A Counterexample in Image Registration

TL;DR

This paper investigates fundamental limits on the accuracy of one-dimensional image registration by studying the estimation of spatially limited piecewise constant signals from noiseless sampling patterns. By analyzing translations of the signal where a discontinuity is set to zero, it derives how the estimation error depends on the reference point and on the completeness of the sampling patterns. The main contributions include Lemmas 3–4 and Theorem 5, which characterize the complete-data case and provide an explicit energy bound for the optimal estimate, as well as Theorem 7 (and Example 6) that address partial data, showing how incomplete information interacts with reference-point choice to shape worst-case errors. Overall, the work reveals a fundamental, reference-point–driven limit on the accuracy of signal reconstruction from samples, with implications for understanding the theoretical bounds in image registration under quantization and sampling constraints.

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.