DART: a robust algorithm for fast reconstruction of three-dimensional grain maps

xrd_simulator: 3D X-ray diffraction simulation software supporting 3D polycrystalline microstructure morphology descriptions

An open source Python package named xrd_simulator, capable of simulating geometrical interactions between a monochromatic X-ray beam and a polycrystalline microstructure, is described and demonstrated. The software can simulate arbitrary intragranular lattice variations of single crystals embedded within a multiphase 3D aggregate by making use of a tetrahedral mesh representation where each element holds an independent lattice. By approximating the X-ray beam as an arbitrary convex polyhedral region in space and letting the sample be moved continuously through arbitrary rigid motions, data from standard and non-standard measurement sequences can be simulated. This implementation is made possible through analytical solutions to a modified, time-dependent version of the Laue equations. The software, which primarily targets three-dimensional X-ray diffraction microscopy (high-energy X-ray diffraction microscopy) type experiments, enables the numerical exploration of which sample quantities can and cannot be reconstructed for a given acquisition scheme. Similarly, xrd_simulator targets investigations of different measurement sequences in relation to optimizing both experimental run times and sampling.

Intragranular three-dimensional stress tensor fields in plastically deformed polycrystals

The failure of polycrystalline materials used in infrastructure and transportation can be catastrophic. Multiscale modeling, which requires multiscale measurements of internal stress fields, is the key to predicting the deformation and failure of alloys. We determined the three-dimensional intragranular stress tensor fields in plastically deformed bulk steel using a high-energy x-ray microbeam. We observed intragranular local stresses that deviated greatly from the grain-averaged stresses and exceeded the macroscopic tensile strength. Even under deformation smaller than the uniform elongation, the intragranular stress fields were in highly triaxial stress states, which cannot be determined from the grain-averaged stresses. The ability to determine intragranular stress tensor fields can facilitate the understanding and prediction of the deformation and failure of materials through multiscale modeling.

poly-DART: A discrete algebraic reconstruction technique for polychromatic X-ray CT

The discrete algebraic reconstruction technique (DART) is a tomographic method to reconstruct images from X-ray projections in which prior knowledge on the number of object materials is exploited. In monochromatic X-ray CT (e.g., synchrotron), DART has been shown to lead to high-quality reconstructions, even with a low number of projections or a limited scanning view. However, most X-ray sources are polychromatic, leading to beam hardening effects, which significantly degrade the performance of DART. In this work, we propose a new discrete tomography algorithm, poly-DART, that exploits sparsity in the attenuation values using DART and simultaneously accounts for the polychromatic nature of the X-ray source. The results show that poly-DART leads to a vastly improved segmentation on polychromatic data obtained from Monte Carlo simulations as well as on experimental data, compared to DART.

A modified discrete tomography for improving the reconstruction of unknown multi-gray-level material in the `missing wedge' situation

Full angular rotational projections cannot always be acquired in tomographic reconstructions because of the limited space in the experimental setup, leading to the `missing wedge' situation. In this paper, a recovering `missing wedge' discrete algebraic reconstruction technique algorithm (rmwDART) has been proposed to solve the `missing wedge' problem and improve the quality of the three-dimensional reconstruction without prior knowledge of the material component's number or the material's values. By using oversegmentation, boundary extraction and mathematical morphological operations, `missing wedge' artifact areas can be located. Then, in the iteration process, by updating the located areas and regions, high-quality reconstructions can be obtained from the simulations, and the reconstructed images based on the rmwDART algorithm can be obtained from soft X-ray nano-computed tomography experiments. The results showed that there is the potential for discrete tomography.

