Inverse Radon transforms through inverse wavelet transforms. (English) Zbl 0751.35049

Holschneider, M.

Inverse Probl. 7, No. 6, 853-861 (1991).

Let \(\Lambda\) be a locally compact group and \(U\) be a representation of \(\Lambda\) in the space \({\mathcal O}(L^ 2(\mathbb{R}^ n))\) of unitary operators acting on \(L^ 2(\mathbb{R}^ n)\). Given the “wavelet function” \(g\) and an arbitrary function (or distribution) \(s\) denote by \(W_ gs(\lambda)=\langle U(\lambda)g\mid s\rangle\), \(\lambda\in\Lambda\), “projections” of \(s\) onto the states \(g_ \lambda=U(\lambda)g\).
The following problem is considered. Given all projections \(W _ gs(\lambda)\), how to recover \(s\)? The solution to this problem is given for the case when \(\Lambda\) is the group of translations, dilations and rotations on the plane \(\mathbb{R}^ 2\) and the wavelet \(g\) may be a distribution. The Radon transform on \(\mathbb{R}^ 2\) can be interpreted as the wavelet transform with the singular analyzing wavelet \(g\) which is a delta distribution on a line. By using this fact the author inverts the Radon data by means of the wavelet-inversion formula.

Reviewer: B.Rubin (Jerusalem)

MSC:

35R30	Inverse problems for PDEs
44A12	Radon transform

Keywords:

wavelet translations; dilations and rotations; wavelet-inversion formula