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

Abstract

Let R ∗ μ := ∫ S 2 μ ( α , α ⋅ x ) d α R^\ast \mu := \int _{S^2} \mu (\alpha , \alpha \cdot x) d\alpha , x ∈ R n x \in {\mathbb {R}}^n , be the backprojection operator. The range of this operator as an operator on non-smooth functions R ∗ : X := L 0 ∞ ( S n − 1 × R ) → L l o c 2 ( R n ) 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 ∗ ) − 1 (R^\ast )^{-1} are derived. It is proved that the operator R ∗ R^\ast is not injective on X X but is injective on the subspace X e X_e of X X which consists of even functions μ ( α , p ) = μ ( − α , − p ) \mu (\alpha , p) = \mu (-\alpha , -p) . Singularities of the function ( R ∗ ) − 1 h (R^\ast )^{-1} h are studied. Here h h is a piecewise-smooth compactly supported function. Conditions for μ \mu to have compact support are given. Some applications are considered.