open ball of radius $r$ centered at $x\in{\mathbf{R}}^d$ is denoted by $B_r(x)$. The $(d-1)$-dimensional Hausdorff measure in ${\mathbf{R}}^d$ is denoted by $\H^{d-1}$. We write $|E|$ to denote the Lebesgue measure of a measurable set $E\subset {\mathbf{R}}^d$. When $\mu$ is a Borel meaure and $E\subset {\mathbf{R}}^d$ is a Borel set, we denote by $\mu{ \!\!\mbox{{ \Large$\llcorner$}} }E$ the measure defined as $\mu{ \!\!\mbox{{ \Large$\llcorner$}} }E(F)=\mu(E\cap F)$.\
Let us remark that from Section \[sec:Comp\] onwards, we work in dimension $d=2$.\
For $O\in\A({\Omega})$, the functional ${\mathcal{F}}_{\varepsilon}(\cdot,\cdot;O)$ is the functional obtained by substuting $O$ for ${\Omega}$ in the definition of ${\mathcal{F}}_{\varepsilon}$ (see ). Similarly we define the local version ${\mathcal{E}}_\a(\cdot,\cdot;O)$ of ${\mathcal{E}}_\a$.

BV(${\Omega}$) functions and Slicing
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BV(${\Omega}$) is the space of functions $u\in L^1({\Omega})$ having as distributional derivative $Du$ a measure with finite total variation. For $u\in BV({\Omega})$, we denote by $S_u$ the complement of the Lebesgue set of $u$, that is $x\not \in S_u$ if and only if $$\lim_{\rho{\rightarrow}0^+}\frac{1}{|B_\rho(x)|}\int_{B_\rho(x)}|u(y)-z|\dy=0$$ for some $z\in{\mathbf{R}}$. We say that $x$ is an approximate jump point of $u$ if there exist $\xi \in \S^{d-1}$ and distinct points $a,b \in {\mathbf{R}}$ such that $$\lim_{\rho{\downarrow}0}\frac{1}{|B^+_\rho(x,\xi)|}\int_{B^+_\rho(x,\xi)}|u(y)-a|\dy=0\qquad\text{and}\qquad\lim_{\rho{\downarrow}0}\frac{1}{|B^-_\rho(x,\xi)|}\int_{B^+_\rho(x,\xi)}|u(y)-b|\dy=0,$$ where $B^\pm_\rho(x,\xi):=\{y\in B_\rho(x)\;:\;\pm\langle y-x,\xi\rangle\geq0\}$. Up to a permutation of $a$ and $b$ and a 