Total variation seminorm on a domain ($\mathrm{TV}(u)$)

For $u\in L^1(\Omega )$, the total variation seminorm isIt extends the integral of the magnitude of the gradient to nonsmooth functions.

For the local flow of a smooth compactly supported vector field and any scalar BV space function $w$, the change of variables formula and polar decomposition of a vector measure giveThe two cofactor matrices are $I\pm tB+O(t^2)$ uniformly. On $|{\sigma }_w|=1$, the linear terms in their Euclidean norm expansions cancel. Integration bounds the remainder by $C{t}^2|Dw|(\Omega )$. The identity includes all components of the decomposition of a BV derivative; it does not require smoothness of $w$ or its jump interfaces.

For an absolutely one-homogeneous functional $J=\mathrm{TV}$ on a real Hilbert space, a subgradient $q$ at $u$ is characterized by $⟨q,v⟩\le J(v)$ for every $v$ and $⟨q,u⟩=J(u)$. A bounded vector field with $|z|\le 1$ and $q=\mathrm{div}z$ proves the inequality through the dual definition, provided cutoff approximation is valid in the ambient space. Equality is called a calibration. It certifies a global minimizer of total variation denoising, rather than only a minimizer within a chosen ansatz.

Total variation flow is the $L^2$ gradient flow of the total variation seminorm on a domain. Formally it is $u_t=\mathrm{div}(\nabla u/|\nabla u|)$, but the subgradient formulation handles zero gradients and jumps. Adding quadratic fidelity gives a strictly convex stationary total variation denoising problem.

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.

For $a<b$ strictly inside a one-dimensional domain, the indicator $u=1_{[a,b]}$ has distributional derivative $Du={\delta }_a-{\delta }_b$, hence $\mathrm{TV}(u)=2$. Directly from the test-function definition, $\int u{\phi }^{\prime }=\phi (b)-\phi (a)\le 2$, and two disjoint normalized smooth bump functions attain the bound. Values at the endpoints themselves do not change the $L^1$ function.

A function $u\in L^1(\Omega )$ has bounded variation when $\mathrm{TV}(u)<\infty$. The space $BV(\Omega )$ is a Banach space with norm $\Vert u{\Vert }_{L^1}+\mathrm{TV}(u)$.

A Cauchy sequence $(u_n)$ in the BV space converges in $L^1$ to $u$. For fixed sufficiently large $n$, lower semicontinuity gives $|D(u_n-u)|(\Omega )\le {\mathrm{liminf}}_m|D(u_n-u_m)|(\Omega )$. The $L^1$ norm of the same difference converges, so the small full-norm Cauchy bound passes to $u_n-u$. The triangle inequality then puts $u$ in the BV space and proves convergence in its full norm. This argument turns completeness of $L^1$ and lower semicontinuity of variation into completeness of the combined norm.

For a BV space function $a$, almost every restriction $a_y$ to a line parallel to $e_j$ has bounded variation on the corresponding domain slice. Directional derivative variation disintegrates as $|D_{j}a|={\mathcal{L}}^{n-1}\otimes |D{a}_y|$. Slice jump values agree with the appropriate oriented BV traces on a hypersurface for almost every crossing of a rectifiable jump surface. Integrating slice jump sums produces a surface integral with projection factor $|\nu \cdot e_j|$. Uniform essential bounds on the slices are not asserted for an unbounded function.

Let $a\in BV(\Omega )$, $b\in BV(\Omega )\cap L^{\infty }(\Omega )$ and ${\delta }_{t}c=c\circ {\Phi }_t-c$, where ${\Phi }_t$ is the local flow of $\phi e_j$ with nonnegative $\phi \in C_c^{\infty }(\Omega )$. The formula uses common oriented BV traces on a hypersurface; the trace difference of $a$ is zero almost everywhere on $J_b\setminus J_a$. It also holds for both negative-time increments with denominator positive $t$.

The BV slicing theorem reduces the calculation to one dimension. With a right-continuous representative of $a$ and $\mu =Da$, Fubini's theorem yieldsThe inner factor tends to $\phi (s)[b](s)$ and is bounded in absolute value by $2\Vert b{\Vert }_{\infty }\Vert \phi {\Vert }_{\infty }$. Dominated convergence against $|\mu |$ leaves only its measure atoms at the countable slice jumps of $b$. That same bound times $|D{a}_y|$ permits integration over the transverse coordinates. No essential bound on $a$ is used, even if its slice values grow without bound. The formula is useful for opposite-flow fidelity identity for quadratic data with unbounded data and a bounded comparison function.

