Funk transform

In the mathematical field of integral geometry, the Funk transform (also known as Minkowski–Funk transform, Funk–Radon 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.
Definition.
The Funk transform is defined as follows. Let "ƒ" be a continuous function on the 2-sphere S2 in R3. 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:
Inversion.
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.
Spherical harmonics.
Every square-integrable function formula_3 on the sphere can be decomposed into spherical harmonics formula_4
Then the Funk transform of "f" reads
where formula_7 for odd values and 
for even values. This result was shown by .
Helgason's inversion formula.
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
Generalization.
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), due to . Suppose that "ƒ" is a homogeneous function of degree −2 on R3. Then, for linearly independent vectors x and y, define a function φ 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, φ satisfies
and so gives a homogeneous function of degree −1 on the exterior square of R3,
The function "Fƒ" : Λ2R3 → 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 Λ2R3 is identified with the space of all circles on the sphere. Alternatively, Λ2R3 can be identified with R3 in an SL(3,R)-invariant manner, and so the Funk transform "F" maps smooth even homogeneous functions of degree −2 on R3\{0} to smooth even homogeneous functions of degree −1 on R3\{0}.
Applications.
The Funk-Radon transform is used in the Q-Ball method for Diffusion MRI introduced in .
It is also related to intersection bodies in convex geometry. 
Let formula_15 be a star body with radial function formula_16 formula_17.
Then the intersection body "IK" of "K" has the radial function formula_18, see .