Algebraic Reconstruction Method Research Articles

Fast Approximation of Algebraic Reconstruction Methods for Tomography

Most reconstruction algorithms for transmission tomography can be subdivided in two classes: variants of Filtered Backprojection (FBP) and iterative algebraic methods. Filtered backprojection is very fast and yields accurate results when a large number of projections are available, with high SNR and a full angular range. Algebraic methods require much more computation time, yet they are more flexible in dealing with limited data problems and noise. In this paper we propose an algorithm that combines the best of these two approaches: for a given linear algebraic method, a filter is computed that can be used within the FBP algorithm. The FBP reconstructions that result from using this filter strongly resemble the algebraic reconstructions and have many of their favorable properties, while the required reconstruction time is similar to standard-FBP. Based on a series of experiments, for both simulation data and experimental data, we demonstrate the merits of the proposed algorithm.

Cone Beam CT using motion-compensated algebraic reconstruction methods with limited data

Cone Beam Computed Tomography (CBCT) is widely used in radiation therapy for verifying treatment areas, since it provides three-dimensional image reconstruction of those tumour regions under inspection. However, organ motion is problematic during the scanning process, it causes motion artefacts on the CBCT image and can lead to mispositioning for the subsequent treatment. Moreover, patient dose is also considerable and there is a need for methods which yield acceptable image quality with as few X-ray images as possible. Although methods have been developed to handle limited projection data, such as the Algebraic Reconstruction Technique (ART); Simultaneous ART (SART); and Ordered-Subset SART (OS-SART), this study applied motion compensation to these reconstruction techniques. Root Mean Square Error (RMSE) of image is calculated to study the convergence of reconstructed images compared with the truth image. When motion was applied to a phantom and the motion compensation was used to account for the motion, the results showed that motion compensation improved the quality of CBCT image, when compared to uncompensated images. Furthermore, the experiments suggested that minimising phase error, for breathing models, was more important than minimising amplitude error.

AIR Tools — A MATLAB package of algebraic iterative reconstruction methods

We present a MATLAB package with implementations of several algebraic iterative reconstruction methods for discretizations of inverse problems. These so-called row action methods rely on semi-convergence for achieving the necessary regularization of the problem. Two classes of methods are implemented: Algebraic Reconstruction Techniques (ART) and Simultaneous Iterative Reconstruction Techniques (SIRT). In addition we provide a few simplified test problems from medical and seismic tomography. For each iterative method, a number of strategies are available for choosing the relaxation parameter and the stopping rule. The relaxation parameter can be fixed, or chosen adaptively in each iteration; in the former case we provide a new “training” algorithm that finds the optimal parameter for a given test problem. The stopping rules provided are the discrepancy principle, the monotone error rule, and the NCP criterion; for the first two methods “training” can be used to find the optimal discrepancy parameter.

Study of the internal structure of lithium fluoride single crystal by laboratory X-ray topo-tomography

Defects in a synthetic LiF crystal have been studied by X-ray topo-tomography on laboratory X-ray sources with a spatial resolution of ∼10 μm. An algebraic reconstruction method was applied to reconstruct the defect 3D structure of the crystal based on the diffraction data. The results presented in this study are in good agreement with the topographic data obtained by the Lang method.

Industrial gamma-ray tomographic scan method for large scale industrial plants

In this paper, a tomographic scan method with fixed installed detectors and a rotating gamma-ray source system is presented to diagnose industrial plants, which were impossible to examine by conventional tomographic systems. Monte Carlo simulations had been performed for two kinds of phantoms. Lab-scale experiments with the same condition as one of the phantoms, have been carried out. Algebraic and statistical reconstruction methods were applied for the reconstruction of simulation and experimental data. The reconstruction results from different algorithms were compared. Simulation results showed that reconstruction from the photopeak counting measurement gave better results than the gross counting measurement, although the photopeak counting measurement had large statistical errors. The statistical algorithm gives better results for tomographic scan methods than the algebraic method for the simulation data. Simulation and experimental data showed that this work demonstrates the feasibility of the fixed detection and rotating source system for scanning of industrial plants. Those results appear to be promising for industrial tomographic applications, especially for petrochemical industries.

