We will discuss recent results on a classic problem in signal processing: multichannel blind deconvolution. We observe the convolution of a signal unknown source vector with M different channel response vectors; from these observations, our task is to estimate the M responses. Traditionally, this is formulated as a null space approximation problem; given that the channel responses have limited length, we can form a matrix from the observations that is guaranteed to have a null space of dimension 1, and knowledge of this null space immediately reveals the channel responses up to a global constant. The stability of this process in the presence of noise depends on the "spectral gap" of this matrix. We will demonstrate the effect that different channel models have on this spectral gap, and derive performance guarantees for two different algorithms to perform the channel estimation. We will also discuss provable guarantees for a more structured bilinear model, where the ensemble of channel responses is modeled as lying in a low-dimensional subspace but with each channel modulated by an independent gain. Under this model, we show how the channel estimates can be found by minimizing a quadratic functional over a non-convex set. We analyze two methods for solving this non-convex program, and provide performance guarantees for each. The work presented in this talk was done in collaboration with Kiryung Lee, Felix Krahmer, and Ning Tian.

This presentation is part of Minisymposium “MS60 - Computational and Compressive Imaging Technologies and Applications (3 parts)”
organized by: Robert Muise (Lockheed Martin) , Richard Baraniuk (Rice University) .

Authors:

Justin Romberg (Georgia Tech)

Keywords:

image deblurring, image reconstruction, inverse problems, nonlinear optimization