  Most of the existing methods for sparse signal recovery assume a static
system: the unknown signal is a finite-length vector for which a fixed set of
linear measurements and a sparse representation basis are available and an
L1-norm minimization program is solved for the reconstruction. However, the
same representation and reconstruction framework is not readily applicable in a
streaming system: the unknown signal changes over time, and it is measured and
reconstructed sequentially over small time intervals.
  In this paper, we discuss two such streaming systems and a homotopy-based
algorithm for quickly solving the associated L1-norm minimization programs: 1)
Recovery of a smooth, time-varying signal for which, instead of using block
transforms, we use lapped orthogonal transforms for sparse representation. 2)
Recovery of a sparse, time-varying signal that follows a linear dynamic model.
For both the systems, we iteratively process measurements over a sliding
interval and estimate sparse coefficients by solving a weighted L1-norm
minimization program. Instead of solving a new L1 program from scratch at every
iteration, we use an available signal estimate as a starting point in a
homotopy formulation. Starting with a warm-start vector, our homotopy algorithm
updates the solution in a small number of computationally inexpensive steps as
the system changes. The homotopy algorithm presented in this paper is highly
versatile as it can update the solution for the L1 problem in a number of
dynamical settings. We demonstrate with numerical experiments that our proposed
streaming recovery framework outperforms the methods that represent and
reconstruct a signal as independent, disjoint blocks, in terms of quality of
reconstruction, and that our proposed homotopy-based updating scheme
outperforms current state-of-the-art solvers in terms of the computation time
and complexity.
