Fundamental Limits in Multi-Image Alignment

Abstract

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 Cramér-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.Index Terms-Multi-image alignment, performance bounds, Cramér-Rao bound, Ziv-Zakai bound, Bayesian Cramér-Rao, maximum likelihood estimator.

A Practical Guide to Multi-Image Alignment

“…The Cramér-Rao bound, under the assumption that the underlying image follows a natural image prior (CRB) was derived. The CRB gives a lower bound on the mean square error (MSE) of any unbiased estimator of the shifts τ τ τ (see [19,Eq. (44)]). Different behaviors for the alignment accuracy are identified, depending on the SNR of the input images.…”

“…For high to moderate SNR, increasing the number of images does improve performance. Interestingly, theory predicts the existence of an SNR threshold below which performance degrades briskly and a lower limit SNR value The CRB is displayed as a performance benchmark [19]. below which alignment is not possible.…”

A Spectral Method for Stable Bispectrum Inversion With Application to Multireference Alignment

“…Consider the case where the number of observations is much larger than the length of the signal, namely M N . In this regime, the invariant features approach has two important advantages over methods that rely on estimating the translations, 1) there will be no need to determine the translations in order to recover the signal hence reducing the computational complexity of the problem, and 2) with high level of noise, given enough samples, it does not suffer from the fundamental limit [3] for the pairwise alignment approach, that depends on the noise variance.…”

Heterogeneous Multireference Alignment for Images With Application to 2D Classification in Single Particle Reconstruction

“…Given an accurate estimate of these variables, the problem becomes trivial: one can cluster the observations into the K class averages, undo the rotations, and average within each class to suppress the noise. However, in the low SNR regime, estimating the labels and rotations becomes challenging, and indeed impossible as the SNR drops to zero; see for instance [21] for analysis in a related model. Notwithstanding, it was shown in a series of papers that in many MRA setups the underlying signal (or signals in our case) can be estimated at any noise level, provided sufficiently many observations are recorded [19,[22][23][24][25][26][27].…”