  Given values of a piecewise smooth function $f$ on a square grid within a
domain $\Omega$, we look for a piecewise adaptive approximation to $f$.
Standard approximation techniques achieve reduced approximation orders near the
boundary of the domain and near curves of jump singularities of the function or
its derivatives. The idea used here is that the behavior near the boundaries,
or near a singularity curve, is fully characterized and identified by the
values of certain differences of the data across the boundary and across the
singularity curve. We refer to these values as the signature of $f$. In this
paper, we aim at using these values in order to define the approximation. That
is, we look for an approximation whose signature is matched to the signature of
$f$. Given function data on a grid, assuming the function is piecewise smooth,
first, the singularity structure of the function is identified. For example in
the 2-D case, we find an approximation to the curves separating between smooth
segments of $f$. Secondly, simultaneously we find the approximations to the
different segments of $f$. A system of equations derived from the principle of
matching the signature of the approximation and the function with respect to
the given grid defines a first stage approximation. An second stage improved
approximation is constructed using a global approximation to the error obtained
in the first stage approximation.
