Comprehensive Analysis of Alpha-Parametric Set for the Calculation of Intersection Lengths of Radiological Ray Path in Siddon's Algorithm Used in 3D Image Reconstruction

Year 2021, Volume: 7 Issue: 2, 172 - 181, 30.06.2021

Adem Polat

https://doi.org/10.28979/jarnas.841993

Cited By: 1

Abstract

The Siddon algorithm is one of the radiological ray path calculation tools used in 3D image reconstruction in medical imaging. In the algorithm, a set of alpha-parametric values is computed containing the length and index values where the voxel array of the x-ray intersects the x-y-z axes. In the alpha-set creation section of the Siddon algorithm, the set elements are sorted from small to large, but some elements have been noticed to have the same value in simulations. These elements are used to calculate which voxels are hit by the ray along the radiological path and at what ratio, but it was recognized that some values of the set were zero, which means some rays did not intersect some voxels at all. This situation may lead to data loss in 3D image reconstructions in medical imaging such as digital breast tomosynthesis (DBT) and computed tomography (CT) especially for huge dimensions such as size up to 800×800×50. Considering the mentioned problems, in this study, the effect of using or eliminating the same repetitive values in the alpha parametric set of the Siddon algorithm on calculations was investigated. To prove our proposal, we performed lossy and lossless 3D image reconstruction (100×100×50) of a synthetic phantom. Using special functions that do not take into account the duplicate values and exclude them in the algorithm solved the stated problems (lossless reconstruction). In this way, data loss that may occur in 3D image reconstruction was reduced since voxel indices and intersection lengths were matched correctly.

Keywords

Siddon algorithm, x-ray, 3D image reconstruction, digital breast tomosynthesis, computed tomography

