  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.
