Fundamental Limits in Multi-image Alignment

TL;DR

This paper investigates the fundamental performance limits of multi-image alignment under various noise conditions, deriving theoretical bounds and analyzing how the number of images and SNR affect alignment accuracy.

Contribution

It derives and analyzes the Cramér-Rao and Ziv-Zakai bounds for multi-image alignment, revealing how SNR and the number of images influence the problem's fundamental limits.

Findings

Increasing the number of images improves alignment only below a certain SNR threshold.

Above the SNR threshold, pairwise maximum likelihood estimation is optimal.

Different behavior zones exist depending on the SNR conditions of the images.

Abstract

The performance of multi-image alignment, bringing different images into one coordinate system, is critical in many applications with varied signal-to-noise ratio (SNR) conditions. A great amount of effort is being invested into developing methods to solve this problem. Several important questions thus arise, including: Which are the fundamental limits in multi-image alignment performance? Does having access to more images improve the alignment? Theoretical bounds provide a fundamental benchmark to compare methods and can help establish whether improvements can be made. In this work, we tackle the problem of finding the performance limits in image registration when multiple shifted and noisy observations are available. We derive and analyze the Cram\'er-Rao and Ziv-Zakai lower bounds under different statistical models for the underlying image. The accuracy of the derived bounds is experimentally assessed through a comparison to the maximum likelihood estimator. We show the existence of different behavior zones depending on the difficulty level of the problem, given by the SNR conditions of the input images. We find that increasing the number of images is only useful below a certain SNR threshold, above which the pairwise MLE estimation proves to be optimal. The analysis we present here brings further insight into the fundamental limitations of the multi-image alignment problem.