  This paper is concerned with the inverse obstacle scattering problem with
phaseless far-field data at a fixed frequency. The main difficulty of this
problem is the so-called translation invariance property of the modulus of the
far-field pattern or the phaseless far-field pattern generated by one plane
wave as the incident field, which means that the location of the obstacle can
not be recovered from such phaseless far-field data at a fixed frequency. It
was recently proved in our previous work \cite{XZZ18} that the obstacle can be
uniquely determined by the phaseless far-field patterns generated by infinitely
many sets of superpositions of two plane waves with different directions at a
fixed frequency if the obstacle is a priori known to be a sound-soft or an
impedance obstacle with real-valued impedance function. The purpose of this
paper is to develop a direct imaging algorithm to reconstruct the location and
shape of the obstacle from the phaseless far-field data corresponding to
infinitely many sets of superpositions of two plane waves with a fixed
frequency as the incident fields. Our imaging algorithm only involves the
calculation of the products of the measurement data with two exponential
functions at each sampling point and is thus fast and easy to implement.
Further, the proposed imaging algorithm does not need to know the type of
boundary conditions on the obstacle in advance and is capable to reconstruct
multiple obstacles with different boundary conditions. Numerical experiments
are also carried out to illustrate that our imaging method is stable, accurate
and robust to noise.
