Sparse blind deconvolution based on scale invariant smoothed ℓ0-norm

In this work, we explore the problem of blind deconvolution in the context of sparse signals. We show that the ℓ
₀
-norm works as a contrast function, if the length of the impulse response of the system is smaller than the shortest distance between two spikes of the input signal. Demonstrating this sufficient condition is our basic theoretical result. However, one of the problems of dealing with the ℓ
₀
-norm in optimization problems is the requirement of exhaustive or combinatorial search methods, since it is a non continuous function. In order to propose an alternative for that, Mohimani et al. (2009) proposed a smoothed and continuous version of the ℓ
₀
-norm. Here, we propose a modification of this criterion in order to make it scale-invariant and, finally, we derive a gradient-based algorithm for the modified criterion. Results with synthetic data suggests that the imposed conditions are sufficient but not strictly necessary.