Image processing in electron tomography

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1 A. Méndez-Vilas and J. Díaz (Eds.) Image processing in electron tomography J.J. Fernández 1,2, J.I. Agulleiro 2, J.R. Bilbao-Castro 2, A. Martínez 2, I. García 2, F.J. Chichón 1, J. Martín-Benito 1, J.L. Carrascosa 1 1 Centro Nacional de Biotecnología- CSIC. Campus de Excelencia UAM+CSIC. Cantoblanco Madrid. Spain. 2 Unidad Asociada al CSIC 'Supercomputación y Algoritmos'. Univ. Almería & Málaga. Spain. Web site: Electron tomography (ET) has emerged as the leading technique for three-dimensional (3D) structural analysis of unique complex biological specimens. ET makes it possible to directly visualize the molecular architectures of organelles, cells and complex viruses at close-to-molecular resolution. ET has been crucial to reveal some of the major breakthroughs in life sciences in the last few years. ET combines electron microscopy with the power of 3D imaging to derive the 3D structure of complex specimens. A number of computational stages are involved in a typical structural study by ET, which range from the image acquisition with the electron microscope to the 3D reconstruction by means of tomographic reconstruction algorithms. Afterwards, a number of computational steps are necessary to facilitate the interpretation of the 3D data: noise reduction, segmentation and analysis of subvolumes. This chapter reviews the different computational stages involved in ET, from image acquisition to interpretation of the 3D reconstruction. Keywords image processing; electron tomography; electron microscopy; three-dimensional reconstruction; structural biology; cell biology 1. Introduction Electron tomography (ET) has emerged as the leading technique for three-dimensional (3D) structural analysis of unique complex biological specimens [1-3]. The great technological advances in the last decade have made it feasible to directly visualize the molecular architectures of organelles, cells and complex viruses as well as in vivo cellular dynamic events at close-to-molecular resolution [4-6]. The resolution attained by ET is rapidly approaching the molecular level (better than 2nm), which allows the study of the 3D organization of the structural components at a detail sufficient for the identification of macromolecular complexes, the analysis of their spatial distribution and their interactions in the native cellular context [7]. Electron tomography has got a unique potential to bridge the gap between Cellular and Molecular biology as it acts as a link between low resolution imaging techniques (light microscopy, X-ray tomography) and the high-resolution structural techniques (X-ray crystallography, nuclear magnetic resonance, electron crystallography, single particle electron microscopy). Therefore, ET has the power to allow the integration of the structural information derived by the different techniques and gathered at multiple levels of the biological complexity so as to provide a comprehensive description of the cellular function in molecular and even pseudo-atomic detail [2]. ET involves essential computational stages to derive and interpret the 3D structure of the complex biological specimens being studied: data acquisition, image alignment, image restoration, 3D reconstruction, post-processing of the 3D volume and, finally, interpretation. However, prior to these computational stages, there is an important stage related to specimen preparation. Due to the vacuum conditions in the microscope, the sensitivity of the biological material and the limited penetrating power of the electrons, the specimen has to be specially prepared before being imaged in the electron microscope. For an in-depth description of specimen preparation techniques, the reader is referred to [3,26]. 2. Data acquisition The transmission electron microscope, whose optical principles are similar to the widely known light microscope, provides projection images of the specimen being imaged. A beam of electrons is shot towards the specimen, and scattered and unscattered electrons emerging from the specimen are then collected by magnetic lenses and focused to form an interference pattern, which constitutes the projection image [3]. These projection images are thus formed by the integration of the 3D information of the specimen along the direction of the electron beam. In other words, the images look like radiographs of the sample. The images are then recorded on a CCD (charge-coupled device) camera. The principle of ET is the 3D reconstruction of a specimen from a series of projection images taken with a transmission electron microscope. In ET, a single individual biological sample is introduced in the electron microscope and a series of images (so-called tilt series) is recorded by tilting the sample around a single-tilt axis at different angles, typically over a tilt range of +/- 60 or 70 degrees and at small increments of 1-2 degrees (Figure 1). Sometimes, for better angular coverage, another tilt series is taken with the specimen rotated by 90 degrees (the double-axis tilting 19

