Algebraic Reconstruction Method Research Articles

High-resolution X-ray fluorescence microtomography of homogeneous samples

First experimental results of fluorescence microtomography with 6 /spl mu/m resolution obtained at ESRF, Grenoble, France, are described. The setup comprises high-quality optics (monochromator, mirror, focusing lenses) coupled to the high-energy/brilliance/coherence of the ID 22 undulator beamline. The tomographic setup allows precise measurements in the "pencil-beam" geometry. The image reconstruction is based on a modification of the algebraic reconstruction method but includes simplifications of the model. The quality and precision of the two-dimensional reconstructed elemental images of a phantom sample are encouraging. The method will be further refined and applied for the analysis of more complex inhomogeneous samples.

Algebraic PET image reconstruction with pre-computed reconstruction operators using subsets of sensitivity functions

Algebraic reconstruction methods have been successfully used to improve quality of positron emission tomography (PET) images by accurate modeling of measurement system, but they have computational burden. The authors propose a fast PET image reconstruction method based on an algebraic technique. In this method, reconstruction operators are pre-computed approximately using subsets of sensitivity functions. The subsets contain the sensitivity functions that contribute significantly to each point to be reconstructed. The proposed method was applied to simulated data and experimental data for the ECAT EXACT HR+ (Siemens/CTI) scanner operating in 2D mode. These results show that the proposed method produces images with almost the same quality as the conventional algebraic methods do and has a similar computation time to the filtered backprojection method.

An Acceleration Algorithm for Image Reconstruction Based on Continuous-Discrete Mapping Model

In image reconstruction for X-ray computed tomography, images reconstructed by filtered backprojection (FBP) include systematic errors because the FBP method does not take into account some of the properties of the actual imaging system such as the divergence of X-ray beam. One solution to this problem is to use algebraic reconstruction methods, such as generalized analytic reconstruction from discrete samples and natural pixel decomposition. However, in the process of reconstruction using these methods, it is necessary to solve the linear algebraic equations which have a large coefficient matrix. In this paper, we propose a method to accelerate the iteration solving these equations by preconditioning the coefficient matrix using a polynomial function. The results of the computer simulations show the effectiveness of the proposed method.

Rapid 3-D cone-beam reconstruction with the simultaneous algebraic reconstruction technique (SART) using 2-D texture mapping hardware.

Algebraic reconstruction methods, such as the algebraic reconstruction technique (ART) and the related simultaneous ART (SART). reconstruct a two-dimensional (2-D) or three-dimensional (3-D) object from its X-ray projections. The algebraic methods have, in certain scenarios, many advantages over the more popular Filtered Backprojection approaches and have also recently been shown to perform well for 3-D cone-beam reconstruction. However, so far the slow speed of these iterative methods have prohibited their routine use in clinical applications. In this paper, we address this shortcoming and investigate the utility of widely available 2-D texture mapping graphics hardware for the purpose of accelerating the 3-D algebraic reconstruction. We find that this hardware allows 3-D cone-beam reconstructions to be obtained at almost interactive speeds, with speed-ups of over 50 with respect to implementations that only use general-purpose CPUs. However, we also find that the reconstruction quality is rather sensitive to the resolution of the framebuffer, and to address this critical issue we propose a scheme that extends the precision of a given framebuffer by 4 bits, using the color channels. With this extension, a 12-bit framebuffer delivers useful reconstructions for 0.5% tissue contrast, while an 8-bit framebuffer requires 4%. Since graphics hardware generates an entire image for each volume projection, it is most appropriately used with an algebraic reconstruction method that performs volume correction at that granularity as well, such as SART or SIRT. We chose SART for its faster convergence properties.

