Parallel-beam Tomography Research Articles

Data processing methods and data acquisition for samples larger than the field of view in parallel-beam tomography.

Parallel-beam tomography systems at synchrotron facilities have limited field of view (FOV) determined by the available beam size and detector system coverage. Scanning the full size of samples bigger than the FOV requires various data acquisition schemes such as grid scan, 360-degree scan with offset center-of-rotation (COR), helical scan, or combinations of these schemes. Though straightforward to implement, these scanning techniques have not often been used due to the lack of software and methods to process such types of data in an easy and automated fashion. The ease of use and automation is critical at synchrotron facilities where using visual inspection in data processing steps such as image stitching, COR determination, or helical data conversion is impractical due to the large size of datasets. Here, we provide methods and their implementations in a Python package, named Algotom, for not only processing such data types but also with the highest quality possible. The efficiency and ease of use of these tools can help to extend applications of parallel-beam tomography systems.

Effect of projections number on the image quality of industrial parallel beam gamma tomography

Gamma tomography technique can provide cross-sectional visualization of an object that needed for investigating pipe scale in geothermal power plants. Parallel beam tomography has advantages such as a simple system so that it is easier to apply in the field, but the scanning duration is relatively long. This paper discusses the effect of projections number on the reconstructed images quality and proposes the number of most effective projections for pipe scale investigation. Geothermal pipe sample (OD = 275 mm, t = 10 mm) with scale has been scanned with a parallel beam gamma tomography system. The system consists of a gamma radiation source (137Cs, 80 mCi), scintillation detector, motorized gantry, control module, data acquisition, and computer. The images were reconstructed with six different projections: 128, 64, 32, 16, 8, and 4 projections and the scanning duration: 530.8, 258.9, 127.9, 63.5, 31.7, and 15.8 minutes, respectively. Visually, the 128 projections data produces the smoothest image, whereas 64 and 32 projections images look almost the same. The 16 and 8 projections images are still able to distinguish between the pipe wall, scale, and void even though the 8 projections image looks very blurry. Then, the 4 projections image is not able to visualize the shape of the object. The gaps between the average pipe wall and void gray-scale pixels value of 128, 64, 32, 16, 8 and 8 projections images are 175, 174, 167, 153, 106, and 45, respectively. Based on the scanning duration, visualization, and the gray contrast, then the number of most effective projections is 32.

Virtual multi-alignment theory of parallel-beam CT image reconstruction for elastic objects

In parallel-beam tomography, the virtual alignment method plays an important role in obtaining an ideally aligned reconstruction of a rigid specimen. Furthermore, the method has been developed for elastic specimens with specific motions such as periodic motion, regular expansion or contraction, and elliptical expansion or contraction to obtain a sinogram with an ideal sinogram pattern by transforming an elastic-type projection image set into a rigid-type projection image set satisfying the Helgason-Ludwig consistency condition. In this article, we present a method to convert a combined elastic specimen to a rigid specimen using the virtual multi-alignment method that allows us to obtain an ideally multi-aligned reconstruction of a combined elastic specimen.

Phase-Based Alignment and Improved Projection Matching of Parallel Beam Tomography Data

Tomography is an imaging technique that works by reconstructing a scene from acquired data in the form of line integrals of the imaging domain. A fundamental underlying assumption in the reconstruction procedure is the precise alignment of the data values, i.e., the relationship between the data values and the paths of the lines of integration is accurately known. In many applications, e.g., electron and X-ray tomography, it is necessary to establish this relationship using software alignment techniques or image registration due to misalignment when rotating the physical specimen. Unfortunately, highly accurate software alignment is still a challenge to achieve in many cases, and improper alignment results in severe loss in the imaging resolution. In this paper, we develop a new approach that considers the alignment problem through a completely different lens, as a problem of recovering phase shifts in Fourier domain within the reconstruction algorithm. The recovery of these phase shifts serves as the data alignment, which is done by calculating discrepancies between the misaligned data and the current reconstruction. In the development of the approach, we investigate proper selection of parameters, and we show that it is fairly flexible and surprisingly accurate. Finally, the analysis of our approach provides insight into why projection matching alignment by cross correlation can be improved through low pass filtering, which we demonstrate. Our methods are validated in a wide range of examples and settings.

