On the Logic Elements Associated with Round-Off Errors and Gaussian Blur in Image Registration: A Simple Case of Commingling

TL;DR

The paper addresses the problem of reconstructing one-dimensional piecewise-constant signals from samples blurred by Gaussian kernels and quantized, focusing on low-blur regimes where blur, sampling, and rounding can be analyzed as a dataflow. It proposes a logic-inspired framework with three primitive behaviors and a commingling phenomenon, and develops a collaborative parsing approach that jointly analyzes two sample sequences via the difference matrix M_D to recover signal amplitudes. It derives structural results and distance bounds (notably a critical 1.5T threshold) and presents an algorithmic pathway for amplitude recovery grounded in the classification of sample-difference patterns and their parsing. The findings offer insights into amplitude recovery and bounds estimation in image registration under blur and quantization, with potential implications for hardware implementations and broader discrete-data processing in imaging.

Abstract

Discrete image registration can be a strategy to reconstruct signals from samples corrupted by blur and noise. We examine superresolution and discrete image registration for one-dimensional spatially-limited piecewise constant functions which are subject to blur which is Gaussian or a mixture of Gaussians as well as to round-off errors. Previous approaches address the signal recovery problem as an optimization problem. We focus on a regime with low blur and suggest that the operations of blur, sampling, and quantization are not unlike the operation of a computer program and have an abstraction that can be studied with a type of logic. When the minimum distance between discontinuity points is between $1.5$ and 2 times the sampling interval, we can encounter the simplest form of a type of interference between discontinuity points that we call ``commingling.'' We describe a way to reason about two sets of samples of the same signal that will often result in the correct recovery of signal amplitudes. We also discuss ways to estimate bounds on the distances between discontinuity points.