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RadonTransform
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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:
  • Assumptions $Assumptionsassumptions on parameters
    GenerateConditions Falsewhether to generate results that involve conditions on parameters
    MethodAutomaticwhat method to use
  • In TraditionalForm, RadonTransform is output using TemplateBox[{{f, (, {x, ,, y}, )}, x, y, p, phi}, RadonTransform].

Examples

open allclose all

Basic Examples  (1)Summary of the most common use cases

Compute the Radon transform of a function:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-ff11wx
Out[1]=1

Plot the function along with the transform:

In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-cd5g5z
Out[2]=2

Scope  (10)Survey of the scope of standard use cases

Basic Uses  (2)

Compute the Radon transform of a function for symbolic parameter values:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-qbzrq
Out[1]=1

Use exact values for the parameters:

In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-ieerd4
Out[2]=2

Use inexact values for the parameters:

In[3]:=3
✖
https://wolfram.com/xid/0n4q94l0xaq-h993gr
Out[3]=3

Obtain the condition for validity of a Radon transform:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-31cta
Out[1]=1

Specify assumptions:

In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-jxia6n
Out[2]=2

Gaussian Functions  (5)

Radon transform of a circular Gaussian function:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-m2fi0h
Out[1]=1

Plot the function along with the transform:

In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-prve1g
Out[2]=2

Radon transform of an elliptic Gaussian function:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-fhbmsm
Out[1]=1

Plot the function along with the transform:

In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-b2ds95
Out[2]=2

Product of a polynomial with a Gaussian function:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-c7mytc
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-bcqjes
Out[2]=2

Product of Hermite polynomials and a Gaussian function:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-jgkko3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-6rv16
Out[2]=2

Products of trigonometric functions with Gaussian functions:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-n664eb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-bsx98x
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0n4q94l0xaq-ibjcpe
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0n4q94l0xaq-d7ncr
Out[4]=4

Piecewise and Generalized Functions  (3)

Radon transform of the characteristic function for the unit disk:

In[3]:=3
✖
https://wolfram.com/xid/0n4q94l0xaq-ef55i3
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0n4q94l0xaq-bmssmh
Out[4]=4

Products of polynomials with the characteristic function for the unit disk:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-cyzo4c
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-nsydu
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0n4q94l0xaq-dlqzy3
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0n4q94l0xaq-f1u6yi
Out[4]=4

Radon transforms for expressions involving DiracDelta:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-93vbt
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-cm9ww0
Out[2]=2

Options  (2)Common values & functionality for each option

Assumptions  (1)

Specify assumptions:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-bzc0xf
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-e2ox2
Out[2]=2

GenerateConditions  (1)

Generate conditions for the validity of the result:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-e6wry2
Out[1]=1
In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-i3w7rk
Out[1]=1

Applications  (2)Sample problems that can be solved with this function

Compute the symbolic Radon transform for the characteristic function of a disk:

In[3]:=3
✖
https://wolfram.com/xid/0n4q94l0xaq-hdvkhg
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0n4q94l0xaq-hb3dmy
Out[4]=4

Obtain the same result using Radon:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-gju1a
In[4]:=4
✖
https://wolfram.com/xid/0n4q94l0xaq-jwqwr4
Out[4]=4

Use the Radon transform to solve a Poisson equation:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-eg8s6u

Apply RadonTransform to the equation:

In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-ct0d6v
Out[2]=2

Solve the ordinary differential equation using DSolveValue:

In[3]:=3
✖
https://wolfram.com/xid/0n4q94l0xaq-ky7r2d
Out[3]=3

Set the arbitrary constants in the solution to 0:

In[4]:=4
✖
https://wolfram.com/xid/0n4q94l0xaq-dlrt5p
Out[4]=4

Obtain the solution for the original equation using InverseRadonTransform:

In[5]:=5
✖
https://wolfram.com/xid/0n4q94l0xaq-huk5g1
Out[5]=5

Verify the solution:

In[6]:=6
✖
https://wolfram.com/xid/0n4q94l0xaq-gabztc
Out[6]=6

Plot the solution:

In[7]:=7
✖
https://wolfram.com/xid/0n4q94l0xaq-gzd1ge
Out[7]=7

Properties & Relations  (10)Properties of the function, and connections to other functions

RadonTransform computes the integral :

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-b6ht2u
Out[1]=1

Obtain the same result using Integrate:

In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-9z2j4
In[3]:=3
✖
https://wolfram.com/xid/0n4q94l0xaq-pftzeq
Out[3]=3

RadonTransform and InverseRadonTransform are mutual inverses:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-ey81a0
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-f0lrqc
Out[2]=2

RadonTransform is a linear operator:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-csnlc
Out[1]=1

The shifting property for RadonTransform:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-lgnp1t
In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-bcvg0z
In[3]:=3
✖
https://wolfram.com/xid/0n4q94l0xaq-iwmj7z
Out[3]=3

The symmetry property for RadonTransform:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-cfvdwk
In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-fivbpl

Express the Radon transform of in terms of a unit vector:

In[3]:=3
✖
https://wolfram.com/xid/0n4q94l0xaq-oekbq
Out[3]=3

Verify the symmetry property:

In[4]:=4
✖
https://wolfram.com/xid/0n4q94l0xaq-ha37ep
Out[4]=4

The homogeneity property for RadonTransform:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-d93mhm
In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-j7c37j

Express the Radon transform of in terms of a unit vector:

In[3]:=3
✖
https://wolfram.com/xid/0n4q94l0xaq-jqaaax
Out[3]=3

Verify the homogeneity property:

In[4]:=4
✖
https://wolfram.com/xid/0n4q94l0xaq-bthe14
In[5]:=5
✖
https://wolfram.com/xid/0n4q94l0xaq-jh681g
Out[5]=5

The scaling property for RadonTransform:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-c97bf
In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-eh49oo

Express the Radon transform of in terms of a unit vector:

In[3]:=3
✖
https://wolfram.com/xid/0n4q94l0xaq-lq6h0
Out[3]=3

Express the Radon transform of in terms of a unit vector:

In[4]:=4
✖
https://wolfram.com/xid/0n4q94l0xaq-bfhyrf
Out[4]=4

Verify the scaling property:

In[5]:=5
✖
https://wolfram.com/xid/0n4q94l0xaq-gqr09
Out[5]=5

RadonTransform of derivatives:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-ihv0f7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-bopv7y
Out[2]=2

RadonTransform of the Laplacian:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-cwymlo
Out[1]=1

RadonTransform can be computed using Fourier transforms:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-1zbsk

Compute the Fourier transform of f in polar coordinates:

In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-hwlt02
Out[2]=2

Compute the inverse Fourier transform to obtain the Radon transform:

In[3]:=3
✖
https://wolfram.com/xid/0n4q94l0xaq-bk5b6r
Out[3]=3

Obtain the same result directly using RadonTransform:

In[4]:=4
✖
https://wolfram.com/xid/0n4q94l0xaq-c8mvr0
Out[4]=4

Neat Examples  (1)Surprising or curious use cases

Create a table of basic Radon transforms:

In[1]:=1
✖
https://wolfram.com/xid/0n4q94l0xaq-chq0g5
In[2]:=2
✖
https://wolfram.com/xid/0n4q94l0xaq-70w5xz
Wolfram Research (2017), RadonTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/RadonTransform.html.
✖
Wolfram Research (2017), RadonTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/RadonTransform.html.

Text

Wolfram Research (2017), RadonTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/RadonTransform.html.

✖
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.

✖
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

✖
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: 18-April-2025 ]}

✖
@misc{reference.wolfram_2025_radontransform, author="Wolfram Research", title="{RadonTransform}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/RadonTransform.html}", note=[Accessed: 18-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_radontransform, organization={Wolfram Research}, title={RadonTransform}, year={2017}, url={https://reference.wolfram.com/language/ref/RadonTransform.html}, note=[Accessed: 18-April-2025 ]}

✖
@online{reference.wolfram_2025_radontransform, organization={Wolfram Research}, title={RadonTransform}, year={2017}, url={https://reference.wolfram.com/language/ref/RadonTransform.html}, note=[Accessed: 18-April-2025 ]}