High resolution optical coherence tomography by ℓ1-optimization

An algebraic optical coherence tomography (OCT) method is presented in this paper for high resolution tomography imaging. The method achieves high resolution by formulating the frequency domain OCT (FD-OCT) into a ℓ1-optimization based algebraic reconstruction problem. The performance of the proposed algebraic method is evaluated by simulation and experiment, and compared to the FD-OCT method. It is shown that the proposed method can deliver high resolution beyond the coherence length and imaging depth bigger than the ones of the FD-OCT method. The computational load is greatly reduced compared to the traditional algebraic reconstruction methods.

Electron density uncertainties in proton computed tomography

An expression is derived for the electron density resolution of proton computed tomography obtained from algebraic reconstruction methods. The dependence of the resolution on system parameters is clarified and methods for achieving the best resolutions possible are suggested. Comparison of our results with previous ones from the literature is included as well as a discussion of the limitations of the results.

GPU-Based Volume Reconstruction from Very Few Arbitrarily Aligned X-Ray Images

This paper presents a three-dimensional GPU-accelerated algebraic reconstruction method in a few-projection cone-beam setting with arbitrary acquisition geometry. To achieve artifact-reduced reconstructions in the challenging case of unconstrained geometry and extremely limited input data, we use linear methods and an artifact-avoiding projection algorithm to provide high reconstruction quality. We apply the conjugate gradient method in the linear case of Tikhonov regularization and the two-point-step-size gradient method in the nonlinear case of total variation regularization to solve the system of equations. By taking advantage of modern graphics hardware we achieve acceleration of up to two orders of magnitude over classical CPU implementations.

On the efficiency of iterative ordered subset reconstruction algorithms for acceleration on GPUs

Expectation Maximization (EM) and the Simultaneous Iterative Reconstruction Technique (SIRT) are two iterative computed tomography reconstruction algorithms often used when the data contain a high amount of statistical noise, have been acquired from a limited angular range, or have a limited number of views. A popular mechanism to increase the rate of convergence of these types of algorithms has been to perform the correctional updates within subsets of the projection data. This has given rise to the method of Ordered Subsets EM (OS-EM) and the Simultaneous Algebraic Reconstruction Technique (SART). Commodity graphics hardware (GPUs) has shown great promise to combat the high computational demands incurred by iterative reconstruction algorithms. However, we find that the special architecture and programming model of GPUs add extra constraints on the real-time performance of ordered subsets algorithms, counteracting the speedup benefits of smaller subsets observed on CPUs. This gives rise to new relationships governing the optimal number of subsets as well as relaxation factor settings for obtaining the smallest wall-clock time for reconstruction—a factor that is likely application-dependent. In this paper we study the generalization of SIRT into Ordered Subsets SIRT and show that this allows one to optimize the computational performance of GPU-accelerated iterative algebraic reconstruction methods.

SU‐FF‐I‐12: A Total Variation Based Iterative Reconstruction Algorithm for Cone Beam Breast CT with Under‐Sampled Projection Views

Purpose: To implement and investigate a total variation based reconstruction algorithm for cone beam breast CT. The algorithm is focused on reconstruction using under‐sampled projection views. Method and Materials: The implementation of the algorithm includes two major steps. In the reconstruction step, an algebraic reconstruction method is adopted. In the next step, the total variation is calculated and minimized. The initial image is updated and the iteration repeats. A segmented breast model was used to test the algorithm in this study. The cone beam CT images of a mastectomy breast specimen were corrected, filtered and segmented to form a breast model which consists of a 3D map of dense tissue structures. Re‐projection of the model was conducted with 9 degree angular increment to generate 40 projection images over 360°. The total variation based algorithm was then used to reconstruct the breast model with the under‐sampled projection views. The reconstructed images were compared with the original breast model and the percentage deviation was computed for the evaluation. Results: Iterative reconstruction with total variation minimization was performed on under‐ sampled re‐projections of the breast model. The structures of the breast were satisfactorily reconstructed, compared to the breast model. The percentage deviation was found to be 7.0%. Further improvement and optimization of the algorithm are under progress. Conclusion: The preliminary reconstruction result of a breast model demonstrates that under‐sampled breast CT projections can be reconstructed using the total variation iterative method, which provides novel opportunity for breast CT study.