On Tomography with Unknown Orientation

We consider the two-dimensional parallel beam Tomography problem in which both the object being imaged and the projection directions are unknown. The angles of projections need not to be uniformly distributed. Our solution combines two known approaches: the Geometric Moment and the Graph Laplacian. After sorting the projections using the Graph Laplacian method we create a one to one moment function of the angles. We then solve for each angle uniquely.

DEIReconstructor: a software for diffraction enhanced imaging processing and tomography reconstruction

Diffraction enhanced imaging (DEI) has been widely applied in many fields, especially when imaging low-Z samples or when the difference in the attenuation coefficient between different regions in the sample is too small to be detected. Recent developments of this technique have presented a need for a new software package for data analysis. Here, the Diffraction Enhanced Image Reconstructor (DEIReconstructor), developed in Matlab, is presented. DEIReconstructor has a user-friendly graphical user interface and runs under any of the 32-bit or 64-bit Microsoft Windows operating systems including XP and Win7. Many of its features are integrated to support imaging preprocessing, extract absorption, refractive and scattering information of diffraction enhanced imaging and allow for parallel-beam tomography reconstruction for DEI-CT. Furthermore, many other useful functions are also implemented in order to simplify the data analysis and the presentation of results. The compiled software package is freely available.

Reliable method for calculating the center of rotation in parallel-beam tomography.

High-throughput processing of parallel-beam X-ray tomography at synchrotron facilities is lacking a reliable and robust method to determine the center of rotation in an automated fashion, i.e. without the need for a human scorer. Well-known techniques based on center of mass calculation, image registration, or reconstruction evaluation work well under favourable conditions but they fail in cases where samples are larger than field of view, when the projections show low signal-to-noise, or when optical defects dominate the contrast. Here we propose an alternative technique which is based on the Fourier analysis of the sinogram. Our technique shows excellent performance particularly on challenging data.

A novel scheme to design the filter for CT reconstruction using FBP algorithm

BackgroundThe Filtered Back-Projection (FBP) algorithm is the most important technique for computerized tomographic (CT) imaging, in which the ramp filter plays a key role. FBP algorithm had been derived using the continuous system model. However, it has to be discretized in practical applications, which necessarily produces distortion in the reconstructed images.MethodsA novel scheme is proposed to design the filters to substitute the standard ramp filter to improve the reconstruction performance for parallel beam tomography. The design scheme is presented under the discrete image model and discrete projection environment. The designs are achieved by constrained optimization procedures. The designed filter can be regarded as the optimal filter for the corresponding parameters in some ways.ResultsSome filters under given parameters (such as image size and scanning angles) have been designed. The performance evaluation of CT reconstruction shows that the designed filters are better than the ramp filter in term of some general criteria.ConclusionsThe 2-D or 3-D FBP algorithms for fan beam tomography used in most CT systems, are obtained by modifying the FBP algorithm for parallel beam tomography. Therefore, the designed filters can be used for fan beam tomography and have potential applications in practical CT systems.

PITRE: software for phase-sensitive X-ray image processing and tomography reconstruction

Synchrotron-radiation computed tomography has been applied in many research fields. Here, PITRE (Phase-sensitive X-ray Image processing and Tomography REconstruction) and PITRE_BM (PITRE Batch Manager) are presented. PITRE supports phase retrieval for propagation-based phase-contrast imaging/tomography (PPCI/PPCT), extracts apparent absorption, refractive and scattering information of diffraction enhanced imaging (DEI), and allows parallel-beam tomography reconstruction for conventional absorption CT data and for PPCT phase retrieved and DEI-CT extracted information. PITRE_BM is a batch processing manager for PITRE: it executes a series of tasks, created via PITRE, without manual intervention. Both PITRE and PITRE_BM are coded in Interactive Data Language (IDL), and have a user-friendly graphical user interface. They are freeware and can run on Microsoft Windows systems via IDL Virtual Machine, which can be downloaded for free and does not require a license. The data-processing principle and some examples of application will be presented.

