Role of positivity in blind deconvolution (Conference Presentation)

Abstract

Blind deconvolution is an important problem arising in many engineering and scientific applications, ranging from imaging, communication to computer vision and machine learning. Classical techniques to solve this highly ill posed problem exploit statistical priors on the signals of interest. In recent times, there has been a renewed interest in deterministic approaches for blind deconvolution, whereby, using the novel idea of lifting, the non-convex blind deconvolution problem can be cast as a semidefinite program. Using suitable subspace assumptions on the unknown signals, precise theoretical guarantees can be derived on the number of measurements needed to perform blind deconvolution. In this paper, we will address the problem of positive sparse blind deconvolution, where the signals of interest exhibit positivity (alongside sparsity) either naturally, or in appropriate transform domains. Important applications of positive blind deconvolution include image deconvolution and positive spike detection. We will show that positivity is a powerful constraint that can be exploited to cast the blind deconvolution problem in terms of a simple linear program that can be theoretically analyzed. We will explore the questions of uniqueness and identifiability, and develop conditions under which the linear program reveals the true positive sparse solution. Numerical results will demonstrate the superior performance of the proposed approach.