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

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.