An algorithm for recovering unknown projection orientations and shifts in 3-D tomography

It is common for example in Cryo-electron microscopy of viruses, that the orientations at which the projections are acquired, are totally unknown. We introduce here a moment based algorithm for recovering them in the three-dimensional parallel beam tomography. In this context, there is likely to be also unknown shifts in the projections. They will be estimated simultaneously. Also stability properties of the algorithm are examined. Our considerations rely on recent results that guarantee a solution to be almost always unique. A similar analysis can also be done in the two-dimensional problem.

Application of the topological gradient method to tomography

A new method for parallel beam tomography is proposed. This method is based on the topological gradient approach. The use of the topological asymptotic analysis for detecting the main edges of the data allows us to filter the noise while inverting the Radon transform. Experimental results obtained on noisy data illustrate the efficiency of this promising approach in the case of Magnetic Resonance Imaging. We also study the sensitivity of the algorithm with respect to several regularization and weight parameters. Une nouvelle méthode de reconstruction pour la tomographie par faisceaux parallèles est proposée. Cette méthode est basée sur l’approche du gradient topologique. En détectant les contours sur les données grâce à l’analyse asymptotique topologique, il est possible de filtrer le bruit dans le processus d’inversion de la transformée de Radon. Des résultats expérimentaux obtenus sur des données bruitées illustrent les possibilités de cette approche prometteuse dans le domaine de traitement d’images IRM. Nous étudions également la sensibilité de l’algorithme par rapport aux différents paramètres de régularisation et pondération.

Quadratic Regularization Design for 2-D CT

Statistical methods for tomographic image reconstruction have improved noise and spatial resolution properties that may improve image quality in X-ray computed tomography (CT). Penalized weighted least squares (PWLS) methods using conventional quadratic regularization lead to nonuniform and anisotropic spatial resolution due to interactions between the weighting, which is necessary for good noise properties, and the regularizer. Previously, we addressed this problem for parallel-beam emission tomography using matrix algebra methods to design data-dependent, shift-variant regularizers that improve resolution uniformity. This paper develops a fast angular integral mostly analytical (AIMA) regularization design method for 2-D fan-beam X-ray CT imaging, for which parallel-beam tomography is a special case. Simulation results demonstrate that the new method for regularization design requires very modest computation and leads to nearly uniform and isotropic spatial resolution in transmission tomography when using quadratic regularization.

Uniqueness of two-dimensional tomography withunknown projection directions

We consider uniqueness of two-dimensional parallel beam tomography in whichboth the object being imaged and the projection directions are unknown. This problem occursin certain practical applications. For example, in magnetic resonance imaging there may beuncertainty in the projection directions due to the involuntary motion of the patient. Thethree-dimensional version of this problem occurs in cryo electron microscopy of viral particles,where the projection directions may be completely unknown due to the random orientations ofthe particles being imaged. We show that the problem is related to some algebraic geometricproperties of a certain system of homogeneous polynomials. We also show that for sufficientlyasymmetric objects, the object is uniquely determined up to an orthogonal transformation bythe projection data from unknown directions.

Unique recovery of unknown projection orientations in three-dimensional tomography

We consider uniqueness of three-dimensional parallel beam tomography in which both the object being imaged and the projection orientations are unknown. This problem occurs in certain practical applications, for example in cryo electron microscopy of viral particles, where the projection orientations may be completely unknown due to the random orientations of the particles being imaged. We show that only three projections are needed to guarantee unique recovery of the unknown projection orientations (up to a common orthogonal transformation), if the object belongs to a certain generic set of objects. In particular, the uniqueness holds for almost all objects. We also show that, if the object belongs to that generic set, $k+1$ projections at unknown orientations suffice to determine uniquely also the geometric moments of the object of order less or equal to $k$. As a consequence, the object belonging to that generic set, is uniquely determined (up to an orthogonal transformation) by almost any infinitely many projections at unknown orientations. We show that the uniqueness problem is related to some properties of certain homogeneous polynomials that depend on the projection orientations and the geometric moments of objects. Here certain theorems of algebraic geometry turn out to be useful. We also provide a system of equations, that is uniquely solvable and gives the desired projection orientations and object's geometric moments.