2 A. Méndez-Vilas and J. Díaz (Eds.) geometry). Typical ET data sets then range from 60 to 280 images. Due to the resolution requirements, the image size typically is 2048 x 2048, 4096 x 4096 or even 8192 x 8192 pixels. The computation of a distortion-free 3D reconstruction from a tilt series would require that a data set from the full tilt range (+/- 90 degrees) be available. Due to physical limitations of microscopes, the angular tilt range is limited and, as a result, tomographic tilt series have a wedge of missing data corresponding to the uncovered angular range. This limitation, which is not present in other disciplines such as medical tomography, causes distortions in 3D reconstructions, as described below. Computer-automated data collection has been crucial for the advent of ET as a structural technique in cellular biology [1,3]. It allows automated tracking, focusing and recording of images under low electron-dose conditions by dividing the maximum tolerable dose over the total number of images, which preserves the specimen from radiation damage. But, as a consequence, the images exhibit a poor signal-to-noise ratio (SNR) around 0.1. During acquisition, the imperfections of the mechanical tilt system and the electron optics produce shifts in the images. The larger component of these shifts is compensated by the automated tracking procedure. However, a more accurate alignment of the images is needed afterwards by computational procedures. Fig. 1 Single-tilt axis data acquisition geometry. The specimen is imaged in the microscope by tilting it over a range typically +/- 60 or 70 degrees in small tilt increments (left). As a result, a set of projection images (so-called tilt series) needed to structure determination is collected (right). 3. Preprocessing: alignment and restoration The computational alignment is intended to mutually set the images to a common coordinate system by correction for the shifts and possible rotations and magnification changes. The standard procedure for the alignment starts at the preparation stage, where colloidal gold particles are included in the biological sample to be used as electron-dense fiducial markers. Once the tilt series is acquired, the coordinates of those markers are determined manually or semiautomatically throughout the images of the tilt series. Using the coordinates of those markers, the images are then mutually aligned by means of a least squares procedure aiming at minimizing the alignment error as a function of the shifts, rotations and other parameters included in the alignment model [3] (see Fig. 2). An alternative automated procedure that does not require the use of fiducial markers is based on cross-correlation, however its sensitivity to noise limits its application to particular high-contrast ET studies. The electron microscope leaves an imprint in the acquired images as a result of its transfer function (so-called CTF, which stands for Contrast Transfer Function). The CTF arises from the aberrations of the lenses and from the defocus used in imaging. The artefacts of the CTF consist in oscillations that produce the attenuation of structural details and, even worse, in contrast reversals at certain spatial resolution ranges. The contrast reversal is especially harmful as the structural details at some resolution ranges are presented as white density over a black background whereas at other resolution ranges they appear as black over white. Correction for the CTF effects is essential for any image to faithfully represent a projection of the specimen at high resolution. So far there have not been urgent needs for that in electron tomography due to the relatively low-resolution limits. However, as the technique improves and the attainable resolution is pushed forward, the need for CTF correction is getting importance. 20

3 A. Méndez-Vilas and J. Díaz (Eds.) Fig. 2 Aligned projection images of Vaccinia virus [8]. Three projection images of a specific virion, corresponding at and 40 o tilts are shown. -60o, 0 o The first issue related to this restoration problem is the determination of the CTF as the low SNR hampers visualization of the oscillatory component of the CTF in the Fourier spectrum. An approach that is getting increasing interest in the field is based on the assumption that the tilt-series is eucentric (i.e. the defocus at the tilt axis is constant during acquisition) [9]. The images of the tilt-series are split into tiles and the spectra of tiles at the same defocus, taken from the images throughout the tilt-series, are averaged. This yields an average spectrum with the CTF oscillations visible and highlighted, and hence an accurate fitting with the theoretical CTF model can be carried out. From this average CTF, the CTF at any point of any image in the tilt series can be easily determined by simple geometry. The major problem of CTF correction in ET is the gradient of the defocus along the direction perpendicular to the tilt axis in tilted images. This turns the restoration problem into a spatially variant one. An approach that succeeds in dealing with this CTF gradient in tilted images decomposes the global restoration problem into multiple local spatially invariant problems confined to strips parallel to the tilt axis [9]. This way, the CTF correction is applied to the aligned tilt series prior to the tomographic reconstruction. 