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Detection of 5-fold Symmetry axes. (a) The scoring function for 5fold symmetry axes is calculated by Eq.(10). Shown here is a scoring map of the inverted and normalized scores. The high contrast peaks indicate clearly the 5-fold symmetry axes. (b) Six sets of critical points were tested. The horizontal axis represents the twelve 5-fold symmetry axes while the vertical axis represents the scores calculated for each detected 5-fold symmetry axes and each set of critical points.
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We present an automatic algorithm to segment all the local and global asymmetric units of a three-dimensional density map of icosahedral viruses. This approach is readily applicable to the structural analysis of a broad range of virus structures that are reconstructed using cryo-electron microscopy (cryo-EM) technique. Our algorithm includes three...
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... in the scoring function map SF(B j ), j = 1, 2, · · · , q for the twelve angular bins with the smallest scores. However, these bins cannot be too close to each other. Otherwise, the one with larger score is discarded and the search continues until all twelve bins are located. As an example, we show in Fig. 4(a) the (inverted and normalized) scoring function of the outer capsid layer of the rice dwarf virus (RDV) map [2]. The "peaks" with high contrast can be clearly seen and hence easily detected by conducting a "peak-searching" procedure. One issue we would like to address here is that in Eq. (9) the deviation measure alone may be ...
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... functions, we test our method on the outer capsid layer of rice dwarf virus (RDV) map [2] with different numbers of critical points. The total number of critical points on this map is 36,161. We choose only a subset of the critical points by requiring that the density at the critical points should be larger than a threshold (e.g., I > t 0 ). In Fig. 4(b), six sets of critical points are tested. For each set, we can detect the twelve best symmetry axes by comparing their scores in order to find the icosahedral (global) symmetry. The scores of the selected symmetry axes are shown in Fig. 4(b). The legend gives the number of critical points and the corresponding threshold for each of the ...
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... by requiring that the density at the critical points should be larger than a threshold (e.g., I > t 0 ). In Fig. 4(b), six sets of critical points are tested. For each set, we can detect the twelve best symmetry axes by comparing their scores in order to find the icosahedral (global) symmetry. The scores of the selected symmetry axes are shown in Fig. 4(b). The legend gives the number of critical points and the corresponding threshold for each of the six tests, provided that the original density map is normalized to [0,255]. From this figure, we can see that the number of critical points does not significantly affect the resulting scoring functions. Interestingly the detected symmetry ...
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Citations
... The Cryo-EM developed to prevent structural deformations that may occur when preparing a sample in a classical EM. Cryo-EM is a powerful tool and it is called a non-crystallography or single-particle reconstruction technique (Yu and Bajaj, 2005). Samples must be cooled by preventing the formation of ice crystals in the Cryo-EM technique. ...
... The main problem with the Cryo-EM is the unexplained displacements of particles floating in the glassy ice environment. Therefore, many of the macromolecules studied with this technique remain dependent on the symmetry of the structures studied (Yu and Bajaj, 2005). In other words, a high concentration of homogeneous virus particles is required to create the image. ...
Viruses are very small physical particles that can not be observed with a normal microscope. Only the largest virus, the poxvirus can be seen in the light microscope. Other tools are required for a detailed examination of viruses in ultrastructural size. Especially for the last 20 years, these advanced and ongoing tools have been used in the investigation of the biological molecules and biological processes of viruses. With the help of virus imaging tools, viruses can be identified in clinical samples, thereby explaining the detailed life cycle of the virus. Thus, these tools help the creation of new and safe vaccines and antiviral medicine. In this review, electron microscopy (EM) tools were given as scanning electron microscopy (SEM), transmission electron microscopy (TEM), and cryo-electron microscopy (Cryo-EM) since generally used tools based on EM. Besides EM-based tools, X-ray crystallography which is the basis for cryo-EM and atomic force microscopy (AFM) that is relatively economic and easy to apply compared to other microscopies were also described.
