Inversion formulas for the spherical Radon transform and the generalized cosine transform

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In this Subject: <a title="Mathematics and Statistics" href="/content/subcat?j_subject=195&j_availability=">Mathematics and Statistics
By this author: Rubin B.

Author: Rubin B.

Source: Advances in Applied Mathematics, Volume 29, Number 3, October 2002 , pp. 471-497(27)

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Abstract:

The k-dimensional totally geodesic Radon transform on the unit sphereSⁿ and the corresponding cosine transform can be regarded as members of the analytic family of intertwining fractional integrals\left( {R^{\scriptscriptstyle{{\alpha }}}f}\right) \left( \xi \right) =\gamma _{\scriptscriptstyle{{n,k}}}\left( \alpha \right) {\underset{\scriptstyle{{S^{\scriptscriptstyle{{n}}}}}}{\int }}f\left( x\right) \left( {\mathrm{{sin}}\left[ {d\left( x,\xi \right) }\right] }\right) ^{\scriptscriptstyle{{\alpha +k-n}}}\hspace{0.2em}\mathrm{{d}}x, d(x,) being the geodesic distance betweenx isin Sⁿ and thek-geodesic . We develop a unified approach to the inversion ofRf for all alpha ges 0, 1 les k les n-1, n ges 2. The cases of smooth f andf isin L^p are considered. A series of new inversion formulas is obtained. The convolution–backprojection method is developed.