2,295 research outputs found

    Sparse non-negative super-resolution -- simplified and stabilised

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    The convolution of a discrete measure, x=∑i=1kaiδtix=\sum_{i=1}^ka_i\delta_{t_i}, with a local window function, ϕ(s−t)\phi(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\{a_i,t_i\}_{i=1}^k with an accuracy beyond the essential support of ϕ(s−t)\phi(s-t), typically from mm samples y(sj)=∑i=1kaiϕ(sj−ti)+ηjy(s_j)=\sum_{i=1}^k a_i\phi(s_j-t_i)+\eta_j, where ηj\eta_j indicates an inexactness in the sample value. We consider the setting of xx being non-negative and seek to characterise all non-negative measures approximately consistent with the samples. We first show that xx is the unique non-negative measure consistent with the samples provided the samples are exact, i.e. ηj=0\eta_j=0, m≥2k+1m\ge 2k+1 samples are available, and ϕ(s−t)\phi(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^\hat{x} consistent with the samples within the bound ∑j=1mηj2≤δ2\sum_{j=1}^m\eta_j^2\le \delta^2. Any such non-negative measure is within O(δ1/7){\mathcal O}(\delta^{1/7}) of the discrete measure xx generating the samples in the generalised Wasserstein distance, converging to one another as δ\delta 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)\phi(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.Comment: 59 pages, 7 figure

    The dual approach to non-negative super-resolution: perturbation analysis

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    We study the problem of super-resolution, where we recover the locations and weights of non-negative point sources from a few samples of their convolution with a Gaussian kernel. It has been shown that exact recovery is possible by minimising the total variation norm of the measure, and a practical way of achieve this is by solving the dual problem. In this paper, we study the stability of solutions with respect to the solutions dual problem, both in the case of exact measurements and in the case of measurements with additive noise. In particular, we establish a relationship between perturbations in the dual variable and perturbations in the primal variable around the optimiser and a similar relationship between perturbations in the dual variable around the optimiser and the magnitude of the additive noise in the measurements. Our analysis is based on a quantitative version of the implicit function theorem.Comment: 35 pages, 5 figure
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