Set-Theoretical Foundation of Imaging

In recent decades, we have observed the flourishing of different microscopic techniques, e.g. super-resolution microscopy and light-sheet microscopy which generate a substantial amount of data. An increasing number of sophisticated mathematical approaches are applied for image processing and data extraction. We propose to build a logically consistent link between imaging and mathematics formalising imaging in microscopy. To achieve this we use the Morse–Kelley (MK) set theory, which was first proposed by Kelley (Kelley, 1975), and construct different mathematical structures over sets, e.g. Von Neumann universes. At the end, we define an image as an indexed set of physical and mathematical objects supplemented with mathematical structures. In addition, to demonstrate the power of the approach, we link biological objects and their hierarchy with Von Neumann universes.