A unified approach to inversion formulae for vector and tensor ray and radon transforms and the Natterer inequality

Abstract

Most derivations of inversion formulae for x-ray or Radon transform are based on the projection theorem, where for fixed direction the Fourier transform of x-ray or Radon transform is calculated and compared with the Fourier transform of the searched-for function. In contrast to this we start here off from the searched-for field, calculate its Fourier transform for fixed direction, which is now a vector or tensor field, that we then expand in a suitable direction dependent basis. The expansion coefficients are recognized as the Fourier transform of longitudinal, transversal or mixed ray transforms or vectorial Radon transform respectively. The inverse Fourier transform of the searched-for field then directly leads to inversion formulae for those transforms applying problem adapted backprojections. When considering the Helmholtz decomposition of the field we immediately find inversion formulae for those transversal or longitudinal transforms. First inversion formulae for the longitudinal ray transform, similar to those given by Natterer (1986 The Mathematics of Computerized Tomography (Teubner and Wiley)) for x-ray tomography, were given by Natterer-Wübbeling in 2001, Natterer and Wübbeling (2001 Mathematical Methods in Image Reconstruction (SIAM)), but then not pursued by other authors. In this paper, we present the above described method and derive in a unified way inversion formulae for the ray transforms treated in Louis (2022 Inverse Problems 38 065008) containing the results from Louis (2022 Inverse Problems 38 065008) as special cases. Additionally we present new inversion formulae for the vectorial Radon transform. As a consequence the inversion formulae directly give Plancherel’s formulae for the vectorial or tensorial transforms. Together with the Natterer inequality, which is independent of the ray or Radon transforms, we present the Natterer stability of those vectorial and tensorial transforms.