A BV space function has two one-sided approximate mean BV traces on a hypersurface at almost every point of an oriented countably rectifiable hypersurface, with respect to Hausdorff measure. They agree outside its approximate discontinuity set. At an approximate jump point, they are the jump values, up to the choice of orientation.

A BV space function is approximately differentiable at almost every point with respect to Lebesgue measure. Its approximate gradient is the density of the absolutely continuous part of its distributional derivative. This does not imply membership in a Sobolev space.

For a BV space function, this is the set where no finite approximate mean limit of a function exists. Its difference from the jump set of a bounded-variation function has zero $(n-1)$-dimensional Hausdorff measure.

Both functions represent the same $L^1$ equivalence class. Fubini's theorem shows that the interval-mass function has derivative $Du$, so its difference from $u$ is constant. Continuity of a finite measure gives left continuity for $u_l$ and right continuity for $u_r$. Their opposite one-sided limits differ by the measure atom $Du(\{t\})$. There are only countably many nonzero measure atoms, since every set of measure atoms above a fixed magnitude threshold is finite.

An approximate mean limit of a function is a value satisfying the displayed condition. Functions in the BV space have such finite limits outside an approximate discontinuity set up to negligible sets in the appropriate measure. Their approximate jump points instead have two distinct half-ball limits.

A scalar function in the BV space has an approximate jump at a point when its mean on each of two half-balls approaches a different finite value. The half-balls are determined by a unit normal ${\nu }_u$. Reversing the normal exchanges the two BV traces on a hypersurface. Their collection is the jump set of a BV function.

For a scalar function in the BV space, derivative variation is the integral of the relative perimeter measures of its superlevel sets. The formula holds on every Borel subset $A$ of the domain, not just for total mass. It converts the continuous ROF denoising problem into linked minimization problems for finite-perimeter sets.

The functional $\phi \mapsto -\int u\mathrm{div}\phi$ is bounded in the uniform norm exactly when $u$ belongs to the BV space. Extend it from compactly supported smooth tests to $C_0$, then apply the Riesz-Markov-Kakutani representation theorem to obtain the unique finite vector Radon measure $Du$. Its dual norm is the defining variation supremum. This representation should be distinguished from the deeper rectifiability results used in the decomposition of a BV derivative.

The Lebesgue decomposition theorem separates the absolutely continuous part from the singular part. BV fine structure then splits the latter into the jump part of a BV derivative and the Cantor part of a BV derivative. The density of the absolutely continuous part is the almost-everywhere approximate gradient. These three mutually singular components distinguish smooth variation, surface discontinuities and diffuse singular variation.

This derivative component is carried by the jump set of a BV function. Swapping the two BV traces on a hypersurface and reversing the normal leaves it unchanged. In one dimension it consists of the point measure atoms, each weighted by the difference between the right and left limits.

For a smooth function, integrate its directional derivative along each segment of length $|h|$ and then integrate in space. This gives the displayed bound with $\int |\nabla w|$. Smooth BV approximation and sequential lower semicontinuity give the same inequality for general functions in the BV space. Averaging it over a mollifier proves the BV mollification error estimate.

For a nonnegative normalized mollifier supported in the radius-$ϵ$ ball, integrate the BV translation estimate with weight ${\rho }_{ϵ}(h)$. The full-space derivative variation controls the full-space error. Replacing it by variation on a smaller domain is invalid without a support condition: a nonconstant bump outside that domain has zero derivative measure inside but a nonzero mollification error.

For a bounded Lipschitz domain, functions can be extended with the displayed bound and common compact support. Reflect across local Lipschitz boundary graphs, combine the reflected functions with a partition of unity, and multiply by a fixed cutoff function. Lipschitz coordinate changes control derivative measures; the product rule bounds cutoff function terms by the $L^1$ norm. Zero extension is also possible with an estimate for BV traces on a hypersurface, but its boundary jump must be included in the variation.

