  Phase recovery from the bispectrum is a central problem in speckle
interferometry which can be posed as an optimization problem minimizing a
weighted nonlinear least-squares objective function. We look at two different
formulations of the phase recovery problem from the literature, both of which
can be minimized with respect to either the recovered phase or the recovered
image. Previously, strategies for solving these formulations have been limited
to first-order optimization methods such as gradient descent or quasi-Newton
methods. This paper explores Gauss-Newton optimization schemes for the problem
of phase recovery from the bispectrum. We implement efficient Gauss-Newton
optimization schemes for all the formulations. For the two of these
formulations which optimize with respect to the recovered image, we also extend
to projected Gauss-Newton to enforce element-wise lower and upper bounds on the
pixel intensities of the recovered image. We show that our efficient
Gauss-Newton schemes result in better image reconstructions with no or limited
additional computational cost compared to previously implemented first-order
optimization schemes for phase recovery from the bispectrum. MATLAB
implementations of all methods and simulations are made publicly available in
the BiBox repository on Github.
