Registration of phase‐contrast images in propagation‐based X‐ray phase tomography

X‐ray phase tomography aims at reconstructing the 3D electron density distribution of an object. It offers enhanced sensitivity compared to attenuation‐based X‐ray absorption tomography. In propagation‐based methods, phase contrast is achieved by letting the beam propagate after interaction with the object. The phase shift is then retrieved at each projection angle, and subsequently used in tomographic reconstruction to obtain the refractive index decrement distribution, which is proportional to the electron density. Accurate phase retrieval is achieved by combining images at different propagation distances. For reconstructions of good quality, the phase‐contrast images recorded at different distances need to be accurately aligned. In this work, we characterise the artefacts related to misalignment of the phase‐contrast images, and investigate the use of different registration algorithms for aligning in‐line phase‐contrast images. The characterisation of artefacts is done by a simulation study and comparison with experimental data. Loss in resolution due to vibrations is found to be comparable to attenuation‐based computed tomography. Further, it is shown that registration of phase‐contrast images is nontrivial due to the difference in contrast between the different images, and the often periodical artefacts present in the phase‐contrast images if multilayer X‐ray optics are used. To address this, we compared two registration algorithms for aligning phase‐contrast images acquired by magnified X‐ray nanotomography: one based on cross‐correlation and one based on mutual information. We found that the mutual information‐based registration algorithm was more robust than a correlation‐based method.     

Formation and reduction of streak artefacts in electron tomography

We have analysed the formation of streak artefacts in the reconstruction based on the filtered back projection algorithm in electron tomography (ET) and accordingly applied an adaptive interpolation technique to artefact reduction. In the adaptive interpolation to recover the missing information, the edge positions in a projection curve were tracked to reduce the interpolation error. A simulation was used to demonstrate the effectiveness of the artefact reduction. Furthermore, image reconstruction of integrated circuit specimens in the ET experiments with the ultra‐high voltage electron microscope show that the strong streak artefacts can be reduced effectively by our artefact reduction technique.     

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.     

Improvements in planar feature reconstructions in atom probe tomography

Standard atom probe tomography spatial reconstruction techniques have been reasonably successful in reproducing single crystal datasets. However, artefacts persist in the reconstructions that can be attributed to the incorrect assumption of a spherical evaporation surface. Using simulated and experimental field evaporation, we examine the expected shape of the evaporating surface and propose the use of a variable point projection position to mitigate to some degree these reconstruction artefacts. We show initial results from an implementation of a variable projection position, illustrating the effect on simulated and experimental data, while still maintaining a spherical projection surface. Specimen shapes during evaporation of model structures with interfaces between regions of low- and high-evaporation-field material are presented. Use of two-and three-dimensional projection-point maps in the reconstruction of more complicated datasets is discussed.     

A practical method to determine the effective resolution in incoherent experimental electron tomography

It is not straightforward to determine resolution for a 3D reconstruction when performing an electron tomography experiment. Different contributions such as missing wedge and misalignment add up and often influence the final resolution in an anisotropic manner. The conventional resolution measures can not be used for all of the reconstruction techniques, especially for iterative techniques which are more commonly used for electron tomography in materials science. Here we define a quantitative resolution measure that determines the resolution in three orthogonal directions of the reconstruction. As an application we use this measure to determine the optimum number of simultaneous iterative reconstruction technique (SIRT) iterations to reconstruct the gold nanoparticles, based on a high angle annular dark field STEM (HAADF-STEM) tilt series.     

The benefit of thresholding carbon layers in electron tomographic tilt series by intensity downshifting

When performing electron tomography, tilt series of images are often acquired from samples that contain unwanted carbonaceous material, such as an embedding resin, a thin carbon support film or hydrocarbon contamination. The presence of such layers can introduce artefacts in reconstructions, obscuring features of interest. Here, we illustrate the benefit of preprocessing a high‐angle annular dark‐field tomographic tilt series by thresholding unwanted low‐density materials using a simple intensity downshifting procedure. The resulting tomograms have fewer artefacts and segmentation can be performed more accurately. We present two representative examples taken from studies of catalyst nanoparticles and amyloid plaque core material from the human brain.     

