On ℓp -norm sparse blind deconvolution

In this work, we demonstrate that, under some hypothesis, the ℓ_p-norms can be seen as contrast functions for the blind deconvolution of sparse signals. This demonstration is based under two major hypothesis about the input signal and the system, which are unknown. First, the input signal s(n) has unit power and is composed of a few spikes of unknown position, signal and magnitude, separated by zero terms. Second, the impulse response of the system is finite, of unit power and length L_g shorter than the shortest distance between two spikes in s(n). These hypothesis can be relaxed in a probabilistic framework if the probability of having more than one spike in the period L_g is small enough. Also, a gradient-based algorithm is derived and results with synthetic and real seismic data are presented.