Compensation of missing wedge effects with sequential statistical reconstruction in electron tomography

Missing Wedge Completion via Unsupervised Learning with Coordinate Networks

Cryogenic electron tomography (cryoET) is a powerful tool in structural biology, enabling detailed 3D imaging of biological specimens at a resolution of nanometers. Despite its potential, cryoET faces challenges such as the missing wedge problem, which limits reconstruction quality due to incomplete data collection angles. Recently, supervised deep learning methods leveraging convolutional neural networks (CNNs) have considerably addressed this issue; however, their pretraining requirements render them susceptible to inaccuracies and artifacts, particularly when representative training data is scarce. To overcome these limitations, we introduce a proof-of-concept unsupervised learning approach using coordinate networks (CNs) that optimizes network weights directly against input projections. This eliminates the need for pretraining, reducing reconstruction runtime by 3-20× compared to supervised methods. Our in silico results show improved shape completion and reduction of missing wedge artifacts, assessed through several voxel-based image quality metrics in real space and a novel directional Fourier Shell Correlation (FSC) metric. Our study illuminates benefits and considerations of both supervised and unsupervised approaches, guiding the development of improved reconstruction strategies.

Missing Wedge Completion via Unsupervised Learning with Coordinate Networks

Cryogenic electron tomography (cryoET) is a powerful tool in structural biology, enabling detailed 3D imaging of biological specimens at a resolution of nanometers. Despite its potential, cryoET faces challenges such as the missing wedge problem, which limits reconstruction quality due to incomplete data collection angles. Recently, supervised deep learning methods leveraging convolutional neural networks (CNNs) have considerably addressed this issue; however, their pretraining requirements render them susceptible to inaccuracies and artifacts, particularly when representative training data is scarce. To overcome these limitations, we introduce a proof-of-concept unsupervised learning approach using coordinate networks (CNs) that optimizes network weights directly against input projections. This eliminates the need for pretraining, reducing reconstruction runtime by 3 - 20× compared to supervised methods. Our in silico results show improved shape completion and reduction of missing wedge artifacts, assessed through several voxel-based image quality metrics in real space and a novel directional Fourier Shell Correlation (FSC) metric. Our study illuminates benefits and considerations of both supervised and unsupervised approaches, guiding the development of improved reconstruction strategies.

Evaluation of Electron Tomography Capabilities for Shale Imaging

Despite the advantageous resolution of electron tomography (ET), reconstruction of three-dimensional (3D) images from multiple two-dimensional (2D) projections presents several challenges, including small signal-to-noise ratios, and a limited projection range. This study evaluates the capabilities of ET for thin sections of shale, a complex nanoporous medium. A numerical phantom with 1.24 nm pixel size is constructed based on the tomographic reconstruction of a Barnett shale. A dataset of 2D projection images is numerically generated from the 3D phantom and studied over a range of conditions. First, common reconstruction techniques are used to reconstruct the shale structure. The reconstruction uncertainty is quantified by comparing overall values of storage and transport metrics, as well as the misclassification of pore voxels compared to the phantom. We then select the most robust reconstruction technique and we vary the acquisition conditions to quantify the effect of artifacts. We find a strong agreement for large pores over the different acquisition workflows, while a wider variability exists for nanometer-scale features. The limited projection range and reconstruction are identified as the main experimental bottlenecks, thereby suggesting that sample thinning, advanced holders, and advanced reconstruction algorithms offer opportunities for improvement.

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.

The unremarkable alveolar epithelial glycocalyx: a thorium dioxide-based electron microscopic comparison after heparinase or pneumolysin treatment

