We consider the problem of non-negative super-resolution, which concerns reconstructing a non-negative signal $x={\sum }_{i=1}^k{a}_i{\delta }_{t_i}$ from m samples of its convolution with a window function $\varphi (s-t)$, of the form $y(s_j)={\sum }_{i=1}^k{a}_i\varphi (s_j-t_i)+{\delta }_j$, where ${\delta }_j$ 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 $\hat{x}$ consistent with the samples within the bound ${\sum }_{j=1}^m{\delta }_j^2\le {\delta }^2$. We show that the integrals of $\hat{x}$ and x over $(t_i-ϵ,t_i+ϵ)$ converge to one another as ϵ and δ approach zero and that x and $\hat{x}$ are similarly close in the generalised Wasserstein distance. Lastly, we make these results precise for $\varphi (s-t)$ 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.

我们考虑了非负超分辨率问题，该问题涉及重建非负信号 $X={\sum }_{\mathrm{一世}=\mathrm{1个}}^{ķ}{\mathrm{一种}}_{\mathrm{一世}}{\delta }_{{Ť}_{\mathrm{一世}}}$从带有窗口函数的m个卷积样本中$\varphi （s-Ť）$，形式为 $ÿ（s_{Ĵ}）={\sum }_{\mathrm{一世}=\mathrm{1个}}^{ķ}{\mathrm{一种}}_{\mathrm{一世}}\varphi （s_{Ĵ}-{Ť}_{\mathrm{一世}}）+{\delta }_{Ĵ}$，在哪里 ${\delta }_{Ĵ}$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$\hat{X}$ 与范围内的样本一致 ${\sum }_{Ĵ=\mathrm{1个}}^{米}{\delta }_{Ĵ}^2\le {\delta }^2$。我们证明了$\hat{X}$和x结束$（{Ť}_{\mathrm{一世}}-ϵ，{Ť}_{\mathrm{一世}}+ϵ）$ϵ和δ趋近于零，x和x彼此收敛。$\hat{X}$在广义Wasserstein距离上相似。最后，我们将这些结果精确地用于$\varphi （s-Ť）$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。