Abstract
The inversion of the celebrated Radon transform in three dimensions involves two-dimensional plane integration. This inversion provides the mathematical foundation of the important field of medical imaging, known as three-dimensional positron emission tomography (3D PET). In this chapter, we present an analytical expression for the inversion of the three-dimensional Radon transform, as well as a novel numerical implementation of this formula, based on piecewise polynomials of the third degree.
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Notes
- 1.
Plane integrals are special cases of surface integrals, where the surface of integration is a plane.
- 2.
Plane integrals are special cases of surface integrals, where the surface of integration is a plane.
References
J. Radon, Über die bestimmung von funktionen durch ihre integralwerte längs gewisser mannigfaltigkeiten. Akad. Wiss. 69, 262–277 (1917)
P. Kuchment, The Radon transform and medical imaging, in CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (2014)
N.E. Protonotarios, G.A. Kastis, A.S. Fokas, A New Approach for the Inversion of the Attenuated Radon Transform (Springer, Cham, 2019), pp. 433–457. https://doi.org/10.1007/978-3-030-31339-5_16
M.-Y. Chiu, H.H. Barrett, R.G. Simpson, Three-dimensional reconstruction from planar projections. JOSA 70(7), 755–762 (1980). https://doi.org/10.1364/JOSA.70.000755
M. Defrise, D. Townsend, R. Clack, Three-dimensional image reconstruction from complete projections. Phys. Med. Biol. 34(5), 573 (1989). https://doi.org/10.1088/0031-9155/34/5/002
A. Averbuch, Y. Shkolnisky, 3d fourier based discrete radon transform. Appl. Comput. Harmon. Anal. 15(1), 33–69 (2003). https://doi.org/10.1016/S1063-5203(03)00030-7
J. Bikowski, K. Knudsen, J.L. Mueller, Direct numerical reconstruction of conductivities in three dimensions using scattering transforms. Inverse Probl. 27(1), 015002 (2010). https://doi.org/10.1088/0266-5611/27/1/015002
F.H. Fahey, Data acquisition in PET imaging. J. Nucl. Med. Technol. 30(2), 39–49 (2002)
P. Kinahan, J. Rogers, Analytic 3D image reconstruction using all detected events. IEEE Trans. Nucl. Sci. 36, 964–968 (1989). https://doi.org/10.1109/23.34585
P. La Rivière, X. Pan, Spline-based inverse radon transform in two and three dimensions. IEEE Trans. Nucl. Sci. 45(4), 2224–2231 (1998). https://doi.org/10.1109/23.708352
M. Defrise, P.E. Kinahan, C.J. Michel, Image reconstruction algorithms in PET, in Positron Emission Tomography (Springer, Berlin, 2005), pp. 63–91. https://doi.org/10.1007/1-84628-007-9_4
J. Gaskill, J.W. Sons, Linear Systems, Fourier Transforms, and Optics. Wiley Series in Pure and Applied Optics (Wiley, New York, 1978)
W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd edn. (Cambridge University, New York, 2007)
Acknowledgements
This work was partially supported by the research programme “Inverse Problems and Medical Imaging” (200/947) of the Research Committee of the Academy of Athens. A.S. Fokas has been supported by EPSRC, UK in the form of a senior fellowship.
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Protonotarios, N.E., Kastis, G.A., Dikaios, N., Fokas, A.S. (2021). Piecewise Polynomial Inversion of the Radon Transform in Three Space Dimensions via Plane Integration and Applications in Positron Emission Tomography. In: Rassias, T.M. (eds) Nonlinear Analysis, Differential Equations, and Applications. Springer Optimization and Its Applications, vol 173. Springer, Cham. https://doi.org/10.1007/978-3-030-72563-1_17
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