Recent investigations analyzed in depth the biochemical and biophysical properties of the endothelial glycocalyx. In comparison, this complex cell-covering structure is largely understudied in alveolar epithelial cells. To better characterize the alveolar glycocalyx ultrastructure, unaffected versus injured human lung tissue explants and mouse lungs were analyzed by transmission electron microscopy. Lung tissue was treated with either heparinase (HEP), known to shed glycocalyx components, or pneumolysin (PLY), the exotoxin of Streptococcus pneumoniae not investigated for structural glycocalyx effects so far. Cationic colloidal thorium dioxide (cThO₂) particles were used for glycocalyx glycosaminoglycan visualization. The level of cThO₂ particles orthogonal to apical cell membranes (≙ stained glycosaminoglycan height) of alveolar epithelial type I (AEI) and type II (AEII) cells was stereologically measured. In addition, cThO₂ particle density was studied by dual-axis electron tomography (≙ stained glycosaminoglycan density in three dimensions). For untreated samples, the average cThO₂ particle level was ≈ 18 nm for human AEI, ≈ 17 nm for mouse AEI, ≈ 44 nm for human AEII and ≈ 35 nm for mouse AEII. Both treatments, HEP and PLY, resulted in a significant reduction of cThO₂ particle levels on human and mouse AEI and AEII. Moreover, a HEP- and PLY-associated reduction in cThO₂ particle density was observed. The present study provides quantitative data on the differential glycocalyx distribution on AEI and AEII based on cThO₂ and demonstrates alveolar glycocalyx shedding in response to HEP or PLY resulting in a structural reduction in both glycosaminoglycan height and density. Future studies should elucidate the underlying alveolar epithelial cell type-specific distribution of glycocalyx subcomponents for better functional understanding.

Cryo-electron tomography related radiation-damage parameters for individual-molecule 3D structure determination

To understand the dynamic structure-function relationship of soft- and biomolecules, the determination of the three-dimensional (3D) structure of each individual molecule (nonaveraged structure) in its native state is sought-after. Cryo-electron tomography (cryo-ET) is a unique tool for imaging an individual object from a series of tilted views. However, due to radiation damage from the incident electron beam, the tolerable electron dose limits image contrast and the signal-to-noise ratio (SNR) of the data, preventing the 3D structure determination of individual molecules, especially at high-resolution. Although recently developed technologies and techniques, such as the direct electron detector, phase plate, and computational algorithms, can partially improve image contrast/SNR at the same electron dose, the high-resolution structure, such as tertiary structure of individual molecules, has not yet been resolved. Here, we review the cryo-electron microscopy (cryo-EM) and cryo-ET experimental parameters to discuss how these parameters affect the extent of radiation damage. This discussion can guide us in optimizing the experimental strategy to increase the imaging dose or improve image SNR without increasing the radiation damage. With a higher dose, a higher image contrast/SNR can be achieved, which is crucial for individual-molecule 3D structure. With 3D structures determined from an ensemble of individual molecules in different conformations, the molecular mechanism through their biochemical reactions, such as self-folding or synthesis, can be elucidated in a straightforward manner.

Structure Detection in Three-Dimensional Cellular Cryoelectron Tomograms by Reconstructing Two-Dimensional Annotated Tilt Series

The revolutionary technique cryoelectron tomography (cryo-ET) enables imaging of cellular structure and organization in a near-native environment at submolecular resolution, which is vital to subsequent data analysis and modeling. The conventional structure detection process first reconstructs the three-dimensional (3D) tomogram from a series of two-dimensional (2D) projections and then directly detects subcellular components found within the tomogram. However, this process is challenging due to potential structural information loss during the tomographic reconstruction and the limited scope of existing methods since most major state-of-the-art object detection methods are designed for 2D rather than 3D images. Therefore, in this article, as an alternative approach to complement the conventional process, we propose a novel 2D-to-3D framework that detects structures within 2D projection images before reconstructing the results back to 3D. We implemented the proposed framework as three specific algorithms for three individual tasks: semantic segmentation, edge detection, and object localization. As experimental validation of the 2D-to-3D framework for cryo-ET data, we applied the algorithms to the segmentation of mitochondrial calcium phosphate granules, detection of spherical edges, and localization of mitochondria. Quantitative and qualitative results show better performance for prediction tasks of segmentation on the 2D projections and promising performance on object localization and edge detection, paving the way for future studies in the exploration of cryo-ET for in situ structural biology.

