The application of the maximum entropy method to electron microscopic tomography

The maximum entropy method has been applied to single axis tilt electron microscopic tomography. Its application requires that the problem be correctly formulated and that the model for the noise in electron micrographs be developed. A suitable noise model was determined empirically. The maximum entropy method was applied to a reconstruction of a test object from projections to which noise had been added. These reconstructions were superior to those obtained by reciprocal space weighted back protection. The method was also robust towards the incorrect specification of the noise, the penalty being an increase in the time required for convergence rather than degradation of the quality of the reconstructed image. In the reconstruction of negatively stained chromatin fibres it was possible to obtain satisfactory images utilizing all the information in the projections, in contrast to conventional methods in which high resolution data are removed by the application of Fourier space filters.     

Three-dimensional reconstruction from electron micrographs of disordered specimens. II. Implementation and results

The computational procedures to implement the method described in the companion paper for three-dimensional reconstruction from projections of a disordered collection of single particles are presented. Computer simulations are used to demonstrate the way the method functions, and practical aspects are discussed in detail. Examples are given of how different symmetries can be exploited by imposing selection rules on the model equations. Applications to negatively stained 50S ribosomes and to cryo-electron micrographs of thin vitrified layers of unstained and unsupported tomato bushy stunt and Semliki Forest viruses are described, and the resulting reconstructions are presented.     

Reconstruction of the rose of directions from a digital microhologram of fibres

Digital holography makes it possible to acquire quickly the interference patterns of objects spread in a volume. The digital processing of the fringes is still too slow to achieve on line analysis of the holograms. We describe a new approach to obtain information on the direction of illuminated objects. The key idea is to avoid reconstruction of the volume followed by classical three-dimensional image processing. The hologram is processed using a global analysis based on autocorrelation. A fundamental property of diffraction patterns leads to an estimate of the mean geometric covariogram of the objects projections. The rose of directions is connected with the mean geometric covariogram through an inverse problem. In the general case, only the two-dimensional rose of the object projections can be reconstructed. The further assumption of unique-size objects gives access with the knowledge of this size to the three-dimensional direction information. An iterative scheme is suggested to reconstruct the three-dimensional rose in this special case. Results are provided on holograms of paper fibres.     

A fast surface reconstruction method for fluorescence molecular tomography based on cross-beam edge back projection

An accurate surface reconstruction method is important to fluorescence molecular tomography (FMT) for it provides boundary information of the domain occupied by the image object which is essential to modeling light propagation in free space and inside the object. In this paper, a method based on cross-beam edge back projection (CEBP) is proposed to achieve fast and three-dimensional (3D) surface reconstruction for FMT. This method consists of a cost effective and easy-to-implement setup; it back-projects the edge of an image object of all projection angles along the actual light propagation path to perform 3D surface reconstruction. Simulation studies and experiments were performed to compare the reconstruction accuracy and computational cost of the CEBP based method and the conventional radon transform (RT) based method. Results demonstrate that the CEBP based method significantly accelerates surface reconstruction compared with the RT based method while keeping similar accuracy.     

Angular reconstitution: a posteriori assignment of projection directions for 3D reconstruction

In computerized tomography as well as in most problems of three-dimensional reconstruction from projections, one knows from the experimental set-up the angular relationships between the projections from which the reconstruction is to be calculated. A serious difficulty is encountered when the angles are not known. In this paper, a method of "angular reconstitution" is described, which allows the a posteriori determination of the relative angular orientations of the projections and thus enables the three-dimensional reconstruction of the object to be calculated. For asymmetric objects, a minimum of three projections is required, which should not be related by a tilt around a single rotation axis. The method can be applied to determine the three-dimensional structure of biological macromolecules based on electron micrographs of randomly oriented individual molecules. Angular reconstitution, in combination with multivariate statistical techniques to classify and average the characteristic views of a molecule forms a complete, self-contained methodology for molecular structure analysis by electron microscopy.     

