  Frame design for phaseless reconstruction is now part of the broader problem
of nonlinear reconstruction and is an emerging topic in harmonic analysis. The
problem of phaseless reconstruction can be simply stated as follows. Given the
magnitudes of the coefficients generated by a linear redundant system (frame),
we want to reconstruct the unknown input. This problem first occurred in X-ray
crystallography starting in the early 20th century. The same nonlinear
reconstruction problem shows up in speech processing, particularly in speech
recognition.
  In this lecture we shall cover existing analysis results as well as stability
bounds for signal recovery including: necessary and sufficient conditions for
injectivity, Lipschitz bounds of the nonlinear map and its left inverses,
stochastic performance bounds, and algorithms for signal recovery.
