We analyze a nonlinear discrete scheme depending on second-order finite differences. This
is the two-dimensional analog of a scheme which in one dimension approximates a
free-discontinuity energy proposed by Blake and Zisserman as a higher-order correction of
the Mumford and Shah functional. In two dimension we give a compactness result showing
that the continuous problem approximating this difference scheme is still defined on
special functions with bounded hessian, and we give an upper and a lower bound in terms of
the Blake and Zisserman energy. We prove a sharp bound by exhibiting the
discrete-to-continuous Γ-limit for a special class of functions, showing
the appearance new ‘shear’ terms in the energy, which are a genuinely two-dimensional
effect.