Validation of a morphometric reconstruction technique applied to a juvenile pelvis

Three-dimensional reconstructions of bone geometry from microCT (computed tomography) data are frequently used in biomechanical and finite element analyses. Digitization of bone models is usually a simple process for specimens with a complete geometry, but in instances of damage or disarticulation it can be very challenging. Subsequent to digitization, further imaging techniques are often required to estimate the geometry of missing bone or connecting cartilage. This paper presents an innovative approach to the reconstruction of incomplete scan data, to reproduce proper anatomical arrangements of bones, including absent connecting cartilaginous elements. Utilizing geometric morphometric tools, the reconstruction technique is validated through comparison of a reconstructed 9 year old pelvis, to the original CT data. A principal component analysis and an overlay of the two pelves provide a measure of the accuracy of the reconstructed model. Future work aims to investigate the biomechanical effects of any minor positional error on the bone's predicted structural properties through the use of finite element analysis.     

Window function-based regularization for electrical capacitance tomography image reconstruction

Image reconstruction for electrical capacitance tomography (ECT) is to retrieve the permittivity distribution of materials inside the sensor from the capacitance measurements outside. It is a typical inverse problem and has long been a challenge for its nonlinearity and ill-posedness. This paper discusses the application of Tikhonov regularization, widely used for ill-posed problems, to the image reconstruction for electrical capacitance tomography. Two methods using different regularizations are investigated, which are the standard Tikhonov regularization and the Tikhonov regularization based on the second order derivative operator. Particularly, a combined method using the linear back projection (LBP) result as the prior constraint for the Tikhonov regularization with the second order derivative operator is suggested. Simulation and experiment results show that this combined method takes advantages from both the linear back projection and the Tikhonov regularization and provides reconstructions better than those from the LBP and the Tikhonov regularization. In addition, considering the essence that the Tikhonov regularization can be described as a spectral filter characterized by its corresponding window function, we propose the possibility of applying other window functions to the ECT image reconstruction, which include the Gauss window, the Hanning window, the Blackman window, and the cosine window. Results also show the feasibility of using window functions as regularization, which presents a new strategy for the regularization of ECT image reconstruction.     

3D ECT reconstruction by an improved Landweber iteration algorithm

Electrical capacitance tomography (ECT) is a relatively mature non-invasive imaging technique that attempts to map dielectric permittivity of materials. Recently, 3D ECT has gained interest because of its potential to generate volumetric images. The study of a fast and accurate image reconstruction algorithm is a challenge task, especially for 3D reconstruction. In this paper, we propose an improved Landweber iteration algorithm. We incorporate an additional acceleration term into the cost function and apply an adaptive threshold operation to the image obtained in each iteration for reducing artefacts. The algorithm proposed is tested by the noise-free and noise-contaminated capacitance data. Sensitivity matrixes and capacitance data of a 3D ECT sensor are obtained by using the finite element (FE) method. Extensive simulations in 3D reconstruction are carried out. The results verify the effectiveness of these improvements. Both the reconstruction time and the artefacts in the reconstructed image are reduced obviously. The experimental results of 3D reconstruction of objects in the shape of letters U and L confirm the effectiveness of the proposed algorithm further.     

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.     

A deep neural networks-based image reconstruction algorithm for a reduced sensor model in large-scale tomography system

Image reconstruction for soft-field tomography is a highly nonlinear and ill-posed inverse problem. Owing to the highly complicated nature of soft-field, the reconstructed images are always poor in quality. One of the factors that affect image quality is the number of sensors in a tomography system. It is commonly assumed that increasing the number of sensors in a tomography system will improve the ill-posed condition in image reconstruction and hence improve image quality. However, as the number of sensors increases, challenges such as more complicated and expensive hardware, slower data acquisition rates, longer image reconstruction times, and larger sensitivity matrices will arise, resulting in a greater ill-posed condition. Since deep learning (DL) is capable of expressing complex nonlinear functions, the majority of research efforts have been directed toward developing a robust DL-based inverse solver for image reconstruction. However, no study has been conducted to solve the inverse problem and improve the quality of the reconstructed image using a reduced sensor model for a large-scale tomography system. This paper proposed an image reconstruction algorithm based on Deep Neural Networks (DNN) to investigate its feasibility in solving the ill-posed inverse problem caused by the reduced sensor model for a large-scale tomography system. The proposed DNN model is based on a supervised, feed-forward, fully connected, backpropagation network. It comprises an input layer, three hidden layers and an output layer. Also, it was trained using large data samples obtained from COMSOL simulation. The relationship between the scattered electromagnetic field measurement and the corresponding true electromagnetic field distribution vector is determined. During the image reconstruction process, the untrained scattered electromagnetic field measurement samples are used as inputs to the trained DNN model, and the model output is an estimate of the electromagnetic field distribution. The results show that the proposed DNN can accurately describe the distribution of electromagnetic field and boundary shape of phantom compared to traditional algorithms (LBP, FBP, Noser and Tikhonov), regardless of the size and number of phantoms within the monitoring area. Hence, the proposed DNN is more robust and has a high degree of generalization.     

