The creation of fast parallel iterative statistical algorithms based on the use of graphics accelerators is an important and urgent task of great scientific and practical importance. An algorithm based on the method of maximizing the mathematical expectation of maximum likelihood (maximum likelihood expectation MLEM) is considered. MLEM is a numerical method for determining maximum likelihood estimates and, since its first application in the field of image reconstruction in 1982, remains one of the most popular statistical methods of image reconstruction, being the foundation for many other approaches. A new version of the MLEM parallel algorithm is proposed, which provides global convergence of the iterative algorithm. To parallelize the algorithm, the texture mapping method is used using the OpenGL graphics library. The parallel algorithm is described in as much detail as possible. Examples of several reconstructions of images of aluminum casting products are given The obtained result can be used for non-destructive testing of various industrial products, including testing of foundry products.

Methods of restoring images and properties of non-destructive testing objects based on solving inverse problems (problems of restoring distribution functions of unknown characteristics of an object based on the results of indirect measurements) are considered. Management methods are based on solving inverse problems and allow you to get the most complete information about the distributed properties of an object. The need to attract additional information imposes serious restrictions on the development of universal applied algorithms for solving incorrectly set tasks. As a rule, individual additional information is available for each specific non-destructive testing task. An effective numerical algorithm for solving an incorrectly posed problem should be focused on taking this information into account at each stage of the solution search. When solving an applied problem, it is also necessary that the algorithm corresponds to both the measuring capabilities and the capabilities of available computing tools. The problem of low-projection X-ray tomography is always associated with a lack of initial data and can only be solved using a priori information. To introduce the necessary additional information into the numerical algorithm, the methods of iterative reconstruction of tomographic images are identified as the most suitable. One of the approaches to the presentation of this kind of information is described. A practical solution to this problem will expand the scope of the X-ray tomography method.

The traditional tomography is an effective remedy for medical diagnostics, nondestructive control of industrial designs and for quality check of industrial products. Tomographic visualization of objects in case of an incomplete viewing angle, limited number of projections and/or the insufficient power of a source of x-ray radiation is strongly incorrect return task. Article is devoted to case research when noise in input data significantly affect convergence and quality of the reconstructed image.

Now one of the most important tasks is development and adaptation of iterative methods for the solution of the superbig rarefied systems of the algebraic equations. The problem of iterative parallel reconstruction of three-dimensional images of industrial facilities leads to such computing tasks. The fact that iterative methods of the solution of computing problems of big dimension are implemented on parallel structures much more effectively, than direct methods of their decision is important.

Few-view X-ray tomography is a typical representative of the class of ill posed problems. A practical solution to this problem will expand the scope of the X-ray tomography method. The problem of few-view X-ray tomography is often a problem with a shortage of initial data and can be solved only with the involvement of a priori information. The methods of iterative reconstruction of tomographic images are most suitable for introducing the necessary additional information into a numerical algorithm. This paper describes one of the approaches to introducing this kind of information.

The pipe wall thickness was estimated based on three-dimensional images of the pipe recovered from several X-ray projections, which were made in a limited angle of view. Since the effects of scattered radiation and beam hardening are up to 50 % of the main radiation, ignoring them leads to blur of the image and inaccuracy in determining dimensions. To restore pipe images from projections, a volume and/or shell representation of the pipe is used, as well as iterative Bayesian methods. Using these methods, the error in estimating the pipe wall thickness from the projection data can be equal to or less than 300 μm. It has been shown that standard X-ray projections on the film or imaging plates used to obtain data can be used to restore pipe wall thickness profiles in factory conditions.

The possibility of an accurate estimation of the pipe wall thickness measured directly from the reconstructed image of the pipe, reconstructed from only a few X-ray projections made in a limited viewing angle, is discussed. Since the effects of radiation scattering and X-ray beam hardening distort up to 50 % of the primary radiation, ignoring these effects leads to blurred images, strong artifacts, and inaccurate sizing. A computerized technique has been developed that takes into account the contribution of scattered radiation and the hardening of the X-ray beam. Iterative Bayesian reconstruction techniques are then used to reconstruct the pipe image using the volumetric and surface-oriented representation of the pipe. Using these methods, the error in estimating the pipe wall thickness can be increased to 300 microns.

