An inversion formula for the attenuated \(X\)-ray transformation
In this paper the author deals with inverse scattering for the attenuated \(X\)-ray equation. Let \(P_a f(x,\theta)=\int_R \exp(-D_a(x+s\theta))f(x+s\theta)\,ds,\) be the attenuated \(X\)-ray transformation, where \(a,f\) are sufficiently regular functions on \(\mathbb{R}^d\), sufficiently rapidly vanishing at infinity, and \(D_a(x)=\int_0^{+\infty} a(x+s\theta)\,ds,\) \(x\in \mathbb{R}^d, \theta\in S^{d-1}\). The problem of finding \(f| _Y\) from \(P_af| _{T {S}^1(Y)}\) is considered under the assumption that \(a| _Y\) is known, where \(Y\) is a two-dimensional plane in \( \mathbb{R}^d\), \(d\geq 2\), and \(T {S}^1(Y)\) is the set of all oriented straight lines lying in \(Y\). This is an inverse scattering problem for the equation \(\sum \theta_i\partial/\partial x_i \psi(x,\theta) +a(x)\psi(x,\theta)=f(x)\). For this problem, the case \(d\geq 3\) is reduced to the case \(d=2\). In this case, under some assumptions on the regularity of \(a\) and \(f\), and under the condition that \(a\) is known, \(P_af\) on \(T {S}^1\) uniquely determines \(f\) on \( \mathbb{R}^2\), and the inversion formulae are explicitly given. This inversion method implies an explicit scheme by which \(P_af\) on \(\mathcal{M} (S)=\bigcup_{y\in X^{\perp}(S)}T {S}^1(X(S)+y)\) determines \(f\) on \(\mathbb{R}^d\), \(d\geq 2\), uniquely where \(S\) is a great circle in \( {S}^{d-1}\) and \(X(S)\) is the linear span of \(S\) in \( \mathbb{R}^d\). Some generalizations, and several interesting subsequent results are also given.
scientific article; zbMATH DE number 1960563