The Radon transform

(Note that the improper integral converges.) Here the integral denotes the standard line integral from vector calculus. In higher dimensions the Radon transform maps a function f to its integrals over hyperplanes. It is defined by the same formula (1), but now θ ∈ Sn−1, and the integration domain θ · x = s is a hyperplane. The Radon transformed is named after the Austrian mathematician Johann Radon, who studied it in a paper that appeared in 1917. In particular he proved several inversion formulas in his paper. The two-dimensional Radon transform has application in medical imaging, in particular in X-ray computed tomography. In an X-ray scanner, a beam of X-ray radiation is generated that passes through an object, see Figure 1. At the other side of the object a line of detectors is located that measures the intensity I of the rays that have passed through the object. The beam and the detectors can be rotated, such that the intensity is measured for all lines passing through the object, hence I = I(θ, s). Depending on the material inside the object, some of the X-ray radiation is absorbed. When an X-ray travels over a small distance ∆x in a medium with absorbtion coefficient f(x), the intensity change is ∆I = −f(x)I∆x. This leads to a differential equation, if x(t) is a parametrization of a line, with ∣∣dx/dt∣∣ = 1, the equation is dI dt = −f(x(t))I with solution I(t) = e− ∫ t I0.