In mathematics, the Radon transform is the integral transform which takes a function f defined on the plane to a function Rf defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line. The transform was introduced in 1917 by Johann Radon,[1] who also provided a formula for the inverse transform. Radon further included formulas for the transform in three dimensions, in which the integral is taken over planes (integrating over lines is known as the X-ray transform). It was later generalized to higher-dimensional Euclidean spaces and more broadly in the context of integral geometry. The complex analogue of the Radon transform is known as the Penrose transform. The Radon transform is widely applicable to tomography, the creation of an image from the projection data associated with cross-sectional scans of an object.
Explanation[edit]
If a function represents an unknown density, then the Radon transform represents the projection data obtained as the output of a tomographic scan. Hence the inverse of the Radon transform can be used to reconstruct the original density from the projection data, and thus it forms the mathematical underpinning for tomographic reconstruction, also known as iterative reconstruction.
The Radon transform data is often called a sinogram because the Radon transform of an off-center point source is a sinusoid. Consequently, the Radon transform of a number of small objects appears graphically as a number of blurred sine waves with different amplitudes and phases.
The Radon transform is useful in computed axial tomography (CAT scan), barcode scanners, electron microscopy of macromolecular assemblies like viruses and protein complexes, reflection seismology and in the solution of hyperbolic partial differential equations.
Definition[edit]
Let be a function that satisfies the three regularity conditions:[3]
- is continuous;
- the double integral , extending over the whole plane, converges;
- for any arbitrary point on the plane it holds that
The Radon transform, , is a function defined on the space of straight lines by the line integral along each such line as:
Relationship with the Fourier transform[edit]
The Radon transform is closely related to the Fourier transform. We define the univariate Fourier transform here as:
Thus the two-dimensional Fourier transform of the initial function along a line at the inclination angle is the one variable Fourier transform of the Radon transform (acquired at angle ) of that function. This fact can be used to compute both the Radon transform and its inverse. The result can be generalized into n dimensions:
Dual transform[edit]
The dual Radon transform is a kind of adjoint to the Radon transform. Beginning with a function g on the space , the dual Radon transform is the function on Rn defined by:
Concretely, for the two-dimensional Radon transform, the dual transform is given by:
Intertwining property[edit]
Let denote the Laplacian on defined by:
Reconstruction approaches[edit]
The process of reconstruction produces the image (or function in the previous section) from its projection data. Reconstruction is an inverse problem.
Radon inversion formula[edit]
In the two-dimensional case, the most commonly used analytical formula to recover from its Radon transform is the Filtered Back-projection Formula or Radon Inversion Formula[9]:
Ill-posedness[edit]
Intuitively, in the filtered back-projection formula, by analogy with differentiation, for which , we see that the filter performs an operation similar to a derivative. Roughly speaking, then, the filter makes objects more singular. A quantitive statement of the ill-posedness of Radon inversion goes as follows:
Iterative reconstruction methods[edit]
Compared with the Filtered Back-projection method, iterative reconstruction costs large computation time, limiting its practical use. However, due to the ill-posedness of Radon Inversion, the Filtered Back-projection method may be infeasible in the presence of discontinuity or noise. Iterative reconstruction methods (e.g. iterative Sparse Asymptotic Minimum Variance[10]) could provide metal artefact reduction, noise and dose reduction for the reconstructed result that attract much research interest around the world.
Inversion formulas[edit]
Explicit and computationally efficient inversion formulas for the Radon transform and its dual are available. The Radon transform in dimensions can be inverted by the formula:[11]
Explicitly, the inversion formula obtained by the latter method is:[4]
Radon transform in algebraic geometry[edit]
In algebraic geometry, a Radon transform (also known as the Brylinski–Radon transform) is constructed as follows.
Write
for the universal hyperplane, i.e., H consists of pairs (x, h) where x is a point in d-dimensional projective space and h is a point in the dual projective space (in other words, x is a line through the origin in (d+1)-dimensional affine space, and h is a hyperplane in that space) such that x is contained in h.
Then the Brylinksi–Radon transform is the functor between appropriate derived categories of étale sheaves
The main theorem about this transform is that this transform induces an equivalence of the categories of perverse sheaves on the projective space and its dual projective space, up to constant sheaves.[14]
See also[edit]
- Periodogram
- Matched filter
- Deconvolution
- X-ray transform
- Funk transform
- The Hough transform, when written in a continuous form, is very similar, if not equivalent, to the Radon transform.[15]
- Cauchy–Crofton theorem is a closely related formula for computing the length of curves in space.
- Fast Fourier transform
Notes[edit]
- ^ Radon 1917.
- ^ Odložilík, Michal (2023-08-31). Detachment tomographic inversion study with fast visible cameras on the COMPASS tokamak (Bachelor's thesis). Czech Technical University in Prague. hdl:10467/111617.
- ^ Radon 1986.
- ^ a b Roerdink 2001.
