Super-resolution estimation is the problem of recovering a stream of spikes (point sources) from the noisy observation of a few numbers of its first trigonometric moments. The performance of super-resolution is recognized to be intimately related to the separation between the spikes to recover. A novel notion of stability of the Fisher information matrix (FIM) of the super-resolution problem is introduced when the minimal eigenvalue of the FIM is not asymptotically vanishing. The regime where the minimal separation is inversely proportional to the number of acquired moments is considered. It is shown that there is a separation threshold above which the eigenvalues of the FIM can be bounded by a quantity that does not depend on the number of moments. The proof relies on characterizing the connection between the stability of the FIM and a generalization of the Beurling-Selberg box approximation problem.
翻译:超分辨率估计是从对第一个三角点数数的杂音观测中恢复一串钉子(点源)的问题。 超分辨率的性能被确认与要恢复的钉子之间的分离密切相关。 当FIM的最小值不是瞬间消失时,就引入了超分辨率问题的Fisher信息矩阵(FIM)稳定性的新概念。 考虑的是最小分离值与获得的时数成反比的制度。 显示有分离阈值, 超过这一阈值, FIM的机能价值可以受不取决于分钟数的数量的约束。 证据取决于FIM的稳定性与Beurling- Selberg箱近距离问题的一般化之间的联系特征。