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.
CITATION STYLE
Ramm, A. (1996). Inversion formula and singularities of the solution for the backprojection operator in tomography. Proceedings of the American Mathematical Society, 124(2), 567–577. https://doi.org/10.1090/s0002-9939-96-03155-3
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