# Multi-target detection with application to cryo-electron microscopy

## TL;DR

This paper introduces an autocorrelation-based method for multi-target signal estimation in noisy measurements, enabling recovery even at high noise levels where traditional detection fails, with applications to cryo-electron microscopy.

## Contribution

It develops a novel autocorrelation analysis approach that allows signal estimation from noisy data without detection or clustering, extending capabilities to extreme noise conditions.

## Key findings

- Method accurately estimates signals at high noise levels.
- Autocorrelation relations enable signal recovery from long measurements.
- Numerical experiments demonstrate effectiveness across various scenarios.

## Abstract

We consider the multi-target detection problem of recovering a set of signals that appear multiple times at unknown locations in a noisy measurement. In the low noise regime, one can estimate the signals by first detecting occurrences, then clustering and averaging them. In the high noise regime however, neither detection nor clustering can be performed reliably, so that strategies along these lines are destined to fail. Notwithstanding, using autocorrelation analysis, we show that the impossibility to detect and cluster signal occurrences in the presence of high noise does not necessarily preclude signal estimation. Specifically, to estimate the signals, we derive simple relations between the autocorrelations of the observation and those of the signals. These autocorrelations can be estimated accurately at any noise level given a sufficiently long measurement. To recover the signals from the observed autocorrelations, we solve a set of polynomial equations through nonlinear least-squares. We provide analysis regarding well-posedness of the task, and demonstrate numerically the effectiveness of the method in a variety of settings. The main goal of this work is to provide theoretical and numerical support for a recently proposed framework to image 3-D structures of biological macromolecules using cryo-electron microscopy in extreme noise levels.