A Counterexample in Image Registration

TL;DR

This paper explores the theoretical limits of image registration accuracy, revealing that the estimation error depends on the choice of reference points, and provides a counterexample illustrating these bounds.

Contribution

It introduces a counterexample demonstrating fundamental limitations in image registration accuracy related to reference point selection.

Findings

Error bounds depend on the reference point in the signal.

Uncertainties in discontinuity positions affect registration accuracy.

Theoretical limits are established for noiseless sampling scenarios.

Abstract

Image registration is a widespread problem which applies models about image transformation or image similarity to align discrete images of the same scene. Nevertheless, the theoretical limits on its accuracy are not understood even in the case of one-dimensional data. Just as Nyquist's sampling theorem states conditions for the perfect reconstruction of signals from samples, there are bounds to the quality of reproductions of quantized functions from sets of ideal, noiseless samples in the absence of additional assumptions. In this work we estimate spatially-limited piecewise constant signals from two or more sets of noiseless sampling patterns. We mainly focus on the energy of the error function and find that the uncertainties of the positions of the discontinuity points of the function depend on the discontinuity point selected as the reference point of the signal. As a consequence, the accuracy of the estimate of the signal depends on the reference point of that signal.