RadonTransform

RadonTransform, named after Johann Radon, is an integral transform that assigns to a function defined on the plane its line integrals along all possible straight lines. In two dimensions, if f is a function on R^2, its Radon transform Rf is defined by Rf(θ, s) = ∫_{−∞}^{∞} f(s cos θ − t sin θ, s sin θ + t cos θ) dt, where θ ∈ [0, π) specifies the line orientation and s ∈ R is the signed distance of the line from the origin. Equivalently, Rf(θ, s) is the integral of f over the line x cos θ + y sin θ = s.

The transform is central to computed tomography, where projections Rf(θ, s) are measured for many angles and used to recover the image f. Under suitable smoothness and decay conditions, f can be reconstructed from Rf by inversion formulas. A common form is f(x, y) = −(1/2π) ∫_{0}^{π} ∂/∂s Rf(θ, s) |_{s = x cos θ + y sin θ} dθ. In practice, reconstruction is often performed by filtered back projection, which applies a ramp filter to Rf in the s-variable and then back-projects over all angles.

The Radon transform satisfies the Fourier slice theorem: the two-dimensional Fourier transform of f is given by the one-dimensional Fourier transform of Rf along radial lines. The transform has well-developed range descriptions and generalizations to higher dimensions, where it integrates over hyperplanes.

Beyond medical imaging, the Radon transform is used in geophysics, astronomy, and materials science, as a mathematical tool for inferring internal structure from projection data. Variants include discrete, limited-angle, and generalized Radon transforms, each with distinct inversion and stability properties.