Total generalized variation (${\mathrm{TGV}}_{\alpha }^2$)

Second-order total generalized variation combines a first derivative with an auxiliary vector field and its symmetric derivative. A standard continuous form isHere $Ew$ 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.

For a quadratic data term and divergence maps $A_1,A_2$, the dual constraint $A_{1}v\in D$ can be split through ${\delta }_D(A_{1}v)={\sup }_w[⟨w,A_{1}v⟩-{\delta }_D^{\ast }(w)]$. For a product of Euclidean row balls of radius $\beta$, its support function is ${\delta }_D^{\ast }(w)=\beta {\sum }_i\Vert w_i{\Vert }_2$. The resulting saddle coupling is $K(u,w)=A_2^{\ast }u-A_1^{\ast }w$. The Chambolle–Pock algorithm then uses a row-ball projection, a quadratic proximal operator and radial soft thresholding, with no projection onto an intersection involving a divergence operator.