Spectral methods for passive imaging: Nonasymptotic performance and robustness

We study the problem of passive imaging through convolutive channels. A scene is illuminated with an unknown, unstructured source, and the measured response is the convolution of this source with multiple channel responses, each of which is time-limited. Spectral methods based on the commutativity of convolution, first proposed and analyzed in the 1990s, provide an elegant mathematical framework for attacking this problem. However, these now classical methods are very sensitive to noise, especially when working from relatively small sample sizes. In this paper, we show that a linear subspace model on the coefficients of the impulse responses of the channels can make this problem well-posed. We derive nonasymptotic error bounds for the generic subspace model by analyzing the spectral gap of the cross-correlation (CC) matrix of the channels relative to the perturbation introduced by noise. Numerical results show that this modified spectral method offers significant improvements over the classical method and outperforms other competing methods for multichannel blind deconvolution.