A Network Flow Algorithm for Reconstructing Binary Images from Continuous X-rays

Tomography is a powerful technique to obtain accurate images of the interior of an object in a nondestructive way. Conventional reconstruction algorithms, such as filtered backprojection, require many projections to obtain high quality reconstructions. If the object of interest is known in advance to consist of only a few different materials, corresponding to known image intensities, the use of this prior knowledge in the reconstruction procedure can dramatically reduce the number of required projections. In previous work we proposed a network flow algorithm for reconstructing a binary image defined on a lattice from its projections. In this paper we propose a new algorithm for the reconstruction of binary images that do not have an intrinsic lattice structure and are defined on a continuous domain, from a small number of their projections. Our algorithm relies on the fact that the problem of reconstructing an image from only two projections can be formulated as a network flow problem in a graph. We derive this formulation for parallel beam and fan beam tomography and show how it can be used to compute binary reconstructions from more than two projections. Our algorithm is capable of computing high quality reconstructions from very few projections. We evaluate its performance on both simulated and real experimental projection data and compare it to other reconstruction algorithms.

Shear-Based Fast Hierarchical Backprojection for Parallel-Beam Tomography

We introduce a family of fast algorithms for 2-D parallel-beam tomographic backprojection. They aggregate the projections in a hierarchical structure involving the shearing and addition of sparsely sampled images. The algorithms achieve a computational cost of O(N(2) log P), when backprojecting an N x N pixel image from P projections. The algorithms provide a systematic means, guided by a Fourier-domain interpretation, to adjust and optimize the tradeoff between computational cost and accuracy. In an example with N = 512 and P = 1458 the algorithms provide high accuracy, with more than an order of magnitude reduction in operation counts.

Compensation of Some Time Dependent Deformations in Tomography

This work concerns 2D + t dynamic tomography. We show that a much larger class of deformations than the affine transforms can be compensated analytically within filtered back projection algorithms in 2D parallel beam and fan beam dynamic tomography. We present numerical experiments on the Shepp and Logan phantom showing that nonaffine deformations can be compensated. A generalization to 3D cone beam tomography is proposed.

Two-Dimensional tomography with unknown view angles

We consider uniqueness of two-dimensional parallel beam tomography with unknown view angles. We show that infinitely many projections at unknown view angles of a sufficiently asymmetric object determine the object uniquely. An explicit expression for the required asymmetry is given in terms of the object's geometric moments. We also show that under certain assumptions finitely many projections guarantee uniqueness for the unknown view angles. Compared to previous results about uniqueness of view angles, our result reduces the minimum number of required projections to approximately half and is applicable to a larger set of objects. Our analysis is based on algebraic geometric properties of a certain system of homogeneous polynomials determined by the Helgason-Ludwig consistency conditions.

Inversion of ionospheric electron density from GPS beacon observations

This paper studies the mathematical foundation of time dependent three-dimensional (3-D) computerized ionospheric tomography (CIT) for reconstructing ionospheric electron density,N e, from ground-based GPS beacon observations. After simplifying the relation betweenN e and time, the time-dependent 3-D inversion in consideration is reduced to a 3-D tomography with incomplete projections. To see clearly the effects of the incompleteness on the quality of reconstruction under 3-D condition, the formula of 3-D parallelbeam tomography is deduced theoretically. After establishing the mathematical foundation, simulations based on actual GPS ray paths with the help of the IRI-90 model are performed, and reasonable time-dependent 3-D distribution images ofN e are obtained when taking proper layout of the netowrk and allowing variable resolutions. The quality of the reconstruction is rather good when compared with the images from the IRI-90 model directly. Therefore, results in this paper demonstrate that imaging of the ionospheric electron density distribution from GPS beacon observations is reasonable in theory and feasible in practice.

High-resolution computed tomography from efficientsampling

This paper investigates high-resolution reconstructions fromefficiently sampled data in parallel-beam tomography, inparticular, local tomography. A class of sampling schemes isdefined and characterized, and it is shown that the standardscheme and the interlaced scheme of Cormack (Cormack A M 1978Phys. Med. Biol. 23 1141-8) are most promising. Anerror analysis for the filtered backprojection algorithm forboth global and local tomography is presented. The analysisprovides insights on how to realize the theoretically superiorresolution of the interlaced scheme in practice. A numericalexperiment with real data indicates the feasibility ofhigh-resolution local tomography using the interlaced scheme.