The ultrastructural heterogeneity of lung surfactant revealed by serial section electron tomography: Insights into the 3D architecture of human tubular myelin

Weibel's hypothetical 3D model in 1966 provided first ultrastructural details into tubular myelin (TM), a unique, complex surfactant subtype found in the hypophase of the alveolar lining layer. Although initial descriptions by electron microscopy (EM) were already published in the 1950s, a uniform morphological differentiation from other intraalveolar surfactant subtypes is still missing and potential structure-function relationships remain enigmatic. Technical developments in volume EM methods now allow a more detailed reinvestigation. To address unanswered ultrastructural questions, we analyzed ultrathin sections of humanized SP-A1/SP-A2 co-expressing mouse as well as human lung samples by conventional transmission EM. We combined these 2D information with 3D analysis of single- and dual-axis electron tomography of serial sections for high z-resolution (in a range of a few nm) and extended volumes of up to 1 µm total z-information. This study reveals that TM constitutes a heterogeneous surfactant organization mainly comprised of distorted parallel membrane planes with local intersections, which are distributed all over the TM substructure. These intersecting membrane planes form, among other various polygons, the well-known 2D "lattice", respectively 3D quadratic tubules, which in many analyzed spots of human alveoli appear to be less abundant than also observed non-concentric 3D lamellae. The additional application of serial section electron tomography to conventional transmission EM demonstrates a high heterogeneity of TM membrane networks, which indicates dynamic transformations between its substructures. Our method provides an ideal basis for further in and ex vivo structural analyses of surfactant under various conditions at nanometer scale.

Parakeet: a digital twin software pipeline to assess the impact of experimental parameters on tomographic reconstructions for cryo-electron tomography

In cryo-electron tomography (cryo-ET) of biological samples, the quality of tomographic reconstructions can vary depending on the transmission electron microscope (TEM) instrument and data acquisition parameters. In this paper, we present Parakeet, a 'digital twin' software pipeline for the assessment of the impact of various TEM experiment parameters on the quality of three-dimensional tomographic reconstructions. The Parakeet digital twin is a digital model that can be used to optimize the performance and utilization of a physical instrument to enable in silico optimization of sample geometries, data acquisition schemes and instrument parameters. The digital twin performs virtual sample generation, TEM image simulation, and tilt series reconstruction and analysis within a convenient software framework. As well as being able to produce physically realistic simulated cryo-ET datasets to aid the development of tomographic reconstruction and subtomogram averaging programs, Parakeet aims to enable convenient assessment of the effects of different microscope parameters and data acquisition parameters on reconstruction quality. To illustrate the use of the software, we present the example of a quantitative analysis of missing wedge artefacts on simulated planar and cylindrical biological samples and discuss how data collection parameters can be modified for cylindrical samples where a full 180° tilt range might be measured.

Correlative imaging using super-resolution fluorescence microscopy and soft X-ray tomography at cryogenic temperatures provides a new way to assess virosome solutions for vaccine development

Active virosomes (AVs) are derivatives of viruses, broadly similar to ‘parent’ pathogens, with an outer envelope that contains a bespoke genome coding for four to five viral proteins capable of eliciting an antigenic response. AVs are essentially novel vaccine formulations that present on their surface selected viral proteins as antigens. Once administered, they elicit an initial ‘anti‐viral’ immune response. AVs are also internalised by host cells where their cargo viral genes are used to express viral antigen(s) intracellularly. These can then be transported to the host cell surface resulting in a second wave of antigen exposure and a more potent immuno‐stimulation. A new 3D correlative microscopy approach is used here to provide a robust analytical method for characterisation of Zika‐ and Chikungunya‐derivatised AV populations including vesicle size distribution and variations in antigen loading. Manufactured batches were compared to assess the extent and nature of batch‐to‐batch variations. We also show preliminary results that verify antigen expression on the surface of host cells. We present here a reliable and efficient high‐resolution 3D imaging regime that allows the evaluation of the microstructure and biochemistry of novel vaccine formulations such as AVs.

An overview of the recent advances in cryo-electron microscopy for life sciences

Cryo-electron microscopy (CryoEM) has superseded X-ray crystallography and NMR to emerge as a popular and effective tool for structure determination in recent times. It has become indispensable for the characterization of large macromolecular assemblies, membrane proteins, or samples that are limited, conformationally heterogeneous, and recalcitrant to crystallization. Besides, it is the only tool capable of elucidating high-resolution structures of macromolecules and biological assemblies in situ. A state-of-the-art electron microscope operable at cryo-temperature helps preserve high-resolution details of the biological sample. The structures can be determined, either in isolation via single-particle analysis (SPA) or helical reconstruction, electron diffraction (ED) or within the cellular environment via cryo-electron tomography (cryoET). All the three streams of SPA, ED, and cryoET (along with subtomogram averaging) have undergone significant advancements in recent times. This has resulted in breaking the boundaries with respect to both the size of the macromolecules/assemblies whose structures could be determined along with the visualization of atomic details at resolutions unprecedented for cryoEM. In addition, the collection of larger datasets combined with the ability to sort and process multiple conformational states from the same sample are providing the much-needed link between the protein structures and their functions. In overview, these developments are helping scientists decipher the molecular mechanism of critical cellular processes, solve structures of macromolecules that were challenging targets for structure determination until now, propelling forward the fields of biology and biomedicine. Here, we summarize recent advances and key contributions of the three cryo-electron microscopy streams of SPA, ED, and cryoET.