Inexact reconstructions from projections

Images of biological objects formed in the electron microscope are of such a nature that we can simplify the problem of three-dimensional reconstruction. One simplification is to make the assumption that the object can be represented as a Boolean three-dimensional matrix. In this paper we examine the problem of reconstructing two-dimensional Boolean matrices from their projections and conclude that surprisingly good reconstructions can be obtained using only four projections.     

Electron-irradiation-induced flattening of negatively stained 2D protein crystals

The thickness of negatively stained 2D crystalline arrays of the bladder membrane does not vary significantly during air drying and exposure to high vacuum. High-dose electron irradiation reduces the thickness to about 60% of the native value. These results, together with the fact that the same behaviour has been observed on another 2D system (gap junctions), indicate that the flattening induced by an electron beam on 2D crystals may be general. The implications for 3D reconstruction of negatively stained objects are discussed.     

Phase contrast electron microscopy of biological materials

The properties of two electron microscope phase contrast imaging methods are compared. The first is the conventional bright-field method in which dark phase contrast is created by defocusing; the second is a phase plate method in which bright phase contrast is created by means of a suitably shaped electric field. Using some negatively stained biological specimens which have a well-known repeating structure as the test object, it is shown that the phase plate method has some important advantages over the bright-field method. Its contrast transfer characteristics are such that it can provide a more faithful representation of the high resolution detail in the object. Moreover, by producing bright, rather than the normal dark, phase contrast it is able to simultaneously enhance the detail in the specimen and weaken the detail in the stain; this latter property enables the method to display information about the specimen that it would not be possible to detect with the bright-field method.     

Phase and amplitude contrast in electron microscopy of stained biological objects.

For biological objects negatively stained with heavy atom material, electron microscope images show best contrast for image detail on the scale of 10--20 A when a small objective aperture is used. In images taken under the optimum phase contrast imaging conditions of Scherzer, the required image detail is lost in unwanted noise. Both of these conditions may be described in terms of phase contrast imaging for a thin phase object. Calculations of image intensities and noise are reported for a model object consisting of heavy and light atoms randomly distributed to simulate a negatively stained protein molecule. The results are consistent with experimental observations.     

Three-dimensional reconstruction of biological macromolecular complexes from in-lens scanning electron micrographs

Two helical samples: F-actin and the bacteriophage T4 tail sheath were reconstructed in three dimensions from contrast enhanced (rotational shadowing and negatively stained) in-lens cryo-field emission scanning electron micrographs, using the iterative real-space helical reconstruction method. The F-actin – and bacteriophage T4 reconstructions compare favourably to an atomic model refined against fibre diffraction data and a cryo-electron microscopy reconstruction, respectively. These results show that single-particle methods, developed for macromolecules imaged in the transmission electron microscope can be applied to cryo-field emission scanning electron micrographs data with appropriate symmetry.     

Strul--a method for 3D alignment of single-particle projections based on common line correlation in Fourier space

A central problem of 3D reconstruction in single-particle electron microscopy is the determination of relative orientations of the individual projections contributing to the reconstruction. This article describes an implementation of the method of common lines correlation in Fourier space that allows generation of common lines between an arbitrary number of projections which might posses an arbitrary point group symmetry. Based on this method, it is possible to optimize rotational and translational alignment parameters for individual single-particle projections. The underlying philosophy and details of implementation are discussed, and as an illustration a 3D reconstruction in ice of peroxisomal alcohol oxidase from Pichia pastoris, an octameric assembly with 422-symmetry and a molecular weight of 592 kDa is presented.     

Non‐rigid alignment in electron tomography in materials science

Electron tomography is a key technique that enables the visualization of an object in three dimensions with a resolution of about a nanometre. High‐quality 3D reconstruction is possible thanks to the latest compressed sensing algorithms and/or better alignment and preprocessing of the 2D projections. Rigid alignment of 2D projections is routine in electron tomography. However, it cannot correct misalignments induced by (i) deformations of the sample due to radiation damage or (ii) drifting of the sample during the acquisition of an image in scanning transmission electron microscope mode. In both cases, those misalignments can give rise to artefacts in the reconstruction. We propose a simple‐to‐implement non‐rigid alignment technique to correct those artefacts. This technique is particularly suited for needle‐shaped samples in materials science. It is initiated by a rigid alignment of the projections and it is then followed by several rigid alignments of different parts of the projections. Piecewise linear deformations are applied to each projection to force them to simultaneously satisfy the rigid alignments of the different parts. The efficiency of this technique is demonstrated on three samples, an intermetallic sample with deformation misalignments due to a high electron dose typical to spectroscopic electron tomography, a porous silicon sample with an extremely thin end particularly sensitive to electron beam and another porous silicon sample that was drifting during image acquisitions.     