OBJECTIVE:To compare image quality and breast density of two reconstruction methods, the widely-used filtered-back projection (FBP) reconstruction and the iterative heuristic Bayesian inference reconstruction (Bayesian inference reconstruction plus the method of total variation applied, HBI). METHODS:Thirty-two clinical DBT data sets with malignant and benign findings, n = 27 and 17, respectively, were reconstructed using FBP and HBI. Three experienced radiologists evaluated the images independently using a 5-point visual grading scale and classified breast density according to the American College of Radiology Breast Imaging-Reporting And Data System Atlas, fifth edition. Image quality metrics included lesion conspicuity, clarity of lesion borders and spicules, noise level, artifacts surrounding the lesion, visibility of parenchyma and breast density. RESULTS:For masses, the image quality of HBI reconstructions was superior to that of FBP in terms of conspicuity,clarity of lesion borders and spicules (p < 0.01). HBI and FBP were not significantly different in calcification conspicuity. Overall, HBI reduced noise and supressed artifacts surrounding the lesions better (p < 0.01). The visibility of fibroglandular parenchyma increased using the HBI method (p < 0.01). On average, five cases per radiologist were downgraded from BI-RADS breast density category C/D to A/B. CONCLUSION:HBI significantly improves lesion visibility compared to FBP. HBI-visibility of breast parenchyma increased, leading to a lower breast density rating. Applying the HBIR algorithm should improve the diagnostic performance of DBT and decrease the need for additional imaging in patients with dense breasts. ADVANCES IN KNOWLEDGE:Iterative heuristic Bayesian inference (HBI) image reconstruction substantially improves the image quality of breast tomosynthesis leading to a better visibility of breast carcinomas and reduction of the perceived breast density compared to the widely-used filtered-back projection (FPB) reconstruction. Applying HBI should improve the accuracy of breast tomosynthesis and reduce the number of unnecessary breast biopsies. It may also reduce the radiation dose for the patients, which is especially important in the screening context.

Одни из наиболее важных вычислительных задач разработка и адаптация итерационных методов для решения сверхбольших разреженных систем алгебраических уравнений. К таким вычислительным задачам приводит задача итерационной параллельной реконструкции трехмерных изображений промышленных изделий. Важно, что итерационные методы решения вычислительных задач большой размерности реализуются на параллельных структурах намного эффективнее, чем прямые методы их решения. В этой работе описан синхронный параллельный алгоритм, основанный на использовании системы MPI для решения задачи реконструкции трехмерных изображений промышленных изделий. Purpose. Currently, one of the most important tasks is the development and adaptation of iterative methods for solving ultra-large sparse systems of algebraic equations. Such computational problems are caused by the iterative parallel reconstruction of three-dimensional images of industrial products. It is important that iterative methods for solving computational problems of large size are implemented on parallel structures much more efficiently than direct methods for solving them. This paper describes a synchronous parallel algorithm based on the MPI system for solving the problem of reconstruction of three-dimensional images of industrial products. Methodology. It is important that iterative methods for solving computational problems of larger dimensionality on parallel structures are implemented more efficiently than the direct ones. Most algorithms based on direct solving methods have a significant hereditary sequential structure and require a large number of processors interactions which cannot be executed in parallel mode. Iterative methods, for the most part, require a significantly smaller number of interactions of this type and are relatively easily mapped onto parallel computational structures. Equally important, in most cases parallel implementations of classical iterative methods are more effective in terms of computational speed. The parallel execution of the algorithm is based on the distribution of the process in some way between various groups of processors. Depending on the interaction method between local processors, two different types of parallel iterative algorithms execution are distinguished: synchronous and asynchronous. In the former case, it is assumed that the processors complete the calculations and exchange all the necessary results before the start of a new iteration. The main disadvantage of synchronous parallel algorithms is that they require synchronization of iterations. This is a very difficult task, especially with large number of processors. In addition, the overall calculation speed is limited by the speed of the slowest processor. At the same time, faster processors spend most of their time in the waiting mode. But, nevertheless, the implementation of these parallel algorithms can be effectively achieved using the MPI standard. Findings. Synchronous parallel computing algorithms and program codes for threedimensional tomographic reconstruction in a conical beam were developed. Program code debugging and numerical calculations were performed on a hybrid cluster based on the OpenPOWER architecture using the MPI system. For designing a parallel threedimensional tomographic reconstruction, a voxel form of parallelism was used. Originality. Implemented parallel iterative technology of three-dimensional images reconstruction has undeniable advantages over traditional sequential iterative tomographic reconstruction. It allows reducing the time of tomographic reconstruction as many as tens of times, provides the ability to reconstruct products with sizes of 5123 to 10243 voxels with simultaneous storage of the submatrix node of the projection matrix in RAM, which eliminates the need for recalculation of matrix coefficients for each new iteration.