Evolution of tip shape during field evaporation of complex multilayer structures

Atom-probe tomography analysis of complex multilayer structures is a promising avenue for studying interfacial properties. However, significant artefacts in the three-dimensional reconstructed data arise due to the field evaporation process. To clarify the origin and impact of these artefacts for a FeCoB/FeCo/MgO/FeCo/IrMn multilayer, tip shapes were observed by transmission electron microscopy and compared to those obtained by finite difference modelling of electric fields and evaporation processes. It was found that the emitter shape is not spherical and its surface morphology evolves during successive evaporation of the different layers. This evolving morphology contributes to the artefacts generally observed in the reconstructed atom-probe data for multilayer structures because algorithms for three-dimensional reconstruction are based on the assumption that the shape of the emitter during field evaporation is spherical. Some proposed improvements to data reconstruction are proposed.     

3-D reconstruction of the atomic positions in a simulated gold nanocrystal based on discrete tomography: prospects of atomic resolution electron tomography

A novel reconstruction procedure is proposed to achieve atomic resolution in electron tomography. The method exploits the fact that crystals are discrete assemblies of atoms (atomicity). This constraint enables us to obtain a three-dimensional (3-D) reconstruction of test structures from less than 10 projections even in the presence of noise and defects. Phase contrast transmission electron microscopy (TEM) images of a gold nanocrystal were simulated in six different zone axes. The discrete number of atoms in every column is determined by application of the channelling theory to reconstructed electron exit waves. The procedure is experimentally validated by experiments with gold samples. Our results show that discrete tomography recovers the shape of the particle as well as the position of its 309 atoms from only three projections.

Experiments on a nanocrystal that contains several missing atoms, both on the surface and in the core of the nanocrystal, while considering a high noise level in each simulated image were performed to prove the stability of the approach to reconstruct defects. The algorithm is well capable of handling structural defects in a highly noisy environment, even if this causes atom count “errors” in the projection data.     

Micro‐CT versus synchrotron radiation phase contrast imaging of human cochlea

High‐resolution images of the cochlea are used to develop atlases to extract anatomical features from low‐resolution clinical computed tomography (CT) images. We compare visualization and contrast of conventional absorption‐based micro‐CT to synchrotron radiation phase contrast imaging (SR‐PCI) images of whole unstained, nondecalcified human cochleae. Three cadaveric cochleae were imaged using SR‐PCI and micro‐CT. Images were visually compared and contrast‐to‐noise ratios (CNRs) were computed from n = 27 regions‐of‐interest (enclosing soft tissue) for quantitative comparisons. Three‐dimensional (3D) models of cochlear internal structures were constructed from SR‐PCI images using a semiautomatic segmentation method. SR‐PCI images provided superior visualization of soft tissue microstructures over conventional micro‐CT images. CNR improved from 7.5 ± 2.5 in micro‐CT images to 18.0 ± 4.3 in SR‐PCI images (p < 0.0001). The semiautomatic segmentations yielded accurate reconstructions of 3D models of the intracochlear anatomy. The improved visualization, contrast and modelling achieved using SR‐PCI images are very promising for developing atlas‐based segmentation methods for postoperative evaluation of cochlear implant surgery.     