Application of the Tomographic Laser Doppler Anemometry (TDLA) to a fuel spray

In some applications, the size of the probe volume of a Laser Doppler Anemometry (LDA) system limits the spatial resolution of the measurement. If the probe volume is far from the LDA probe due to limited optical access, the probe volume cannot be assumed as a point. It becomes an ellipsoid, which allows no spatial correlation of a registered burst. In this work, a tomographic algorithm is presented to locate registered bursts along an extended LDA probe volume by means of a tomographic reconstruction with the inverse Radon transformation. The integral information of velocity and particle number within the probe volume is analyzed to obtain sub-volume resolution, which is only limited by the increment of traversing the probe volume through the flow field, the width of the probe volume and the number of projections. The algorithm is used to reconstruct the velocity field of a fuel spray from LDA measurements at different injection pressures.     

Iterative Region-of-Interest Reconstruction from Limited Data Using Prior Information

In practice, computed tomography and computed laminography applications suffer from incomplete data. In particular, when inspecting large objects with extremely different diameters in longitudinal and transversal directions or when high resolution reconstructions are desired, the physical conditions of the scanning system lead to restricted data and truncated projections, also known as the interior or region-of-interest (ROI) problem. To recover the searched-for density function of the inspected object, we derive a semi-discrete model of the ROI problem that inherently allows the incorporation of geometrical prior information in an abstract Hilbert space setting for bounded linear operators. Assuming that the attenuation inside the object is approximately constant, as for fibre reinforced plastics parts or homogeneous objects where one is interested in locating defects like cracks or porosities, we apply the semi-discrete Landweber–Kaczmarz method to recover the inner structure of the object inside the ROI from the measured data resulting in a semi-discrete iteration method. Finally, numerical experiments for three-dimensional tomographic applications with both an inherent restricted source and ROI problem are provided to verify the proposed method for the ROI reconstruction.     

Implementing a back-projection tomographic reconstruction algorithm using a fan-shaped beam

A back-projection tomographic reconstruction algorithm using a fan-shaped beam is considered. The freeware that implements the method of back projections is mainly applicable only in the case of parallel radiation. Distinctions in how this approach is applied when using parallel and fan-shaped beams are described in part that is related to the geometry of an experiment. The implementation of the algorithm with allowance for the peculiarities of reconstruction with a divergent beam is presented as a flow chart. The results provided in the paper demonstrate the working capacity of the algorithm.     

Volume reconstruction of large tissue specimens from serial physical sections using confocal microscopy and correction of cutting deformations by elastic registration

A set of methods leading to volume reconstruction of biological specimens larger than the field of view of a confocal laser scanning microscope (CLSM) is presented. Large tissue specimens are cut into thin physical slices and volume data sets are captured from all studied physical slices by CLSM. Overlapping spatial tiles of the same physical slice are stitched in horizontal direction. Image volumes of successive physical slices are linked in axial direction by applying an elastic registration algorithm to compensate for deformations because of cutting the specimen. We present a method enabling us to keep true object morphology using a priori information about the shape and size of the specimen, available from images of the cutting planes captured by a USB light microscope immediately before cutting the specimen by a microtome. The errors introduced by elastic registration are evaluated using a stereological point counting method and the Procrustes distance. Finally, the images are enhanced to compensate for the effect of the light attenuation with depth and visualized by a hardware accelerated volume rendering. Algorithmic steps of the reconstruction, namely elastic registration, object morphology preservation, image enhancement, and volume visualization, are implemented in a new Rapid3D software package. Because confocal microscopes get more and more frequently used in scientific laboratories, the described volume reconstruction may become an easy‐to‐apply tool to study large biological objects, tissues, and organs in histology, embryology, evolution biology, and developmental biology. In this work, we demonstrate the reconstruction using a postcranial part of a 17‐day‐old laboratory Wistar rat embryo. Microsc. Res. Tech., 2009. © 2008 Wiley‐Liss, Inc.     

