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 &fnof; be a continuous function on the 2-sphere S2 in R3. Then, for a unit vector x, let


 * $$Ff(\mathbf{x}) = \int_{\mathbf{u}\in C(\mathbf{x})} f(\mathbf{u})\,ds(\mathbf{u})$$

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:


 * $$C(\mathbf{x}) = \{\mathbf{u}\in S^2\mid \mathbf{u}\cdot\mathbf{x}=0\}.$$

Inversion
The Funk transform annihilates all odd functions, and so it is natural to confine attention to the case when &fnof; 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 $$f\in L^2 (S^2)$$ on the sphere can be decomposed into spherical harmonics $$Y_n^k$$


 * $$f = \sum_{n=0}^{\infty} \sum_{k=-n}^n \hat f (n,k) Y_n^k.$$

Then the Funk transform of f reads


 * $$F f = \sum_{n=0}^{\infty} \sum_{k=-n}^n P_n(0) \hat f (n,k) Y_n^k$$

where $$P_{2n+1}(0)=0$$ for odd values and
 * $$P_{2n}(0) = (-1)^n\, \frac{1\cdot 3\cdot 5\cdots 2n-1}{2\cdot 4\cdot 6 \cdots 2n}

= (-1)^n\, \frac{(2n-1)!!}{(2n)!!}$$ 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


 * $$(F^*f)(p,\mathbf{x}) = \frac{1}{2\pi\cos p}\int_{\|\mathbf{u}\|=1,\mathbf{x}\cdot\mathbf{u}=\sin p} f(\mathbf{u})\,|d\mathbf{u}|.$$

This is the average value of the circle function &fnof; over circles of arc distance p from the point x. The inverse transform is given by


 * $$f(\mathbf{x}) = \frac{1}{2\pi}\left\{\frac{d}{du}\int_0^u F^*(Ff)(\cos^{-1}v,\mathbf{x})v(u^2-v^2)^{-1/2}\,dv\right\}_{u=1}.$$

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). Suppose that &fnof; is a homogeneous function of degree &minus;2 on R3. Then, for linearly independent vectors x and y, define a function φ by the line integral


 * $$\varphi(\mathbf{x},\mathbf{y}) = \frac{1}{2\pi}\oint f(u\mathbf{x} + v\mathbf{y})(u\,dv-v\,du)$$

taken over a simple closed curve encircling the origin once. The differential form


 * $$f(u\mathbf{x} + v\mathbf{y})(u\,dv-v\,du)$$

is closed, which follows by the homogeneity of &fnof;. By a change of variables, φ satisfies


 * $$\phi(a\mathbf{x}+b\mathbf{y},c\mathbf{x}+d\mathbf{y}) = \frac{1}{|ad-bc|}\phi(\mathbf{x},\mathbf{y}),$$

and so gives a homogeneous function of degree &minus;1 on the exterior square of R3,


 * $$Ff(\mathbf{x}\wedge\mathbf{y}) = \phi(\mathbf{x},\mathbf{y}).$$

The function F&fnof; : Λ2R3 → R agrees with the Funk transform when &fnof; is the degree &minus;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 &minus;2 on R3\{0} to smooth even homogeneous functions of degree &minus;1 on R3\{0}.

Applications
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 $$K\subset \mathbb R^d$$ be a star body with radial function $$\rho_K(\boldsymbol x)=\max\{t:t\boldsymbol x\in K\},$$ $$x\in S^{d-1}$$. Then the intersection body IK of K has the radial function $$\rho_{IK}=F\rho_K$$.