Chemistry - Radon transform

In mathematics, the Radon transform in two dimensions is the integral transform

$\mathcal{R} \left\{ f(x,y) \right\} = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,y) \delta(y-(mx+b)) \, dx \, dy.$

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

$P_m: b \mapsto \mathcal{R} \left\{ f(x,y) \right\}(m,b)$

is the 1D projection of $f$ along the direction

y = m x

and we want to reconstruct the 2D image $f$ from all the 1D projections $P m$ .

A less computationally-intensive algorithm for reconstructing from the sinogram is the filtered back-projection.

External link

MathWorld page

Categories: Integral geometry | Integral transforms

Radon transform

See also

External link