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We investigate the problems of 1-D and 2-D signal recovery from subsampledHadamard measurements using Haar wavelet sparsity prior. These problems are ofinterest in, e.g., computational imaging applications relying on opticalmultiplexing or single-pixel imaging. However, the realization of suchmodalities is often hindered by the coherence between the Hadamard and Haarbases. The variable and multilevel density sampling strategies solve this issueby 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-Haarsystems as well as uniform and non-uniform recovery guarantees; a seeminglymissing result in the related literature. We explore the faithfulness of thenumerical simulations to the theoretical results and show in a practicallyrelevant instance, e.g., single-pixel camera, that the target signal can beobtained from a few Hadamard measurements.

Close Encounters of the Binary Kind: Signal Reconstruction Guarantees for Compressive Hadamard Sampling with Haar Wavelet Basis