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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 $A^{\ast }$ be its exact adjoint operator for the chosen difference and boundary conventions. The closed dual ball and its image signal areThe 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 givesthe support function of $C$.

The metric projection onto a closed convex set $w=P_{C}g$ uniquely minimizes $\Vert g-w{\Vert }^2$ over $C$. Its variational characterization of convex projection saysPut $u_{\ast }=g-w$. This inequality says precisely that ${\sigma }_C(u_{\ast })=⟨u_{\ast },w⟩$. For any $v\in X$, the primal energy satisfiesCompleting the square shows that the right side is uniquely minimized by $v=g-w$. At that point the inequality is equality, soThis 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 $g-u_{\ast }$, not generally $u_{\ast }$ 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 $P_{C}g$ by solving the convex dual least-squares problemIts gradient is $\nabla f(p)=A(A^{\ast }p-g)$ and has Lipschitz constant $L=\Vert A{\Vert }^2$. Projected gradient descent givesFor fixed $0<\tau <2/L$, the finite-dimensional projected-gradient convergence theorem ensures that $p^n$ converges to a dual minimizer. Reconstruct $u^n=g-A^{\ast }{p}^n$; 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, $\Vert A{\Vert }^2\le 8$, 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 useThis is again a compact convex image signal of a Euclidean ball, now under the second-difference adjoint. In suitable conventions $H^{\ast }$ 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.

Write $D=\nabla$ and use the real pixelwise inner products. Let $D^{\ast }$ be the adjoint operator, defined by $⟨Du,p⟩=⟨u,D^{\ast }p⟩$. For each nontrivial grid direction its one-dimensional formula iswith the analogous formula in $j$ for $D_y^{+\ast }{p}^2$. Their sum is $D^{\ast }p$; unused components $p_{N,j}^1$ and $p_{i,N}^2$ do not contribute. For $N=1$, $D=D^{\ast }=0$. If discrete divergence is defined, its sign convention is $\mathrm{div}=-D^{\ast }$, as in the adjoint of a discrete forward gradient.

Define the compact convex set $P_{\lambda }=\{p\in X^2:|p_{i,j}{|}_2\le \lambda \}$ and its image $C=D^{\ast }{P}_{\lambda }$. The image is compact and convex, hence closed, and contains zero. The finite-dimensional Euclidean duality formula $\lambda |a{|}_2={\max }_{|p{|}_2\le \lambda }a\cdot p$, applied independently at every pixel, givesThus the penalty is the support function ${\sigma }_C$. Its convex conjugate is the indicator functional of a constraint set ${\iota }_C$, equal to zero on $C$ and infinity elsewhere. For $w\in C$, the conjugate supremum is zero; for $w\notin C$, the separation theorem supplies a direction $u$ with $⟨u,w⟩>{\sigma }_C(u)$, and scaling that direction makes the supremum infinite.

The proximal operator is ${\mathrm{prox}}_{{\sigma }_C}(g)=\arg {\min }_u\{{\sigma }_C(u)+\frac{1}{2}\Vert u-g{\Vert }_2^2\}$. It exists uniquely because the objective is continuous, coercive and strictly convex. The Moreau decomposition for a proper closed convex function gives ${\mathrm{prox}}_{{\sigma }_C}(g)+{\mathrm{prox}}_{{\iota }_C}(g)=g$. The second term is the Euclidean projection onto a convex set, so the projection residual for discrete total variation isAlternatively, the variational characterization of convex projection says $w=P_{C}g$ exactly when $⟨g-w,z-w⟩\le 0$ for every $z\in C$. Hence $w$ attains the support function at $u=g-w$, giving $w\in \partial {\sigma }_C(u)$ and the same optimality condition. This supplies a direct verification of the Moreau decomposition step in this instance.

The printed wording needs “the residual after a projection”, rather than “the projection itself”. In general this denoising map is not a projection onto any fixed closed convex set. A metric projection onto a closed convex set is idempotent, while denoising a sufficiently large jump twice shrinks it twice. Concretely, for $N=2$ and data with rows $(0,0)$ and $(6\lambda ,6\lambda )$, the minimizer has rows $(\lambda ,\lambda )$ and $(5\lambda ,5\lambda )$. Reapplying the map gives rows $(2\lambda ,2\lambda )$ and $(4\lambda ,4\lambda )$, so it is not idempotent. These formulas have a global certificate: take $p_{1,j}^1=\lambda$, all other dual entries zero, giving $D^{\ast }p$ with rows $(-\lambda ,-\lambda )$ and $(\lambda ,\lambda )$; it saturates every positive row difference and yields the primal optimality condition. The qualification is therefore not merely a consequence of a restricted two-level ansatz.

Assume $\pi \in C$ andfor every $z\in C$. Expanding the squared euclidean distance givesHence $\pi$ minimizes the distance from $x$ over $C$, so $\pi ={\pi }_C(x)$. This proves the sufficient direction of the variational characterization of convex projection.

The candidate $\Pi$ is tacitly required to belong to $S$, as in the usual variational characterization of convex projection; this feasibility holds for the candidate constructed in part (b). Use the Frobenius inner productFor any $Z\in S$,and henceThus $\Pi$ minimizes the squared Frobenius distance over $S$. Since that objective is strictly convex,