... Noise reduction techniques include wavelet transforms (Reichel et al., 2001), median filters (Sandberg, 2007;van der Heide et al., 2007), bilateral filters (Jiang et al., 2003;Pantelic et al., 2006), anisotropic (Fernandez, 2009;Fernandez and Carrascosa, 2010;Fernandez et al., 2008) and non-anisotropic diffusion filters (Volkmann, 2010;Yamashita et al., 2007). Segmentation algorithms include drawing and interpolation tools (Alber et al., 2008;Noske et al., 2008), thresholding algorithms (John, 1986;Shapiro and Linda, 2002), ridge detectors (Cardenes et al., 2017) gradient-based edge detectors (Gonzalez 2002a,b;Prewitt, 1970;Roberts, 1963), snake algorithms (Kang et al., 2015;Kass et al., 1988), watershed transforms (Adiga et al., 2004;Roerdink and Meijster, 2001;Sijbers et al., 1997;Volkmann, 2010), bilateral edge filters (Gonzalez 2002a,b;Marr and Hildreth, 1980;Pantelic et al., 2007), Laplacian of Gaussian filters (Marr and Hildreth, 1980), fast marching methods Baker et al., 2006), the 3D recursive filter (Monga et al., 1991;Yu and Bajaj, 2005), template matching techniques (Comolli et al., 2009;Frangakis et al., 2002;Lebbink et al., 2007), correlation approaches (Zhu et al., 2003) and machine learning approaches (Luengo et al., 2017;Mallick et al., 2004;Moussavi et al., 2010). These tools along with the advantages and drawbacks of each are reviewed more comprehensively elsewhere (Ali, 2016). ...
Resolving the 3D architecture of cells to atomic resolution is one of the most ambitious challenges of cellular and structural biology. Central to this process is the ability to automate tomogram segmentation to identify sub-cellular components, facilitate molecular docking and annotate detected objects with associated metadata. Here we demonstrate that RAZA (Rapid 3D z-crossings algorithm) provides a robust, accurate, intuitive, fast, and generally applicable segmentation algorithm capable of detecting organelles, membranes, macromolecular assemblies and extrinsic membrane protein domains. RAZA defines each continuous contour within a tomogram as a discrete object and extracts a set of 3D structural fingerprints (major, middle and minor axes, surface area and volume), enabling selective, semi-automated segmentation and object extraction. RAZA takes advantage of the fact that the underlying algorithm is a true 3D edge detector, allowing the axes of a detected object to be defined, independent of its random orientation within a cellular tomogram. The selectivity of object segmentation and extraction can be controlled by specifying a user-defined detection tolerance threshold for each fingerprint parameter, within which segmented objects must fall and/or by altering the number of search parameters, to define morphologically similar structures. We demonstrate the capability of RAZA to selectively extract subgroups of organelles (mitochondria) and macromolecular assemblies (ribosomes) from cellular tomograms. Furthermore, the ability of RAZA to define objects and their contours, provides a basis for molecular docking and rapid tomogram annotation.
... Currently, this step is often performed manually [8,16] with the help of software packages like ImageJ, Bsoft, Imaris, Amira, which demands a great amount of time. Much work has been dedicated towards automating segmentation, most of which is targeted to deal with EM data, such as watershed [17], fast marching [18,19], template matching [20], adaptive shape and points clustering [21], active contours [22], Gaussianlike membrane models [23], ridge-based membrane detector [24,25] and tensor voting approach [26]. In general, software packages developed for EM tomography are suboptimal when used on TXM data, and there is a need for software dedicated to TXM so as to exploit better these unique characteristics [14,27]. ...
Structural analysis of biological membranes is important for understanding cell and sub-cellular organelle function as well as their interaction with the surrounding environment. Imaging of whole cells in three dimension at high spatial resolution remains a significant challenge, particularly for thick cells. Cryo-transmission soft X-ray microscopy (cryo-TXM) has recently gained popularity to image, in 3D, intact thick cells (∼10μm) with details of sub-cellular architecture and organization in near-native state. This paper reports a new tool to segment and quantify structural changes of biological membranes in 3D from cryo-TXM images by tracking an initial 2D contour along the third axis of the microscope, through a multi-scale ridge detection followed by an active contours-based model, with a subsequent refinement along the other two axes. A quantitative metric that assesses the grayscale profiles perpendicular to the membrane surfaces is introduced and shown to be linearly related to the membrane thickness. Our methodology has been validated on synthetic phantoms using realistic microscope properties and structure dimensions, as well as on real cryo-TXM data. Results demonstrate the validity of our algorithms for cryo-TXM data analysis.
