On the Logic Elements Associated with Round-Off Errors and Gaussian Blur in Image Registration: A Simple Case of Commingling
This work addresses signal reconstruction in image registration for researchers in computational imaging, but it is incremental as it builds on existing optimization approaches by introducing a logical abstraction for a specific low-blur regime.
The paper tackles the problem of recovering one-dimensional piecewise constant signals from samples corrupted by Gaussian blur and round-off errors in image registration, showing that when the minimum distance between discontinuity points is 1.5 to 2 times the sampling interval, a method based on logical reasoning can often correctly recover signal amplitudes.
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.