On Image Registration and Subpixel Estimation

TL;DR

This paper addresses subpixel estimation in one-dimensional image registration under ideal sampling and quantization by formulating a precise model of a spatially limited, piecewise-constant signal with $m$ regions, where region lengths are $R_i=(n_i-f_i)T$ and sampling occurs at intervals of $T$ with offsets $\Delta_i\in[0,T)$. It develops a rigorous theory linking observable region-counts $\eta_i$ and offsets to the unknown integers $n_i$ and fractions $f_i$, using a sequence of results (Propositions 1–4) and explicit extension rules (Theorems 5–6) to propagate partial observations to full vectors $\{\eta_i\}$. With a complete set of $m+1$ observation vectors, the $f_i$ lie in one of $m!$ partitions of the unit hypercube, effectively resolving subinterval information; partial data still yields valuable subinterval information for certain patterns. The work also discusses how noise would transform the problem into a detection/hypothesis-testing framework and suggests future directions, including joint processing of multiple measurements and potential deep-learning approaches for robust, task-oriented registration.

Abstract

Image registration is a classical problem in machine vision which seeks methods to align discrete images of the same scene to subpixel accuracy in general situations. As with all estimation problems, the underlying difficulty is the partial information available about the ground truth. We consider a basic and idealized one-dimensional image registration problem motivated by questions about measurement and about quantization, and we demonstrate that the extent to which subinterval/subpixel inferences can be made in this setting depends on a type of complexity associated with the function of interest, the relationship between the function and the pixel size, and the number of distinct sampling count observations available.