... The threefold symmetry detection is currently claimed in various areas of crystallography [9,10], virology [11], analysis of electron microscope images [12], ...
This paper considers an algebraic method for symmetry analysis of hexagonally sampled images, based on the interpretation of such images as functions on “Eisenstein fields”. These are finite fields \(\mathbb {GF}(p^2)\) of special characteristics \(p=12k+5\), where \(k>0\) is an integer. Some properties of such fields are studied; in particular, it is shown that its elements may be considered as “discrete Eisenstein numbers” and are in natural correspondence with hexagons in a \((p\times p)\)-diamond-shaped fragment of a regular plane tiling. The concept of logarithm in Eisenstein fields is introduced and used to define a “log-polar”-representation of hexagonal images. Next, an algorithm for threefold symmetry detection in gray-level images is proposed.
... The input 3D multi-material classification map is often obtained by multi-domain segmentation or classifiers using normalized graph cut operating on a voxellized 3D reconstructed image of the MMO [29,30]. Softwares such as AsymSeg and SymSeg [31,32] implemented and publicly available from VolRover [33] and Segger [34] can be used to generate such material classification maps. Generally, if a face is shared by two voxels with the same material ID i, then the face will be removed and the two voxels are clustered into the same i material region. ...
We present a simple but effective algorithm for generating topologically and geometrically consistent quality triangular surface meshing of compactly packed multiple heterogeneous domains in 3. By compact packing we imply that adjacent homogeneous domains or materials share some 0, 1, and/or 2 dimensional boundary. Such packed multiple material (or multi-material) solids arise naturally from classification/partitioning/segmentation of homogeneous domains in 3 into different sub-regions. The multi-materials may also represent separate functionally classified sections or just be multiple component copies tightly fused together as perhaps by layered manufacturing processes. The input to our algorithm is a geometric representation of the entire multi- material solid, and a volumetric classification map identifying the individual materials. As output, each individual material region is represented by a triangulated 2-manifold boundary, with adjacent material regions having shared boundaries. Our algorithm has been implemented, and applied to different multi-material solids, and the results are additionally presented with quantitative analysis of detection and cure of non-manifold interfaces as well as spurious small components. These meshes are useful for combined boundary element analysis, however these simulation results are not presented.
... Definition: Given a 2D scalar image I, the marching distance between two points A and B in the function domain is defined by [16]: ...
This paper describes a new method for image edge enhancement and boundary segmentation. Like many interactive graph-based segmentation methods, users are asked to provide some foreground (or object) and background seeds. A set of randomly generated points representing the foreground are paired with another set of random points representing the background. The corresponding shortest paths of all such pairs are found and accumulated. These paths tend to go through the boundaries of the object of interest. Therefore, the accumulated paths can enhance the object edges, from which the final segmentation is obtained. Several experiments are provided to demonstrate the effectiveness of the proposed approach.
... The watershed diffusion method of Pintillie et al. provides fast segmentations given a small amount of user guidance (4,5). Bajaj's multiseeded fast-matching method likewise produces reliable segmentations when provided with symmetry information (6)(7)(8). An unsupervised method is thus necessary for those cases where such prior structural information is not available. ...
... By visual inspection, segmentations by Nhs capture multi-scale structural organization of macromolecules. However, the shape-match scores of our segmentations seem low when compared to segmentation scores received by semi-supervised methods (4)(5)(6)(7)(8). It is important to note that many previously reported segmentation scores were for simulated cryo-EM maps, as well as experimental maps. ...