Necessary and Sufficient Convergence Conditions for Algebraic Image Reconstruction Algorithms

The Landweber scheme is an algebraic reconstruction method and includes several important algorithms as its special cases. The convergence of the Landweber scheme is of both theoretical and practical importance. Using the singular value decomposition (SVD), we derive an iterative representation formula for the Landweber scheme and consequently establish the necessary and sufficient conditions for its convergence. In addition to verifying the necessity and sufficiency of known convergent conditions, we find new convergence conditions allowing relaxation coefficients in an interval not covered by known results. Moreover, it is found that the Landweber scheme can converge within finite iterations when the relaxation coefficients are chosen to be the inverses of squares of the nonzero singular values. Furthermore, the limits of the Landweber scheme in all convergence cases are shown to be the sum of the minimum norm solution of a weighted least-squares problem and an oblique projection of the initial image onto the null space of the system matrix.

Application of radial basis function neural network for information processing in fiber optical distributed measuring systems

The paper discusses tomography reconstruction of distributed physical fields. The problem is shown to be solved by using distributed measuring networks based on optical fibre sensors. Special attention is paid to tomography measuring networks based on measuring elements with integrated sensitivity. The advantages of radial basis function neural networks (RBFNN) for data processing of signals in the distributed fiber optical measuring systems are studied. RBFNN specifics which enhance the efficiency of computations of physical fields and technical and technological objects under reconstruction are key issues. Comparative analysis of the operating efficiency of RBFNN method and standard analytical and algebraic method for fiber-optical tomography reconstruction is reported.

Reconstruction of the number and positions of dipoles and quadrupoles using an algebraic method

Localization of dipoles and quadrupoles is important in inverse potential analysis, since they can effectively express spatially extended sources with a small number of parmeters. This paper proposes an algebraic method for reconstruction of pole positions as well as the number of dipole-quadrupoles without providing an initial parameter guess or iterative computing forward solutions. It is also shown that a magnetoencephalography inverse problem with a source model of dipole-quadrupoles in 3D space is reduced into the same problem as in 2D space.

Optimized algebraic reconstruction technique for generation of grain maps based on three-dimensional x-ray diffraction (3DXRD)

Recently, an algebraic reconstruction method has been presented for generation of three-dimensional (3D) maps of the grain boundaries within polycrystals. The grains are mapped layer by layer in a nondestructive way by diffraction with hard x rays. We optimize the algorithm by means of simulations and discuss ways to automate the analysis. The use of generalized Kaiser-Bessel functions as basis functions is shown to be superior to a conventional discretization in terms of square pixels. The algorithm is reformulated as a block-iterative method in order to incorporate the instrumental point-spread function and, at the same time, to avoid the need to store the set of equations. The first reconstruction of a full layer and two neighboring 3D grains from experimental data are demonstrated.

Pinhole SPECT imaging: compact projection/backprojection operator for efficient algebraic reconstruction

We describe the efficient algebraic reconstruction (EAR) method, which applies to cone-beam tomographic reconstruction problems with a circular symmetry. Three independant steps/stages are presented, which use two symmetries and a factorization of the point spread functions (PSFs), each reducing computing times and eventually storage in memory or hard drive. In the case of pinhole single photon emission computed tomography (SPECT), we show how the EAR method can incorporate most of the physical and geometrical effects which change the PSF compared to the Dirac function assumed in analytical methods, thus showing improvements on reconstructed images. We also compare results obtained by the EAR method with a cubic grid implementation of an algebraic method and modeling of the PSF and we show that there is no significant loss of quality, despite the use of a noncubic grid for voxels in the EAR method. Data from a phantom, reconstructed with the EAR method, demonstrate 1.08-mm spatial tomographic resolution despite the use of a 1.5-mm pinhole SPECT device and several applications in rat and mouse imaging are shown. Finally, we discuss the conditions of application of the method when symmetries are broken, by considering the different parameters of the calibration and nonsymmetric physical effects such as attenuation.

