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In the [[mathematics|mathematical]] field of [[integral geometry]], the '''Funk transform''' (also called '''Minkowski–Funk transform''') is an [[integral transform]] defined by integrating a [[function (mathematics)|function]] on [[great circle]]s of the [[sphere]].  It was introduced by [[Paul Funk]] in 1916, based on the work of {{harvtxt|Minkowski|1904}}.  It is closely related to the [[Radon transform]].  The original motivation for studying the Funk transform was to describe [[Zoll surface|Zoll metrics]] on the sphere.


Classically, the Funk transform is defined as follows.  Let ''&fnof;'' be a [[continuous function]] on the 2-sphere in '''R'''<sup>3</sup>.  Then, for a [[unit vector]] '''x''', let


:<math>Ff(\mathbf{x}) = \int_{\mathbf{u}\in C(\mathbf{x})} f(\mathbf{u})\,ds(\mathbf{u})</math>
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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'':
 
:<math>C(\mathbf{x}) = \{\mathbf{u}\in S^2\mid \mathbf{u}\cdot\mathbf{x}=0\}.</math>
 
Clearly, the Funk transform annihilates all [[odd function]]s, 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.  As with the Radon transform, the inversion formula relies on the dual transform, defined by
 
:<math>(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}|.</math>
 
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
 
:<math>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}.</math>
 
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 {{harv|Bailey|Eastwood|Gover|Mason|2003}}. Suppose that ''&fnof;'' is a [[homogeneous function]] of degree &minus;2 on '''R'''<sup>3</sup>.  Then, for [[linear independence|linearly independent]] vectors '''x''' and '''y''', define a function φ by the [[line integral]]
 
:<math>\varphi(\mathbf{x},\mathbf{y}) = \frac{1}{2\pi}\oint f(u\mathbf{x} + v\mathbf{y})(u\,dv-v\,du)</math>
 
taken over a simple closed curve encircling the origin once. The [[differential form]]
 
:<math>f(u\mathbf{x} + v\mathbf{y})(u\,dv-v\,du)</math>
 
is [[closed and exact differential forms|closed]], which follows by the homogeneity of ''&fnof;''.  By a [[integration by substitution|change of variables]], φ satisfies
 
:<math>\phi(a\mathbf{x}+b\mathbf{y},c\mathbf{x}+d\mathbf{y}) = \frac{1}{|ad-bc|}\phi(\mathbf{x},\mathbf{y}),</math>
 
and so gives a homogeneous function of degree &minus;1 on the [[exterior algebra|exterior square]] of '''R'''<sup>3</sup>,
 
:<math>Ff(\mathbf{x}\wedge\mathbf{y}) = \phi(\mathbf{x},\mathbf{y}).</math>
 
The function ''F&fnof;''&nbsp;:&nbsp;Λ<sup>2</sup>'''R'''<sup>3</sup>&nbsp;→&nbsp;'''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 Λ<sup>2</sup>'''R'''<sup>3</sup> is identified with the space of all circles on the sphere. Alternatively, Λ<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 &minus;2 on '''R'''<sup>3</sup>\{0} to smooth even homogeneous functions of degree &minus;1 on '''R'''<sup>3</sup>\{0}.
 
==See also==
* [[Spherical mean]]
 
==References==
*{{Citation | doi=10.4134/JKMS.2003.40.4.577 | last1=Bailey | first1=T. N. | last2=Eastwood | first2=Michael G. | last3=Gover | first3=A. Rod | last4=Mason | first4=L. J. | title=Complex analysis and the Funk transform | mr=1995065 | year=2003 | journal=Journal of the Korean Mathematical Society | volume=40 | issue=4 | pages=577–593|url=http://mathnet.kaist.ac.kr/mathnet/kms_tex/980815.pdf}}
*{{Citation | last1=Dann | first1=Susanna | title=On the Minkowski-Funk Transform | arxiv=1003.5565 | year=2010}}
*{{Citation | doi=10.1007/BF01456824 | last1=Funk | first1=Paul | title=Über eine geometrische Anwendung der Abelschen Integralgleichung | mr=1511851 | year=1915 | journal=[[Mathematische Annalen]] | volume=77 | issue=1 | pages=129–135}}.
*{{Citation | doi=10.1016/0001-8708(76)90139-0 | last1=Guillemin | first1=Victor | title=The Radon transform on Zoll surfaces | mr=0426063 | year=1976 | journal=Advances in Mathematics | volume=22 | issue=1 | pages=85–119}}.
*{{Citation | last1=Helgason | first1=Sigurdur | authorlink1=Sigurdur Helgason (mathematician)|title=The Radon transform | publisher=Birkhäuser Boston | location=Boston, MA | edition=2nd | series=Progress in Mathematics | isbn=978-0-8176-4109-2 | mr=1723736 | year=1999 | volume=5}}.
*{{Citation | last1=Minkowski | first1=Hermann | author1-link=Hermann Minkowski | title=About bodies of constant width | year=1904 | journal=Mathematics Sbornik | volume=25 | pages=505–508}}
 
[[Category:Integral geometry]]
[[Category:Integral transforms]]

Revision as of 03:47, 17 February 2014


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