Cryo-electron microscopy (cryo-EM) experiments yield low-resolution (3-30 Å) 3D-density maps of macromolecules. These density maps are segmented to identify structurally distinct proteins, protein domains, and subunits. Such partitioning aids the inference of protein motions and guides fitting of high-resolution atomistic structures. Cryo-EM density map segmentation has traditionally required tedious and subjective manual partitioning or semisupervised computational methods, whereas validation of resulting segmentations has remained an open problem in this field. We introduce a network-based hierarchical segmentation (Nhs) method, that provides a multi-scale partitioning, reflecting local and global clustering, while requiring no user input. This approach models each map as a graph, where map voxels constitute nodes and weighted edges connect neighboring voxels. Nhs initiates Markov diffusion (or random walk) on the weighted graph. As Markov probabilities homogenize through diffusion, an intrinsic segmentation emerges. We validate the segmentations with ground-truth maps based on atomistic models. When implemented on density maps in the 2010 Cryo-EM Modeling Challenge, Nhs efficiently and objectively partitions macromolecules into structurally and functionally relevant subregions at multiple scales.
... The algorithmic steps for detection of secondary structures from an intermediate resolution 3D EM map are: 1) A monomeric protein 3D map Γ is segmented from the overall 3D map of the macromolecular assembly. The description of this step is skipped in this paper as we use the implementation of the algorithms developed by Yu and Bajaj (Yu and Bajaj, 2005) in the publicly available software tool VolRover (Bajaj, 2003). 2) A suitable molecular surface Σ is extracted from 3D map Γ of the protein under investigation and the medial axis of this surface is computed (Section 3.1 for definitions and details). ...
Recent advances in three-dimensional electron microscopy (3D EM) have enabled the quantitative visualization of the structural building blocks of proteins at improved resolutions. We provide algorithms to detect the secondary structures (α-helices and β-sheets) from proteins for which the volumetric maps are reconstructed at 6-10Å resolution. Additionally, we show that when the resolution is coarser than 10Å, some of the supersecondary structures can be detected from 3D EM maps. For both these algorithms, we employ tools from computational geometry and differential topology, specifically the computation of stable/unstable manifolds of certain critical points of the distance function induced by the molecular surface. Our results connect mathematically well-defined constructions with bio-chemically induced structures observed in proteins.
... In [4], a variant of the level set method was described to segment features in an image. This method, called the multi-seeded fast marching method, is used in conjunction with seed classification and region growing and merging [4,11]. ...
... , nt vector in 3D s the angle bet vectors [11]. O detect the sour point outwards ed points in ou -seeded fast m tation (Fig. 1C onstrained Grap p is an over-seg ions need to be e below has thr r so-called p usion and the f ser-guided gra ) a graphical us to guide t e fully automa al user input th res in the ima mentation as a ed as a "superp formed betwee dy each edge nce and averag ries between tw as the shape o region, may unction [19]. ...
In this paper we present a high-fidelity method for 2D and 3D image boundary segmentation. The algorithm is a novel combination of graph-cuts and initial image segmentation. The pre-segmentation using anisotropic vector diffusion and the fast marching method is employed so that the size of the graph being considered is significantly reduced. To further improve the segmentation accuracy, some user guidance is taken into account in finding the minimal graph cut. To this end, a user-friendly graphical user interface (GUI) is developed not only for visualization purposes but for user input and editing as well. The approaches and tools developed are validated on a number of 2D/3D biomedical imaging data, showing the high efficiency and effectiveness of our method.
... SSEHunter [8] uses pseudoatoms to reduce the data complexity, each pseudoatom is assigned a composite score based on several criterions, finally all the pseudoatoms with high composite scores are manually linked together to represent the helices. The tensor based method [9] selects local critical points with cylinder-like eigenvalues as the seed points. From each seed point, the method traces helix structure in two opposite directions based on 3×3 local structure tensor. ...
Accurate identification of protein secondary structures is beneficial to the structural interpretation of low-resolution X-ray and EM electron density maps. Existing alpha helix identification methods mainly focus on locally voxelwise classification and then link the helix-voxels based on post-processing processes. In this paper, a novel alpha helix identification approach, named as SSEPredictor, based on Metropolis-Hastings sampling is proposed, which can provide both locally and globally optimized prediction for alpha helical structures in low-resolution electron density maps. The approach has been tested on X-ray crystallographic electron density maps at 8Å resolution. The experimental results show that the identification accuracy is promising.