Optical tomography of the aurora and EISCAT

Abstract. Tomographic reconstruction of the three-dimensional auroral arc emission is used to obtain vertical and horizontal distributions of the optical auroral emission. Under the given experimental conditions with a very limited angular range and a small number of observers, algebraic reconstruction methods generally yield better results than transform techniques. Different algebraic reconstruction methods are tested with an auroral arc model and the best results are obtained with an iterative least-square method adapted from emission-computed tomography. The observation geometry used during a campaign in Norway in 1995 is tested with the arc model and root-mean-square errors, to be expected under the given geometrical conditions, are calculated. Although optimum geometry was not used, root-mean-square errors of less than 2% for the images and of the order of 30% for the distribution could be obtained. The method is applied to images from real observations. The correspondence of original pictures and projections of the reconstructed volume is discussed, and emission profiles along magnetic field lines through the three-dimensionally reconstructed arc are calibrated into electron density profiles with additional EISCAT measurements. Including a background profile and the temporal changes of the electron density due to recombination, good agreement can be obtained between measured profiles and the time-sequence of calculated profiles. These profiles are used to estimate the conductivity distribution in the vicinity of the EISCAT site. While the radar can only probe the ionosphere along the radar beam, the three-dimensional tomography enables conductivity estimates in a large area around the radar site.Key words. Tomography · Aurora · EISCAT · Ionosphere · Conductivity

Algebraic reconstruction for magnetic resonance imaging under B0 inhomogeneity.

In magnetic resonance imaging, spatial localization is usually achieved using Fourier encoding which is realized by applying a magnetic field gradient along the dimension of interest to create a linear correspondence between the resonance frequency and spatial location following the Larmor equation. In the presence of B0 inhomogeneities along this dimension, the linear mapping does not hold and spatial distortions arise in the acquired images. In this paper, the problem of image reconstruction under an inhomogeneous field is formulated as an inverse problem of a linear Fredholm equation of the first kind. The operators in these problems are estimated using field mapping and the k-space trajectory of the imaging sequence. Since such inverse problems are known to be ill-posed in general, robust solvers, singular value decomposition and conjugate gradient method, are employed to obtain corrected images that are optimal in the Frobenius norm sense. Based on this formulation, the choice of the imaging sequence for well-conditioned matrix operators is discussed, and it is shown that nonlinear k-space trajectories provide better results. The reconstruction technique is applied to sequences where the distortion is more severe along one of the image dimensions and the two-dimensional reconstruction problem becomes equivalent to a set of independent one-dimensional problems. Experimental results demonstrate the performance and stability of the algebraic reconstruction methods.

Position Alignment in Algebraic Reconstruction Method by Using Center of Gravity

The algebraic method was applied for transmission electron microscope (TEM) images taken from four directions. The reconstruction accuracy of this method was then evaluated for a 3-dimensional specimen structure. The reconstruction error was simulated by using various images which contain projection error, dark current noise, and position error. A 10% projection error and 20-dB dark current noise in the projection image resulted in reconstruction errors of 15% and 13%, respectively. On the other hand, the position error influence was so extreme that 58% reconstruction error was induced by displacement of only ±4-pixels between the projection images. It was found that a position alignment of images within ±1 pixel was necessary in order to achieve proper reconstruction. A new position alignment method using the center of gravity of a projection image is applied for TEM images of uranyl acetate. Using this alignment method, a 3-dimensional structure was successfully reconstructed from four images.

Texture analysis of differently reconstructed PET images

The insufficient number of projections and the statistics of radioactive decay are supposed to lead to different textures in the images reconstructed by the diverse algorithms used in positron emission tomography (PET). In our study, four reconstruction methods were examined, including the filtered backprojection and three algebraic approaches. Grey-level morphological operators were used to extract the texture. To quantify the observations, we used the texture parameters directionality, fractal dimension, and lacunarity. By increasing the number of iteration steps, each of the algebraic reconstruction methods evolve the texture from coarse to fine granularities, but, surprisingly, during the iteration process all methods follow a quite similar `trail' of possible textures within the active zone. The texture of the termination points of the movement have been compared with the result that the `hills' and `valleys' of the activity distribution are located in the same positions. The texture derived from the filtered backprojection is included in the set of textures produced by the algebraic methods. Artefacts are restricted to zones without activity. We found that the texture is independent of the number of counts when more than ten counts are registered in each pixel within the examined region of interest. A high reliability in the reconstructed images is given. All methods allow texture based tissue classification.