Discrete tomography in an in vivo small animal bone study

This study aimed at assessing the feasibility of a discrete algebraic reconstruction technique (DART) to be used in in vivo small animal bone studies. The advantage of discrete tomography is the possibility to reduce the amount of X-ray projection images, which makes scans faster and implies also a significant reduction of radiation dose, without compromising the reconstruction results. Bone studies are ideal for being performed with discrete tomography, due to the relatively small number of attenuation coefficients contained in the image [namely three: background (air), soft tissue and bone]. In this paper, a validation is made by comparing trabecular bone morphometric parameters calculated from images obtained by using DART and the commonly used standard filtered back-projection (FBP). Female rats were divided into an ovariectomized (OVX) and a sham-operated group. In vivo micro-CT scanning of the tibia was done at baseline and at 2, 4, 8 and 12 weeks after surgery. The cross-section images were reconstructed using first the full set of projection images and afterwards reducing them in number to a quarter and one-sixth (248, 62, 42 projection images, respectively). For both reconstruction methods, similar changes in morphometric parameters were observed over time: bone loss for OVX and bone growth for sham-operated rats, although for DART the actual values were systematically higher (bone volume fraction) or lower (structure model index) compared to FBP, depending on the morphometric parameter. The DART algorithm was, however, more robust when using fewer projection images, where the standard FBP reconstruction was more prone to noise, showing a significantly bigger deviation from the morphometric parameters obtained using all projection images. This study supports the use of DART as a potential alternative method to FBP in X-ray micro-CT animal studies, in particular, when the number of projections has to be drastically minimized, which directly reduces scanning time and dose.

A modified discrete algebraic reconstruction technique for multiple grey image reconstruction for limited angle range tomography

The `missing wedge', which is due to a restricted rotation range, is a major challenge for quantitative analysis of an object using tomography. With prior knowledge of the grey levels, the discrete algebraic reconstruction technique (DART) is able to reconstruct objects accurately with projections in a limited angle range. However, the quality of the reconstructions declines as the number of grey levels increases. In this paper, a modified DART (MDART) was proposed, in which each independent region of homogeneous material was chosen as a research object, instead of the grey values. The grey values of each discrete region were estimated according to the solution of the linear projection equations. The iterative process of boundary pixels updating and correcting the grey values of each region was executed alternately. Simulation experiments of binary phantoms as well as multiple grey phantoms show that MDART is capable of achieving high-quality reconstructions with projections in a limited angle range. The interesting advancement of MDART is that neither prior knowledge of the grey values nor the number of grey levels is necessary.

Polycrystal orientation mapping using scanning three-dimensional X-ray diffraction microscopy

A modified three-dimensional X-ray diffraction (3DXRD) technique is proposed as a solution to the main problem with 3DXRD-type experiments, namely, polycrystalline diffraction spot overlap. The modified method, termed scanning 3DXRD, enables three-dimensional crystallographic orientation mapping in polycrystals using a narrow incident X-ray beam with a beam size sufficiently smaller than the average grain size. This method can potentially allow one to apply a 3DXRD-type technique to specimens with a larger number of grains. Moreover, because of the use of a far-field area detector, scanning 3DXRD provides spacious specimen surroundings for equipment such as stress rigs, which are not feasible in 3DXRD methods using a near-field detector. As a first demonstration, a three-dimensional orientation map was obtained by an experiment using a 20 × 20 µm beam and a well annealed iron specimen with an average grain size of 60 µm. Scanning 3DXRD compared reasonably well with orientation image microscopy by electron backscatter diffraction (EBSD), considering the influence of the beam size in the case of scanning 3DXRD. The spatial resolution was estimated to be about twice the incident beam size from a scanning 3DXRD reconstruction simulation using an orientation map modeled on the EBSD orientation image of the specimen.

Super-resolution for computed tomography based on discrete tomography

In computed tomography (CT), partial volume effects impede accurate segmentation of structures that are small with respect to the pixel size. In this paper, it is shown that for objects consisting of a small number of homogeneous materials, the reconstruction resolution can be substantially increased without altering the acquisition process. A super-resolution reconstruction approach is introduced that is based on discrete tomography, in which prior knowledge about the materials in the object is assumed. Discrete tomography has already been used to create reconstructions from a low number of projection angles, but in this paper, it is demonstrated that it can also be applied to increase the reconstruction resolution. Experiments on simulated and real μCT data of bone and foam structures show that the proposed method indeed leads to significantly improved structure segmentation and quantification compared with what can be achieved from conventional reconstructions.

