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Sparse non-negative super-resolution — simplified and stabilised
Applied and Computational Harmonic Analysis ( IF 2.6 ) Pub Date : 2019-08-13 , DOI: 10.1016/j.acha.2019.08.004
Armin Eftekhari , Jared Tanner , Andrew Thompson , Bogdan Toader , Hemant Tyagi

We consider the problem of non-negative super-resolution, which concerns reconstructing a non-negative signal x=i=1kaiδti from m samples of its convolution with a window function ϕ(st), of the form y(sj)=i=1kaiϕ(sjti)+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 j=1mδj2δ2. We show that the integrals of xˆ and x over (tiϵ,ti+ϵ) converge to one another as ϵ 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(st) 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.



中文翻译:

稀疏的非负超分辨率-简化和稳定

我们考虑了非负超分辨率问题,该问题涉及重建非负信号 X=一世=1个ķ一种一世δŤ一世从带有窗口函数的m个卷积样本中ϕs-Ť,形式为 ÿsĴ=一世=1个ķ一种一世ϕsĴ-Ť一世+δĴ,在哪里 δĴ表示样本值不精确。我们首先证明x是与样本一致的唯一非负度量,前提是样本精确。此外,我们刻画非负解的特征Xˆ 与范围内的样本一致 Ĵ=1个δĴ2δ2。我们证明了Xˆx结束Ť一世-ϵŤ一世+ϵϵδ趋近于零,xx彼此收敛。Xˆ在广义Wasserstein距离上相似。最后,我们将这些结果精确地用于ϕs-Ť高斯。主要的创新之处在于,非负性足以定位点源,并且在非负环境中不需要诸如总变化之类的正则化器。

更新日期:2019-08-13
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