Nanoparticle shape and configuration analysis by transmission electron tomography

Tomographic reconstruction by transmission electron microscopy is used to reveal three‐dimensional nanoparticle shapes and the stacking configurations of nanoparticle ensembles. Reconstructions are generated from bright‐field image tilt series, with a sample tilt range up to ± 70°, using single or dual tilt axes. We demonstrate the feasibility of this technique for the analysis of nanomaterials, using appropriate acquisition conditions. Tomography reveals both cubic and hexagonal close‐packing configurations in multi‐layered arrays of size‐selected In nanospheres. By tomography and phase‐contrast lattice imaging, we relate the three‐dimensional shape of PbSe octahedral nanoparticles to the underlying crystal structure. We also confirm simple‐cubic packing in multi‐layers of PbSe nanocubes and see evidence that the particle shapes have cubic symmetry. The shapes of TiO₂ nanorod bundles are shown by tomographic reconstruction to resemble flattened ellipsoids.     

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.     

Reduction of the scanning time by total variation minimization reconstruction for X‐ray tomography in a SEM

Total variation minimization is applied to the particular case of X‐ray tomography in a scanning electron microscope. To prove the efficiency of this reconstruction method, noise‐free and noisy data based on the Shepp & Logan phantom have been simulated. These simulations confirm that Total variation minimization‐reconstruction algorithm better manages data containing low number of projections with respect to simultaneous iterative reconstruction technique or filtered backprojection, even in the presence of noise. The algorithm has been applied to real data sets, with a low angular sampling and a high level of noise. Two samples containing micro‐interconnections have been analyzed and 3D reconstructions show that Total variation minimization‐based algorithm performs well even with 60 projections in order to properly recover a 500 nm diameter void inside a copper interconnection.     

Removing noises caused by motion artefacts in microcirculation maps of human skin in vivo

This paper presents a zero‐padding and cross‐correlation technique‐based correlation mapping optical coherence tomography (ZPCC‐cmOCT) to reconstruct microcirculation maps of human skin in vivo, which can remove the background decorrelation noise caused by motion artefacts. In conventional correlation mapping optical coherence tomography method, the correlation degree of static tissue may be lowered by the motion artefacts due to cardiac and respiratory motion, resulting in background decorrelation noise in microcirculation maps. In zero‐padding and cross‐correlation technique‐based correlation mapping optical coherence tomography method, structural images are first obtained by performing Fourier transform on zero‐padded interference fringes, and then cross‐correlation‐based image registration is utilized to align local areas in two adjacent structural images. Finally, correlation mapping optical coherence tomography method is performed to generate microcirculation maps. Both phantom experiments and in vivo experiments were implemented and the results demonstrate that the proposed method is capable of providing microcirculation maps with the background decorrelation noise removed.     

Total fractional-order variation regularization based image reconstruction method for capacitively coupled electrical resistance tomography

Compared with electrical resistance tomography, capacitively coupled electrical resistance tomography (CCERT) is preferred since it avoids problems of electrode corrosion and electrode polarization. However, reconstruction of conductivity distribution is still a great challenge for CCERT. To improve reconstruction quality, this work proposes a novel image reconstruction method based on total fractional-order variation regularization. Simulation work is conducted and reconstruction of several typical models is studied. Robustness of the proposed method to noise is also conducted. Additionally, the performance of the proposed reconstruction method is quantitatively evaluated. We have also carried out phantom experiment to further verify the effectiveness of the proposed method. The results demonstrate that the quality of reconstruction has been largely improved when compared with the images reconstructed by Landweber, Newton-Raphson and Tikhonov methods. The inclusion is more accurately reconstructed and the background is much clearer even under the impact of noise.     

变电压CT重建的灰度加权算法

常规固定电压的CT重建,因成像系统动态范围受限,投影数据易出现过曝光和欠曝光共存现象,造成信息缺失多,成像质量差,为此提出变电压CT重建。通过变电压获得跟工件有效厚度相匹配的有效投影序列,在ART迭代图像的基础上,调整全变差使其最小化,从而优化重建。在重建过程中,依据灰度加权,把低电压的重建图像作为初值,应用在相邻高电压有效投影重建中,得到相邻高电压的重建图像,依次类推直至最高电压。至此,工件的全部结构信息重建完毕。仿真结果表明,灰度加权算法不仅实现了变电压图像信息的完整重建,而且像素值更加稳定。