Total variation denoising (${\min }_u\{\mathrm{TV}(u)+\frac{1}{2}\Vert u-g{\Vert }_2^2\}$)

For $g\in L^2$ on a bounded domain, total variation denoising has a unique minimizer. A bounded minimizing sequence has a weakly convergent $L^2$ subsequence; variation is a supremum of weakly continuous test-function pairings and the fidelity has weak lower semicontinuity. The strictly convex fidelity proves uniqueness. Bounded-variation contraction under clipping proves that data range bounds pass to the minimizer.

This finite-image convex optimization model combines robust L1 norm fidelity with discrete isotropic total variation and pixel-range constraints. An exact second-order cone program uses two orthant fidelity slacks per pixel, two box slacks per pixel and a three-dimensional Lorentz cone for each gradient magnitude. The product canonical barrier has parameter $6n$ for $n$ pixels.

For scalar total variation denoising with $f\in BV(\Omega )\cap L^2(\Omega )$ on a bounded Lipschitz domain in arbitrary dimension, the inequality holds with common oriented BV traces on a hypersurface. It says that every surviving jump has the same direction as its data jump and no larger amplitude. Reversing the normal changes both differences and preserves the inequality.

Use residual-preserving clipping of an ROF minimizer and the jump-amplitude inequality for a bounded ROF minimizer. At each finite jump, choose an integer clipping level above both output traces. The clipped function has those same traces, and its residual-adjusted data have the same traces as $f$. Taking a countable union over the integer clipping levels proves the result. This gives the no-new-jumps property of total variation denoising under the full $L^2\cap BV$ assumptions, without asserting regularity of unbounded-forcing perimeter minimizers.

For scalar total variation denoising with $f\in BV(\Omega )\cap L^2(\Omega )$ on a bounded Lipschitz domain in arbitrary dimension, the reconstructed jump set of a bounded-variation function lies in the data jump set of a bounded-variation function up to a surface-null set. The jump-amplitude inequality for total variation denoising proves this: outside $J_f$, the two BV traces on a hypersurface of $f$ agree, so $[u]([f]-[u])\ge 0$ forces $[u]=0$. Surviving jumps cannot be stronger than the corresponding data jumps and must have the same orientation. Existing jumps may disappear.

The proof uses scalar total variation splitting under clipping, residual-preserving clipping of an ROF minimizer, total variation under opposite smooth flows and the BV jump-product limit with one bounded factor. It requires neither bounded data nor bounded output. The bounded-forcing geometric proof through noncontact of ROF level boundaries remains an alternative for bounded images; it is not used to infer regularity under unbounded forcing.

Assume $w\in BV\cap L^{\infty }$ minimizes scalar total variation denoising for $g\in BV\cap L^2$. The data $g$ may be unbounded. For the local flow of $\phi e_j$, use $(1-\theta )w+\theta w\circ {\Phi }_{\pm t}$ as two competitors. The total variation under opposite smooth flows and convexity of the total variation seminorm bound the sum of regularizer changes by $O(t^2)$. Minimality and the opposite-flow fidelity identity for quadratic data, evaluated using the BV jump-product limit with one bounded factor, implyLet $\theta \downarrow 0$. The integrands define finite signed Radon measures, since $w$ is bounded and the jump variation of $g$ is finite. Arbitrary nonnegative smooth $\phi$ make their densities nonnegative. Testing every coordinate direction removes the factor $|{\nu }_w\cdot e_j|$ and proves the stated inequality.

If $u$ minimizes scalar total variation denoising with $f\in BV\cap L^2$, set $w=T_{M}u$, $r=u-w$ and $g=f-r$. The scalar total variation splitting under clipping gives $\mathrm{TV}(u)=\mathrm{TV}(w)+\mathrm{TV}(r)$. Compare $u$ with $v+r$ and use $\mathrm{TV}(v+r)\le \mathrm{TV}(v)+\mathrm{TV}(r)$; cancelling the tail shows that $w$ minimizes the same model for $g$. It is bounded while $g$ may be unbounded, and $w-g=u-f$ exactly. This reduction concerns the actual minimizer, rather than convergence of minimizers for truncated data.

The squared-error term $\frac{1}{2}\Vert u-f{\Vert }_2^2$ is a strictly convex cost measuring agreement with data.

Let $w\in BV\cap L^{\infty }$ and $g\in BV\cap L^2$ on a bounded domain, $F_g(v)=\frac{1}{2}\Vert v-g{\Vert }_2^2$, and $w_{\pm t}=(1-\theta )w+\theta w\circ {\Phi }_{\pm t}$ for a smooth compactly supported local flow. The displayed formula is exact to an $o(t)$ remainder, and $g$ need not be bounded. Writing $J_t=\det D{\Phi }_t$, its cross-term identity isThe remainder is $o(t)$ because $1-J_{-t}=O(t)$ and weighted $L^1$ convergence follows fromFirst let $t\to 0$, then $K\to \infty$. The mass change of $w^2$ under opposite flows is $O(t^2)\Vert w{\Vert }_2^2$. Combined with the BV jump-product limit with one bounded factor, this evaluates the jump contribution without a bounded fidelity derivative.

