Full-space, Dirichlet, and Neumann TV conventions

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a bounded Lipschitz domain, meaning a bounded open set whose boundary has a finite cover by coordinate neighborhoods in which it is the graph of a Lipschitz function and $\mathrm{\Omega }$ lies on one side of that graph. For $u\in BV(\mathrm{\Omega })$ define the Neumann total variation by $T{V}_N(u;\mathrm{\Omega }):=\mathrm{\mid }Du\mathrm{\mid }(\mathrm{\Omega })$. Define its zero extension $\tilde{u}\in L^1({\mathbb{R}}^n)$ by $\tilde{u}=u$ almost everywhere on $\mathrm{\Omega }$ and $\tilde{u}=0$ almost everywhere on ${\mathbb{R}}^n\setminus \mathrm{\Omega }$. Define the Dirichlet total variation by $T{V}_D(u;\mathrm{\Omega }):=TV(\tilde{u};{\mathbb{R}}^n)$ when $\tilde{u}\in BV({\mathbb{R}}^n)$ and by $T{V}_D(u;\mathrm{\Omega }):=+\mathrm{\infty }$ otherwise. For $v\in BV({\mathbb{R}}^n)$ define the full-space total variation by $T{V}_F(v):=TV(v;{\mathbb{R}}^n)$. Thus Neumann variation counts only the distributional derivative inside $\mathrm{\Omega }$, while Dirichlet variation also counts any jump created by zero extension across $\mathrm{\partial }\mathrm{\Omega }$.

Spatial total variation is the mass of the partition variation of the distributional derivative.

Iglesias, Mercier, and Scherzer distinguish Neumann variation in the open domain from full-space variation of a zero extension.