Compressed Sensing Electron Tomography for Determining Biological Structure

Electron tomography (ET), as performed in the transmission electron microscope (TEM) or the scanning transmission electron microscope (STEM), has the unique capability of providing three-dimensional ultrastructure of cells and tissues in a native context, thus revealing important constituents such as membranes, cytoskeletal fibers, and protein complexes, on a macromolecular scale. These 3D visualizations are obtained from multiple 2D projections of a biological specimen when it is tilted through a wide range of angles relative to the incident beam direction. Currently, most ET reconstructions of cells are obtained using the weighted back-projection (WBP) algorithm, or the simultaneous iterative reconstruction technique (SIRT) algorithm. With the continuing development of electron tomographic techniques and their widespread adoption by structural and cell biologists, there has been an impetus to enhance the quality of 3D visualizations to discern smaller structures within the complex milieu of the cell, and to do so with an improved signal-to-noise ratio. Development of more optimal 3D reconstruction algorithms offers one potential route to better visualization of cellular ultrastructure. It has been well established that the quality of a tomographic reconstruction can be improved through the incorporation of prior knowledge about the specimen, i.e. through regularized image reconstruction. More recently compressed sensing (CS), which exploits signal structure to reconstruct a signal from undersampled measurements via regularized recovery, has attracted increasing attention for a number of data processing applications. The success of CS and related mathematical techniques in medical imaging — particularly magnetic resonance imaging (MRI)1,2,3,4, and x-ray computed tomography5 — have led to a growing interest in using CS methods in the field of electron tomography (ET)6,7,8,9,10. Until now, most reports of CS applications in ET have involved the imaging of inorganic materials, e.g. STEM dark-field tomography of nanoparticles6,9. Recently, however, the CS approach has been applied to STEM dark-field tomography of needle-shaped embedded biological specimens, which are tilted through an angular range of ±90° to avoid missing wedge effects10. In addition, there have been studies aimed at applying regularization and CS methods to cryo-TEM11,12. Here, we consider the application of CS to improve the quality of 3D tomographic reconstructions from microtomed cells and tissues, which have been prepared using heavy atom staining to enhance contrast. This type of specimen preparation technique is the one most widely used among cell biologists because it enables the analysis of large eukaryotic cells. It is well known that compressed sensing reconstruction methods rely on prior assumptions about the statistical properties of the specimen structure being imaged13,14,15,16. In this paper, we consider the extent to which the greater structural complexity of biological systems relative to nanoscale materials affects CS-ET performance in these very different categories of specimen. There is an important distinction between applications of CS methods to MRI and their application to ET, which is crucially important for the mathematical theory of CS-ET. The MRI sampling procedure allows for mechanical modifications to create pseudo-random sampling of a specimen in Fourier space, a process compatible with the theory of randomized sampling in CS. The instrument limitations of ET restrict feasible sampling procedures to projections of the specimen, tilted to an angle within a mechanically allowable range. This has the effect of sampling planes through the origin in Fourier space, a procedure not compatible with any existing CS sampling theory. This obstruction has yet to be adequately resolved and remains an important open problem in the theory of CS-ET. In this paper, we perform numerical simulations of a simple randomized variant of the traditional ET sampling process, wherein projection angles are chosen at random instead of being spaced uniformly in the feasible range. Theoretical obstructions remain for this variant, and simulation results indicate that this method performs worse than traditional tomographic sampling. Thus, there is still a need to develop further mathematical formalism to connect the theoretical foundation with the experimental setting. The use of scanning transmission electron microscopes (STEM) has become increasingly commonplace in the physical and life sciences for characterizing structures from atomic to macromolecular scales. Under suitable conditions17, both bright field (BF) STEM and dark field (DF) STEM measurements may be modeled as linear projections of the 3D electron density in the imaged specimen. Electron tomography can be applied to biological specimens prepared by rapid freezing in a vitrified frozen hydrated state, or by fixation with cross-linking agents followed by plastic-embedding and staining with heavy metals to enhance ultrastructural contrast. The choice of specimen preparation depends on the structures that need to be visualized: whereas it is generally preferable to image bacteria and small eukaryotic cells in a frozen hydrated state using cryo-TEM18,19,20, larger eukaryotic cells and tissues are often best visualized in specimens that have been stained with heavy atom contrast agents, due to limitations in cryo-preparation of large samples21,22,23. ET can be performed on both types of specimen but with different imaging modes: phase contrast TEM for frozen hydrated specimens; and amplitude contrast TEM, bright-field STEM, or dark-field STEM for stained specimens. In this work, we apply CS recovery algorithms to data sets obtained from STEM tomography, which can be applied to thicker specimens and therefore larger 3D volumes23,24,25. To assess the potential advantage of the technique for determining 3D cellular ultrastructure, we have simulated electron tomographic tilt series from membrane-bound compartments within cells with statistical properties that ensure a high degree of compressibility in the total variation, wavelet, and identity domains, and then perform CS-ET reconstructions from these simulated data sets. We also make comparisons with CS-ET recovery from simulations of nanoscale inorganic materials, and show that the approach gives results that are consistent with previous work by Saghi et al.6,9. Under such conditions, the advantages of CS-ET relative to WBP or SIRT reconstructions are substantial. Reconstructions from a membrane phantom, comprising randomly distributed ellipsoids and spheres at multiple scales, eccentricities, and orientations are used to characterize the performance of CS-ET under a range of noise and measurement conditions. In these simulated membrane reconstructions, we find high a correlation between sparsity and reconstruction error, significant performance advantages of CS-ET relative to WBP, and consistent superiority of uniformly-sampled CS-ET to random-angle CS-ET. We subsequently analyze experimental STEM tomographic tilt series acquired from plastic-embedded, heavy metal stained sections of fixed cells. Although CS reconstructions substantially outperform the commonly used WBP and SIRT methods for highly sparse objects, our simulations and experimental results indicate that CS generally provides a lesser advantage for reconstructions of biological structures. We attribute this disparity in performance to differences in sparsity between simple phantoms and complex biological structures. We discover that the relationship between data sparsity and the performance of CS reconstruction on the one hand, and between specimen structure and data sparsity on the other, can complicate a thorough understanding of the effectiveness of CS-ET under all potential imaging conditions. However, in application domains for which accurate sparse signal models can be established, there is strong evidence for the efficacy of CS-ET in both fully sampled and undersampled tomogram recovery.