Tuesday, July 02, 2013

On Embedding The Angles Between Signals, Sparse Signal Reconstruction from Phase-only Measurements

Petros just sent me the following from Berlin at SampTA (SPARS2013 is next week!): 


There are two new papers on my website, which your readers might find interesting. They show that the phase of complex measurements is sufficient to reconstruct a signal robustly. This generalizes the sign of 1-bit CS, in the sense that the phase of a complex number generalizes the sign of a real number. They also show that we can use that method to design an embedding that preserves angles better than the BeSE, assuming the same number of measurements (of course, since the phase measurements are continuous, the comparison is slightly unfair to the BeSE, especially if bits matter in the application).....

Thanks Petros ! Normally, the literature is slanted toward phase retrieval, i.e. given the magnitude of the signal, find back the phase. Here the phase is given and connected to 1-bit compressive sensing.  I am sure that in acoustic, there will be use for that. Here are the two intriguing papers:

Petros Boufounos, On Embedding The Angles Between Signals,

Abstract—The phase of randomized complex-valued projections of real signals preserves information about the angle, i.e., the correlation, between signals. This information can be exploited to design angle preserving embeddings, which represent such correlations. These embeddings generalize known results on binary embeddings and 1-bit compressive sensing and reduce the embedding uncertainty.

Petros BoufounosSparse Signal Reconstruction from Phase-only Measurements

Abstract—We demonstrate that the phase of complex linear measurements of signals preserves significant information about the angles between those signals. We provide stable angle embedding guarantees, akin to the restricted isometry property in classical compressive sensing, that characterize how well the angle information is preserved. They also suggest that a number of measurements linear in the sparsity and logarithmic in the dimensionality of the signal contains sufficient information to acquire and reconstruct a sparse signal within a positive scalar factor. We further show that the reconstruction can be formulated and solved using standard convex and greedy algorithms taken directly from the CS literature. Even though the theoretical results only provide approximate reconstruction guarantees, our experiments suggest that exact reconstruction is possible.

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