Now one of the most important tasks is development and adaptation of iterative methods for the solution of the big rarefied systems of the algebraic equations. The problem of iterative parallel reconstruction of three-dimensional images of industrial facilities leads to such computing tasks. The fact that iterative methods of the solution of computing problems of big dimension are implemented on parallel structures much more effectively is important than direct methods of their decision. In this work the synchronous parallel algorithm is described, based on use of the MPI system for the solution of a problem of reconstruction of three-dimensional images of industrial facilities.

The technology of three-dimensional Bayesian tomographic reconstruction of homogeneous objects with high-density inclusions is developed. The approach is based on preliminary correction of projections by extracting the data corresponding to X-rays passing through a high-density region, and replacing it with synthesized data obtained by two-dimensional interpolation. An original method for selecting interpolation points is proposed and a mathematical algorithm is described that ensures the implementation of two-dimensional interpolation correction of projections.

The statistical maximum likelihood (EM) method and the algebraic reconstruction method with simultaneous iterations (SART) are two methods of iterative tomographic reconstruction. These algorithms are often used when the projection data contains a large amount of statistical noise or has been obtained from a limited range of angles. One of the popular approaches used to increase the rate of convergence of these algorithms is to perform a correction of the current approximation of the reconstructed object on subsets of the projection data. The desire to increase the convergence rate of the iterative methods led to the use of ordered subsets of projections for both the maximum likelihood method of EM (OS-EM) and for the algebraic reconstruction method with simultaneous iterations of SART (OS-SART). The efficiency of using ordered subsets of projections was first established for sequential programs that run on the central processor of the computer (CPU). In this work, both these methods have been accelerated by using the OpenGL graphics library by mirroring them on the graphics processor architecture of the video card.

Computed tomography is still being intensively studied and widely used to solve a number of industrial and medical applications. The simultaneous algebraic reconstruction technique (SART) and Bayesian inference reconstruction (BIR) are considered as advantageous iteration methods that are most suitable for improving the quality of the reconstructed 3D-images. The paper deals with the parallel iterative algorithms to ensure the reconstruction of threedimensional images of the breast, recovered from a limited set of noisy X-ray projections. Algebraic method of reconstruction with simultaneous iterations – SART and iterative method for statistical reconstruction of BIR are deemed to be the most preferred iterative methods. We believe that these methods are particularly useful for improving the quality of breast reconstructed image. We use the graphics processor (GPU) to accelerate the process of reconstruction. Preliminary results show that all investigated methods are useful in breast reconstruction layered images. However, it was found that the method of classical tomosynthesis SAA is less efficient than iterative methods SART and BIR as the worst suppress the anatomical noise. Despite the fact that the estimated ratio of the contrast / noise ratio in the presence of internal structures with low contrast is higher for classical tomosynthesis method the SAA, its effectiveness in the presence of highly structured background is low. In our opinion the best results can be achieved using statistical iterative reconstruction BIR.

Creating a fast parallel iterative tomographic algorithms based on the use of graphics accelerators, which simultaneously provide the minimization of residual and total variation of the reconstructed image is an important and urgent task, which is of great scientific and practical importance. Such algorithms can be used, for example, in the implementation of radiation therapy patients, because it is always done pre-computed tomography of patients in order to better identify areas which can then be subjected to radiation exposure.

