Nonlocal inversion formulas for the X-ray transform
type: Item
instance of: scholarly article
title: Nonlocal inversion formulas for the X-ray transform
zbMATH Open document ID: 0668.44004
DOI: 10.1215/S0012-7094-89-05811-0
author: Allan Greenleaf
author: Gunther Uhlmann
published in: Duke Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: 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 \(L\subset C\) i…
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 47Gxx
Mathematics Subject Classification ID: 46F12
Mathematics Subject Classification ID: 42A38
zbMATH DE Number: 4094126
zbMATH Keywords: nonlocal inversion formulas
zbMATH Keywords: function reconstruction
zbMATH Keywords: double filtration
zbMATH Keywords: X-ray transform
zbMATH Keywords: Fourier integral operator
zbMATH Keywords: wave front set
zbMATH Keywords: relative left parametrix
zbMATH Keywords: distributions
reviewed by: Wolfram Luther
MaRDI profile type: Publication
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full work available at URL: s0012-7094-89-05811-0
OpenAlex ID: W2001541232
description: scientific article; zbMATH DE number 4094126