A Technical Note on Hilbert Spaces as Mathematical Tools to Probe and Process Cryo-EM Images – An Architectural and Computational Point of View Using Higher Order Logic (HOL)/Scala/Java/JVM Software Environment

‘Hilbert space is where wave functions dwell,’ according to David Hilbert. Therefore, it is both an interesting and inspiring statement to consider and use Hilbert spaces as mathematical tools to process Cryo-electron microscopy (Cryo-EM) images using novel informatics and computing methodologies. In general, Hilbert spaces and their implementation mechanisms could be used in a wider range of science and technological domains. Here, it is proposed to consider HS (Hilbert Spaces), from an imaging processing viewpoint using higher order logic (HOL)/Scala/Java/JVM Software Environment. In this context, a general informatics framework and its computing aspects are discussed. Though Cryo-EM image processing is focused in this technical note, the concepts apply to other domains like transmission electron microscopy/scanning electron microscopy and others. Hilbert spaces are an excellent form of promising mathematical anticipatory tool to process signals. HOL is a widely used tool for creating formal specifications of systems. HOL has been used in both the industry and academia to support formal reasoning in various domains of science and technology, in addition to hardware and software verification. In principle, they could be used to support any project which needs to be defined on a sound mathematical basis. Given the scope of Cryo-EM Imaging Software Systems, it is very much logical to start a computational framework using HOL as the prototyping platform.