References

• Andersen, A. H., & Kak, A. C. (1984). Simultaneous Algebraic Reconstruction Technique (SART): A Superior Implementation of the Art Algorithm. Ultrasonic Imaging 6(1), 81–94. DOI: https://doi.org/10.1016/0161-7346(84)90008-7
• Biguri, A., Dosanjh, M., Hancock, S., & Soleimani, M. (2017). A general method for motion compensation in x-ray computed tomography. Physics in Medicine & Biology, 62(16), 6532. Retrieved from: https://iopscience.iop.org/article/10.1088/1361-6560/aa7675/meta
• Bracewell, R. N., & Riddle, A. C. (1967). Inversion of Fan-Beam Scans in Radio Astronomy. The Astrophysical Journal 150:427. Retrieved from: http://adsabs.harvard.edu/full/1967ApJ...150..427B
• Dekker, K. H., Battista, J. J., & Jordan, K. J. (2017). Evaluation of an iterative reconstruction algorithm for optical CT radiation dosimetry. Medical physics, 44(12), 6678-6689. DOI: https://doi.org/10.1002/mp.12635
• Gao, Hao. (2012). Fast Parallel Algorithms for the X-Ray Transform and Its Adjoint. Medical Physics 39(11), 7110–20. DOI: https://doi.org/10.1118/1.4761867
• Helvie, M. A. (2010). Digital Mammography Imaging: Breast Tomosynthesis and Advanced Applications. Radiologic Clinics of North America 48(5), 917–29. DOI: https://dx.doi.org/10.1016%2Fj.rcl.2010.06.009
• Jacobs, F., Sundermann, E., Sutter, B. D., Christiaens, M., & Lemahieu, I. (1998). A Fast Algorithm to Calculate the Exact Radiological Path through a Pixel or Voxel Space. Journal of Computing and Information Technology 6(1), 89–94. Retrieved from: https://hrcak.srce.hr/index.php?show=clanak&id_clanak_jezik=221195&lang=en
• Kaczmarz, S. (1937). Angenäherte Auflösung von Systemen Linearer Gleichungen (English Translation by Jason Stockmann: Approximate Solution of Systems of Linear Equations). Bulletin International de l’Académie Polonaise Des Sciences et Des Lettres. 35, 355–357. Retrieved from: https://ntrl.ntis.gov/NTRL/dashboard/searchResults/titleDetail/UCRLTRANS10985.xhtml
• Kak, A. C., Slaney, M., & Wang, G. (2002). Principles of Computerized Tomographic Imaging. Medical Physics 29(1), 107–107. DOI: https://doi.org/10.1118/1.1455742 Klose, A. D., & Hielscher, A. H. (1999). Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer. Medical Physics, 26(8), 1698-1707. DOI: https://doi.org/10.1118/1.598661
• Klose, A. D., & Hielscher, A. H. (1999). Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer. Medical Physics, 26(8), 1698-1707. DOI: https://doi.org/10.1118/1.598661
• Kopans, D. B., Meyer, J. E., & Sadowsky, N. (1984). Breast Imaging. New England Journal of Medicine 310(15), 960–67. DOI: 10.1056/NEJM198404123101506. Retrieved from: https://www.nejm.org/doi/pdf/10.1056/NEJM198404123101506
• Li, N., Zhao, H. X., Cho, S. H., Choi, J. G. & Kim, M. H. (2008). A Fast Algorithm for Voxel-Based Deterministic Simulation of X-Ray Imaging. Computer Physics Communications 178(7), 518–23. DOI: https://doi.org/10.1016/j.cpc.2007.11.008
• Mercan, T., Sevim, G., Kazancı, H. Ö., Üncü, Y. A., & Canpolat, M. (2017). Comparison of Images Produced by Diffuse Optical Tomography with Two Different Backscatter Techniques. In 2017 21st National Biomedical Engineering Meeting (BIYOMUT) (pp. i-iv). IEEE. Retrieved from: https://ieeexplore.ieee.org/abstract/document/8479038
• Mercan, T., Sevim, G., Üncü, Y. A., Serkan, U. S. L. U., Kazancı, H. Ö., & Canpolat, M. (2019). The Comparison of Reconstruction Algorithms for Diffuse Optical Tomography. Süleyman Demirel Üniversitesi Fen Edebiyat Fakültesi Fen Dergisi, 14(2), 285-295. Retrieved from: https://dergipark.org.tr/en/pub/sdufeffd/issue/50336/549528
• Niklason, L. T., Christian, B. T., Niklason, L. E., Kopans, D. B., Castleberry, D. E., Opsahl-Ong, B. H., .... & Wirth, R. F. (1997). Digital Tomosynthesis in Breast Imaging. Radiology 205(2), 399–406. Retrieved from: https://pubs.rsna.org/doi/pdf/10.1148/radiology.205.2.9356620
• Nobel Media AB. 2014. “The Official Website of the Nobel Prize - NobelPrize.Org.” Godfrey N. Hounsfield – Biographical. Nobel Media AB. Retrieved from: https://www.nobelprize.org/
