Sparse non-negative super-resolution — simplified and stabilised

Journal article

We consider the problem of non-negative super-resolution, which concerns reconstructing a non-negative signal x = ki=1 aiδti from m samples of its convolution with a window function φ(s−t), of the form y(sj ) = ki=1 aiφ(sj −ti)+δj , where δj indicates an inexactness in the sample value. We first show that x is the unique non-negative measure consistent with the samples, provided the samples are exact. Moreover, we characterise non-negative solutions xˆ consistent with the samples within the bound mj=1 δ2j ≤ δ2. We show that the integrals of xˆ and x over (ti − ,ti + ) converge to one another as  and δ approach zero and that x and xˆ are similarly close in the generalised Wasserstein distance. Lastly, we make these results precise for φ(s − t) Gaussian. The main innovation is that non-negativity is sufficient to localise point sources and that regularisers such as total variation are not required in the non-negative setting.

Authors: Eftekhari, A; Tanner, J; Thompson, A; Toader, B; Tyagi, H

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Applied and Computational Harmonic Analysis
• Volume: 50
• Pages: 216-280
• Publication date: 2019-08-13
• Acceptance date: 2019-08-05
• DOI: 10.1016/j.acha.2019.08.004
• EISSN: 1096-603X
• ISSN: 1063-5203
• Language: English
• Keywords: grid-free compressed sensing; FFR; T-systems; sparse deconvolution; super-resolution