2018/04/04 by Eftekhari, Armin, Tanner, Jared, Thompson, Andrew +2 · 1 citation
#FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.1804.01490
The convolution of a discrete measure, x=∑i=1kaiδti, with a local window function, ϕ(s-t), is a common model for a measurement device whose resolution is substantially lower than that of the objects being observed. Super-resolution concerns localising the point sources \ai,ti\i=1k with an accuracy beyond the essential support of ϕ(s-t), typically from m samples y(sj)=∑i=1k aiϕ(sj-ti)+ηj, where ηj indicates an inexactness in the sample value. We consider the setting of x being non-negative and seek to characterise all non-negative measures approximately consistent with the samples. We first show that x is the unique non-negative measure consistent with the samples provided the samples are exact, i.e. ηj=0, m≥ 2k+1 samples are available, and ϕ(s-t) generates a Chebyshev system. This is independent of how close the sample locations are and \em does not rely on any regulariser beyond non-negativity; as such, it extends and clarifies the work by Schiebinger et al. and De Castro et al., who achieve the same results but require a total variation regulariser, which we show is unnecessary. Moreover, we characterise non-negative solutions x consistent with the samples within the bound ∑j=1mηj2≤ δ2. Any such non-negative measure is within \mathcal O(δ1/7) of the discrete measure x generating the samples in the generalised Wasserstein distance, converging to one another as δ approaches zero. We also show how to make these general results, for windows that form a Chebyshev system, precise for the case of ϕ(s-t) being a Gaussian window. The main innovation of these results is that non-negativity alone is sufficient to localise point sources beyond the essential sensor resolution.