Noise propagation in linear and nonlinear inverse scattering.

The propagation of noise from the data to the reconstructed speed of sound image by inverse scattering within the framework of the Lippmann-Schwinger integral equation of scattering is discussed. The inversion algorithm that was used consisted in minimizing a Tikhonov functional in the unknown speed of sound. The gradient of the objective functional was computed by the method of the adjoint fields. An analytical expression for the inverse scattering covariance matrix of the image noise was derived. It was shown that the covariance matrix in the linear x-ray computed tomography is a special case of the inverse scattering matrix derived in this paper. The matrix was also analyzed in the limit of the linearized Born approximation, and the results were found to be in qualitative agreement with those recently reported in the literature for Born inversion using filtered backpropagation algorithm. Finally, the applicability of the analysis reported here to the obstacle problem and the physical optics approximation was discussed.