Second-order approximation of free-discontinuity problems with linear growth

Motivated by applications to image denoising, we propose an approximation of functionals of the form F (u) = integral(Omega) vertical bar del u vertical bar dx + integral(Su) g (vertical bar u(+)- u(-)vertical bar) d'Hn-1 + vertical bar D(c)u vertical bar (Omega), u is an element of BV(Omega), with g : [0, +infinity) -> [0, + infinity) increasing and bounded. The approximating functionals are of Ambrosio-Tortorelli type and depend on the Hessian or on the Laplacian of the edge variable v which thus belongs to W-2,W-2(Omega). When the space dimension is equal to two and three v is then continuous and this improved regularity leads to a sequence of approximating functionals which are ready to be used for numerical simulations.