Deconvolution with the l 1 norm

Abstract

Given a wavelet w and a noisy trace t = s * w + n, an approximation s of the spike train s can be obtained using the l 1 norm. This extraction has the advantage of preserving isolated spikes in s. On some types of data the spike train s can represent s as a sparse series of spikes, which may be sampled at a rate higher than the sample rate of the data trace t. The extracted spike train s may be qualitatively much different than those commonly extracted using the l 2 norm.The l 1 norm can also be used to extract a wavelet w from a trace t when a spike train s is known. This wavelet extraction can be constrained to give a smooth wavelet which integrates to zero and goes to zero at the ends.Given a trace t and an initial approximation for either s or w, it is possible to alternately extract spike trains and wavelets to improve the representation of trace t.Although special algorithms have been developed to solve l 1 problems, all of the calculations can be performed using a general linear programming system. Proper weighting procedures allow these methods to be used on ungained data.