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