论文标题
多元超分辨率无分离
Multivariate Super-Resolution without Separation
论文作者
论文摘要
在本文中，我们研究了高维超分辨率成像问题。在这里，我们得到了许多点光源的图像，其位置和强度未知。图像被像素化，并通过成像设备引起的已知点传播函数模糊。我们通过非负措施编码未知点源及其强度，并提出了一个凸优化程序来找到它。假设该设备的点传播函数是可以分解的，我们表明，最佳解决方案是无噪声情况下的真实度量，并且在嘈杂的情况下，相对于广义的Wasserstein距离，它在嘈杂的情况下近似于真实度量。我们的主要假设是，点传播函数的组件在嘈杂的情况下形成了tchebychev系统（$ t $ - 系统），在嘈杂的情况下，$ t^*$ - 系统，由高斯点传播功能满足的温和条件。我们的工作是对工作的所有维度的概括（Eftekhari，Bendory和Tang，2021），其中在2个维度上进行了相同的分析。我们解决了（Schiebinger，Robeva和Recht，2018）在点传播函数分解的情况下提出的一个开放问题。
In this paper we study the high-dimensional super-resolution imaging problem. Here we are given an image of a number of point sources of light whose locations and intensities are unknown. The image is pixelized and is blurred by a known point-spread function arising from the imaging device. We encode the unknown point sources and their intensities via a nonnegative measure and we propose a convex optimization program to find it. Assuming the device's point-spread function is component-wise decomposable, we show that the optimal solution is the true measure in the noiseless case, and it approximates the true measure well in the noisy case with respect to the generalized Wasserstein distance. Our main assumption is that the components of the point-spread function form a Tchebychev system ($T$-system) in the noiseless case and a $T^*$-system in the noisy case, mild conditions that are satisfied by Gaussian point-spread functions. Our work is a generalization to all dimensions of the work (Eftekhari, Bendory, & Tang, 2021) where the same analysis is carried out in 2 dimensions. We resolve an open problem posed in (Schiebinger, Robeva, & Recht, 2018) in the case when the point-spread function decomposes.