4. Tomographic reconstruction The reconstruction problem in ET is to obtain the 3D structure of the specimen from the set of aligned, and possibly CTF-corrected, images in the tilt series. The mathematical principles of tomographic reconstruction are based upon the central section theorem, which states that the Fourier transform (FT) of a 2D projection of a 3D object is a central section of the 3D FT of the object [3]. Therefore, the 3D FT of the specimen can be computed by assembling the 2D FTs of the images in the tilt series, which yields the 3D structure of the specimen by an inverse FT. One problem of this approach is related to the interpolation in Fourier space. The standard method for tomographic reconstruction is Weighted Backprojection (WBP), which essentially is equivalent to the Fourier approach just described but working in real space [3]. WBP assumes that the projection images represent the amount of mass density encountered by imaging rays. The method simply distributes the known specimen mass present in projection images evenly over computed backprojection rays. This way, the specimen mass is projected back into a reconstruction volume (i.e., backprojected). When this process is repeated for all the projection images in the tilt series, backprojection rays from the different images intersect and reinforce each other at the points where mass is found in the original structure. Therefore, the 3D mass of the specimen is reconstructed from a series of 2D projection images. The backprojection process involves an implicit low-pass filtering that makes reconstructed volumes strongly blurred. In practice, in order to compensate the transfer function of the backprojection process, a previous high-pass filter (i.e., weighting) is applied to the projection images, hence the term ``weighted backprojection''. This weighting is necessary to properly represent the high frequency information in the reconstruction. Fig. 3 shows a sketch of the backprojection process to 3D reconstruction from projections. For a detailed description of the method, refer to [3]. Figure 4 left shows the 3D reconstruction of a Vaccinia virion with WBP [8]. The relevance of WBP in ET mainly stems from the linearity and the computational simplicity of the method. The main disadvantages of WBP are that (i) the results may be strongly affected by limited tilt angle data obtained with the microscope, and (ii) WBP does not implicitly take into account the transfer function of the microscope or the noise conditions. As a consequence of the latter, a posteriori regularization techniques (such as low-pass filtering) may be needed to attenuate the effects of the noise. 21

4 A. Méndez-Vilas and J. Díaz (Eds.) Fig. 3 Three-dimensional reconstruction from projections. In Backprojection (left), the projection images in the tilt-series are projected back into the volume to be reconstructed. In iterative methods (right), the reconstruction is progressively refined by minimizing the error between the experimental and the calculated projections. There exist alternative real-space reconstruction algorithms that formulate the 3D reconstruction problem as a large system of linear equations to be solved by iterative methods [10]. These methods iteratively refine the reconstruction by minimizing the error between the experimental projections and the projections calculated from the current volume. These alternative iterative algorithms are getting increasing interest in the field because their robustness against the experimental conditions found in ET [1] (Fig. 4 right shows the 3D reconstruction of a Vaccinia virion with 100 iterations of SIRT). These methods have not been used extensively in this field so far because of their computational demands. However, high performance computing and the advent of modern computing platforms such as multi-core computers or GPUs (graphics processors) are making them viable alternatives to WBP [11-13]. The standard high performance computing approach to tomographic reconstruction is based on the decomposition of the 3D reconstruction problem into a set of independent 3D reconstruction sub-problems corresponding to slabs of slices perpendicular to the tilt axis (see Fig. 5; the concept of slice is sketched in Fig. 1). Those subproblems can be solved in parallel with clusters of computers, supercomputers or multicore computers [11], or with standard computers using