Nonlocal inversion formulas for the X-ray transform
The authors consider the problem of reconstructing a function f on \({\mathbb{R}}^ n\) from its integrals over k-dimensional planes. Let \(M_{k,n}\) be the \((k+1)(n-k)\)-dimensional bundle of affine k-planes in \({\mathbb{R}}^ n\), and define the k-plane transform \({\mathcal R}_{k,n}f(\pi)=\int_{\pi}f(x)d\mu_{\pi}(x),\) \(\pi \in M_{k,n}\), with the Lebesgue measure \(\mu\) on \(\pi\). Because of the overdeterminedness of \({\mathcal R}_{k,n}\) for \(1\leq k<n-1\), they ask which n-dimensional submanifolds \({\mathcal C}\subset M_{k,n}\) allow an inversion giving f in terms of \({\mathcal R}_{k,n}| {\mathcal C}\). A partial solution to this problem is given in the case \(k=1\). The authors use the framework of double filtration and its microlocal analogue and look at two diagrams for \({\mathcal C}\) with the natural projections \(\pi\) and \(\rho\), the point line relation \(Z\subset {\mathbb{R}}^ n\times {\mathcal C}\) and the conormal bundle \(C'=N^*Z:* \pi: Z\to {\mathbb{R}}^ n\), \(\rho\) : \(Z\to {\mathcal C}\), \(\pi\) : \(C\to T^*{\mathbb{R}}^ n\), \(\rho\) : \(C\to T^*{\mathcal C}\). The restricted X-ray transform \({\mathcal R}_{{\mathcal C}}\) is a Fourier integral operator associated with the 2n-dimensional canonical relation C. For certain admissible geodesic complexes \({\mathcal C}\) the main theorem states the existence of a relative left parametric for \({\mathcal R}_{{\mathcal C}}\) modulo \(C^{\infty}\) and an error term B. Examples for these complexes are given and an analogue of Kirillov's theorem is found. However, there exists an important class of line complexes in \({\mathbb{R}}^ n\) that fall outside the scope of the main theorem. In order to obtain a left inverse for the authors consider the operator \({\mathcal R}_{{\mathcal C}}t\circ {\mathcal R}_{{\mathcal C}}\). This operator has wave front set contained in the union of the diagonal \(\Delta\) of \(T^*{\mathbb{R}}^ n\) and the flowout \(\Lambda\) of \(\pi\) (L) where \(L\subset C\) is the fold. \({\mathcal R}^ t_{{\mathcal C}}\circ {\mathcal R}_{{\mathcal C}}\) is an elliptic element of the class I(\(\Delta\),\(\Lambda)\) of Fourier integral distributions, which allows to obtain a relative left parametrix for \({\mathcal R}^ t_{{\mathcal C}}\circ {\mathcal R}_{{\mathcal C}}\), and hence for \({\mathcal R}_{{\mathcal C}}\), modulo a classical Fourier integral operator in \(I^{-1/2}(\Lambda)\), acting on distributions with wave front set contained in \(\pi\) (C). Finally, extensions to \(S^ n\) are indicated.
scientific article; zbMATH DE number 4094126
scientific article