# Relationship between the Radon transform and Twistor spaces

I have often heard that the theory of Twistor spaces is a complex analogue" of the Radon transform. What is the precise connection ?

The correspondence between the Radon transform (from a space of real-valued functions on $\mathbb{R}^2$ to the space of functions on the manifold of straight lines in $\mathbb{R}^2$) and the Penrose transform (from $\mathbb{CP}_2$ to the dual projective space $\mathbb{CP}^\ast_2$) is easiest to work out when lines in the Radon transform are replaced by great circles on a sphere. This special spherical case of the Radon transform is called the Funk transform, and it corresponds to the Penrose transform when $\mathbb{CP}_2$ is restricted to $\mathbb{RP}_2$, see The Funk transform as a Penrose transform (1999).