Mathematics for computer tomography

Computerized tomography requires not only fast computers, but also analysis of mathematical models and construction of numerical algorithms. Classical mathematical theory is combined with modern numerical analysis to form the basis for efficient implementation on fast computers.The solution of the inverse problem of finding the image from given x-ray projections is theoretically obtained by the inverse Radon transform. Since only a finite number of projections are available, some approximation must be found, and this leads to a discrete counterpart of the continuous problem. There are three major classes of numerical solution methods: the Algebraic Reconstruction Method, the Filtered Back projection Method and the Direct Fourier Method. Much research is devoted to making the methods faster and more robust. The first one was used for the original tomography machine, the second one is used on almost all current machines in use. The third one has great potential for the future, since almost all computation is done by using the fast discrete Fourier transform.We shall describe the basic mathematical problem in computer tomography and the computational methods mentioned above for solving it. In particular we shall emphasize the special difficulties that are built into the problem. However, this is not a review article. Instead, it is intended to describe the influence of modern numerical methods on a fundamental problem of great significance for the society. We shall also indicate how computerized tomography has initiated new important research in central fields of numerical analysis, that can be used for problems in many other applications.

Evaluation of Optical Source-Detector Configurations for Tomographic Reconstruction of Chemical Concentrations in Indoor Air

This article reports on numerical studies to evaluate and compare optical remote sensing configurations for tomographically reconstructing pollutant concentrations in indoor air. With a remote sensing/computed tomography system, two-dimensional maps of pollutant concentrations with good spatial resolution can be created for an entire room. The successful use of such a system for exposure assessment, ventilation assessment, or source monitoring depends on the remote sensing configuration. A systematic method was developed to evaluate the performance of 10 configurations. One hundred and twenty test maps were reconstructed with an algebraic reconstruction method using all 10 configurations; reconstruction quality was evaluated using 4 criteria. Reconstruction quality was related to the number and location of detectors in the room and the complexity of the test maps. Configurations using the same number of detectors placed in different locations resulted in reconstructions that differed in quality. The effect of reducing the number density of rays on reconstruction quality was studied. Based on these simulations, two configurations that used four detectors to scan the room were selected, and their performance was evaluated in the presence of various levels of measurement noise. Two configurations that used four detectors were most suited for exposure assessment. It was found that when designing a configuration, the number and independence of rays should be maximized. Results underscored the need to thoroughly test configurations through numerical studies prior to field implementation; a wide variety of concentration maps, relevant to the application, should be tested under both ideal and nonideal sampling conditions.

Using Layer Replacement to Improve Velocity Analysis Beneath Near-Surface Reefs

Exploration in the Timor Sea is hampered by the presence of many near-surface limestone reefs. Reef structures form complex velocity anomalies because of their internal velocity structure and because they produce a rapidly varying sea-floor profile. Distortion of seismic data recorded over such reefs comprises event pull-up and poor imaging. Conventional common mid-point processing is not adequate to interpret prospective targets below the reefs. Seismic distortion associated with reefs can be reduced using replacement techniques of varying complexity. Wave-equation datuming is an accurate method but computationally expensive, and replacement statics is an efficient method but only approximate. All replacement methods require an estimate of the velocity structure inside the reef itself, although this is not normally attempted in production processing. The seismic image beneath surface reefs can be improved by integrating layer replacement with velocity analysis. The seismic data are first modified by replacing the water velocity with an average velocity of the reef. This involves two steps: downward continuing to the sea floor, and then upward continuing to the surface. A sufficiently accurate value for an average velocity can usually be obtained using conventional velocity analysis. This approach removes most of the velocity inhomogeneity associated with the water/reef interface and renders subsequent velocity analysis more accurate. A model of a typical reef from Browse Basin, which includes strong lateral velocity inhomogeneity on the reef flanks, shows that velocity analysis post-replacement gives much more accurate RMS velocities inside and immediately below the reef. Events below the reef are imaged more faithfully. Conventional velocity analysis is not accurate enough to give interval velocities for the reef because the CMP method cannot image such strong lateral velocity changes. The same model can be analyzed using reflection tomography based on the algebraic reconstruction method (ART). ART is sufficiently accurate to give useful estimates of the interval velocities within the model reef. Comparing results before and after replacement indicates that replacing the water layer prior to ray tracing gives closer estimates of the high-velocity reef flanks and reduces velocity smearing across the sea floor to water interface. Estimates of the interval velocities inside the reef are important for accurate imaging. These can be used to continue downward through the reef structure for a second pass of wave-equation replacement, or for applying a variety of pre-stack depth migration methods.

