reconstructions. A 3D model of the object is used to simulate the projections that include the missing information inside the MW and outside the RoI. Such information augments the experimental projections, thereby drastically improving the reconstruction results. An X-ray CT dataset of a selected object is modified to mimic various degrees of RoI and MW problems. The results are evaluated in comparison to a standard FBP reconstruction of the complete dataset. In all cases, the reconstruction quality is significantly improved. Small inclusions present in the scanned object are better localized and quantified. The proposed method has the potential to improve the results of any CT reconstruction algorithm.

Computational methods for in situ structural studies with cryogenic electron tomography.

Cryo-electron tomography (cryo-ET) plays a critical role in imaging microorganisms in situ in terms of further analyzing the working mechanisms of viruses and drug exploitation, among others. A data processing workflow for cryo-ET has been developed to reconstruct three-dimensional density maps and further build atomic models from a tilt series of two-dimensional projections. Low signal-to-noise ratio (SNR) and missing wedge are two major factors that make the reconstruction procedure challenging. Because only few near-atomic resolution structures have been reconstructed in cryo-ET, there is still much room to design new approaches to improve universal reconstruction resolutions. This review summarizes classical mathematical models and deep learning methods among general reconstruction steps. Moreover, we also discuss current limitations and prospects. This review can provide software and methods for each step of the entire procedure from tilt series by cryo-ET to 3D atomic structures. In addition, it can also help more experts in various fields comprehend a recent research trend in cryo-ET. Furthermore, we hope that more researchers can collaborate in developing computational methods and mathematical models for high-resolution three-dimensional structures from cryo-ET datasets.

Graph identification of proteins in tomograms (GRIP-Tomo).

In this study, we present a method of pattern mining based on network theory that enables the identification of protein structures or complexes from synthetic volume densities, without the knowledge of predefined templates or human biases for refinement. We hypothesized that the topological connectivity of protein structures is invariant, and they are distinctive for the purpose of protein identification from distorted data presented in volume densities. Three-dimensional densities of a protein or a complex from simulated tomographic volumes were transformed into mathematical graphs as observables. We systematically introduced data distortion or defects such as missing fullness of data, the tumbling effect, and the missing wedge effect into the simulated volumes, and varied the distance cutoffs in pixels to capture the varying connectivity between the density cluster centroids in the presence of defects. A similarity score between the graphs from the simulated volumes and the graphs transformed from the physical protein structures in point data was calculated by comparing their network theory order parameters including node degrees, betweenness centrality, and graph densities. By capturing the essential topological features defining the heterogeneous morphologies of a network, we were able to accurately identify proteins and homo-multimeric complexes from 10 topologically distinctive samples without realistic noise added. Our approach empowers future developments of tomogram processing by providing pattern mining with interpretability, to enable the classification of single-domain protein native topologies as well as distinct single-domain proteins from multimeric complexes within noisy volumes.

Precisely Picking Nanoparticles by a "Nano-Scalpel" for 360° Electron Tomography.

Electron tomography (ET) has gained increasing attention for the 3D characterization of nanoparticles. However, the missing wedge problem due to a limited tilt angle range is still the main challenge for accurate reconstruction in most experimental TEM setups. Advanced algorithms could in-paint or compensate to some extent the missing wedge artifacts, but cannot recover the missing structural information completely. 360° ET provides an option to solve this problem by tilting a needle-shaped specimen over the full tilt range and thus filling the missing information. However, sample preparation especially for fine powders to perform full-range ET is still challenging, thus limiting its application. In this work, we propose a new universal sample preparation method that enables the transfer of selected individual nanoparticle or a few separated nanoparticles by cutting a piece of carbon film supporting the specimen particles and mounting them onto the full-range tomography holder tip with the help of an easily prepared sharp tungsten tip. This method is demonstrated by 360° ET of Pt@TiO2 hollow cage catalyst showing high quality reconstruction without missing wedge.

VP-Detector: A 3D multi-scale dense convolutional neural network for macromolecule localization and classification in cryo-electron tomograms

