Radon Transform
The Radon transform is an integral transform whose inverse is used to reconstruct images from medical CT scans. A technique for
using Radon transforms to reconstruct a map of a planet's polar regions using a spacecraft
in a polar orbit has also been devised (Roulston and Muhleman 1997).
The Radon and inverse Radon transforms are implemented in the Wolfram Language as RadonTransform
and InverseRadonTransform,
respectively.
The Radon transform can be defined by
where 
is the slope
of a line, 
is its intercept, and 
is the
delta function. The inverse Radon transform is
![f(x,y)=1/(2pi)int_(-infty)^inftyd/(dy)H[U(p,y-px)]dp,](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/RadonTransform/NumberedEquation1.gif) |
(4)
|
where 
is a Hilbert
transform. The transform can also be defined by
![R^'(r,alpha)[f(x,y)]=int_(-infty)^inftyint_(-infty)^inftyf(x,y)delta(r-xcosalpha-ysinalpha)dxdy,](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/RadonTransform/NumberedEquation2.gif) |
(5)
|
where 
is the perpendicular
distance from a line to the origin and 
is the angle
formed by the distance vector.
Using the identity
![F_(omega,alpha)[R[f(omega,alpha)]](x,y)=F_(u,v)^2[f(u,v)](x,y),](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/RadonTransform/NumberedEquation3.gif) |
(6)
|
where 
is the Fourier
transform, gives the inversion formula
![f(x,y)=cint_0^piint_(-infty)^inftyF_(omega,alpha)[R[f(omega,alpha)]]|omega|e^(iomega(xcosalpha+ysinalpha))domegadalpha.](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/RadonTransform/NumberedEquation4.gif) |
(7)
|
The Fourier transform can be eliminated by writing
![f(x,y)=int_0^piint_(-infty)^inftyR[f(r,alpha)]W(r,alpha,x,y)drdalpha,](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/RadonTransform/NumberedEquation5.gif) |
(8)
|
where 
is a weighting
function such as
Nievergelt (1986) uses the inverse formula
![f(x,y)=1/pilim_(c->0)int_0^piint_(-infty)^inftyR[f(r+xcosalpha+ysinalpha,alpha)]G_c(r)drdalpha,](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/RadonTransform/NumberedEquation6.gif) |
(11)
|
where
 |
(12)
|
Ludwig's inversion formula expresses a function in terms of its Radon transform. 
and
are related by
The Radon transform satisfies superposition
![R(p,tau)[f_1(x,y)+f_2(x,y)]=U_1(p,tau)+U_2(p,tau),](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/RadonTransform/NumberedEquation8.gif) |
(15)
|
linearity
![R(p,tau)[af(x,y)]=aU(p,tau),](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/RadonTransform/NumberedEquation9.gif) |
(16)
|
scaling
![R(p,tau)[f(x/a,y/b)]=|a|U(pa/b,tau/b),](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/RadonTransform/NumberedEquation10.gif) |
(17)
|
rotation, with 
rotation
by angle 
![R(p,tau)[R_phif(x,y)]=1/(|cosphi+psinphi|)U((p-tanphi)/(1+ptanphi),tau/(cosphi+psinphi)),](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/RadonTransform/NumberedEquation11.gif) |
(18)
|
and skewing
![R(p,tau)[f(ax+by,cx+dy)]=1/(|a+bp|)U[(c+dp)/(a+bp),(tau(ab+bc))/(a+bp)]](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/RadonTransform/NumberedEquation12.gif) |
(19)
|
(Durrani and Bisset 1984; correction in Durrani and Bisset 1985).
