Signal Recovery in Compressive Sensing via Multiple Sparsifying Bases

Compressive sensing theory asserts that, under certain conditions, a high dimensional but compressible signal can be recovered from a small number of random linear projections by utilizing computationally efficient algorithms. The a priori knowledge of the basis in which the signal of interest is sparse is the key assumption utilized by such algorithms. However, the basis in which the signal is the sparsest is unknown for many natural signals of interest. Instead there may exist multiple bases which lead to a compressible representation of the signal: e.g., an image is compressible in different wavelet transforms. We show that a significant performance improvement can be achieved by utilizing multiple estimates of the signal using sparsifying bases in the context of signal reconstruction from compressive samples. Further, we derive a customized interior-point method to jointly obtain multiple estimates of a 2-D signal (image) from compressive measurements utilizing multiple sparsifying bases as well as the fact that the images usually have a sparse gradient.