Inversion of a generalized Radon transform by algebraic iterative methods. (English) Zbl 1540.44009
Summary: Inversion of a generalized Radon transform (GRT) of seismic type is investigated by employing the algebraic iterative methods ART and SIRT, which require discrete formulations of the reconstruction problem. To this aim, two discretization procedures are proposed: a direct discretization and a discretization via the relation of GRT with the regular Radon transform (RT). The feasibility and the semi-convergence behavior of the proposed methods are analyzed and compared by discussing the effect of noisy data.
{© 2022 John Wiley & Sons, Ltd.}
{© 2022 John Wiley & Sons, Ltd.}
MSC:
| 44A12 | Radon transform |
| 53C65 | Integral geometry |
| 65F10 | Iterative numerical methods for linear systems |
Keywords:
algebraic reconstruction; generalized Radon transform; simultaneous iterative reconstructionReferences:
| [1] | RadonJ. Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber Verh Sä chs Akad. 1917;69:262‐277. · JFM 46.0436.02 |
| [2] | CormackAM. Representation of a function by its line integrals, with some radiological applications. J Appl Phys. 1963;34:2722‐2727. · Zbl 0117.32303 |
| [3] | DeansSR. The Radon Transform and Some of its Applications. Dover Publications; 2007. · Zbl 1121.44004 |
| [4] | DebnathL, BhattaD. Integral Transforms and Their Applications. CRC Press; 2007. |
| [5] | NattererF. The Mathematics of Computerized Tomography. Wiley; 1986. · Zbl 0617.92001 |
| [6] | NattererF, WubbelingF. Mathematical Methods in Image Reconstruction, Monographs on Mathematical Modeling and Computation, vol. 5: SIAM; 2001. · Zbl 0974.92016 |
| [7] | BeisterM, KolditzD, KalenderW. Iterative reconstruction methods in X‐ray CT. Physica Medica. 2012;28:94‐108. |
| [8] | JiangM, WangG. Development of iterative algorithms for image reconstruction. J X‐ray Sci Technol. 2001;10:77‐86. |
| [9] | BeylkinG. The inversion problem and applications of the generalized Radon transform. Comm Pure Appl Math. 1984;37:579‐599. · Zbl 0577.44003 |
| [10] | ToftP. The Radon Transform: Theory and Implementation. Technical University of Denmark; 1996. PhD thesis. |
| [11] | UstaogluZ. On the inversion of a generalized Radon transform of seismic type. J Math Anal Appl. 2017;453:287‐303. · Zbl 1372.65336 |
| [12] | MoonS. Inversion of the seismic parabolic Radon transform and the seismic hyperbolic Radon transform. Inverse Probl Sci Eng. 2016;24:317‐327. |
| [13] | ChiharaH. Inversion of seismic‐type Radon transforms on the plane. Integr Transf Spec F. 2020;31:998‐1009. · Zbl 1471.44004 |
| [14] | EpsteinC. Introduction to the Mathematics of Medical Imaging. SIAM; 2008. · Zbl 1305.92002 |
| [15] | JiangM, WangG. Convergence studies on iterative algorithms for image reconstruction. IEEE Trans Med Imaging. 2003;22:569‐579. |
| [16] | ElfvingT, HansenPC, NikazadT. Semi‐convergence properties of Kaczmarz’s method. Inverse Probl. 2014;30(5):055007. · Zbl 1296.65054 |
| [17] | ElfvingT, NikazadT, HansenPC. Semi‐convergence and relaxation parameters for a class of SIRT algorithms. Electron Trans Numer Anal. 2010;37:321‐336. · Zbl 1205.65148 |
| [18] | HansenPC, Saxild‐HansenM. AIR Tools—a Matlab package of algebraic iterative reconstruction methods. J Comput Appl Math. 2012;236(8):2167‐2178. · Zbl 1241.65042 |
| [19] | GordonR, BenderR, HermanGT. Algebraic reconstruction techniques for 3 dimensional electron microscopy and X‐ray photograph.J Theor Biol. 1970;29:471‐481. |
| [20] | HounsfieldGN. Computerized transverse axial scanning (tomography): Part 1. Description of system. Br J Radiol. 1973;46:1016‐1022. |
| [21] | KaczmarzS. Angenäherte Auflösung von Systemen linearer Gleichungen. Bull Acad Pol Sci Lett. 1937;A35:355‐357. · JFM 63.0524.02 |
| [22] | BjörckÅ, ElfvingT. Accelerated projection methods for computing pseudoinverse solutions of systems of linear equations. BIT. 1979;19:145‐163. · Zbl 0409.65022 |
| [23] | StrohmerT, VershyninR. A randomized Kaczmarz algorithm for linear systems with exponential convergence. J Fourier Anal Appl. 2009;15:262‐278. · Zbl 1169.68052 |
| [24] | CimminoG. Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari. La Ric Sci, XVI, Ser II, Anno IX. 1938;1:326‐333. · Zbl 0018.41802 |
| [25] | LandweberL. An iteration formula for Fredholm integral of the first kind. Amer J Math. 1951;73:615‐624. · Zbl 0043.10602 |
| [26] | AndersenAH, KakAC. Simultaneous algebraic reconstruction (SART): a superior implementation of the ART algorithm. Ultrason Imaging. 1984;6:81‐94. |
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