RIT — A new robust iterative technique for image reconstruction in emission tomography

Emission tomography is a reliable tool for testing nuclear fuel elements. The conventional algebraic reconstruction methods of computer tomography are non-robust and can be fatally affected by outliers in the input data, i.e. by data having unexpectedly high errors due to some unpredictable effects. In the present paper, a robust iterative technique (RIT) for emission tomography is described. RIT is based on robust M-estimation methods and on a new algorithm for computing the M-estimates. RIT needs no filtering of the input data. It's computational expenses do not exceed those of SIRT method. RIT is included in the TOMODAT program and is being used for testing the fuel elements after irradiation in the reactors. The fuel distributions reconstructed by RIT are practically not affected by outliers in the input data, while the ART, SIRT and MENT give quite unstable results for the same spoiled data.

Reconstruction of an incomplete ultrasonic image: Maximum entropy approach

The convolution backprojection and the algebraic reconstruction methods are two of the most popular techniques used for the reconstruction of a soft tissue attenuation image by ultrasound. However, where data have been truncated due to physical limitations, these techniques do not produce satisfactory results. One-dimensional linear prediction estimators by which the missing data can be estimated based on known projection information have been developed. In this paper, the missing data are extrapolated using a two-dimensional maximum entropy method, and the backprojection algorithms are applied to reconstruct the completed projection data set. There is a significant improvement in the quality and reliability of the reconstructed image as compared to one-dimensional linear prediction approach. There is also a relative improvement in the computation time as compared to the conventional methods. The techniques are developed based on a soft tissue phantom. Computer simulation results and comparison with conventional methods are presented.

Tomography transform and inverse in geometrical optics

Geometrical optics is applied in a modeling of the forward-propagation process for transmission tomography with diffracting sources. The forward process is described mathematically as an optical tomography transform of an object function into a set of measured projection data representing line integrals of that parameter function. A recursive scheme is devised for the inverse process of reconstruction of the object parameter function from its forward projections. Ultimately, a new computational scheme is presented for the fast and accurate reconstruction of the object function. Ray-tracing methods lead to a representation of the forward-propagation process as a system of linear equations. Simultaneous algebraic reconstruction methods are applied in the discretized inverse process.

Reconstruction from NMR Data Acquired with Imaging Gradients Having Arbitrary Time Dependence

An algebraic reconstruction method for NMR imaging from data acquired with imaging gradients having essentially arbitrary time dependence is presented. With reasonable discretization, an overdetermined linear system is obtained for which there exist simple factorized 1-inverses of the coefficient matrices; a computationally efficient reconstruction scheme then follows. Some numerical simulations are also described.

Reconstruction from Limited Angular Projection Data Using Constrained Optimization

In this paper, a new type of pixel decomposition is introduced to suppress modeling noise and then an algebraic method is developed for reconstruction from limited angular projection data. The basic idea with this method is to directly control the tradeoff between resolution and noise, taking into account the uncertainty of the discrepancy criterion generally used in an algebraic method for reconstruction. The results of a simulation using a simple mathematical phantom are also presented, demonstrating the effectiveness of this method.

Two-Dimensional Void Reconstruction by Neutron Transmission

Contemporary algebraic reconstruction methods are utilized in investigating the two-dimensional void distribution in a water analog from neutron transmission measurements. It is sought to ultimately apply these techniques to the determination of time-averaged void distribution in two-phase flow systems as well as for potential usage in neutron radiography. Initially, projection data were obtained from a digitized model of a hypothetical two-phase representation and later from neutron beam traverses across a voided methacrylate plastic model. From 10 to 15 views were incorporated, and decoupling of overlapped measurements was utilized to afford greater resolution. In general, the additive Algebraic Reconstruction Technique yielded the best reconstructions, with others showing promise for noisy data. Results indicate the need for some further development of the method in interpreting real data.