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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 totalvariation seminorm. Our method also yields an alternative proof of previouslyknown 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.

Representation of the total variation as a $\Gamma$-limit of $BMO$-type seminorms