Radon Transform - Dual Transform

Dual Transform

The dual Radon transform is a kind of adjoint to the Radon transform. Beginning with a function g on the space Σ_n, the dual Radon transform is the function R∗g on Rn defined by

The integral here is taken over the set of all lines incident with the point x ∈ Rn, and the measure dμ is the unique probability measure on the set invariant under rotations about the point x.

Concretely, for the two-dimensional Radon transform, the dual transform is given by

In the context of image processing, the dual transform is commonly called backprojection as it takes a function defined on each line in the plane and 'smears' or projects it back over the line to produce an image. Computationally efficient inversion formulas reconstruct the image from the points where the back-projection lines meet.