Radiation imaging techniques are important tools in many fields of basic and applied sciences, from medical analyses to industrial applications. In addition to the conventional photon transmission tomography, methods based on the Compton scattering and X-ray dim-action have been developed in the last ten years. In this paper, we discuss some inverse problems relevant to the image recovering of scattering and diffraction tomography. In particular, the problem of recovering the X-ray spectrum from measured data is discussed in detail and it is shown that the original signal from both Nal(Tl) and HP-Ge spectrometers characterized by very different energy resolution, can be uniquely reconstructed. © Springer-Verlag Berlin Heidelberg 1997.