Expectation maximization (EM) algorithms using polar symmetriesfor computed tomography(CT) image reconstruction

We suggest a symmetric-polar pixellation scheme which makes possible a reduction of the computational cost for expectation maximization (EM) iterative algorithms. The proposed symmetric-polar pixellation allows us to deal with 3D images as a whole problem without dividing the 3D problem into 2D slices approach. Performance evaluation of each approach in terms of stability and image quality is presented. Exhaustive comparisons between all approaches were conducted in a 2D based image reconstruction model. From these 2D approaches, that showing the best performances were finally implemented and evaluated in a 3D based image reconstruction model. Comparison to 3D images reconstructed with FBP is also presented. Although the algorithm is presented in the context of computed tomography (CT) image reconstruction, it can be applied to any other tomographic technique as well, due to the fact that the only requirement is a scanning geometry involving measurements of an object under different projection angles. Real data have been acquired with a small animal (CT) scanner to verify the proposed mathematical description of the CT system.This work was supported by the Spanish Plan Nacional de Investigacion Cientifica, Desarrollo e Innovacion Tecnologica (I+D+I) under Grant, FIS2010-21216-CO2-01, Valencian Local Government under Grant Nos. PROMETEO 2008/114 and APOSTD/2010/012. The authors would like to thank Brennan Holt for checking and correcting the text.Rodríguez Álvarez, MJ.; Soriano Asensi, A.; Iborra Carreres, A.; Sánchez Martínez, F.; González Martínez, AJ.; Conde, P.; Hernández Hernández, L.... (2013). Expectation maximization (EM) algorithms using polar symmetriesfor computed tomography(CT) image reconstruction. Computers in Biology and Medicine. 43(8):1053-1061. https://doi.org/10.1016/j.compbiomed.2013.04.015S1053106143

New pixellation scheme for CT algebraic reconstruction to exploit matrix symmetries

In this article we propose a new pixellation scheme which makes it possible to speed up the time of reconstruction. This proposal consists in splitting the field of view of the scanner into as many circular sectors as rotation positions of the detector. The sectors are pixellated using circular pixels whose size is always smaller than the resolution needed. The geometry of the pixels and the arrangement on circular sectors make it possible to compute the entire matrix from only one position of the scanner. Therefore, the size of the matrix decreases as many times as the number of rotations. This results in a significant reduction of the system matrix which allows algebraic methods to be applied within a reasonable time of reconstruction and speeds up the time of matrix generation. The new model is studied by means of analytical CT simulations which are reconstructed using the Maximum Likelihood Emission Maximization algorithm for transmission tomography and is compared to the cartesian pixellation model. Therefore, two different grids of pixels were developed for the same scanner geometry: one that employs circular pixels within a cartesian grid and another that divides the field of view into a polar grid which is composed by identical sectors, with circular pixels too. The results of both models are that polar matrix is built in a few seconds and the cartesian one needs several hours, the size of the matrix is significantly smaller than the circular one, and the time of reconstruction per iteration using the same iterative method is less in the polar pixel model than in the square pixel model. Several figures of merit have been computed in order to compare the original phantom with the reconstructed images. Finally, we can conclude that both reconstructions have been proved to have enough quality but, the polar pixel model is more efficient than the square pixel model. © 2008 Elsevier Ltd. All rights reserved.Mora Mora, MTC.; Rodríguez Álvarez, MJ.; Romero Bauset, JV. (2008). New pixellation scheme for CT algebraic reconstruction to exploit matrix symmetries. Computers and Mathematics with Applications. 56(3):717-726. doi:10.1016/j.camwa.2008.02.019S71772656

Itera- tive Reconstruction Framework for High-Resolution X-ray CT Data

Small animal medical imaging has become an important tool for researchers as it allows noninvasively screening animal models for pathologies as well as monitoring dis- ease progression and therapy response. Currently, clinical CT scanners typically use a Filtered Backprojection (FBP) based method for image reconstruction. This algorithm is fast and generally produces acceptable results, but has several drawbacks. Firstly, it is based upon line integrals, which do not accurately describe the process of X-ray attenuation. Secondly, noise in the projection data is not properly modeled with FBP. On the other hand, iterative algorithms allow the integration of more complicated sys- tem models as well as robust scatter and noise correction techniques. Unfortunately, the iterative algorithms also have much greater computational demands than their FBP counterparts. In this thesis, we develop a framework to support iterative reconstruc- tions of high-resolution X-ray CT data. This includes exploring various system models and algorithms as well as developing techniques to manage the significant computa- tional and system storage requirements of the iterative algorithms. Issues related to the development of this framework as well as preliminary results are presented

Efficient methodologies for system matrix modelling in iterative image reconstruction for rotating high-resolution PET

