Radon transform
From Academic Kids
In mathematics, the Radon transform in two dimensions is the integral transform
- <math>\mathcal{R} \left\{ f(x,y) \right\} = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,y) \delta(y-(mx+b)) \, dx \, dy.<math>
The Radon transform integrates a function over lines in the plane, mapping a function of position to a function of the slope and the y-intercept.
This transform in two dimensions and three dimensions (where a function is integrated over planes) was introduced in a 1917 paper by Johann Radon, who provided formulae for the inverse transform (reconstruction problem). It was later generalised, in the context of integral geometry.
A discrete Radon transform is a Hough transform.
The Radon transform is useful in computed axial tomography (CAT scan). In the 2D case
- <math>P_m: b \mapsto \mathcal{R} \left\{ f(x,y) \right\}(m,b)<math>
is the 1D projection of <math>f<math> along the direction
- <math>y=mx<math>
and we want to reconstruct the 2D image <math>f<math> from all the 1D projections <math>P_m<math>.
A less computationally-intensive algorithm for reconstructing from the sinogram is the filtered back-projection.
See also
External link
- MathWorld page (http://mathworld.wolfram.com/RadonTransform.html)de:Radon-Transformation
