  We address a question raised by Ambrosio, Bourgain, Brezis, and Figalli,
proving that the $\Gamma$-limit, with respect to the $L^1_{\rm loc}$ topology,
of a family of $BMO$-type seminorms is given by $\tfrac14$ times the total
variation seminorm. Our method also yields an alternative proof of previously
known lower bounds for the pointwise limit and conveys a compactness result in
$L^1_{\rm loc}$ in terms of the boundedness of the $BMO$-type seminorms.