The scalar ROF denoising minimizer has perimeter-plus-forcing minimizing superlevel sets at almost every height. Conversely any finite-energy function with that property minimizes the full energy. The signed layer-cake identity for quadratic fidelity and coarea formula for BV functions prove sufficiency. To prove necessity, construct nested minimizing sets using comparison of perimeter minimizers with ordered forcing, reconstruct a function from rational levels, clip to establish energy bounds, and use uniqueness from strict convexity of the squared fidelity.

For bounded BV space data and $s<t$, nested ROF level-set formulation minimizing sets cannot share an interface of positive surface measure away from data jumps. On regular graph patches, upward and downward first variations use the BV traces on a hypersurface. At contacts outside $J_f$ the forcing BV trace on a hypersurface agrees, whereas the two levels differ by $t-s$. Graph regularity and contact differentiation give the contradiction. Bounded forcing is a hypothesis of this argument; arbitrary unbounded $L^2\cap BV$ data are not covered by this statement.

For positive $u$, integrate $t-f$ from zero to $u$; for negative $u$, reverse the integral from $u$ to zero. This proves the identity without a positivity restriction on the data. For $u,f\in L^2$, the absolute integral is bounded by $u^2/2+|fu|$, so Fubini's theorem applies. The zero-function baseline removes the divergence from the negative-height tail.

For finite convex total variation and a positive squared-error budget, the Slater condition supplies a multiplier $\lambda \ge 0$ such that every constrained minimizer also minimizes $\mathrm{TV}(u)+\lambda \Vert u-g{\Vert }^2$. Conversely, a minimizer of that penalized objective with $\lambda >0$ minimizes total variation under the budget equal to its own squared error. Complementary slackness proves the first implication; a comparison of objective values proves the second. This does not assert uniqueness of the multiplier or equality of all minimizer sets when $\lambda =0$.

The graph total variation measures absolute variation of a vertex signal across edges. It is independent of the orientation used for the oriented incidence matrix. It vanishes exactly on signals constant on each connected component of a graph. A graph total variation denoising estimator combines this penalty with squared error, favouring signals that are piecewise constant on connected regions.

Graph total variation denoising is a penalized least-squares estimator for noisy vertex signals on a graph. The squared loss is strictly convex, so the fitted signal is unique. Its basic inequality for a penalized least-squares estimator separates a noise inner product from the change in penalty. The incidence pseudoinverse decomposition controls the nonconstant noise through the geometry of the graph.

For data $g={\chi }_{B(0,R)}$ on ${\mathbb{R}}^2$, the minimizer of $\alpha \mathrm{TV}(u)+\frac{1}{2}\Vert u-g{\Vert }_2^2$ is the displayed amplitude shrinkage, including extinction at $\alpha =R/2$. The total variation calibration uses $z(x)=x/R$ inside the disk and $z(x)=Rx/|x{|}^2$ outside. Its continuous normal component creates no boundary measure, and $\mathrm{div}z=(2/R){\chi }_{B(0,R)}$. This subgradient certifies the positive branch; scaling it down certifies the zero branch. The quadratic fidelity is strictly convex, ensuring uniqueness.

Staircasing is the formation of approximately constant plateaux in reconstructed data, even when the original data vary smoothly. The linear growth of the total variation seminorm on a domain favors sparse spatial gradients, so preserving jumps can come at the cost of flattening gradual transitions.

With zero forward differences at the grid boundary, discrete isotropic total variation is the sum of the Euclidean lengths of the two-component forward gradients. Its dual constraint is the product of unit Euclidean balls, and minimizing $\frac{1}{2}\Vert g-D^{\ast }p{\Vert }_2^2$ over that product yields $u_{\ast }=g-D^{\ast }{p}_{\ast }$. The exact difference adjoint operator is essential. A dual proximal gradient method projects $p+\tau D(g-D^{\ast }p)$ pointwise onto unit balls; $0<\tau \le 1/8$ is safe on an unscaled square grid.

For $C=D^{\ast }\{p:|p_i{|}_2\le \lambda \}$, discrete total variation denoising is the proximal operator of the support function of $C$. The Moreau decomposition gives the displayed projection residual. It is generally not itself a metric projection onto a closed convex set: such a projection is idempotent, but TV denoising shrinks a sufficiently large two-level jump again when reapplied.

Minimize $F(p)=\frac{1}{2}\Vert g-D^{\ast }p{\Vert }_2^2$ over $P=\{p:|p_i{|}_2\le \lambda \}$ usingThe pointwise Euclidean projection onto a convex set sends a block $r$ to $r/\max (1,|r{|}_2/\lambda )$. The gradient of $F$ has Lipschitz constant $\Vert D{\Vert }^2$, so finite-dimensional projected gradient descent converges for $0<\tau <2/\Vert D{\Vert }^2$. On an unscaled square grid with $N>1$, $0<\tau <1/4$ is safe. Reconstruct $u^k=g-D^{\ast }{p}^k$; unused dual directions may make the minimizing $p$ nonunique while the primal minimizer is unique.