Noise in linear inverse problems

Abstract: We study how noise in the data affects the noise in the reconstruction, for linear inverse problems, more precisely when the operator we have to invert is a Fourier Integral Operator. We apply the results to the Radon transform in the plane in parallel and in fan-bean coordinates. In this talk, we concentrate on additive noise, assuming that it is white but the methods apply to non-white noise as well. We propose the microlocal defect measure as a measure of the spectral power of the noise in the phase space. We show that one can compute the spectral power of the noise in the reconstruction, including its standard deviation, as a function of the known statistical characteristics of the input noise. For the Radon transform in parallel geometry, we show that the induced noise is position independent, isotropic, and “blue”. In fan-bean coordinates, the noise varies with position and it is not isotropic anymore but still “blue”. This dependence is weak however and the standard deviation which we compute, still gives a good characterization of the strength of the induced noise. This is a joint project, still in progress, with Samy Tindel, Purdue.

Mathematics

Audience: researchers in the topic

International Zoom Inverse Problems Seminar, UC Irvine