 Homogeneous bounded-variation space

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.