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Write $A_1={\mathrm{Div}}_1$ and $A_2={\mathrm{Div}}_2$, with Euclidean adjoints $A_1^{\ast },A_2^{\ast }$ determined by the chosen discretization and boundary conditions. Take the usual positive total generalized variation weights ${\alpha }_1,{\alpha }_2$. Introduce a primal variable $w\in {\mathbb{R}}^{n\times 2}$ throughThe support function of the row-ball product is a sum of row norms. Convex duality gives the equivalent augmented saddle problemThe equality follows by dualizing the row-ball constraint. With positive radii, $v=0$ strictly satisfies both row constraints, supplying the finite-dimensional qualification for this splitting. The original feasible dual set is compact and nonempty, and the quadratic primal term is coercive; saddle points exist. The TGV divergence splitting avoids the difficult projection onto $\{v:v\in C,A_{1}v\in D\}$.

Use the Chambolle–Pock algorithm. Choose $\tau ,\sigma >0$ with $\tau \sigma \Vert K{\Vert }^2<1$; the sufficient bound $\tau \sigma (\Vert A_2{\Vert }^2+\Vert A_1{\Vert }^2)<1$ is convenient. Initialize $u^0,w^0,v^0$ and ${\bar{u}}^0=u^0$, ${\bar{w}}^0=w^0$. For $k=0,1,\dots$, computeThe dual update is a Euclidean projection onto a convex set onto $C$; the two primal updates are the quadratic proximal operator and radial soft thresholding. Their signs follow from $K^{\ast }v=(A_{2}v,-A_{1}v)$. All substeps are closed form, and the standard finite-dimensional primal-dual convergence result applies to this saddle problem with the stated step-size condition. The iterates $v^k$ satisfy $C$; the additional $D$ constraint is enforced through the splitting at convergence, not claimed for every intermediate iterate. If a weight is zero, the corresponding row projection or support-function proximal step is interpreted directly rather than by division by zero.

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.