The line integral along 
is
 |
(20)
|
The analog of the one-dimensional convolution
theorem is
![R(p,tau)[f(x,y)*g(y)]=U(p,tau)*g(tau),](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/RadonTransform/NumberedEquation14.gif) |
(21)
|
the analog of Plancherel's theorem is
 |
(22)
|
and the analog of Parseval's theorem is
![int_(-infty)^inftyR(p,tau)[f(x,y)]^2dtau=int_(-infty)^inftyint_(-infty)^inftyf^2(x,y)dxdy.](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/RadonTransform/NumberedEquation16.gif) |
(23)
|
If 
is a continuous function on 
, integrable with
respect to a plane Lebesgue measure, and
 |
(24)
|
for every (doubly) infinite line 
where 
is the length measure,
then 
must be identically zero. However, if
the global integrability condition is removed, this result fails (Zalcman 1982, Goldstein
1993).
SEE ALSO: Hammer's X-Ray Problems,
Inverse Radon Transform,
Radon
Transform--Cylinder,
Radon Transform--Delta
Function,
Radon Transform--Gaussian,
Radon Transform--Square,
Tomography
REFERENCES:
Anger, B. and Portenier, C. Radon
Integrals. Boston, MA: Birkhäuser, 1992.
Armitage, D. H. and Goldstein, M. "Nonuniqueness for the Radon Transform."
Proc. Amer. Math. Soc. 117, 175-178, 1993.
Deans, S. R. The
Radon Transform and Some of Its Applications. New York: Wiley, 1983.
Durrani, T. S. and Bisset, D. "The Radon Transform and its Properties."
Geophys. 49, 1180-1187, 1984.
Durrani, T. S. and Bisset, D. "Erratum to: The Radon Transform and Its
Properties." Geophys. 50, 884-886, 1985.
Esser, P. D. (Ed.). Emission Computed Tomography: Current Trends. New York: Society of Nuclear Medicine,
1983.
Gindikin, S. (Ed.). Applied
Problems of Radon Transform. Providence, RI: Amer. Math. Soc., 1994.
Goldstein, H. Classical
Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 1980.
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press,
2000.
Helgason, S. The
Radon Transform. Boston, MA: Birkhäuser, 1980.
Hungerbühler, N. "Singular Filters for the Radon Backprojection."
J. Appl. Analysis 5, 17-33, 1998.
Kak, A. C. and Slaney, M. Principles
of Computerized Tomographic Imaging. IEEE Press, 1988.
Kunyansky, L. A. "Generalized and Attenuated Radon Transforms: Restorative Approach to the Numerical Inversion." Inverse Problems 8, 809-819,
1992.
Nievergelt, Y. "Elementary Inversion of Radon's Transform." SIAM Rev. 28,
79-84, 1986.
Rann, A. G. and Katsevich, A. I. The
Radon Transform and Local Tomography. Boca Raton, FL: CRC Press, 1996.
Robinson, E. A. "Spectral Approach to Geophysical Inversion Problems by Lorentz, Fourier, and Radon Transforms." Proc. Inst. Electr. Electron. Eng. 70,
1039-1053, 1982.
Roulston, M. S. and Muhleman, D. O. "Synthesizing Radar Maps of Polar Regions with a Doppler-Only Method." Appl. Opt. 36, 3912-3919,
1997.
Shepp, L. A. and Kruskal, J. B. "Computerized Tomography: The New
Medical X-Ray Technology." Amer. Math. Monthly 85, 420-439, 1978.
Strichartz, R. S. "Radon Inversion--Variation on a Theme." Amer.
Math. Monthly 89, 377-384 and 420-423, 1982.
Weisstein, E. W. "Books about Radon Transforms." http://www.ericweisstein.com/encyclopedias/books/RadonTransforms.html.
Zalcman, L. "Uniqueness and Nonuniqueness for the Radon Transform." Bull.
London Math. Soc. 14, 241-245, 1982.
Referenced on Wolfram|Alpha:
Radon Transform
CITE THIS AS:
Weisstein, Eric W. "Radon Transform."
From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RadonTransform.html
![R(p,tau)[f(x,y)]](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/RadonTransform/Inline1.gif)
![int_(-infty)^inftyint_(-infty)^inftyf(x,y)delta[y-(tau+px)]dydx](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/RadonTransform/Inline6.gif)
![F^(-1)[|omega|].](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/RadonTransform/Inline23.gif)
Contact the MathWorld Team
[3,8)
