A Counterexample in Image Registration
This addresses a foundational theoretical gap in image registration for researchers, but it is incremental as it builds on existing sampling theory without introducing new methods.
The paper tackles the problem of theoretical accuracy limits in image registration, showing that the error in estimating piecewise constant signals from noiseless samples depends on the choice of reference discontinuity point, with uncertainties in discontinuity positions affecting signal accuracy.
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.