Cramer-Rao Lower Bound for Point Based Image Registration with Heteroscedastic Error Model for Application in Single Molecule Microscopy

TL;DR

This paper derives the Cramer-Rao lower bound for affine transformation estimation in heteroscedastic errors-in-variables models, specifically applied to feature-based image registration in fluorescence microscopy, providing theoretical limits and validation through simulations and experiments.

Contribution

It introduces a novel Cramer-Rao lower bound for heteroscedastic error models in image registration, tailored for fluorescence microscopy applications.

Findings

Derived bounds for affine transformation parameters in heteroscedastic models.

Simplified bounds applicable when localization errors are scalar multiples of a common matrix.

Theoretical bounds match asymptotic distributions of estimators under certain conditions.

Abstract

The Cramer-Rao lower bound for the estimation of the affine transformation parameters in a multivariate heteroscedastic errors-in-variables model is derived. The model is suitable for feature-based image registration in which both sets of control points are localized with errors whose covariance matrices vary from point to point. With focus given to the registration of fluorescence microscopy images, the Cramer-Rao lower bound for the estimation of a feature's position (e.g. of a single molecule) in a registered image is also derived. In the particular case where all covariance matrices for the localization errors are scalar multiples of a common positive definite matrix (e.g. the identity matrix), as can be assumed in fluorescence microscopy, then simplified expressions for the Cramer-Rao lower bound are given. Under certain simplifying assumptions these expressions are shown to match asymptotic distributions for a previously presented set of estimators. Theoretical results are verified with simulations and experimental data.