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 (F_max), minimum Feret diameter (F_min), volume-equivalent diameter (D_eq) and aspect ratio (F_rat) of the NPs. The results of the five labs were compared with the results obtained at NRC-NANO. The maximum disagreement in measurements of F_min and F_max obtained by the participating labs did not exceed 7 %. The measured D_eq 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.

LoTToR: An Algorithm for Missing-Wedge Correction of the Low-Tilt Tomographic 3D Reconstruction of a Single-Molecule Structure

A single-molecule three-dimensional (3D) structure is essential for understanding the thermal vibrations and dynamics as well as the conformational changes during the chemical reaction of macromolecules. Individual-particle electron tomography (IPET) is an approach for obtaining a snap-shot 3D structure of an individual macromolecule particle by aligning the tilt series of electron tomographic (ET) images of a targeted particle through a focused iterative 3D reconstruction method. The method can reduce the influence on the 3D reconstruction from large-scale image distortion and deformation. Due to the mechanical tilt limitation, 3D reconstruction often contains missing-wedge artifacts, presented as elongation and an anisotropic resolution. Here, we report a post-processing method to correct the missing-wedge artifact. This low-tilt tomographic reconstruction (LoTToR) method contains a model-free iteration process under a set of constraints in real and reciprocal spaces. A proof of concept is conducted by using the LoTToR on a phantom, i.e., a simulated 3D reconstruction from a low-tilt series of images, including that within a tilt range of ±15°. The method is validated by using both negative-staining (NS) and cryo-electron tomography (cryo-ET) experimental data. A significantly reduced missing-wedge artifact verifies the capability of LoTToR, suggesting a new tool to support the future study of macromolecular dynamics, fluctuation and chemical activity from the viewpoint of single-molecule 3D structure determination.

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.

Towards a mechanistic understanding of cellular processes by cryoEM

A series of recent hardware and software developments have transformed cryo-electron microscopy (cryoEM) from a niche tool into a method that has become indispensable in structural and functional biology. Samples that are rapidly frozen are encased in a near-native state inside a layer of amorphous ice, and then imaged in an electron microscope cooled to cryogenic temperatures. Despite being conceptually simple, cryoEM owns its success to a plethora of technological developments from numerous research groups. Here, we review the key technologies that have made this astonishing transformation possible and highlight recent trends with a focus on cryo-electron tomography. Additionally, we discuss how correlated microscopy is an exciting and perpendicular development route forward in this already rapidly growing field. We specifically discuss microscopy techniques that allow to complement time-dependent information of dynamic processes to the unique high resolution obtained in cryoEM.

Single-Molecule 3D Images of "Hole-Hole" IgG1 Homodimers by Individual-Particle Electron Tomography

The engineering of immunoglobulin-G molecules (IgGs) is of wide interest for improving therapeutics, for example by modulating the activity or multiplexing the specificity of IgGs to recognize more than one antigen. Optimization of engineered IgG requires knowledge of three-dimensional (3D) structure of synthetic IgG. However, due to flexible nature of the molecules, their structural characterization is challenging. Here, we use our reported individual-particle electron tomography (IPET) method with optimized negative-staining (OpNS) for direct 3D reconstruction of individual IgG hole-hole homodimer molecules. The hole-hole homodimer is an undesired variant generated during the production of a bispecific antibody using the knob-into-hole heterodimer technology. A total of 64 IPET 3D density maps at ~15 Å resolutions were reconstructed from 64 individual molecules, revealing 64 unique conformations. In addition to the known Y-shaped conformation, we also observed an unusual X-shaped conformation. The 3D structure of the X-shaped conformation contributes to our understanding of the structural details of the interaction between two heavy chains in the Fc domain. The IPET approach, as an orthogonal technique to characterize the 3D structure of therapeutic antibodies, provides insight into the 3D structural variety and dynamics of heterogeneous IgG molecules.

