A Counterexample in Cross-Correlation Template Matching

TL;DR

This paper presents a counterexample demonstrating limitations of cross-correlation in noisy 1D image registration, and proposes alternative strategies involving difference sequences, thresholding, and dynamic programming for improved alignment and segmentation.

Contribution

The paper provides a specific counterexample showing the failure of cross-correlation in noisy conditions and introduces a novel approach using established techniques for robust registration.

Findings

Cross-correlation can perform poorly on noisy data for certain functions.

Difference sequences, thresholding, and dynamic programming can improve alignment and segmentation.

The paper offers theoretical insights and conditions for successful data sequence alignment.

Abstract

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.