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

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.

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

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.

A Counterexample in Cross-Correlation Template Matching

Sampling and quantization are standard practices in signal and image processing, but a theoretical understanding of their impact is incomplete. We consider discrete image registration when the underlying function is a one-dimensional spatially-limited piecewise constant function. For ideal noiseless sampling the number of samples from each region of the support of the function generally depends on the placement of the sampling grid. Therefore, if the samples of the function are noisy, then image registration requires alignment and segmentation of the data sequences. One popular strategy for aligning images is selecting the maximum from cross-correlation template matching. To motivate more robust and accurate approaches which also address segmentation, we provide an example of a one-dimensional spatially-limited piecewise constant function for which the cross-correlation technique can perform poorly on noisy samples. While earlier approaches to improve the method involve normalization, our example suggests a novel strategy in our setting. Difference sequences, thresholding, and dynamic programming are well-known techniques in image processing. We prove that they are tools to correctly align and segment noisy data sequences under some conditions on the noise. We also address some of the potential difficulties that could arise in a more general case.

A Counterexample in Image Registration

Image registration is a widespread problem which applies models about image transformation or image similarity to align discrete images of the same scene. Nevertheless, the theoretical limits on its accuracy are not understood even in the case of one-dimensional data. Just as Nyquist's sampling theorem states conditions for the perfect reconstruction of signals from samples, there are bounds to the quality of reproductions of quantized functions from sets of ideal, noiseless samples in the absence of additional assumptions. In this work we estimate spatially-limited piecewise constant signals from two or more sets of noiseless sampling patterns. We mainly focus on the energy of the error function and find that the uncertainties of the positions of the discontinuity points of the function depend on the discontinuity point selected as the reference point of the signal. As a consequence, the accuracy of the estimate of the signal depends on the reference point of that signal.

On Image Registration and Subpixel Estimation

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.

On the Construction of Distribution-Free Prediction Intervals for an Image Regression Problem in Semiconductor Manufacturing

The high-volume manufacturing of the next generation of semiconductor devices requires advances in measurement signal analysis. Many in the semiconductor manufacturing community have reservations about the adoption of deep learning; they instead prefer other model-based approaches for some image regression problems, and according to the 2021 IEEE International Roadmap for Devices and Systems (IRDS) report on Metrology a SEMI standardization committee may endorse this philosophy. The semiconductor manufacturing community does, however, communicate a need for state-of-the-art statistical analyses to reduce measurement uncertainty. Prediction intervals which characterize the reliability of the predictive performance of regression models can impact decisions, build trust in machine learning, and be applied to other regression models. However, we are not aware of effective and sufficiently simple distribution-free approaches that offer valid coverage for important classes of image data, so we consider the distribution-free conformal prediction and conformalized quantile regression framework.The image regression problem that is the focus of this paper pertains to line edge roughness (LER) estimation from noisy scanning electron microscopy images. LER affects semiconductor device performance and reliability as well as the yield of the manufacturing process; the 2021 IRDS emphasizes the crucial importance of LER by devoting a white paper to it in addition to mentioning or discussing it in the reports of multiple international focus teams. It is not immediately apparent how to effectively use normalized conformal prediction and quantile regression for LER estimation. The modeling techniques we apply appear to be novel for finding distribution-free prediction intervals for image data and will be presented at the 2022 SEMI Advanced Semiconductor Manufacturing Conference.