- ^ Helgason 1984, Lemma I.2.1.
- ^ Lax, P. D.; Philips, R. S. (1964). "Scattering theory". Bull. Amer. Math. Soc. 70 (1): 130–142. doi:10.1090/s0002-9904-1964-11051-x.
- ^ Bonneel, N.; Rabin, J.; Peyre, G.; Pfister, H. (2015). "Sliced and Radon Wasserstein Barycenters of Measures". Journal of Mathematical Imaging and Vision. 51 (1): 22–25. doi:10.1007/s10851-014-0506-3. S2CID 1907942.
- ^ Rim, D. (2018). "Dimensional Splitting of Hyperbolic Partial Differential Equations Using the Radon Transform". SIAM J. Sci. Comput. 40 (6): A4184–A4207. arXiv:1705.03609. Bibcode:2018SJSC...40A4184R. doi:10.1137/17m1135633. S2CID 115193737.
- ^ a b c Candès 2016b.
- ^ Abeida, Habti; Zhang, Qilin; Li, Jian; Merabtine, Nadjim (2013). "Iterative Sparse Asymptotic Minimum Variance Based Approaches for Array Processing" (PDF). IEEE Transactions on Signal Processing. 61 (4). IEEE: 933–944. arXiv:1802.03070. Bibcode:2013ITSP...61..933A. doi:10.1109/tsp.2012.2231676. ISSN 1053-587X. S2CID 16276001.
- ^ Helgason 1984, Theorem I.2.13.
- ^ Helgason 1984, Theorem I.2.16.
- ^ Nygren 1997.
- ^ Kiehl & Weissauer (2001, Ch. IV, Cor. 2.4)
- ^ van Ginkel, Hendricks & van Vliet 2004.
References[edit]
- Kiehl, Reinhardt; Weissauer, Rainer (2001), Weil conjectures, perverse sheaves and l'adic Fourier transform, Springer, doi:10.1007/978-3-662-04576-3, ISBN 3-540-41457-6, MR 1855066
- Radon, Johann (1917), "Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten", Berichte über die Verhandlungen der Königlich-Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse [Reports on the Proceedings of the Royal Saxonian Academy of Sciences at Leipzig, Mathematical and Physical Section] (69), Leipzig: Teubner: 262–277;
Translation: Radon, J. (December 1986), "On the determination of functions from their integral values along certain manifolds", IEEE Transactions on Medical Imaging, 5 (4), translated by Parks, P.C.: 170–176, doi:10.1109/TMI.1986.4307775, PMID 18244009, S2CID 26553287. - Roerdink, J.B.T.M. (2001) [1994], "Tomography", Encyclopedia of Mathematics, EMS Press.
- Helgason, Sigurdur (1984), Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Academic Press, ISBN 0-12-338301-3.
- Candès, Emmanuel (February 2, 2016a). "Applied Fourier Analysis and Elements of Modern Signal Processing – Lecture 9" (PDF).
- Candès, Emmanuel (February 4, 2016b). "Applied Fourier Analysis and Elements of Modern Signal Processing – Lecture 10" (PDF).
- Nygren, Anders J. (1997). "Filtered Back Projection". Tomographic Reconstruction of SPECT Data.
- van Ginkel, M.; Hendricks, C.L. Luengo; van Vliet, L.J. (2004). "A short introduction to the Radon and Hough transforms and how they relate to each other" (PDF). Archived (PDF) from the original on 2016-07-29.
Further reading[edit]
- Lokenath Debnath; Dambaru Bhatta (19 April 2016). Integral Transforms and Their Applications. CRC Press. ISBN 978-1-4200-1091-6.
- Deans, Stanley R. (1983), The Radon Transform and Some of Its Applications, New York: John Wiley & Sons
- Helgason, Sigurdur (2008), Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, vol. 39 (2nd ed.), Providence, R.I.: American Mathematical Society, doi:10.1090/surv/039, ISBN 978-0-8218-4530-1, MR 2463854
- Herman, Gabor T. (2009), Fundamentals of Computerized Tomography: Image Reconstruction from Projections (2nd ed.), Springer, ISBN 978-1-85233-617-2
- Minlos, R.A. (2001) [1994], "Radon transform", Encyclopedia of Mathematics, EMS Press
- Natterer, Frank (June 2001), The Mathematics of Computerized Tomography, Classics in Applied Mathematics, vol. 32, Society for Industrial and Applied Mathematics, ISBN 0-89871-493-1
- Natterer, Frank; Wübbeling, Frank (2001), Mathematical Methods in Image Reconstruction, Society for Industrial and Applied Mathematics, ISBN 0-89871-472-9
External links[edit]
- Weisstein, Eric W. "Radon Transform". MathWorld.
- Analytical projection (the Radon transform) (video). Part of the "Computed Tomography and the ASTRA Toolbox" course. University of Antwerp. September 10, 2015.