# Semiparametric estimation for incoherent optical imaging

## TL;DR

This paper applies semiparametric estimation theory to incoherent optical imaging, deriving bounds and estimators that demonstrate the advantages of SPADE over direct imaging under diffraction and noise.

## Contribution

It introduces a semiparametric framework for optical imaging, deriving exact bounds and efficient estimators for both classical and quantum-inspired measurement methods.

## Key findings

- SPADE outperforms direct imaging in estimating moments.
- Exact semiparametric Cramér-Rao bounds are derived for the imaging problem.
- SPADE remains superior even with limited prior information.

## Abstract

The theory of semiparametric estimation offers an elegant way of computing the Cram\'er-Rao bound for a parameter of interest in the midst of infinitely many nuisance parameters. Here I apply the theory to the problem of moment estimation for incoherent imaging under the effects of diffraction and photon shot noise. Using a Hilbert-space formalism designed for Poisson processes, I derive exact semiparametric Cram\'er-Rao bounds and efficient estimators for both direct imaging and a quantum-inspired measurement method called spatial-mode demultiplexing (SPADE). The results establish the superiority of SPADE even when little prior information about the object is available.