# Spatio-angular fluorescence microscopy I. Basic theory

## TL;DR

This paper develops a foundational spatio-angular theoretical framework for fluorescence microscopy, modeling imaging systems as linear operators and introducing the dipole spatio-angular transfer function to unify analysis of single and ensemble dipole imaging.

## Contribution

It introduces a novel spatio-angular theory and a dipole transfer function, providing a unified mathematical framework for fluorescence microscopy analysis.

## Key findings

- Modeling fluorescence microscopy as a linear Hilbert-space operator.
- Introduction of the dipole spatio-angular transfer function.
- Framework applicable to both single and ensemble dipole imaging.

## Abstract

We introduce the basic elements of a spatio-angular theory of fluorescence microscopy, providing a unified framework for analyzing systems that image single fluorescent dipoles and ensembles of overlapping dipoles that label biological molecules. We model an aplanatic microscope imaging an ensemble of in-focus fluorescent dipoles as a linear Hilbert-space operator, and we show that the operator takes a particularly convenient form when expressed in a basis of complex exponentials and spherical harmonics$-$a form we call the dipole spatio-angular transfer function. We discuss the implications of our analysis for all quantitative fluorescence microscopy studies and lay out a path towards a complete theory.