• Oliveira, N., Mota, A. M., Matela, N., Janeiro, L., & Almeida, P. (2016). Dynamic relaxation in algebraic reconstruction technique (ART) for breast tomosynthesis imaging. Computer methods and programs in biomedicine, 132, 189-196. Retrieved from: https://www.sciencedirect.com/science/article/pii/S0169260715301590
• Paltauf, G., Viator, J. A., Prahl, S. A., & Jacques, S. L. (2002). Iterative reconstruction algorithm for optoacoustic imaging. The Journal of the Acoustical Society of America, 112(4), 1536-1544. DOI: https://doi.org/10.1121/1.1501898
• Polat, A., & Yildirim, I. (2018). An Iterative Reconstruction Algorithm for Digital Breast Tomosynthesis Imaging Using Real Data at Three Radiation Doses. Journal of X-Ray Science and Technology 26(3), 347–60. DOI: 10.3233/XST-17320. Retrieved from: https://content.iospress.com/articles/journal-of-x-ray-science-and-technology/xst17320
• Polat, A., Matela N., Dinler, A., Zhang, Y. S. & Yildirim, I. (2019a). Digital Breast Tomosynthesis Imaging Using Compressed Sensing Based Reconstruction for 10 Radiation Doses Real Data. Biomedical Signal Processing and Control 48, 26–34. DOI: https://doi.org/10.1016/j.bspc.2018.08.036
• Polat, A., Hassan, S., Yildirim, I., Oliver, L. E., Mostafaei, M., Kumar, S., ... & Zhang, Y. S. (2019b). A miniaturized optical tomography platform for volumetric imaging of engineered living systems. Lab on a Chip, 19(4), 550-561. DOI: 10.1039/C8LC01190G. Retrieved from: https://pubs.rsc.org/no/content/articlehtml/2019/lc/c8lc01190g
• Raju, T. N. (1999). The Nobel Chronicles. 1979: Allan MacLeod Cormack (b 1924); and Sir Godfrey Newbold Hounsfield (b 1919). Lancet 354(9190), 1653. DOI: https://doi.org/10.1016/S0140-6736(05)77147-6
• Ramachandran, G. N., & Lakshminarayanan, A. V. (1971). Three-Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolutions Instead of Fourier Transforms. Proc. Nat. Acad. Sci. 68(9), 2236-2240. DOI: https://doi.org/10.1073/pnas.68.9.2236
• Sevim, G., Merçan, T., Uncu, Y. A., & Canpolat, M. (2017). A new reconstruction technique used in Diffuse Optical Tomography System. In 2017 21st National Biomedical Engineering Meeting (BIYOMUT) (pp. i-iv). IEEE. Retrieved from: https://ieeexplore.ieee.org/abstract/document/8478965
• Siddon, R. L. (1985). Fast Calculation of the Exact Radiological Path for a Three‐dimensional CT Array. Medical Physics 12(2), 252–55. DOI: https://doi.org/10.1118/1.595715
• Sidky, E. Y., Kao, C. M., & Pan, X. (2006). Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT. Journal of X-ray Science and Technology, 14(2), 119-139. Retrieved from: https://content.iospress.com/articles/journal-of-x-ray-science-and-technology/xst00155
• Üncü, Y. A., Merçan, T., Canpolat, M., & Sevim, G. (2017). A new approach to image processing in diffuse optical tomography and 3-D image. In 2017 25th Signal Processing and Communications Applications Conference (SIU) (pp. 1-4). IEEE. Retrieved from: https://ieeexplore.ieee.org/abstract/document/7960192
• Wang, K., Su, R., Oraevsky, A. A., & Anastasio, M. A. (2012). Investigation of iterative image reconstruction in three-dimensional optoacoustic tomography. Physics in Medicine & Biology, 57(17), 5399. Retrieved from: https://iopscience.iop.org/article/10.1088/0031-9155/57/17/5399/pdf
• Wu, T., Moore, R. H., Rafferty, E. A., & Kopans, D. B. (2004). A Comparison of Reconstruction Algorithms for Breast Tomosynthesis. Medical Physics 31(9), 2636–47. DOI: https://doi.org/10.1118/1.1786692
• Xue, Z., Zhang, L., & Pan, J. (2011). A New Algorithm for Calculating the Radiological Path in CT Image Reconstruction. Proceedings of 2011 International Conference on Electronic and Mechanical Engineering and Information Technology (pp. 4527–30). Harbin. DOI: 10.1109/EMEIT.2011.6024036. Retrieved from: https://ieeexplore.ieee.org/abstract/document/6024036
• Zhao, H., & Reader, A.J. (2003). Fast Ray-Tracing Technique to Calculate Line Integral Paths in Voxel Arrays. IEEE Nuclear Science Symposium Conference Record (IEEE Cat. No.03CH37515). (pp. 2808–12). Portland, OR, USA. DOI: 10.1109/NSSMIC.2003.1352469. Retrieved from: https://ieeexplore.ieee.org/abstract/document/1352469