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.

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