Special function of bounded variation ($SBV(\Omega )$)

A special function of bounded variation is a function of bounded variation on a domain whose distributional derivative has no diffuse singular part. Writing the derivative measure as an absolutely continuous part, a jump part and a remaining Cantor part,the defining condition is $D^{c}u=0$. The space is denoted $SBV(\Omega )$. Here $J_u$ is the jump set and $u^{\pm }$ are the two traces across it; ${\mathcal{L}}^n$ and ${\mathcal{H}}^{n-1}$ are Lebesgue measure and Hausdorff measure.

On an interval containing zero, $u=1_{(0,\infty )}$ has derivative a Dirac measure at zero and belongs to $SBV$: jumps are allowed. The Cantor function instead has a diffuse singular derivative and is not in $SBV$. Thus “special” separates sharp interfaces from more distributed singular variation. This makes $SBV$ useful for image segmentation and variational models with both bulk and interface energies.^[ref]