Validation of three-dimensional diffraction contrast tomography reconstructions by means of electron backscatter diffraction characterization

Microstructure reconstructions resulting from diffraction contrast tomography data of polycrystalline bulk strontium titanate were reinvestigated by means of electron backscatter diffraction (EBSD) characterization. Corresponding two-dimensional grain maps from the two characterization methods were aligned and compared, focusing on the spatial resolution at the internal interfaces. The compared grain boundary networks show a remarkably good agreement both morphologically and in crystallographic orientation. Deviations are critically assessed and discussed in the context of diffraction data reconstruction and EBSD data collection techniques.

Advances in X-ray diffraction contrast tomography: flexibility in the setup geometry and application to multiphase materials

Diffraction contrast tomography is a near-field diffraction-based imaging technique that provides high-resolution grain maps of polycrystalline materials simultaneously with the orientation and average elastic strain tensor components of the individual grains with an accuracy of a few times 10−4. Recent improvements that have been introduced into the data analysis are described. The ability to process data from arbitrary detector positions allows for optimization of the experimental setup for higher spatial or strain resolution, including high Bragg angles (0 < 2θ < 180°). The geometry refinement, grain indexing and strain analysis are based on Friedel pairs of diffraction spots and can handle thousands of grains in single- or multiphase materials. The grain reconstruction is performed with a simultaneous iterative reconstruction technique using three-dimensional oblique angle projections and GPU acceleration. The improvements are demonstrated with the following experimental examples: (1) uranium oxide mapped at high spatial resolution (300 nm voxel size); (2) combined grain mapping and section topography at high Bragg angles of an Al–Li alloy; (3) ferrite and austenite crystals in a dual-phase steel; (4) grain mapping and elastic strains of a commercially pure titanium sample containing 1755 grains.

Diffraction/scattering computed tomography for three-dimensional characterization of multi-phase crystalline and amorphous materials

The three-dimensional characterization method described herein is based on diffraction and scattering techniques combined with tomography and uses the variation of these signals to reconstruct a two-dimensional/three-dimensional structural image. To emphasize the capability of the method in discriminating between different poorly ordered phases, it is named diffraction/scattering computed tomography (DSCT). This combination not only allows structural imaging but also yields an enhancement of the weak signals coming from minor phases, thereby increasing the sensitivity of structural probes. This article reports the suitability of the method for discrimination of polycrystalline and amorphous phases and for extraction of their selective local patterns with a contrast sensitivity of about 0.1% in weight of minor phases relative to the matrix. The required background in tomography is given and then the selectivity of scattering signal, the efficiency of the method, reconstruction artefacts and limitations are addressed. The approach is illustrated through different examples covering a large range of applications based on recent literature, showing the potential of DSCT in crystallography and materials science, particularly when functional and/or precious samples with sub-micrometre features have to be investigated in a nondestructive way.

Automatic parameter estimation for the discrete algebraic reconstruction technique (DART)

Computed tomography (CT) is a technique for noninvasive imaging of physical objects. In the discrete algebraic reconstruction technique (DART), prior knowledge about the material's densities is exploited to obtain high quality reconstructed images from a limited number of its projections. In practice, this prior knowledge is typically not readily available. Here, a fully automatic method, called projection distance minimization DART (PDM-DART), is proposed in which the optimal grey level parameters are adaptively estimated during the reconstruction process. To apply PDM-DART, only the number of different grey levels should be known in advance. Simulation as well as real μCT experiments show that PDM-DART is capable of computing reconstructed images of which the quality is similar to reconstructions computed by conventional DART based on exact prior knowledge, thereby eliminating the need for tedious and error-prone user interaction.

An introduction to three-dimensional X-ray diffraction microscopy

Three-dimensional X-ray diffraction microscopy is a fast and nondestructive structural characterization technique aimed at studies of the individual crystalline elements (grains or subgrains) within millimetre-sized polycrystalline specimens. It is based on two principles: the use of highly penetrating hard X-rays from a synchrotron source and the application of `tomographic' reconstruction algorithms for the analysis of the diffraction data. In favourable cases, the position, morphology, phase and crystallographic orientation can be derived for up to 1000 elements simultaneously. For each grain its average strain tensor may also be derived, from which the type II stresses can be inferred. Furthermore, the dynamics of the individual elements can be monitored during typical processes such as deformation or annealing. A review of the field is provided, with a viewpoint from materials science.