  With appropriate modifications, the Hoffman--Ribak algorithm that constructs
constrained realizations of Gaussian random fields having the correct ensemble
properties can also be used to construct constrained realizations of those
non-Gaussian random fields that are obtained by transformations of an
underlying Gaussian field. For example, constrained realizations of lognormal,
generalized Rayleigh, and chi-squared fields having $n$ degrees of freedom
constructed this way will have the correct ensemble properties. The lognormal
field is considered in detail. For reconstructing Gaussian random fields,
constrained realization techniques are similar to reconstructions obtained
using minimum variance techniques. A comparison of this constrained realization
approach with minimum variance, Wiener filter reconstruction techniques, in the
context of lognormal random fields, is also included. The resulting
prescriptions for constructing constrained realizations as well as minimum
variance reconstructions of lognormal random fields are useful for
reconstructing masked regions in galaxy catalogues on smaller scales than
previously possible, for assessing the statistical significance of small-scale
features in the microwave background radiation, and for generating certain
non-Gaussian initial conditions for $N$-body simulations.
