Multi-Target Detection With an Arbitrary Spacing Distribution

Motivated by the structure reconstruction problem in single-particle cryo-electron microscopy, we consider the multitarget detection model, where multiple copies of a target signal occur at unknown locations in a long measurement, further corrupted by additive Gaussian noise. At lownoise levels, one can easily detect the signal occurrences and estimate the signal by averaging. However, in the presence of high noise, which is the focus of this paper, detection is impossible. Here, we propose two approachesautocorrelation analysis and an approximate expectation maximization algorithm-to reconstruct the signal without the need to detect signal occurrences in the measurement. In particular, our methods apply to an arbitrary spacing distribution of signal occurrences. We demonstrate reconstructions with synthetic data and empirically show that the sample complexity of both methods scales as SNR-3 in the low SNR regime.