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
    GenerateConditionsFalsewhether 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

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Basic 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:

Introduced in 2017
 (11.2)