Background and objective: Cryo-electron tomography (cryo-ET) with subtomogram averaging (STA) is indispensable when studying macromolecule structures and functions in their native environments. Due to the low signal-to-noise ratio, the missing wedge artifacts in tomographic reconstructions, and multiple macromolecules of varied shapes and sizes, macromolecule localization and classification remain challenging. To tackle this bottleneck problem for structural determination by STA, we design an accurate macromolecule localization and classification method named voxelwise particle detector (VP-Detector).Methods: VP-Detector is a two-stage particle detection method based on a 3D multiscale dense convolutional neural network (3D MSDNet). The proposed network uses 3D hybrid dilated convolution (3D HDC) to avoid the resolution loss caused by scaling operations. Meanwhile, it uses 3D dense connectivity to encourage the reuse of feature maps to reduce trainable parameters. In addition, the weighted focal loss is proposed to focus more attention on difficult samples and rare classes, which relieves the class imbalance caused by multiple particles of various sizes. The performance of VP-Detector is evaluated on both simulated and real-world tomograms, and it shows that VP-Detector outperforms state-of-the-art methods.Results: The experiments show that VP-Detector outperforms the state-of-the-art methods on particle localization with an F1-score of 0.951 and a precision of 0.978. In addition, VP-Detector can replace manual particle picking in experiment on the real-world tomograms. Furthermore, it performs well in classifying large-, medium-, and small-weight proteins with accuracies of 1, 0.95, and 0.82, respectively. Finally, ablation studies demonstrate the effectiveness of 3D HDC, 3D dense connectivity, weighted focal loss, and training on small training sets.Conclusions: VP-Detector can achieve high accuracy in particle detection with few trainable parameters and support training on small datasets. It can also relieve the class imbalance caused by multiple particles with various shapes and sizes.

NUDIM: A non-uniform fast Fourier transform based dual-space constraint iterative reconstruction method in biological electron tomography

Electron tomography, a powerful imaging tool for studying 3D structures of macromolecular assemblies, always suffers from imperfect reconstruction with limited resolution due to the intrinsic low signal-to-noise ratio (SNR) and inaccessibility to certain tilt angles induced by radiation damage or mechanical limitation. In order to compensate for such insufficient data with low SNR and further improve imaging resolution, prior knowledge constraints about the objects in both real space and reciprocal space are thus exploited during tomographic reconstruction. However, direct Fast Fourier transform (FFT) between real space and reciprocal space remains extraordinarily challenging owing to their inconsistent grid sampling modes, e.g. regular and uniform grid sampling in real space whereas radial or polar grid sampling in reciprocal space. In order to solve such problem, a technique of non-uniform fast Fourier transform (NFFT) has been developed to transform efficiently between non-uniformly sampled grids in real and reciprocal space with sufficient accuracy. In this work, a Non-Uniform fast Fourier transform based Dual-space constraint Iterative reconstruction Method (NUDIM) applicable to biological electron tomography is proposed with a combination of basic concepts from equally sloped tomography (EST) and NFFT based reconstruction. In NUDIM, the use of NFFT can circumvent such grid sampling inconsistency and thus alleviate the stringent equally-sloped sampling requirement in EST reconstruction, while the dual-space constraint iterative procedure can dramatically enhance reconstruction quality. In comparison with conventional reconstruction methods, NUDIM is numerically and experimentally demonstrated to produce superior reconstruction quality with higher contrast, less noise and reduced missing wedge artifacts. More importantly, it is also capable of retrieving part of missing information from a limited number of projections.

Dual-axis Volta phase plate cryo-electron tomography of Ebola virus-like particles reveals actin-VP40 interactions

Cryo-electron tomography (cryo-ET) is a pivotal imaging technique for studying the structure of pleomorphic enveloped viruses and their interactions with the host at native conditions. Owing to the limited tilting range of samples with a slab geometry, electron tomograms suffer from so-called missing wedge information in Fourier space. In dual-axis cryo-ET, two tomograms reconstructed from orthogonally oriented tilt series are combined into a tomogram with improved resolution as the missing wedge information is reduced to a pyramid. Volta phase plate (VPP) allows to perform in-focus cryo-ET with high contrast transfer at low-resolution frequencies and thus its application may improve the quality of dual-axis tomograms. Here, we compare dual-axis cryo-ET with and without VPP on Ebola virus-like particles to visualize and segment viral and host cell proteins within the membrane-enveloped filamentous particles. Dual-axis VPP cryo-ET reduces the missing wedge information and ray artifacts arising from the weighted back-projection during tomogram reconstruction, thereby minimizing ambiguity in the analysis of crowded environments and facilitating 3D segmentation. We show that dual-axis VPP tomograms provide a comprehensive description of macromolecular organizations such as nucleocapsid assembly states, the distribution of glycoproteins on the viral envelope and asymmetric arrangements of the VP40 layer in non-filamentous regions of virus-like particles. Our data reveal actin filaments within virus-like particles in close proximity to the viral VP40 scaffold, suggesting a direct interaction between VP40 and actin filaments. Dual-axis VPP cryo-ET provides more complete 3D information at high contrast and allows for better interpretation of macromolecule interactions and pleomorphic organizations.

