Homogeneous bounded-variation space ($\{u\in L^2({\mathbb{R}}^2):|Du|({\mathbb{R}}^2)<\infty \}$)

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.

For $1<a\le 2$, the radial function $f(x)=(1+|x|{)}^{-a}$ belongs to $L^2({\mathbb{R}}^2)$ and has finite distributional total variation seminorm on a domain, but is not in $L^1({\mathbb{R}}^2)$. Cutting it off outside radius $T$ gives bounded-variation space functions tending to $f$ in $L^2$, with variation tending to that of $f$: the added jump costs $2\pi T(1+T{)}^{-a}\to 0$. Thus the penalty equal to TV on $BV\cap L^2$ and infinity elsewhere is not sequentially lower semicontinuous in $L^2$. The closed homogeneous bounded-variation space extension avoids this domain issue.