  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.