A fully 3D iterative image reconstruction algorithm has been developed for high-resolution PET cameras composed of pixelated scintillator crystal arrays and rotating planar detectors, based on the ordered subsets approach. The associated system matrix is precalculated with Monte Carlo methods that incorporate physical effects not included in analytical models, such as positron range effects and interaction of the incident gammas with the scintillator material. Custom Monte Carlo methodologies have been developed and optimized for modelling of system matrices for fast iterative image reconstruction adapted to specific scanner geometries, without redundant calculations. According to the methodology proposed here, only one-eighth of the voxels within two central transaxial slices need to be modelled in detail. The rest of the system matrix elements can be obtained with the aid of axial symmetries and redundancies, as well as in-plane symmetries within transaxial slices. Sparse matrix techniques for the non-zero system matrix elements are employed, allowing for fast execution of the image reconstruction process. This 3D image reconstruction scheme has been compared in terms of image quality to a 2D fast implementation of the OSEM algorithm combined with Fourier rebinning approaches. This work confirms the superiority of fully 3D OSEM in terms of spatial resolution, contrast recovery and noise reduction as compared to conventional 2D approaches based on rebinning schemes. At the same time it demonstrates that fully 3D methodologies can be efficiently applied to the image reconstruction problem for high-resolution rotational PET cameras by applying accurate pre-calculated system models and taking advantage of the system's symmetries

Incorporating accurate statistical modeling in PET: reconstruction for whole-body imaging

Tese de doutoramento em Biofísica, apresentada à Universidade de Lisboa através da Faculdade de Ciências, 2007The thesis is devoted to image reconstruction in 3D whole-body PET imaging. OSEM ( Ordered Subsets Expectation maximization ) is a statistical algorithm that assumes Poisson data. However, corrections for physical effects (attenuation, scattered and random coincidences) and detector efficiency remove the Poisson characteristics of these data. The Fourier Rebinning (FORE), that combines 3D imaging with fast 2D reconstructions, requires corrected data. Thus, if it will be used or whenever data are corrected prior to OSEM, the need to restore the Poisson-like characteristics is present. Restoring Poisson-like data, i.e., making the variance equal to the mean, was achieved through the use of weighted OSEM algorithms. One of them is the NECOSEM, relying on the NEC weighting transformation. The distinctive feature of this algorithm is the NEC multiplicative factor, defined as the ratio between the mean and the variance. With real clinical data this is critical, since there is only one value collected for each bin the data value itself. For simulated data, if we keep track of the values for these two statistical moments, the exact values for the NEC weights can be calculated. We have compared the performance of five different weighted algorithms (FORE+AWOSEM, FORE+NECOSEM, ANWOSEM3D, SPOSEM3D and NECOSEM3D) on the basis of tumor detectablity. The comparison was done for simulated and clinical data. In the former case an analytical simulator was used. This is the ideal situation, since all the weighting factors can be exactly determined. For comparing the performance of the algorithms, we used the Non-Prewhitening Matched Filter (NPWMF) numerical observer. With some knowledge obtained from the simulation study we proceeded to the reconstruction of clinical data. In that case, it was necessary to devise a strategy for estimating the NEC weighting factors. The comparison between reconstructed images was done by a physician largely familiar with whole-body PET imaging

QR-Factorization Algorithm for Computed Tomography (CT): Comparison With FDK and Conjugate Gradient (CG) Algorithms

[EN] Even though QR-factorization of the system matrix for tomographic devices has been already used for medical imaging, to date, no satisfactory solution has been found for solving large linear systems, such as those used in computed tomography (CT) (in the order of 106 equations). In CT, the Feldkamp, Davis, and Kress back projection algorithm (FDK) and iterative methods like conjugate gradient (CG) are the standard methods used for image reconstruction. As the image reconstruction problem can be modeled by a large linear system of equations, QR-factorization of the system matrix could be used to solve this system. Current advances in computer science enable the use of direct methods for solving such a large linear system. The QR-factorization is a numerically stable direct method for solving linear systems of equations, which is beginning to emerge as an alternative to traditional methods, bringing together the best from traditional methods. QR-factorization was chosen because the core of the algorithm, from the computational cost point of view, is precalculated and stored only once for a given CT system, and from then on, each image reconstruction only involves a backward substitution process and the product of a vector by a matrix. Image quality assessment was performed comparing contrast to noise ratio and noise power spectrum; performances regarding sharpness were evaluated by the reconstruction of small structures using data measured from a small animal 3-D CT. Comparisons of QR-factorization with FDK and CG methods show that QR-factorization is able to reconstruct more detailed images for a fixed voxel size.This work was supported by the Spanish Government under Grant TEC2016-79884-C2 and Grant RTC-2016-5186-1.Rodríguez-Álvarez, M.; Sánchez, F.; Soriano Asensi, A.; Moliner Martínez, L.; Sánchez Góez, S.; Benlloch Baviera, JM. (2018). QR-Factorization Algorithm for Computed Tomography (CT): Comparison With FDK and Conjugate Gradient (CG) Algorithms. IEEE Transactions on Radiation and Plasma Medical Sciences. 2(5):459-469. https://doi.org/10.1109/TRPMS.2018.2843803S4594692

Positron-Emission Tomography

We review positron-emission tomography (PET), which has inherent advantages that avoid the shortcomings of other nuclear medicine imaging methods. PET image reconstruction methods with origins in signal and image processing are discussed, including the potential problems of these methods. A summary of statistical image reconstruction methods, which can yield improved image quality, is also presented.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85853/1/Fessler95.pd

