November  2010, 4(4): 631-637. doi: 10.3934/ipi.2010.4.631

A local uniqueness theorem for weighted Radon transforms

1. 

Department of Mathematics, Stockholm University, SE-10691 Stockholm

Received  July 2009 Revised  November 2009 Published  November 2009

We consider a weighted Radon transform in the plane, $R_m(\xi, \eta) = \int_{\R} f(x, \xi x + \eta) m(x,\xi,\eta) dx$, where $m(x,\xi,\eta)$ is a smooth, positive function. Using an extension of an argument of Strichartz we prove a local injectivity theorem for $R_m$ for essentially the same class of $m(x,\xi,\eta)$ that was considered by Gindikin in his article in this issue.
Citation: Jan Boman. A local uniqueness theorem for weighted Radon transforms. Inverse Problems & Imaging, 2010, 4 (4) : 631-637. doi: 10.3934/ipi.2010.4.631
References:
[1]

E. V. Arbuzov, A. L. Bukhgeim and S. G. Kazantsev, Two-dimensional tomography problems and the theory of $A$-analytic functions,, Siberian Adv. Math., 8 (1998), 1.   Google Scholar

[2]

G. Bal, On the attenuated Radon transform with full and partial measurements,, Inverse Problems, 20 (2004), 399.  doi: doi:10.1088/0266-5611/20/2/006.  Google Scholar

[3]

J. Boman and J.-O. Strömberg, Novikov's inversion formula for the attenuated Radon transform-A new approach,, J. Geom. Anal., 14 (2004), 185.   Google Scholar

[4]

S. Gindikin, A remark on the weighted Radon transform on the plane,, Inverse Probl. Imaging, ().   Google Scholar

[5]

F. Natterer, Inversion of the attenuated Radon transform,, Inverse Problems, 17 (2001), 113.  doi: doi:10.1088/0266-5611/17/1/309.  Google Scholar

[6]

R. G. Novikov, An inversion formula for the attenuated X-ray transform,, Ark. Mat., 40 (2002), 145.  doi: doi:10.1007/BF02384507.  Google Scholar

[7]

V. Palamodov, "Reconstructive Integral Geometry,", Birkhäuser, (2004).   Google Scholar

[8]

R. S. Strichartz, Radon inversion-variations on a theme,, Amer. Math. Monthly, 89 (1982), 377.  doi: doi:10.2307/2321649.  Google Scholar

show all references

References:
[1]

E. V. Arbuzov, A. L. Bukhgeim and S. G. Kazantsev, Two-dimensional tomography problems and the theory of $A$-analytic functions,, Siberian Adv. Math., 8 (1998), 1.   Google Scholar

[2]

G. Bal, On the attenuated Radon transform with full and partial measurements,, Inverse Problems, 20 (2004), 399.  doi: doi:10.1088/0266-5611/20/2/006.  Google Scholar

[3]

J. Boman and J.-O. Strömberg, Novikov's inversion formula for the attenuated Radon transform-A new approach,, J. Geom. Anal., 14 (2004), 185.   Google Scholar

[4]

S. Gindikin, A remark on the weighted Radon transform on the plane,, Inverse Probl. Imaging, ().   Google Scholar

[5]

F. Natterer, Inversion of the attenuated Radon transform,, Inverse Problems, 17 (2001), 113.  doi: doi:10.1088/0266-5611/17/1/309.  Google Scholar

[6]

R. G. Novikov, An inversion formula for the attenuated X-ray transform,, Ark. Mat., 40 (2002), 145.  doi: doi:10.1007/BF02384507.  Google Scholar

[7]

V. Palamodov, "Reconstructive Integral Geometry,", Birkhäuser, (2004).   Google Scholar

[8]

R. S. Strichartz, Radon inversion-variations on a theme,, Amer. Math. Monthly, 89 (1982), 377.  doi: doi:10.2307/2321649.  Google Scholar

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