MBIR: A cryo-ET 3D reconstruction method that effectively minimizes missing wedge artifacts and restores missing information

Cryo-Electron Tomography (cryo-ET) has become an essential technique in revealing cellular and macromolecular assembly structures in their native states. However, due to radiation damage and the limited tilt range, cryo-ET suffers from low contrast and missing wedge artifacts, which limits the tomograms to low resolution and hinders further biological interpretation. In this study, we applied the Model-Based Iterative Reconstruction (MBIR) method to obtain tomographic 3D reconstructions of experimental cryo-ET datasets and demonstrated the advantages of MBIR in contrast improvement, missing wedge artifacts reduction, missing information restoration, and subtomogram averaging compared with other reconstruction approaches. Considering the outstanding reconstruction quality, MBIR has a great potential in the determination of high resolution biological structures with cryo-ET.

Protocols for Subtomogram Averaging of Membrane Proteins in the Dynamo Software Package

Cryo-electron tomography allows low-resolution three-dimensional (3D) viewing of cellular organelles and macromolecular complexes present as multiple copies within a tomogram. These structures are computationally extracted and averaged in order to obtain high-resolution 3D structures, and provide a map of their spatial distribution and interaction with their biological microenvironment. To do so, we apply the user-friendly Dynamo software package on a tomographic data set. Dynamo acts as a modular toolbox adaptable to different biological scenarios, allowing a custom designed pipeline for subtomogram averaging. Here, we use as a textbook example the mitochondrial docking site of the positive-strand RNA Flock house nodavirus (FHV) to describe how Dynamo coordinates several basic steps in the subtomogram averaging workflow. Our framework covers specific strategies to deal with additional issues in subtomogram averaging as tomographic data management, 3D surface visualization, automatic assignment of asymmetry and inherent loss of Fourier information in presence of preferential views.

Three-dimensional structural dynamics of DNA origami Bennett linkages using individual-particle electron tomography

Scaffolded DNA origami has proven to be a powerful and efficient technique to fabricate functional nanomachines by programming the folding of a single-stranded DNA template strand into three-dimensional (3D) nanostructures, designed to be precisely motion-controlled. Although two-dimensional (2D) imaging of DNA nanomachines using transmission electron microscopy and atomic force microscopy suggested these nanomachines are dynamic in 3D, geometric analysis based on 2D imaging was insufficient to uncover the exact motion in 3D. Here we use the individual-particle electron tomography method and reconstruct 129 density maps from 129 individual DNA origami Bennett linkage mechanisms at ~ 6-14 nm resolution. The statistical analyses of these conformations lead to understanding the 3D structural dynamics of Bennett linkage mechanisms. Moreover, our effort provides experimental verification of a theoretical kinematics model of DNA origami, which can be used as feedback to improve the design and control of motion via optimized DNA sequences and routing.

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.

A Bayesian approach for suppression of limited angular sampling artifacts in single particle 3D reconstruction

In the single particle reconstruction, the initial 3D structure often suffers from the limited angular sampling artifact. Selecting 2D class averages of particle images generally improves the accuracy and efficiency of the reference-free 3D angle estimation, but causes an insufficient angular sampling to fill the information of the target object in the 3D frequency space. Similarly, the initial 3D structure by the random-conical tilt reconstruction has the well-known "missing cone" artifact. Here, we attempted to solve the limited angular sampling problem by sequentially applying maximum a posteriori estimate with expectation maximization algorithm (sMAP-EM). Using both simulated and experimental cryo-electron microscope images, the sMAP-EM was compared to the direct Fourier method on the basis of reconstruction error and resolution. To establish selection criteria of the final regularization weight for the sMAP-EM, the effects of noise level and sampling sparseness on the reconstructions were examined with evenly distributed sampling simulations. The frequency information filled in the missing cone of the conical tilt sampling simulations was assessed by developing new quantitative measurements. All the results of visual and numerical evaluations showed the sMAP-EM performed better than the direct Fourier method, regardless of the sampling method, noise level, and sampling sparseness. Furthermore, the frequency domain analysis demonstrated that the sMAP-EM can fill the meaningful information in the unmeasured angular space without detailed a priori knowledge of the objects. The current research demonstrated that the sMAP-EM has a high potential to facilitate the determination of 3D protein structures at near atomic-resolution.