= Total generalized variation
{title2=$\operatorname{TGV}^2_\alpha$}
= TGV
{c}
{synonym}
Second-order total generalized variation combines a first derivative with an auxiliary vector field and its symmetric derivative. A standard continuous form is
$$
\operatorname{TGV}_{\alpha}^{2}(u)=\inf_w\{\alpha_1\|Du-w\|_{\mathcal M}+\alpha_0\|Ew\|_{\mathcal M}\},\qquad\alpha_0,\alpha_1>0.
$$
Here $Ew$ denotes the symmetric distributional derivative and the norms are total variations of the corresponding measures. Discrete variants replace these operators by linear maps and use sums of row norms. Unlike first-order <total variation>, this regularizer accommodates piecewise-affine behavior. A <TGV divergence splitting> makes its constrained divergence dual amenable to explicit <proximal operators>.
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