Convergence in the displayed sense combines strong function convergence with weak-star convergence of derivative vector measures against $C_0$ tests. Strong $L^1$ convergence and a uniform variation bound imply the derivative convergence by integration by parts and uniform test-function approximation. Conversely the Uniform boundedness principle bounds the derivative measures of a convergent sequence.

Strict convergence combines strong convergence in $L^1$ with convergence of the total variation seminorm. It implies weak-star convergence in BV but does not make jump sets stable: mollifications of an interior step can converge strictly to the step while every approximant has an empty jump set of a bounded-variation function.

A measurable set $E$ has finite perimeter in $\Omega$ when ${\chi }_E$ belongs to the BV space. Its relative perimeter is $\mathrm{Per}(E;\Omega )=|D{\chi }_E|(\Omega )$. Relative perimeter excludes the exterior image signal boundary and measures the interior interface in Caccioppoli partitions.

The perimeter of a measurable set relative to an open domain is the total variation seminorm of its indicator function there. For a smooth region it equals the Hausdorff measure of the boundary lying inside the domain. Global perimeter uses the entire space as the domain; an internal interface shared by two regions appears in both relative perimeters and must be counted only once in a segmentation energy.

Here the sets of finite perimeter $E,F$ differ only compactly inside $U$. A minimizing set with bounded volume forcing satisfies this inequality. Interior regularity theorems give graph patches outside a singular set negligible for surface Hausdorff measure; bounded forcing allows the graph estimates used in the noncontact of ROF level boundaries.

The total variation seminorm of the indicator function within an open set counts only its interior interface. The perimeter of a zero extension may add a boundary contribution.

The BV lattice inequality $|D\min (v,w)|+|D\max (v,w)|\le |Dv|+|Dw|$ applied to two indicator functions gives the perimeter inequality. For smooth approximations, minima and maxima partition the two gradients; sequential lower semicontinuity passes the estimate to BV. It prevents a crossing of minimizers with strictly ordered set forcing.

Suppose $E_g$ minimizes $\mathrm{Per}(E)-{\int }_{E}g$ and $E_h$ minimizes $\mathrm{Per}(E)-{\int }_{E}h$. Compare them with their intersection and union. Adding their optimality inequalities and using submodularity of relative perimeter gives ${\int }_{E_g\setminus E_h}(h-g)\le 0$. Strict ordering forces that set to be null. Applied to $g=(f-t)/\alpha$, this yields nested ROF denoising superlevel sets.

A Caccioppoli partition is a countable measurable partition, up to null sets, into sets of finite perimeter with finite sum of relative perimeters. Its interior interfaces are counted twice by that sum. It represents regions of a piecewise constant image signal in the piecewise-constant Mumford–Shah problem.

The derivative measure of a BV space function decomposes into its absolutely continuous, jump and Cantor parts. The Cantor part of a bounded-variation derivative is singular but not a jump measure on a rectifiable hypersurface. The special bounded-variation space excludes it; finite-length image signal interfaces alone cannot describe it.

The jump set consists of points where a BV space function has distinct one-sided approximate values across an approximate normal. In the plane its derivative contribution is $[u]{\nu }_u{\mathcal{H}}^1\lfloor J_u$. The Mumford–Shah functional penalizes its length rather than the jump amplitude.

A BV space function is special when its derivative has no Cantor part of a bounded-variation derivative: $Du=\nabla udx+[u]{\nu }_u{\mathcal{H}}^{n-1}\lfloor J_u$. The remaining singular part is supported on the jump set of a bounded-variation function. This space accommodates sharp interfaces and is the natural setting for the Mumford–Shah functional.

For uniformly bounded values, an $L^p$ gradient bound with $p>1$, and bounded $(n-1)$-dimensional jump measure, a sequence in the SBV space admits an $L^1$-convergent subsequence whose limit remains special. Gradients converge weakly in $L^p$, and jump measure is lower semicontinuous. These hypotheses prevent diffuse singular derivatives from replacing the controlled interfaces in a minimizing limit.

For a BV space function with $Du=\nabla udx+D^{s}u$, relaxed graph area is $A(u)=\int \sqrt{1+|\nabla u{|}^2}dx+|D^{s}u|(\Omega )$. Its dual supremum uses the coupled constraint ${\phi }_0^2+|\phi {|}^2\le 1$ in the pairing $\int ({\phi }_0+u\mathrm{div}\phi )$. It is convex and lower semicontinuous in $L^1$. Independent bounds on the two test fields define a different functional.

