In the mathematical field of integral geometry, the Funk transform (also known as MinkowskiâÂÂFunk transform, FunkâÂÂRadon transform or spherical Radon transform) is an integral transform defined by integrating a function on great circles of the sphere. It was introduced by Paul Funk in 1911, based on the work of . It is closely related to the Radon transform. The original motivation for studying the Funk transform was to describe Zoll metrics on the sphere.
The Funk transform is defined as follows. Let ƒ be a continuous function on the d-1-sphere S<sup>d-1</sup> in R<sup>d</sup>. Then, for a unit vector x, let
where the integral is carried out with respect to the arclength ds of the great circle C(x) consisting of all unit vectors perpendicular to x:
The Funk transform annihilates all odd functions, and so it is natural to confine attention to the case when ƒ is even. In that case, the Funk transform takes even (continuous) functions to even continuous functions, and is furthermore invertible.
Every square-integrable function on the sphere can be decomposed into spherical harmonics
Then the Funk transform of f reads
where for odd values and
for even values. This result was shown by .
Another inversion formula is due to . As with the Radon transform, the inversion formula relies on the dual transform F* defined by
This is the average value of the circle function ƒ over circles of arc distance p from the point x. The inverse transform is given by
The classical formulation is invariant under the rotation group SO(3). It is also possible to formulate the Funk transform in a manner that makes it invariant under the special linear group SL(3,R) . Suppose that ƒ is a homogeneous function of degree −2 on R<sup>3</sup>. Then, for linearly independent vectors x and y, define a function ÃÂ by the line integral
taken over a simple closed curve encircling the origin once. The differential form
is closed, which follows by the homogeneity of ƒ. By a change of variables, ÃÂ satisfies
and so gives a homogeneous function of degree −1 on the exterior square of R<sup>3</sup>,
The function Fƒ : ÃÂ<sup>2</sup>R<sup>3</sup> â R agrees with the Funk transform when ƒ is the degree −2 homogeneous extension of a function on the sphere and the projective space associated to ÃÂ<sup>2</sup>R<sup>3</sup> is identified with the space of all circles on the sphere. Alternatively, ÃÂ<sup>2</sup>R<sup>3</sup> can be identified with R<sup>3</sup> in an SL(3,R)-invariant manner, and so the Funk transform F maps smooth even homogeneous functions of degree −2 on R<sup>3</sup>\{0} to smooth even homogeneous functions of degree −1 on R<sup>3</sup>\{0}.
The Funk-Radon transform is used in the Q-Ball method for Diffusion MRI introduced by . It is also related to intersection bodies in convex geometry. Let be a star body with radial function . Then the intersection body IK of K has the radial function .