Sampling linear inverse problems with noise

We study the effect of additive noise to the inversion of FIOs associated to a diffeomorphic canonical relation. We use the microlocal defect measures to measure the power spectrum of the noise in the phase space and analyze how that power spectrum is transformed under the inversion. In general, white noise, for example, is mapped to noise depending on the position and on the direction. In particular, we compute the standard deviation, locally, of the noise added to the inversion as a function of the standard deviation of the noise added to the data. As an example, we study the Radon transform in the plane in parallel and fan-beam coordinates, and present numerical examples.