= Homogeneous bounded-variation space
{title2=$\{u\in L^2(\mathbb R^2):|Du|(\mathbb R^2)<\infty\}$}
In the $L^2$ formulation of <total variation denoising> on the entire plane, one may allow finite distributional <total variation seminorm on a domain> without requiring global $L^1$ membership. Its extended penalty is a supremum of <continuous> test-function pairings and hence is <sequentially lower semicontinuous>. This domain can be larger than $BV(\mathbb R^2)\cap L^2$, since the usual <bounded-variation space> includes an $L^1$ condition.
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