A fast sparse block circulant matrix vector product

In the context of computed tomography (CT), iterative image reconstruction techniques are gaining attention because high-quality images are becoming computationally feasible. They involve the solution of large systems of equations, whose cost is dominated by the sparse matrix vector product (SpMV). Our work considers the case of the sparse matrices being block circulant, which arises when taking advantage of the rotational symmetry in the tomographic system. Besides the straightforward storage saving, we exploit the circulant structure to rewrite the poor-performance SpMVs into a high-performance product between sparse and dense matrices. This paper describes the implementations developed for multi-core CPUs and GPUs, and presents experimental results with typical CT matrices. The presented approach is up to ten times faster than without exploiting the circulant structure.Romero Alcalde, E.; Tomás Domínguez, AE.; Soriano Asensi, A.; Blanquer Espert, I. (2014). A fast sparse block circulant matrix vector product. En Euro-Par 2014 Parallel Processing. Springer. 548-559. doi:10.1007/978-3-319-09873-9_46S548559Bian, J., Siewerdsen, J.H., Han, X., Sidky, E.Y., Prince, J.L., Pelizzari, C.A., Pal, X.: Evaluation of sparse-view reconstruction from flat-panel-detector cone-beam ct. Physics in Medicine and Biology 55, 6575–6599 (2010)Dalton, S., Bell, N.: CUSP: A C++ templated sparse matrix library version 0.4.0 (2014), http://cusplibrary.github.com/Feldkamp, L., Davis, L., Kress, J.: Practical cone-beam algorithm. Journal of the Optical Society of America 1, 612–619 (1984)Ganine, V., Legrand, M., Michalska, H., Pierre, C.: A sparse preconditioned iterative method for vibration analysis of geometrically mistuned bladed disks. Computers & Structures 87(5-6), 342–354 (2009)Hara, A.K., Paden, R.G., Silva, A.C., Kujak, J.L., Lawder, H.J., Pavlicek, W.: Iterative reconstruction technique for reducing body radiation dose at CT: Feasibility study. 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EM tomographic image reconstruction using polarvoxels

[EN] The splitting of the field of view (FOV) in polar voxels is proposed in this work in order to obtain an efficient description of a cone-beam computed tomography (CT) scanner. The proposed symmetric-polar pixelation makes it possible to deal with the 3D iterative reconstruction considering a number of projections and voxel sizes typical in CT preclinical imaging. The performance comparison, between the filtered backprojection (FBP) and 3D maximum likelihood expectation maximization (MLEM) reconstruction algorithm for CT, is presented. It is feasible to achieve the hardware spatial resolution limit with the considered pixelation. The image quality achieved with MLEM and FBP have been analyzed. The results obtained with both algorithms in clinical images have been compared too. Although the polar-symmetric pixelation is presented in the context of CT imaging, it can be applied to any other tomographic technique as long as the scan comprises the measurement of an object under several projection angles.This work was supported by the Spanish Plan Nacional de Investigaci´on Cient´ıfica, Desarrollo e Innovaci´on Tecnol´ogica (I+D+I) under Grant No. FIS2010-21216-CO2-01 and Valencian Local Government under Grants PROMETEO/2008/114, ISIC/2012/013 and APOSTD/2010/012.Soriano Asensi, A.; Rodríguez Álvarez, MJ.; Iborra Carreres, A.; Sánchez Martínez, F.; Carles Fariña, M.; Conde Castellanos, PE.; González Martínez, AJ.... (2013). EM tomographic image reconstruction using polarvoxels. Journal of Instrumentation. 8(12):1-7. https://doi.org/10.1088/1748-0221/8/01/C01004S1781

GPU-based fast iterative reconstruction of fully 3-D PET sinograms

This work presents a graphics processing unit (GPU)- based implementation of a fully 3-D PET iterative reconstruction code, FIRST (Fast Iterative Reconstruction Software for [PET] Tomography), which was developed by our group. We describe the main steps followed to convert the FIRST code (which can run on several CPUs using the message passing interface [MPI] protocol) into a code where the main time-consuming parts of the reconstruction process (forward and backward projection) are massively parallelized on a GPU. Our objective was to obtain significant acceleration of the reconstruction without compromising the image quality or the flexibility of the CPU implementation. Therefore, we implemented a GPU version using an abstraction layer for the GPU, namely, CUDA C. The code reconstructs images from sinogram data, and with the same System Response Matrix obtained from Monte Carlo simulations than the CPU version. The use of memory was optimized to ensure good performance in the GPU. The code was adapted for the VrPET small-animal PET scanner. The CUDA version is more than 70 times faster than the original code running in a single core of a high-end CPU, with no loss of accuracy.This work was supported in part by AMIT Project funded by CDTI (CENIT Programme), UCM (Grupos UCM, 910059), CPAN (Consolider-Ingenio 2010, CSPD-2007-00042), RECAVA- RETIC network, Comunidad de Madrid (ARTEMIS S2009/DPI-1802), Ministerio de Ciencia e Innovación, Spanish Government (ENTEPRASE grant, PSE-300000-2009-5 and TEC2007-64731/TCM), and European Regional funds.Publicad