X-ray transform
In mathematics, the X-ray transform (also called ray transform[1] or John transform) is an integral transform introduced by Fritz John in 1938[2] that is one of the cornerstones of modern integral geometry. It is very closely related to the Radon transform, and coincides with it in two dimensions. In higher dimensions, the X-ray transform of a function is defined by integrating over lines rather than over hyperplanes as in the Radon transform. The X-ray transform derives its name from X-ray tomography (used in CT scans) because the X-ray transform of a function ƒ represents the attenuation data of a tomographic scan through an inhomogeneous medium whose density is represented by the function ƒ. Inversion of the X-ray transform is therefore of practical importance because it allows one to reconstruct an unknown density ƒ from its known attenuation data.
In detail, if ƒ is a compactly supported continuous function on the Euclidean space Rn, then the X-ray transform of ƒ is the function Xƒ defined on the set of all lines in Rn by
- [math]\displaystyle{ Xf(L) = \int_L f = \int_{\mathbf{R}} f(x_0+t\theta)dt }[/math]
where x0 is an initial point on the line and θ is a unit vector in Rn giving the direction of the line L. The latter integral is not regarded in the oriented sense: it is the integral with respect to the 1-dimensional Lebesgue measure on the Euclidean line L.
The X-ray transform satisfies an ultrahyperbolic wave equation called John's equation.
The Gauss hypergeometric function can be written as an X-ray transform (Gelfand Gindikin).
References
- ↑ Natterer, Frank; Wübbeling, Frank (2001). Mathematical Methods in Image Reconstruction. Philadelphia: SIAM. doi:10.1137/1.9780898718324.fm.
- ↑ Fritz, John (1938). "The ultrahyperbolic differential equation with four independent variables". Duke Mathematical Journal 4: 300–322. doi:10.1215/S0012-7094-38-00423-5. http://projecteuclid.org/euclid.dmj/1077490637. Retrieved 23 January 2013.
- Hazewinkel, Michiel, ed. (2001), "X-ray transform", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=X/x120030.
- Gelfand, I. M.; Gindikin, S. G.; Graev, M. I. (2003), Selected topics in integral geometry, Translations of Mathematical Monographs, 220, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2932-5, https://books.google.com/books?isbn=0821829327
- Helgason, Sigurdur (2008), Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, 39 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4530-1
- Helgason, Sigurdur (1999), The Radon Transform, Progress in Mathematics (2nd ed.), Boston, M.A.: Birkhauser, http://www-math.mit.edu/~helgason/Radonbook.pdf
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