  Given a unirational parameterization of a surface, we present a general
algorithm to determine a birational parameterization without using
parameterization algorithms. Additionally, if the surface is assumed to have a
birational parametrization with empty base locus, and the input parametrization
is transversal, the degree of the solution is determined in advance and the
dimension of the space of solutions is reduced. As a consequence, for these
cases, we present a second faster algorithm.
