# Multi-target Detection with an Arbitrary Spacing Distribution

## TL;DR

This paper introduces autocorrelation and EM-based methods for reconstructing signals in noisy multi-target detection scenarios with arbitrary spacing, especially effective at high noise levels where detection fails.

## Contribution

It presents novel approaches that enable signal reconstruction without detection, applicable to arbitrary spacing distributions, and analyzes their sample complexity in high noise conditions.

## Key findings

- Reconstruction feasible without detecting individual signals.
- Sample complexity scales as 1/SNR^3 at low SNR.
- Methods work with arbitrary spacing distributions.

## Abstract

Motivated by the structure reconstruction problem in single-particle cryo-electron microscopy, we consider the multi-target detection model, where multiple copies of a target signal occur at unknown locations in a long measurement, further corrupted by additive Gaussian noise. At low noise 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 approaches---autocorrelation 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 1/SNR^3 in the low SNR regime.