Inversion formula and singularities of the solution for the backprojection operator in tomography

Author:
A. G. Ramm

Journal:
Proc. Amer. Math. Soc. **124** (1996), 567-577

MSC (1991):
Primary 44A15, 45P05

DOI:
https://doi.org/10.1090/S0002-9939-96-03155-3

MathSciNet review:
1301044

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Abstract | References | Similar Articles | Additional Information

Abstract: Let $R^\ast \mu := \int _{S^2} \mu (\alpha , \alpha \cdot x) d\alpha$, $x \in {\mathbb {R}}^n$, be the backprojection operator. The range of this operator as an operator on non-smooth functions $R^\ast : X:=L^\infty _0 (S^{n-1} \times {\mathbb {R}}) \to L_{\mathrm {loc}}^2 ({\mathbb {R}}^n)$ is described and formulas for $(R^\ast )^{-1}$ are derived. It is proved that the operator $R^\ast$ is not injective on $X$ but is injective on the subspace $X_e$ of $X$ which consists of even functions $\mu (\alpha , p) = \mu (-\alpha , -p)$. Singularities of the function $(R^\ast )^{-1} h$ are studied. Here $h$ is a piecewise-smooth compactly supported function. Conditions for $\mu$ to have compact support are given. Some applications are considered.

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Additional Information

**A. G. Ramm**

Affiliation:
Department of Mathematics, Kansas State University, Manhattan, Kansas 66506-2602;
Los Alamos National Laboratory, Los Alamos, New Mexico 87545

Email:
ramm@math.ksu.edu

Keywords:
Tomography,
range,
inversion formulas,
backprojection

Received by editor(s):
May 10, 1994

Received by editor(s) in revised form:
September 12, 1994

Additional Notes:
The author thanks NSF and LANL for support, Complutense University, Madrid, for hospitality, and A. Katsevich for discussions.

Communicated by:
David Sharp

Article copyright:
© Copyright 1996
American Mathematical Society