Wiener filter unifies Hilbert and Zernike phase plates in electron microscopy

We report on two key discoveries resulting from the combination of the Hilbert phase plate (HPP) and the Wiener filter: firstly, the resolution of the HPP's mixed image problem through a one-step experiment, and secondly, the unification of the Zernike phase plate (ZPP) and the HPP. When the phase of the HPP is reduced to less than π, it produces a mixed image comprising both the normal and the differential images. The HPPU (left-right unified HPP), proposed to address this issue, required a two-step experimental process. However, during our efforts to resolve the mixed image problem using either the left or right HPP, we discovered that the Wiener filtering process not only addresses this issue but also facilitates the unification of the ZPP and HPP. We will discuss the theoretical development behind these discoveries and their verification through simulations of three phase contrast methods: the Scherzer, ZPP, and HPP methods.