Transaxial system models for jPET-D4 image reconstruction

A high-performance brain PET scanner, jPET-D4, which provides four-layer depth-of-interaction (DOI) information, is being developed to achieve not only high spatial resolution, but also high scanner sensitivity. One technical issue to be dealt with is the data dimensions which increase in proportion to the square of the number of DOI layers. It is, therefore, difficult to apply algebraic or statistical image reconstruction methods directly to DOI-PET, though they improve image quality through accurate system modelling. The process that requires the most computational time and storage space is the calculation of the huge number of system matrix elements. The DOI compression (DOIC) method, which we have previously proposed, reduces data dimensions by a factor of 1/5. In this paper, we propose a transaxial imaging system model optimized for jPET-D4 with the DOIC method. The proposed model assumes that detector response functions (DRFs) are uniform along line-of-responses (LORs). Then each element of the system matrix is calculated as the summed intersection lengths between a pixel and sub-LORs weighted by a value from the DRF look-up-table. 2D numerical simulation results showed that the proposed model cut the calculation time by a factor of several hundred while keeping image quality, compared with the accurate system model. A 3D image reconstruction with the on-the-fly calculation of the system matrix is within the practical limitations by incorporating the proposed model and the DOIC method with one-pass accelerated iterative methods.

Non-destructive mapping of grains in three dimensions

A 3D map of an interior grain and its crystallographic orientation in a single-phase bulk sample has been obtained non-destructively, for the first time, by 3DXRD microscopy using an algebraic reconstruction method. The method is new and has potentials for in situ 3D mapping of structural changes during for example annealing.

An adapted fan volume sampling scheme for 3-D algebraic reconstruction in linear tomosynthesis

We study the reconstruction process when the X-ray source translates along a finite straight line, the detector moving or not. This process, called linear tomosynthesis, induces a limited angle of view, which causes the vertical spatial resolution to be poor. To improve this resolution, we use iterative algebraic reconstruction methods, which are commonly used for tomographic reconstruction from a reduced number of projections. With noisy projections, such algorithms produce poor quality reconstructions. To prevent this, we use a first object prior knowledge, consisting of piecewise smoothness constraint. To reduce the computation time associated with both reconstruction and regularization processes, we introduce a second geometrical prior knowledge, based on the linear trajectory of the X-ray source. This linear source trajectory allows us to reconstruct a series of two-dimensional (2-D) planes in a fan organization of the volume. Using this adapted fan volume sampling scheme, we reduce the computation time by transforming the initial three-dimensional (3-D) problem into a series of 2-D problems. Obviously, the algorithm becomes directly parallelizable. Focusing on a particular region of interest becomes easier too. The regularization process can easily be implemented with this scheme. We test the algorithm using experimental projections. The quality of the reconstructed object is conserved, while the computation time is considerably reduced, even without any parallelization of the algorithm.

X-ray microtome by fluorescence tomography

The X-ray fluorescence microtomography method is presented, which is capable of virtually slicing samples to obtain cross-sections of their inner structure. High precision experimental results of fluo-tomography in “pencil-beam” geometry with up to 1.2 μm resolution are described. Image reconstructions are based on either a simplified algebraic reconstruction method (ART) or the filtered back-projection method (FBP). Phantoms of inhomogenous test objects as well as biological samples are successfully analyzed.

Algebraic 2D PET image reconstruction using depth-of-interaction information

Recently a high-performance PET scanner, which measures depth-of-interaction (DOI) information, is being developed for molecular imaging. DOI measurement of multi-layered thin crystals can improve spatial resolution and scanner sensitivity simultaneously. In this paper, we apply an algebraic image reconstruction method to 2-dimensional (2D) DOI-PET scanners using accurate system modeling, in order to evaluate the effects of using DOI information on PET image quality. Algebraic image reconstruction methods have been successfully used to improve PET image quality, compared with the conventional filtered backprojection method. The proposed method is applied to simulated data for a small 2D DOI-PET scanner. The results show that accurate system modeling improves spatial resolution without noise emphasis, and that DOI information improves uniformity of spatial resolution.