Monotonicity of the jump set and jump amplitudes in one-dimensional TV denoising

TL;DR

This work proves that in one-dimensional ROF denoising, the jump set $J_{u_\alpha}$ and the jump amplitudes of the minimizers decrease (are monotone) as the regularization parameter $\alpha$ increases, for inputs $f$ in $L^\infty(0,1)$ with left/right approximate limits. The authors exploit the taut-string reformulation to link ROF to a constrained length-minimization problem and rely on universal minimality to connect variational choices across convex costs. By deriving optimality conditions at the taut-string contact sets and proving that these contact sets $C_\alpha^\pm$ are nonincreasing in $\alpha$, they obtain inclusions for jump sets and, consequently, monotonicity of jump amplitudes. This extends prior BV-based monotonicity results to a broader class of inputs and emphasizes the robustness of edge-preserving smoothing under increasing regularization, with potential extensions to vector-valued settings.

Abstract

We revisit the classical problem of denoising a one-dimensional scalar-valued function by minimizing the sum of an $L^2$ fidelity term and the total variation, scaled by a regularization parameter. This study focuses on proving that the jump set of solutions, corresponding to discontinuities or edges, as well as the amplitude of the jumps are nonincreasing as the regularization parameter increases. Compared with previous works, our results apply to a strictly larger class of input functions, extending beyond the traditional setting of functions of bounded variation to any input in $L^\infty$ with left and right approximate limits everywhere. The proof leverages competitor constructions and convexity properties of the taut string problem, a well-known equivalent formulation of the TV model. This monotonicity property reflects that the extent to which geometric and topological features of the original signal are preserved is consistent with the amount of smoothing desired when formulating the denoising method.

Figures (8)

• Figure 1: ROF solution $u_\alpha$, constraints $F-\alpha$, $F+\alpha$ and taut string solution $U_\alpha$ with $f(x)=\sin(4\pi(x+1/4))+\cos(\pi/2(x+1/4))/2$ and $\alpha = 0.03$.
• Figure 2: Competitor construction from GraObe08. If $x_\ell, x_r$ are in the same connected component of $(0,1)\setminus C_\alpha^-$ and close enough to each other, $\overline U^{x_\ell,x_r}_\alpha$ is admissible for the taut string problem and minimality of $U_\alpha$ against $\overline U^{x_\ell,x_r}_\alpha$ certifies the convexity inequality $U_\alpha((1-\lambda) x_\ell + \lambda x_r) \leqslant (1-\lambda) U_\alpha(x_\ell) + \lambda U_\alpha(x_r)$ for all $\lambda \in (0,1)$.
• Figure 3: The different cases of the result of Theorem \ref{['thm:optcond']}.
• Figure 4: Construction of the competitor $\widetilde{U}_\alpha$ for $\beta^+>0$, exploiting the inequality $\beta^- < \beta^+$ to obtain $\mathcal{J}[\widetilde{U}_\alpha] < \mathcal{J}[U_\alpha]$, a contradiction.
• Figure 5: The setup of the proof of Theorem \ref{['thm:contact_set_inclusion']} for $f$ as in Figure \ref{['fig:rofvstaut']}, $\alpha = 0.03$ and $\bar{\alpha} = 0.06$.

Theorems & Definitions (26)

• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Theorem 2.2: cf. Gra07
• Theorem 2.3: Universal minimality condition
• proof
• Remark 2.4
• Definition 2.5
• Remark 2.6
• Lemma 2.7