BMO-type functionals, total variation, and $\mathrm{\Gamma }$-convergence

We study the BMO-type functional ${\kappa }_{\epsilon }(f,{\mathbb{R}}^n)$, which can be used to characterize BV functions $f\in BV({\mathbb{R}}^n)$. The $\mathrm{\Gamma }$-limit of this functional, taken with respect to $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$-convergence, is known to be $\frac{1}{4}\mathrm{\mid }Df\mathrm{\mid }({\mathbb{R}}^n)$. We show that the $\mathrm{\Gamma }$-limit with respect to $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}$-convergence is $$\frac{1}{4}\mathrm{\mid }{D}^{a}f\mathrm{\mid }({\mathbb{R}}^n)+\frac{1}{4}\mathrm{\mid }{D}^{c}f\mathrm{\mid }({\mathbb{R}}^n)+\frac{1}{2}\mathrm{\mid }{D}^{j}f\mathrm{\mid }({\mathbb{R}}^n),$$ which agrees with the ``pointwise’’ limit in the case of SBV functions.