Nanoparticle size and 3D shape measurement by electron tomography: An Inter-Laboratory Comparison

Electron tomography (ET) has been used for quantitative measurement of shape and size of objects in three dimensions (3D) for many years. However, systematic investigation of repeatability and reproducibility of ET has not been evaluated in detail. To assess the reproducibility and repeatability of a protocol for measuring size and three-dimensional (3D) shape parameters for nanoparticles (NPs) by ET, an inter-laboratory comparison (ILC) has been performed. The ILC included six laboratories and six instruments models from three instrument manufacturers following a standard measurement protocol.A technical specification describing the normative steps of the protocol is published by the International Standards Organization (ISO). Gold NPs with 30 nm nominal diameter contained within a rod-shaped carbon support were measured. The use of a rod-shaped sample support eliminated the missing wedge effect in the experimental tilt series of projected images for improved quantification. A total of 443 NPs were initially measured by NRC-NANO and then 115 out of the 443 NPs were measured by five other labs to compare measurands such as the Volume (V), maximum Feret diameter (Fmax), minimum Feret diameter (Fmin), volume-equivalent diameter (Deq) and aspect ratio (Frat) of the NPs. The results of the five labs were compared with the results obtained at NRC-NANO. The maximum disagreement in measurements of Fmin and Fmax obtained by the participating labs did not exceed 7 %. The measured Deq was between 27.5 nm and 30.3 nm in agreement with the NP manufacturer’s specification (28 nm–32 nm). In addition to the above, the influence of the missing wedge effect and beam-induced NP movement was quantified based on the differences of the results between labs.

Versatile application of a modern scanning electron microscope for materials characterization

Scanning electron microscopy (SEM) is an indispensable characterization technique for materials science. More recently, scanning electron microscopes can be equipped with scanning transmission electron microscopy (STEM) detectors, which considerably extend their capabilities. It is demonstrated in this work that the correlative application of SEM and STEM imaging techniques provides comprehensive sample information on nanomaterials. This is highlighted by the use of a modern scanning electron microscope, which is equipped with in-lens and in-column detectors, a double-tilt holder for electron transparent specimens and a CCD camera for the acquisition of on-axis diffraction patterns. Using multi-walled carbon nanotubes and Pt/Al2O3 powder samples we will show that a complete characterization can be achieved by combining STEM (mass-thickness and diffraction) contrast and SEM (topography and materials) contrast. This is not possible in a standard transmission electron microscope where topography information cannot be routinely obtained. We also exploit the large tilt angle range of the specimen holder to perform 180 degrees STEM tomography on multi-walled carbon nanotubes, which avoids the missing wedge artifacts.

Three-dimensional tomography reveals distinct morphological and optical properties of soot aggregates from coal-fired residential stoves in China

Electron tomography (ET) is used to reconstruct the exact 3-dimensional morphologies of fractal-like soot aggregates sampled from a household heating stove commonly used in China. Conventional ET techniques suffer from “the missing wedge” problem caused by unreachable tilt angles, leading to noisy reconstructed tomograms. We overcame this problem by implementing a high-resolution object-edge identification method coupled with a novel voxel-filling algorithm to improve the reconstruction quality. Our reconstructed micron-length aggregates highlight the local non-idealities present throughout a particle's surface; these characteristics are almost impossible to account for in existing computational simulation exercises. Q-space analysis predicts the fractal dimension of our ET reconstructed aggregates to be in the range between 2.2 and 2.6, which deviates significantly from the universal value of 1.8 obtained using the widely adopted diffusion limited cluster-cluster aggregation (DLCA) model. The optical properties of our ET reconstructed aggregates are compared with those built with a DLCA model and equivalent spheres . The most striking optical characteristics of the ET reconstructed aggregates are their wavelength invariant mass absorption cross-sections of ~3.5 m2/g and single scattering albedo of ~0.5. The sample size investigated in this work is constrained by the extremely time-consuming object-edge identification process of electron tomography. This issue necessitates the development of more efficient computer vision algorithms for future research.

A compact design of four-degree-of-freedom transmission electron microscope holder for quasi-four-dimensional characterization

Electron tomography (ET) has been demonstrated to be a powerful tool in addressing challenging problems, such as understanding 3D interactions among various microstructures. Advancing ET to broader applications requires novel instrumentation design to break the bottlenecks both in theory and in practice. In this work, we built a compact four-degree-of-freedom (three-directional positionings plus self-rotation) nano-manipulator dedicated to ET applications, which is called X-Nano transmission electron microscope (TEM) holder. All the movements of the four degrees of freedom are precisely driven by built-in piezoelectric actuators, minimizing the artefacts due to the vibration and drifting of the TEM stage. Full 360o rotation is realized with an accuracy of 0.05o in the whole range, which solves the missing wedge problem. Meanwhile, the specimen can move to the rotation axis with an integrated 3D nano-manipulator, greatly reducing the effort in tracking sample locations during tilting. Meanwhile, in-situ stimulation function can be seamlessly integrated into the X-Nano TEM holder so that dynamic information can be uncovered. We expect that more delicate researches, such as those about 3D microstructural evolution, can be carried out extensively by means of this holder in the near future.

