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In [[mathematics]], the '''X-ray transform''' (also called '''John transform''') is an [[integral transform]] introduced by Fritz John in 1938<ref>{{cite journal|last=Fritz|first=John|title=The ultrahyperbolic differential equation with four independent variables|journal=Duke Mathematical Journal|year=1938|volume=4|pages=300–322|doi=10.1215/S0012-7094-38-00423-5|url=http://projecteuclid.org/euclid.dmj/1077490637|accessdate=23 January 2013}}</ref> 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 [[line (geometry)|line]]s rather than over [[hyperplane]]s as in the Radon transform. The X-ray transform derives its name from X-ray [[tomography]] because the X-ray transform of a function ''ƒ'' represents the [[scattering]] 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 scattering data. | |||
In detail, if ''ƒ'' is a [[compact support|compactly supported]] [[continuous function]] on the [[Euclidean space]] '''R'''<sup>''n''</sup>, then the X-ray transform of ''ƒ'' is the function ''Xƒ'' defined on the set of all lines in '''R'''<sup>''n''</sup> by | |||
:<math>Xf(L) = \int_L f = \int_{\mathbf{R}} f(x_0+t\theta)dt</math> | |||
where ''x''<sub>0</sub> is an initial point on the line and θ is a unit vector 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 {{harv|Gelfand|Gindikin|Graev|2003|loc=2.1.2}}. | |||
==References== | |||
{{Reflist}} | |||
*{{springer|first=Carlos A.|last=Berenstein|title=X-ray transform|id=X/x120030}}. | |||
*{{Citation | last1=Gelfand | first1=I. M. | last2=Gindikin | first2=S. G. | last3=Graev | first3=M. I. | title=Selected topics in integral geometry | origyear=2000 | url=http://books.google.com/books?isbn=0821829327 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Translations of Mathematical Monographs | isbn=978-0-8218-2932-5 | mr=2000133 | year=2003 | volume=220}} | |||
*{{Citation | last1=Helgason | first1=Sigurdur | title=Geometric analysis on symmetric spaces | publisher=[[American Mathematical Society]] | location=Providence, R.I. | edition=2nd | series=Mathematical Surveys and Monographs | isbn=978-0-8218-4530-1 | mr=2463854 | year=2008 | volume=39}} | |||
*{{Citation | last1=Helgason | first1=Sigurdur | url=http://www-math.mit.edu/~helgason/Radonbook.pdf | title=The Radon Transform | publisher=[[Birkhauser]] | location=Boston, M.A. | edition=2nd | series=Progress in Mathematics | year=1999}} | |||
[[Category:Integral geometry]] | |||
[[Category:Integral transforms]] | |||
Latest revision as of 16:07, 17 November 2013
In mathematics, the X-ray transform (also called John transform) is an integral transform introduced by Fritz John in 1938[1] 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 because the X-ray transform of a function ƒ represents the scattering 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 scattering 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
where x0 is an initial point on the line and θ is a unit vector 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 Template:Harv.
References
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In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
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