On Round-Off Errors and Gaussian Blur in Superresolution and in Image Registration

TL;DR

This work analyzes how round-off errors and Gaussian blur affect superresolution and image registration for one-dimensional, spatially limited piecewise-constant signals. It introduces a signal-dependent measurement framework with a deformation matrix $ ilde{M}$ and a quantized counterpart $M$, along with a difference matrix $M_D$ to capture information about discontinuities. The authors establish regimes under small blur where each column of $M$ has at most one critical value, and they show that with spacing between discontinuities exceeding $2T$, a dynamic-programming scheme can reliably align two noisy sequences and identify the first samples after each discontinuity, even in the presence of blur and quantization. The results challenge the effectiveness of cross-correlation template matching and provide a constructive method for robust inference, with implications for fusing multiple low-resolution measurements in precision-limited sensing.

Abstract

Superresolution theory and techniques seek to recover signals from samples in the presence of blur and noise. Discrete image registration can be an approach to fuse information from different sets of samples of the same signal. Quantization errors in the spatial domain are inherent to digital images. We consider 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. We describe a signal-dependent measurement matrix which captures both types of effects. For this setting we show that the difficulties in determining the discontinuity points from two sets of samples even in the absence of other types of noise. If the samples are also subject to statistical noise, then it is necessary to align and segment the data sequences to make the most effective inferences about the amplitudes and discontinuity points. Under some conditions on the blur, the noise, and the distance between discontinuity points, we prove that we can correctly align and determine the first samples following each discontinuity point in two data sequences with an approach based on dynamic programming.