Correction of Missing-Wedge Artifacts in Filamentous Tomograms by Template-Based Constrained Deconvolution.

Cryo-electron tomography maps often exhibit considerable noise and anisotropic resolution, due to the low-dose requirements and the missing wedge in Fourier space. These spurious features are visually unappealing and, more importantly, prevent an automated segmentation of geometric shapes, requiring a subjective and labor-intensive manual tracing. We developed a novel computational strategy for objectively denoising and correcting missing-wedge artifacts in homogeneous specimen areas of tomograms, where it is assumed that a template repeats itself across the volume under consideration, as happens in the case of filaments. In our deconvolution approach, we use a template and a map of corresponding template locations, allowing us to compensate for the information lost in the missing wedge. We applied the method to tomograms of actin-filament bundles of inner-ear stereocilia, which are critical for the senses of hearing and balance. In addition, we demonstrate that our method can be used for cell membrane detection.

Automated Segmentation and Correction of Missing-Wedge Artifacts in Cryo-Electron Tomography Maps by Shape-Constrained Deconvolution

Cryo-electron tomography maps often exhibit considerable noise and anisotropic resolution, due to the low-dose requirements and the missing wedge in Fourier space. These spurious features are visually unappealing and, more importantly, prevent an automated segmentation of geometric shapes, often requiring a highly subjective, labor-intensive manual tracing. We continue the development of a computational strategy for objectively denoising and correcting missing-wedge artifacts in the case of repetitive basic shapes, such as filamentous structures, membranes, and other sub-tomogram features. In this approach, we use a template and a non-negative “location map” to constrain the deconvolution scheme, allowing us to recover, to a considerable degree, the information lost in the missing wedge. Whereas a previous implementation using constrained optimization showed the merit of the approach, the numerical methods were time-consuming and relied on proprietary libraries, preventing a wider adoption of our methods. Here, we explore a novel implementation based on iterative Richardson Lucy Deconvolution that can be GPU-accelerated and freely disseminated. We apply our method to cellular tomograms from collaborating laboratories and find a better overlap with the experimental map than manual tracing of cytoskeletal filaments was able to achieve. In addition, we demonstrate that our method can also be used for membrane detection and for the detection of sub-tomogram features.

A Monte Carlo framework for missing wedge restoration and noise removal in cryo-electron tomography

We propose a statistical method to address an important issue in cryo-electron tomography image analysis: reduction of a high amount of noise and artifacts due to the presence of a missing wedge (MW) in the spectral domain. The method takes as an input a 3D tomogram derived from limited-angle tomography, and gives as an output a 3D denoised and artifact compensated volume. The artifact compensation is achieved by filling up the MW with meaningful information. To address this inverse problem, we compute a Minimum Mean Square Error (MMSE) estimator of the uncorrupted image. The underlying high-dimensional integral is computed by applying a dedicated Markov Chain Monte-Carlo (MCMC) sampling procedure based on the Metropolis-Hasting (MH) algorithm. The proposed MWR (Missing Wedge Restoration) algorithm can be used to enhance visualization or as a pre-processing step for image analysis, including segmentation and classification of macromolecules. Results are presented for both synthetic data and real 3D cryo-electron images.

A novel constrained reconstruction model towards high-resolution subtomogram averaging.

Electron tomography (ET) offers a unique capacity to image biological structures in situ. However, the resolution of ET reconstructed tomograms is not comparable to that of the single-particle cryo-EM. If many copies of the object of interest are present in the tomograms, their structures can be reconstructed in the tomogram, picked, aligned and averaged to increase the signal-to-noise ratio and improve the resolution, which is known as the subtomogram averaging. To date, the resolution improvement of the subtomogram averaging is still limited because each reconstructed subtomogram is of low reconstruction quality due to the missing wedge issue. In this article, we propose a novel computational model, the constrained reconstruction model (CRM), to better recover the information from the multiple subtomograms and compensate for the missing wedge issue in each of them. CRM is supposed to produce a refined reconstruction in the final turn of subtomogram averaging after alignment, instead of directly taking the average. We first formulate the averaging method and our CRM as linear systems, and prove that the solution space of CRM is no larger, and in practice much smaller, than that of the averaging method. We then propose a sparse Kaczmarz algorithm to solve the formulated CRM, and further extend the solution to the simultaneous algebraic reconstruction technique (SART). Experimental results demonstrate that CRM can significantly alleviate the missing wedge issue and improve the final reconstruction quality. In addition, our model is robust to the number of images in each tilt series, the tilt range and the noise level. The codes of CRM-SIRT and CRM-SART are available at https://github.com/icthrm/CRM. Supplementary data are available at Bioinformatics online.