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

TL;DR

This paper explores how logic can model round-off errors and Gaussian blur in 1D image registration, revealing conditions for accurate signal recovery amidst interference and proposing bounds estimation methods.

Contribution

It introduces a logical framework to analyze the effects of blur and round-off errors on signal recovery, focusing on the commingling interference phenomenon.

Findings

Identifies conditions where signal discontinuities can be correctly recovered.

Describes a logical approach to analyze sample interference.

Proposes methods to estimate bounds on discontinuity distances.

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.