Radon Transform

In mathematics, the Radon transform in two dimensions, named after the Austrian mathematician Johann Radon, is the integral transform consisting of the integral of a function over straight lines. The transform was introduced by Johann Radon (1917), who also provided a formula for the inverse transform. Radon further included formulas for the transform in three-dimensions, in which the integral is taken over planes. It was later generalised to higher-dimensional Euclidean spaces, and more broadly in the context of integral geometry. The complex analog of the Radon transform is known as the Penrose transform.

The Radon transform is widely applicable to tomography, the creation of an image from the scattering data associated with cross-sectional scans of an object. If a function ƒ represents an unknown density, then the Radon transform represents the scattering data obtained as the output of a tomographic scan. Hence the inverse of the Radon transform can be used to reconstruct the original density from the scattering data, and thus it forms the mathematical underpinning for tomographic reconstruction, also known as image reconstruction. The Radon transform data is often called a sinogram because the Radon transform of a Dirac delta function is a distribution supported on the graph of a sine wave. Consequently the Radon transform of a number of small objects appears graphically as a number of blurred sine waves with different amplitudes and phases. The Radon transform is useful in computed axial tomography (CAT scan), barcode scanners, electron microscopy of macromolecular assemblies like viruses and protein complexes, reflection seismology and in the solution of hyperbolic partial differential equations.

Let ƒ(x) = ƒ(x,y) be a continuous function vanishing outside some large disc in the Euclidean plane R2. The Radon transform, Rƒ, is a function defined on the space of straight lines L in R2 by the line integral along each such line:

Concretely, any straight line L can be parametrized by

where s is the distance of L from the origin and is the angle the normal vector to L makes with the x axis (see also Hough space). It follows that the quantities (α,s) can be considered as coordinates on the space of all lines in R2, and the Radon transform can be expressed in these coordinates by

More generally, in the n-dimensional Euclidean space Rn, the Radon transform of a compactly supported continuous function ƒ is a function Rƒ on the space Σ_n of all hyperplanes in Rn. It is defined by

for ξ ∈Σ_n, where the integral is taken with respect to the natural hypersurface measure, dσ (generalizing the |dx| term from the 2-dimensional case). Observe that any element of Σ_n is characterized as the solution locus of an equation

where α ∈ Sn−1 is a unit vector and s ∈ R. Thus the n-dimensional Radon transform may be rewritten as a function on Sn−1×R via

It is also possible to generalize the Radon transform still further by integrating instead over k-dimensional affine subspaces of Rn. The X-ray transform is the most widely used special case of this construction, and is obtained by integrating over straight lines.