From weighted least squares estimation to sparse CS reconstruction

This paper describes a recursive ℓ₁-minimizing approach to CS reconstruction by Kalman filtering. Unlike other approaches using sparsity enforcing a priory density distributions, we consider the ℓ₁-norm as an explicit constraint, formulated as a nonlinear observation of some state to be estimated, which we can additionally (re-)weight, either according to confidence levels or with respect to reweighted ℓ₁-minimization. Interpreting a sparse vector to be estimated as a state which is observed from erroneous (even undersampled) measurements we can easily address time- and space-variant sparsity, any kind of a priori information and also easily address nonstationary error influences in the measurements available. Inherently in our approach we move slightly away from one of the classical RIP based approaches to a more intuitive understanding of the structure of the null space which is implicitly related to the well understood engineering concepts of deterministic and stochastic observability in estimation theory.