Recovery of sparse transformation-invariant signals with continuous basis pursuit
We study the problem of signal decomposition where the signal is a noisy superposition of template features. Each template can occur multiple times in the signal, and associated with each instance is an unknown amount of transformation that the template undergoes. The templates and transformation types are assumed to be known, but the number of instances and associated amounts of transformation with each must be recovered from the signal. In this setting, current methods construct a dictionary containing several transformed copies of each template and employ approximate methods to solve a sparse linear inverse problem. We propose to use a set of basis functions that can interpolate the template under any small amount of transformation(s). Both the amplitude of the feature and the amount of transformation is encoded in the basis coefficients in a way depending on the interpolation scheme used. We construct a dictionary containing transformed copies of these basis functions, where the copies are spaced as far out as the interpolation is accurate. The coefficients are obtained by solving a constrained sparse linear inverse problem where the sparsity penalty is applied across, but not within these groups. We compare our method with standard basis pursuit on a sparse deconvolution task. We find that our method outperforms these methods in that they yield sparser solutions while still having lower reconstruction error.