Accurate stochastic reconstruction of heterogeneous microstructures by limited x‐ray tomographic projections

An accurate knowledge of the complex microstructure of a heterogeneous material is crucial for its performance prediction, prognosis and optimization. X‐ray tomography has provided a nondestructive means for microstructure characterization in 3D and 4D (i.e. structural evolution over time), in which a material is typically reconstructed from a large number of tomographic projections using filtered‐back‐projection (FBP) method or algebraic reconstruction techniques (ART). Here, we present in detail a stochastic optimization procedure that enables one to accurately reconstruct material microstructure from a small number of absorption contrast x‐ray tomographic projections. This discrete tomography reconstruction procedure is in contrast to the commonly used FBP and ART, which usually requires thousands of projections for accurate microstructure rendition. The utility of our stochastic procedure is first demonstrated by reconstructing a wide class of two‐phase heterogeneous materials including sandstone and hard‐particle packing from simulated limited‐angle projections in both cone‐beam and parallel beam projection geometry. It is then applied to reconstruct tailored Sn‐sphere‐clay‐matrix systems from limited‐angle cone‐beam data obtained via a lab‐scale tomography facility at Arizona State University and parallel‐beam synchrotron data obtained at Advanced Photon Source, Argonne National Laboratory. In addition, we examine the information content of tomography data by successively incorporating larger number of projections and quantifying the accuracy of the reconstructions. We show that only a small number of projections (e.g. 20–40, depending on the complexity of the microstructure of interest and desired resolution) are necessary for accurate material reconstructions via our stochastic procedure, which indicates its high efficiency in using limited structural information. The ramifications of the stochastic reconstruction procedure in 4D materials science are also discussed.     

A software tool for tomographic axial superresolution in STED microscopy

A method for generating three‐dimensional tomograms from multiple three‐dimensional axial projections in STimulated Emission Depletion (STED) superresolution microscopy is introduced. Our STED< method, based on the use of a micromirror placed on top of a standard microscopic sample, is used to record a three‐dimensional projection at an oblique angle in relation to the main optical axis. Combining the STED< projection with the regular STED image into a single view by tomographic reconstruction, is shown to result in a tomogram with three‐to‐four‐fold improved apparent axial resolution. Registration of the different projections is based on the use of a mutual‐information histogram similarity metric. Fusion of the projections into a single view is based on Richardson‐Lucy iterative deconvolution algorithm, modified to work with multiple projections. Our tomographic reconstruction method is demonstrated to work with real biological STED superresolution images, including a data set with a limited signal‐to‐noise ratio (SNR); the reconstruction software (SuperTomo) and its source code will be released under BSD open‐source license.     

获取物体三维形状的一种方法

本文提出了利用M系列编码光栅投影获取物体的三维形状的新方法。通过实验证明了本方法的有效性。本方法只需要两张图像,能够快速地获取物体形状。     

Three-dimensional measurement and visualization of morphogenesis applied to cardiac embryology

Volume growth and proliferation are key processes in heart morphogenesis, yet their regionalization during development of the heart has been described only anecdotally. To study the contribution of cardiomyocyte proliferation to heart development, a quantitative reconstruction method was designed, allowing the local mapping of this morphogenetic process. First, a morphological surface reconstruction is made of the heart, using sections stained specifically for cardiomyocytes. Then, by a comprehensive series of image processing steps, local three-dimensional (3D) information of proliferation is obtained. These local quantitative data are then mapped onto the morphological surface reconstruction, resulting in a reconstruction that not only provides morphological information (qualitative), but also displays local information on proliferation rate (quantitative). The resulting 3D quantitative reconstructions revealed novel observations regarding the morphogenesis of the heart.