  We consider the approximation of the total variation of a function by the
family of non-local and non-convex functionals introduced by H. Brezis and
H.-M. Nguyen in a recent paper. The approximating functionals are defined
through double integrals in which every pair of points contributes according to
some interaction law.
  In this paper we answer two open questions concerning the dependence of the
Gamma-limit on the interaction law. In the first result, we show that the
Gamma-limit depends on the full shape of the interaction law, and not only on
the values in a neighborhood of the origin. In the second result, we show that
there do exist interaction laws for which the Gamma-limit coincides with the
pointwise limit on smooth functions.
  The key argument is that for some special classes of interaction laws the
computation of the Gamma-limit can be reduced to studying the asymptotic
behavior of suitable multi-variable minimum problems.
