= Solution
Interpret the printed <Euclidean norm> literally on the full vector array $X^2$. It is a single global <norm>, not the sum of pixelwise <gradient> lengths in <discrete isotropic total variation>. Set $A=\nabla$ and let $A^*$ be its exact <adjoint operator> for the chosen difference and boundary conventions. The closed dual ball and its <image signal> are
$$
B_\alpha=\{p\in X^2:\|p\|_2\leq\alpha\},\qquad
C=A^*B_\alpha.
$$
The set $C$ is nonempty, convex and compact, hence closed, because it is a linear <image signal> of a compact ball in finite dimensions. <Euclidean norm duality> gives
$$
\alpha\|Au\|_2=\sup_{p\in B_\alpha}\langle Au,p\rangle
=\sup_{w\in C}\langle u,w\rangle=\sigma_C(u),
$$
the <support function> of $C$.
The <metric projection onto a closed convex set> $w=P_Cg$ uniquely minimizes $\|g-w\|^2$ over $C$. Its <variational characterization of convex projection> says
$$
\langle g-w,z-w\rangle\leq0\quad(z\in C).
$$
Put $u_*=g-w$. This inequality says precisely that $\sigma_C(u_*)=\langle u_*,w\rangle$. For any $v\in X$, the primal energy satisfies
$$
\sigma_C(v)+\frac12\|v-g\|^2
\geq\langle v,w\rangle+\frac12\|v-g\|^2.
$$
Completing the square shows that the right side is uniquely minimized by $v=g-w$. At that point the inequality is equality, so
$$
\boxed{u_*=g-P_Cg,\qquad C=\nabla^*B_\alpha.}
$$
This also follows from the <proximal operator of a support function> and <Moreau decomposition>: the <convex conjugate> of $\sigma_C$ is the <indicator functional of a constraint set> for $C$. The squared fidelity is <strictly convex> and coercive, so the primal <minimizer> exists and is unique.
The <convex projection> in this formula is the removed component $g-u_*$, not generally $u_*$ itself. A metric <convex projection> onto one fixed closed <convex set> is idempotent. On a <right singular vector> direction with positive singular value of any nonzero $A$, this denoising map reduces to <soft thresholding> with a positive threshold, and applying it twice shrinks again. It therefore cannot be such a <convex projection> for all data. This qualifies the printed <convex projection> wording while giving the required <global gradient-norm projection residual>.
Compute $P_Cg$ by solving the convex dual least-squares problem
$$
\min_{p\in B_\alpha}f(p),\qquad
f(p)=\frac12\|g-A^*p\|^2.
$$
Its <gradient> is $\nabla f(p)=A(A^*p-g)$ and has <Lipschitz constant> $L=\|A\|^2$. <Projected gradient descent> gives
$$
\boxed{p^{n+1}=P_{B_\alpha}\big[p^n+\tau A(g-A^*p^n)\big],\qquad
P_{B_\alpha}(r)=\frac{r}{\max(1,\|r\|_2/\alpha)}.}
$$
For fixed $0<\tau<2/L$, the finite-dimensional projected-gradient convergence theorem ensures that $p^n$ converges to a dual <minimizer>. Reconstruct $u^n=g-A^*p^n$; its limit is the unique primal solution even if dual <minimizers> are nonunique. If $A=0$, the primal solution is simply $g$ and no iteration is needed. For unit-grid forward differences with periodic or zero-difference boundaries, $\|A\|^2\leq8$, so $0<\tau<1/4$ is sufficient. Grid-spacing factors or other boundary stencils change this bound.
The normalization here is global. If a pixelwise sum of <gradient> lengths had instead been intended, the dual feasible set would be a product of pixelwise balls and <convex projection> would normalize each block separately; it is a different regularizer and should not be silently substituted.
For the <discrete Hessian-norm denoising> variant, write $H=\nabla^2:X\to X^4$ and use
$$
\boxed{C_H=H^*\{p\in X^4:\|p\|_2\leq\alpha\},\qquad
u_*=g-P_{C_H}g.}
$$
This is again a compact convex <image signal> of a Euclidean ball, now under the second-difference adjoint. In suitable conventions $H^*$ is a discrete double divergence, but its exact boundary adjoint is what defines the set. It lies in $(\ker H)^\perp$, so components in $\ker H$ are preserved by denoising. Interior second <derivatives> annihilate affine <image signals>; whether all such <image signals> remain in the kernel depends on the boundary convention. With a pixelwise Hessian <norm>, the corresponding balls would instead be four-component blocks.
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