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Bispectrum Unbiasing for Dilation-Invariant Multi-reference Alignment

Liping Yin, Anna Little, Matthew Hirn

TL;DR

The paper tackles multi-reference alignment in the presence of random dilations by developing a data-driven unbiasing procedure for the bispectrum under known dilation distribution. It derives an exact inversion formula for the noiseless case and provides a finite-sample estimator with provable mean-squared error decay, then extends to the noisy dilation setting with a noise-unbiasing step and smoothing, culminating in a convex optimization problem for practical computation. The authors demonstrate both bispectrum recovery and subsequent hidden-signal reconstruction, showing robustness to unknown noise and dilation parameters and quantifying convergence behavior. These results yield a dilation-invariant path to full signal recovery, with potential impact on cryo-EM and other imaging domains where scale variation complicates standard MRA approaches.

Abstract

Motivated by modern data applications such as cryo-electron microscopy, the goal of classic multi-reference alignment (MRA) is to recover an unknown signal $f: \mathbb{R} \to \mathbb{R}$ from many observations that have been randomly translated and corrupted by additive noise. We consider a generalization of classic MRA where signals are also corrupted by a random scale change, i.e. dilation. We propose a novel data-driven unbiasing procedure which can recover an unbiased estimator of the bispectrum of the unknown signal, given knowledge of the dilation distribution. Lastly, we invert the recovered bispectrum to achieve full signal recovery, and validate our methodology on a set of synthetic signals.

Bispectrum Unbiasing for Dilation-Invariant Multi-reference Alignment

TL;DR

The paper tackles multi-reference alignment in the presence of random dilations by developing a data-driven unbiasing procedure for the bispectrum under known dilation distribution. It derives an exact inversion formula for the noiseless case and provides a finite-sample estimator with provable mean-squared error decay, then extends to the noisy dilation setting with a noise-unbiasing step and smoothing, culminating in a convex optimization problem for practical computation. The authors demonstrate both bispectrum recovery and subsequent hidden-signal reconstruction, showing robustness to unknown noise and dilation parameters and quantifying convergence behavior. These results yield a dilation-invariant path to full signal recovery, with potential impact on cryo-EM and other imaging domains where scale variation complicates standard MRA approaches.

Abstract

Motivated by modern data applications such as cryo-electron microscopy, the goal of classic multi-reference alignment (MRA) is to recover an unknown signal from many observations that have been randomly translated and corrupted by additive noise. We consider a generalization of classic MRA where signals are also corrupted by a random scale change, i.e. dilation. We propose a novel data-driven unbiasing procedure which can recover an unbiased estimator of the bispectrum of the unknown signal, given knowledge of the dilation distribution. Lastly, we invert the recovered bispectrum to achieve full signal recovery, and validate our methodology on a set of synthetic signals.
Paper Structure (10 sections, 9 theorems, 98 equations, 9 figures)

This paper contains 10 sections, 9 theorems, 98 equations, 9 figures.

Table of Contents

  1. Introduction to MRA
  2. Notation and preliminaries
  3. Bispectrum Recovery for Model 3
  4. Bispectrum Recovery for Model 2
  5. Numerical Implementation of Bispectrum Recovery
  6. Numerical Experiments for Bispectrum Recovery
  7. Hidden Signal Recovery
  8. Conclusion
  9. Proof of Theorem 2
  10. Proof of Lemma \ref{['lem:noise_bound']}

Key Result

Theorem 1

Assume that $Bf \in C^1(\mathbb{R}^2)$. Then $Bf$ can be recovered from $g_\eta$, namely:

Figures (9)

  • Figure 1: Left Column: three ground truth signals that have been translated without any additive noise. Middle Left Column: Dilating each of the signals. Middle Right Column: Adding Gaussian noise with $\sigma^2 = 0.5$ to each of the signals. Right Column: Adding Gaussian noise with $\sigma^2 = 2$ to each of the signals. Source: hirn2023power.
  • Figure 2: Bispectrum recovery error with oracle $\sigma,\eta$.
  • Figure 3: Bispectrum recovery error with unknown $\sigma,\eta$.
  • Figure 4: Example of bispectrum recovery for $f_2$ with $\eta = 12^{-1/2}$ and $\sigma = 1.0$. Left: ground truth signal. Middle: average of bispectra after additive noise unbiasing. Right: recovered bispectrum using unbiasing procedure.
  • Figure 5: $\eta$ estimation plots with standard error bars.
  • ...and 4 more figures

Theorems & Definitions (18)

  • Theorem 1
  • proof
  • Lemma 1
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 8 more