SIMD (single instruction, multiple data) instructions [13] (see Fig. 5 right). With GPUs, the parallelism-grain is usually much finer, processing multiple individual voxels in parallel [12]. The limited tilt range in ET results in a region empty of information in the Fourier space of the 3D reconstruction (so-called missing wedge ). The resolution of the reconstruction is thus anisotropic (i.e., direction-dependent). In real space, it produces artefacts as blurring of the spatial features in the beam direction, making some features appear as elongated in that direction (i.e. there is a significant loss of resolution in the Z-direction), features oriented perpendicular to the tilt axis tend to fade from view, and others are not resolved at all (see Fig. 4). A +/- 70 tilt range involves that 22% of the information is missing. The use of double-tilt axis acquisition geometry significantly reduces the missing information (down to 7% in the case of +/- 70 tilt range). The angular sampling, that is, the interval between successive tilt images, is another point affecting the resolution of the reconstruction. The more projection images, the better the angular sampling, and as a consequence the better the resolution. 5. Post-processing and interpretation of tomograms Although tomographic reconstructions (also known as tomograms in the field) contain a wealth of information, their interpretation is complicated due to a number of factors: the artefacts due to the missing wedge, the low SNR and the inherent biological complexity. Significant efforts are spent to facilitate the interpretation by several stages of postprocessing of the tomograms: noise reduction, segmentation, analysis of macromolecular complexes and resolution assessment. 22

5 A. Méndez-Vilas and J. Díaz (Eds.) Fig. 4 Three-dimensional reconstruction of Vaccinia virus. Left: Result with WBP. Right: 100 iterations of the iterative method called SIRT. The XZ, XY and ZY planes are shown at top-left, bottom-left and bottom-right panels, respectively. The planes containing the Z axis clearly show the effect of the missing wedge, in the form of severe blurring, elongation and fading-out of features. Furthermore, it is clearly seen that the contrast in the SIRT reconstruction is much better. Fig. 5 High performance tomographic reconstruction. Left: Decomposition of a volume into slices or slabs of slices. The reconstruction of the slabs can be then carried out in parallel in clusters of computers, multicore computers or using SIMD instructions. Right: Computation times and speedups of 3D reconstructions with 30 iterations of the iterative method named SIRT with size of 1024x1024xT, with T=128,256,512,1024 on standard computers using code optimization and SIMD instructions. Seq. denotes the sequential version of the program. SSE denotes the version exploiting the SIMD instructions in modern processors. It is clearly seen that the speedup factor is consistently around 3.3 and, more importantly, a standard reconstruction of 1024x1024x128 from 60 images with 30 iterations take less than 5 minutes on a standard computer based on a Intel Core 2 processor. Tomograms are significantly corrupted by noise, which precludes their visualization and analysis. Standard linear filtering techniques based on local averages, Gaussian kernels or low pass Fourier filtering succeed in reducing the noise, but at the expense of blurring edges and features. This has led to the development of more sophisticated noise reduction techniques that preserve the structural features of interest. Iterative median filtering [14] is a nonlinear technique that succeeds in preserving the structures by, for some rounds, substituting a voxel by the median of its neighbours. It is well suited for high contrast tomograms. The Beltrami flow [15] is another nonlinear method that considers the 3D volume as a 3-manifold embedded into a 4D space, with the latest dimension being the density. This method manages to tune the strength of the filtering according to an edge indicator based on geometry operators, hence preserving structures of interest. Bilateral filtering [16] is an anisotropic nonlinear technique that filters noise by applying Gaussian kernels on the spatial domain and on the density domain. Here a voxel is substituted by the weighted mean of the neighbours with a weight reflecting the similarity in terms of density and the spatial relationships. Anisotropic nonlinear diffusion is by far the standard noise reduction algorithm in the field [17,18]. It achieves feature 23

