Sampling conditions of the exponential ray transform

It has been long shown in 2D, using the principle of stationary phase that the attenuated ray transform is negligible outside of the famous bow tie shape of the support of the Fourier transform of the X-ray transform. However, this result in not strong enough to state sampling conditions for the attenuated ray transform and to control the Fourier interpolation error. In this paper, the authors concentrate on the exponential ray transform (constant attenuation). They give a sketch of the formal proof that the integral of the Fourier transform absolute value of the exponential ray transform outside the bow tie shape is negligible. The results is based on the Debye's formula for the Bessel functions of the first kind and on Natterer's approach. It needs only slightly different assumptions compared to the classical X-ray transform. It can be extended to 3D tomography for parallel geometry.