The key problem in increasing efficiency of radiotherapy of malignant tumors in brain and other dangerous neoplasms is the problem of increasing the quality of 3D positioning of a patient before radiotherapy. We consider the principles of development of fast parallel iterative algorithms based on graphic accelerators and the OpenGL library. The proposed approaches provide simultaneous residual minimization for the sought solution and total variation of the reconstructed 3D image. In this case, the number of required initial data, i.e., conic X-ray projections, can be reduced several times, and therefore, the radiation load on the patient can also be accordingly reduced with preservation of the necessary contrast and spatial resolution of the 3D image of the patient. The new heuristic iterative algorithm can be used as an alternative to the known 3D Feldkamp algorithm.

Computed tomography is still being intensively studied and widely used to solve a number of industrial and medical applications. The algebraic reconstruction method with simultaneous iterations SART considered in this work as one of the most promising of iterative methods, suitable for the tomographic problems. Graphics processor is used to accelerate the speed of the reconstruction. The method of minimizing the total variation (TV) is used as a priori support for the regularization of the iterative process and to overcome the incompleteness of the information.

The further development of the new iterative reconstruction algorithms to improve three-dimensional breast images quality restored from incomplete and noisy mammograms, is provided. The algebraic reconstruction method with simultaneous iterations - Simultaneous Algebraic Reconstruction Technique (SART) and the iterative method of statistical reconstruction Bayesian Iterative Reconstruction (BIR) are referred here as the preferable iterative methods suitable to improve the image quality. For better processing we use the Graphics Processing Unit (GPU). Method of minimizing the Total Variation (TV) is used as a priori support for regularization of iteration process and to reduce the level of noise in the reconstructed image. Preliminary results with physical phantoms show that all examined methods are capable to reconstruct structures layer-by-layer and to separate layers which images are overlapped in the Z-direction. It was found that the method of traditional Shift-And-Add tomosynthesis (SAA) is worse than iterative methods SART and BIR in terms of suppression of the anatomical noise and image blurring in between the adjacent layers. Despite of the fact that the measured contrast/noise ratio in the presence of low contrast internal structures is higher for the method of tomosynthesis SAA than for SART and BIR methods, its effectiveness in the presence of structured background is rather poor. In our opinion the optimal results can be achieved using Bayesian iterative reconstruction BIR.

New parallel iteration algorithms that provide real-time reconstruction of the 3D breast images restored from an incomplete set of noisy mammograms are studied. The simultaneous algebraic reconstruction technique (SART) and Bayesian inference reconstruction (BIR) are considered as advantageous iteration methods that are most suitable for improving the quality of the reconstructed 3D images. The graphics processing unit (GPU) is used to accelerate the reconstruction. The minimization of total variation (TV) is used as a priori support for the regularization of the iteration process and decrease of the noise level in the reconstructed images. Preliminary results for medical physical phantoms show that all the methods are sufficient for the layer-by-layer reconstruction of medical model objects and separation of layers whose images are overlapped on a mammogram that corresponds to vertical transmission (direction along the OZ axis). The traditional shift-and-add (SAA) tomosynthesis is established to be less efficient than SART and BIR in terms of the anatomical-noise reduction and blurring of reconstructed 3D images between conjugate layers. Despite the fact that the estimated contrast-noise ratio, given internal structures with low contrast, is higher for SAA as compared to SART and BIR, its efficiency is very low given the highly structured background. In our opinion, optimal results can be achieved using BIR.

Conventional X-ray tomography is an effective tool of medical diagnostics, nondestructive testing of engineering structures, and technical diagnostics of manufactured products. Tomographic reconstruction of objects is a strongly incorrect inverse problem in the case of limited vision angle, small number of projections, and/or insufficient X-ray source energy. Hull and hull-voxel methods of tomographic reconstruction of piecewise-uniform objects are considered in this paper. These approaches are quite effective for image reconstruction of industrial and biological objects under conditions of sparse input data and for analyzing binary and piecewise-uniform images. It will be a priori assumed that the object being reconstructed consists of areas with constant attenuation coefficients.

Effective synchronous parallel computational algorithms are developed for 3D tomographic reconstruction in a conic beam using the MPI (Message Passing Interface) system.