6 A. Méndez-Vilas and J. Díaz (Eds.) preservation and enhancement as the strength and direction of the smoothing are adaptively tuned to the local structures around each voxel. Figure 6 sketches the fundamentals of some nonlinear and anisotropic filtering procedures. Fig. 6 Approaches for noise filtering. (a) The Beltrami flow considers a 3D volume as a 3-manifold embedded into a 4D space. This sketch shows the 2D case, where an image with a white square over black background (top) is viewed as a surface in a 3D space (bottom). The edges are seen as cliffs in the Z direction. At each point of the surface, the projection of the normal (arrows in blue) to the Z direction (arrows in black) acts as an edge indicator, yielding little value at sharp edges where the smoothing should be attenuated or cancelled out. (b) In anisotropic nonlinear filtering, the local structure around each voxel is determined based on an eigen-analysis that provides three orthogonal eigenvectors (v) and their corresponding eigenvalues (µ). This analysis allows identification of basic geometric figures, as shown, and the strength and direction of the smoothing are adaptively tuned so that the edges of these structures are preserved and enhanced. (a) (b) (c) Fig. 7 Performance of denoising methods. (a) tomogram of HIV virions (taken from EM Data Bank, code: 1155) ; original (left), denoised with Beltrami flow (center) and with median filtering (right). (b) tomogram of Vaccinia virion [8], original (left), denoised with Beltrami flow (center) and with median filtering (right). (c) Tomogram of another Vaccinia virion; original (left), denoised with anisotropic nonlinear diffusion (center), which made it possible to visualize it in 3D with isosurface (right). 24

7 A. Méndez-Vilas and J. Díaz (Eds.) Segmentation intends to decompose the tomogram into its structural components by identifying the sets of voxels that constitute them. Though tedious, manual segmentation is the simplest and the most common approach, which consists in that the user assigns the structural features using visualization tools. Several automatic or semi-automatic approaches have been proposed in the field (some of them reviewed in [3,19]). There exist methods based on simple density thresholds or more sophisticated optimal thresholding [8], the Watershed transform extended to 3D, eigenvector analysis of an affinity matrix, eigenanalysis of the structure tensor (similar to what is done with anisotropic nonlinear diffusion, see Fig. 8), oriented filters, fuzzy logic. In addition, template matching with simple 3D geometric templates has been proposed for high contrast tomograms. However, none of these methods has stood out as a general applicable method yet, and manual segmentation still remains the prevalent method. Figure 9 shows a procedure devised to segment Vaccinia virions. Figure 10 shows the result of another segmentation procedure based on optimal thresholding and manual tracing. This segmentation procedure revealed that the Vaccinia virus is a brick-shaped structure with fixed dimensions of 360 x 270 x 250 nm, composed by an outer membrane (yellow), and an inner core (blue) where the genetic material is located. A membrane that has a palisade of spikes surrounds the core. Systematic inspection of the denoised and segmented tomograms allowed discovery of pores through which the genetic material presumably can pass during infection. There also exist lateral densities (orange) behind the outer membrane and there is room to speculate that they may be special areas for the storage of viral components required for the infection process. Fig. 8 Segmentation by exploiting the information provided by the eigen-analysis of the structure tensor (A. Martinez et al., in preparation). (left) Actin filaments and membranes from a Dictyostelium discoideum tomogram kindly provided by Dr. O. Medalia. (right) HIV virions segmented out from the background (see original tomogram in Fig. 7 top). Fig. 9 This procedure combines Gaussian filtering and anisotropic nonlinear diffusion (AND) to segment Vaccinia virions from the background. The AND result in combination to thresholding and morphological operations allowed generation of a mask. This mask was then applied to the Gaussian-filtered map in order to segment it and obtain a clean view of the details of the virus. This masking procedure was useful to produce the 3D views as that shown in Fig. 7(c). 25

8 A. Méndez-Vilas and J. Díaz (Eds.) Fig. 10 Segmentation of a Vaccinia virion by manual tracing and optimal thresholding. The virus comprises an outer membrane (yellow) that encloses a core (blue) where the genetic material is. The core has a membrane and a palisade of spikes. Lateral densities (orange) are speculated to be material to assist the infection process. Electron tomograms of the cell environment are crowded of macromolecular complexes. The high-resolution structure of many of these complexes are usually available from techniques such as X-ray crystallography or single particle electron microscopy. The current resolution levels in ET allow detection of these complexes in the in situ cell environment using these available structures as search-templates against the tomogram. This allows the analysis of the spatial distribution of these macromolecular complexes and their interactions in the native cellular context, an