RadonTransform [expr,{x,y},{p,ϕ}]
gives the Radon transform of expr.
RadonTransform
RadonTransform [expr,{x,y},{p,ϕ}]
gives the Radon transform of expr.
Details and Options
- The Radon transform of a function is defined to be .
- Geometrically, the Radon transform represents the integral of along a line given in normal form by the equation , with -∞<p<∞ and -π/2<ϕ<π/2.
- The following options can be given:
-
- In TraditionalForm , RadonTransform is output using TemplateBox[{{f, (, {x, ,, y}, )}, x, y, p, phi}, RadonTransform].
Examples
open all close allBasic Examples (1)
Compute the Radon transform of a function:
Plot the function along with the transform:
Scope (10)
Basic Uses (2)
Compute the Radon transform of a function for symbolic parameter values:
Use exact values for the parameters:
Use inexact values for the parameters:
Obtain the condition for validity of a Radon transform:
Specify assumptions:
Gaussian Functions (5)
Radon transform of a circular Gaussian function:
Plot the function along with the transform:
Radon transform of an elliptic Gaussian function:
Plot the function along with the transform:
Product of a polynomial with a Gaussian function:
Product of Hermite polynomials and a Gaussian function:
Products of trigonometric functions with Gaussian functions:
Piecewise and Generalized Functions (3)
Radon transform of the characteristic function for the unit disk:
Products of polynomials with the characteristic function for the unit disk:
Radon transforms for expressions involving DiracDelta :
Options (2)
Assumptions (1)
Specify assumptions:
GenerateConditions (1)
Generate conditions for the validity of the result:
Applications (2)
Compute the symbolic Radon transform for the characteristic function of a disk:
Obtain the same result using Radon :
Use the Radon transform to solve a Poisson equation:
Apply RadonTransform to the equation:
Solve the ordinary differential equation using DSolveValue :
Set the arbitrary constants in the solution to 0:
Obtain the solution for the original equation using InverseRadonTransform :
Verify the solution:
Plot the solution:
Properties & Relations (10)
RadonTransform computes the integral :
Obtain the same result using Integrate :
RadonTransform and InverseRadonTransform are mutual inverses:
RadonTransform is a linear operator:
The shifting property for RadonTransform :
The symmetry property for RadonTransform :
Express the Radon transform of in terms of a unit vector:
Verify the symmetry property:
The homogeneity property for RadonTransform :
Express the Radon transform of in terms of a unit vector:
Verify the homogeneity property:
The scaling property for RadonTransform :
Express the Radon transform of in terms of a unit vector:
Express the Radon transform of in terms of a unit vector:
Verify the scaling property:
RadonTransform of derivatives:
RadonTransform of the Laplacian :
RadonTransform can be computed using Fourier transforms:
Compute the Fourier transform of f in polar coordinates:
Compute the inverse Fourier transform to obtain the Radon transform:
Obtain the same result directly using RadonTransform :
Neat Examples (1)
Create a table of basic Radon transforms:
Related Guides
History
Text
Wolfram Research (2017), RadonTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/RadonTransform.html.
CMS
Wolfram Language. 2017. "RadonTransform." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RadonTransform.html.
APA
Wolfram Language. (2017). RadonTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RadonTransform.html
BibTeX
@misc{reference.wolfram_2025_radontransform, author="Wolfram Research", title="{RadonTransform}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/RadonTransform.html}", note=[Accessed: 08-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_radontransform, organization={Wolfram Research}, title={RadonTransform}, year={2017}, url={https://reference.wolfram.com/language/ref/RadonTransform.html}, note=[Accessed: 08-January-2026]}