Already since the work by Abbe and Rayleigh the difficulty of super resolution where one wants to recover a collection of point sources from low-resolved microscopy measurements is thought to be dependent on whether the distance between the sources is below or above a certain resolution or diffraction limit. Even though there has been a number of approaches to define this limit more rigorously, there is still a gap between the situation where the task is known to be hard and scenarios where the task is provably simpler. For instance, an interesting approach for the univariate case using the size of the Cram\'er-Rao lower bound was introduced in a recent work by Ferreira Da Costa and Mitra. In this paper, we prove their conjecture on the transition point between good and worse tractability of super resolution and extend it to higher dimensions. Specifically, the bivariate statistical analysis allows to link the findings by the Cram\'er-Rao lower bound to the classical Rayleigh limit.

翻译：自阿贝和瑞利的研究以来，超分辨率（即从低分辨率显微测量中恢复点源集合）的难度一直被认为取决于源间距是否低于或高于某一分辨率或衍射极限。尽管已有多种方法试图更严格地定义这一极限，但在任务已知困难的情形与任务可证明更简单的情形之间仍存在差距。例如，Ferreira Da Costa和Mitra近期的一项工作中提出了一种有趣的方法，利用克拉美-拉奥下界的尺度来分析单变量情况。本文证明了他们关于超分辨率可处理性优劣转换点的猜想，并将其推广至更高维度。具体而言，双变量统计分析使得克拉美-拉奥下界的发现与经典瑞利极限之间建立了联系。