approach that is referred to as 'visual proteomics' [7,20]. Detection is carried out by a template-matching procedure based on cross-correlation function (CCF) on the three coordinates and the three orientation parameters of the putative occurrences in the tomogram. In order to account for the limited tilt conditions, the missing wedge is taken into account during the CCF computation [21]. The numerous false positives have to be discarded by visual inspection or statistics [22]. The putative complexes are then extracted from the tomogram, mutually aligned and averaged, with consideration of the missing wedge [23]. Finally, once the location and the orientation of the positive complexes are known, it is then possible to locate them back in the tomogram space with the proper orientation, thereby creating a macromolecular atlas of the cell that allows the analysis of their 3D distribution and their interactions. Figure 11 shows that the visualization of the macromolecular organization of thick eukaryotic organisms such as yeast Saccharomyces cerevisiae is possible thanks to these techniques. Ribosomes in the cell environment were automatically detected by means of template matching techniques, using as a template a previous ribosome density map obtained by single particle EM. The ribosomes were aligned and an averaged was computed. Finally, the ribosomes were put back into the 3D space in their location with the determined orientations, thus building an atlas of the 80s ribosome of S. Cerevisiae [24]. Fig. 11 Atlas of the 80s ribosome (right) built from a section of S. Cerevisiae (left) after detection with template matching, alignment and averaging. 26

9 A. Méndez-Vilas and J. Díaz (Eds.) Resolution estimation aims to find out the level of detail that is undoubtedly present and reliable in the tomogram. An approach that is getting interest in the field is inspired in the Fourier Shell Correlation (FSC) commonly used in other electron microscopy modalities like single particle electron microscopy [2]. This approach, known as FSCe/o, splits the tilt series into two halves (containing the even and odd projection images, respectively). From these half tilt series, two independent tomograms are computed that are then compared in Fourier space to find out the maximum spatial resolution up to which both are mutually consistent [25] (Fig. 12 left). Another approach, more time-consuming, is known as noise-compensated leave one out (NLOO). Here, for each projection image, a tomogram is computed from the whole tilt series except that projection image. Then, a projection is calculated from the tomogram at the direction of the missing projection, and a comparison in Fourier space (according to the 2D version of the FSC) between the calculated and the missing projection is carried out [25]. Therefore, this method estimates the quality of the tomogram as the resolution up to which it is possible to predict a missing projection. For small tilt intervals (1 degree) FSCe/o and NLOO yield similar results, otherwise NLOO is more reliable. On the other hand, in cases where averaging of substructures or macromolecular complexes is involved, the resolution is estimated from the FSC of two independent averages computed from half-sets of substructures (Fig. 12 right). Fig. 12 Resolution assessment. Vaccinia virus studied by electron tomography was solved at 13.4nm resolution according to the FSCe/o criterion. The average ribosome from a section of S. Cerevisiae was solved at 5.6nm resolution according to the FSC criterion. 6. Conclusion Electron tomography has matured as a structural technique in biology, making it possible to visualize the molecular architectures of organelles, cells and complex viruses at near molecular resolution. ET has got a unique potential to bridge the gap between cellular and molecular biology, allowing the integration of the multi-resolution structural information derived by the different structural techniques and thus a comprehensive description of the cellular function. ET combines electron microscopy with the power of 3D imaging and relies on essential computational image processing stages to derive and interpret the 3D structure of the complex biological specimens: data acquisition, image alignment, image restoration, 3D reconstruction, post-processing and interpretation of the structure. In this chapter, the computational methods and technologies involved in experimental ET studies have been reviewed. These methods are expected to play a key role to help ET scientists explore untrodden paths in cell biology and at a level of detail ever imagined. Acknowledgements The authors wish to thank Dr. O. Medalia for the D. discoideum dataset. The HIV dataset was taken from the EM DataBank. The support by the Spanish Ministry of Science (MCI-TIN , MCI-BFU /BMC), CSIC (PIE I075) and Junta de Andalucía (JA-P06-TIC-01428) is gratefully acknowledged. 27

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