  When waves propagate through a strongly scattering medium the energy is
transferred to the incoherent wave part by scattering. The wave intensity then
forms a random speckle pattern seemingly without much useful information.
However, a number of recent physical experiments show how one can extract
useful information from this speckle pattern. Here we present the mathematical
analysis that explains the quite stunning performance of such a scheme for
speckle imaging. Our analysis identifies a scaling regime where the scheme
works well. This regime is the white-noise paraxial regime, which leads to the
Ito-Schrodinger model for the wave amplitude. The results presented in this
paper conform with the sophisticated physical intuition that has motivated
these schemes, but give a more detailed characterization of the performance.
  The analysis gives a description of (i) the information that can be extracted
and with what resolution (ii) the statistical stability or signal-to-noise
ratio with which the information can be extracted.
