  We investigate the problems of 1-D and 2-D signal recovery from subsampled
Hadamard measurements using Haar wavelet sparsity prior. These problems are of
interest in, e.g., computational imaging applications relying on optical
multiplexing or single-pixel imaging. However, the realization of such
modalities is often hindered by the coherence between the Hadamard and Haar
bases. The variable and multilevel density sampling strategies solve this issue
by adjusting the subsampling process to the local and multilevel coherence,
respectively, between the two bases; hence enabling successful signal recovery.
In this work, we compute an explicit sample-complexity bound for Hadamard-Haar
systems as well as uniform and non-uniform recovery guarantees; a seemingly
missing result in the related literature. We explore the faithfulness of the
numerical simulations to the theoretical results and show in a practically
relevant instance, e.g., single-pixel camera, that the target signal can be
obtained from a few Hadamard measurements.