For an assumed $W^{1,1}$ minimizer of $\alpha A(u)+\frac{1}{2}\Vert u-g{\Vert }_2^2$, compactly supported variations give $u-g-\alpha \mathrm{div}(\nabla u/\sqrt{1+|\nabla u{|}^2})=0$ in distributions. If the regularizer is instead the total variation seminorm, a bounded calibration field replaces the quotient and handles zero gradients.

The independent constraints $|{\phi }_0|\le 1$ and $|\phi |\le 1$ separate the dual supremum into $|\Omega |+|Du|(\Omega )$. They do not define square-root graph area. For $|\nabla u|=1$ the box-constrained value is $2|\Omega |$, while the relaxed graph-area functional is $\sqrt{2}|\Omega |$.

On a bounded Lipschitz domain, a sequence with bounded $L^1$ norms and bounded total variation seminorm has a subsequence converging strongly in $L^1$. The limit is in the BV space, and its variation does not exceed the limit inferior. This supplies the compactness step of the direct method in the calculus of variations for linearly growing gradient energies.

In the $L^2$ formulation of total variation denoising on the entire plane, one may allow finite distributional total variation seminorm on a domain without requiring global $L^1$ membership. Its extended penalty is a supremum of continuous test-function pairings and hence is sequentially lower semicontinuous. This domain can be larger than $BV({\mathbb{R}}^2)\cap L^2$, since the usual bounded-variation space includes an $L^1$ condition.

For $1<a\le 2$, the radial function $f(x)=(1+|x|{)}^{-a}$ belongs to $L^2({\mathbb{R}}^2)$ and has finite distributional total variation seminorm on a domain, but is not in $L^1({\mathbb{R}}^2)$. Cutting it off outside radius $T$ gives bounded-variation space functions tending to $f$ in $L^2$, with variation tending to that of $f$: the added jump costs $2\pi T(1+T{)}^{-a}\to 0$. Thus the penalty equal to TV on $BV\cap L^2$ and infinity elsewhere is not sequentially lower semicontinuous in $L^2$. The closed homogeneous bounded-variation space extension avoids this domain issue.

The distributional derivative of this indicator function is $e_1{\mathcal{H}}^1$ on the vertical segment $x_1=1/2$, so its total variation seminorm on a domain is one. This is singular with respect to planar Lebesgue measure, whereas Sobolev space weak derivatives in $W^{1,1}$ have integrable densities. Its bounded-variation space norm is $1/2+1=3/2$.

For $T(t)=\min (b,\max (a,t))$ and $u\in BV(\Omega )$, $T\circ u\in BV(\Omega )$ and $|D(T\circ u)|(\Omega )\le |Du|(\Omega )$. This is the contraction property of a one-Lipschitz scalar composition, obtainable by smooth approximation and sequential lower semicontinuity. It proves range preservation for total variation denoising when the data lie in $[a,b]$.

For a scalar BV space function, clipping to $[-M,M]$ partitions its coarea formula for BV functions into middle and tail levels. The clipped function has the original superlevel sets at heights in $(-M,M)$. The positive tail levels of $u-T_{M}u$ correspond to original heights above $M$, and its negative tail levels to heights below $-M$. Thus their total variation seminorms add exactly. This is stronger than the triangle inequality and is specific to scalar monotone clipping. It supports residual-preserving clipping of an ROF minimizer.

On a bounded domain of positive measure, the constant functions $u_k\equiv k$ satisfy $\mathrm{TV}(u_k)=0$ while $\Vert u_k{\Vert }_{BV}=k|\Omega |\to \infty$. Thus the total variation seminorm on a domain is neither coercive in the $BV$ norm nor strictly convex. Additional control of the mean can remove this obstruction; the Poincaré inequality for total variation supplies the needed bound on suitable connected domains.

The zero-mean bounded-variation space isOn a bounded connected Lipschitz domain, removing the constant nullspace makes total variation coercive on this subspace by the Poincaré inequality for total variation.

On a bounded connected Lipschitz domain,It reduces to $\Vert u{\Vert }_{L^1}\le C_{\Omega }\mathrm{TV}(u)$ on $B{V}_0(\Omega )$.