  As a revolutionary observation tool in life science, biomedical, and material
science, optical microscopy allows imaging of samples with high spatial
resolution and a wide field of view. However, conventional microscopy methods
are limited to single imaging and cannot accomplish real-time image processing.
The edge detection, image enhancement and phase visualization schemes have
attracted great interest with the rapid development of optical analog
computing. The two main physical mechanisms that enable optical analog
computing originate from two geometric phases: the spin-redirection
Rytov-Vlasimirskii-Berry (RVB) phase and the Pancharatnam-Berry (PB) phase.
Here, we review the basic principles and recent research progress of the RVB
phase and PB phase based optical differentiators. Then we focus on the
innovative and emerging applications of optical analog computing in microscopic
imaging. Optical analog computing is accelerating the transformation of
information processing from classical imaging to quantum techniques. Its
intersection with optical microscopy opens opportunities for the development of
versatile and compact optical microscopy systems.
