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

TL;DR

This paper proves that in one-dimensional TV denoising, the number and size of jumps in the solution decrease monotonically as the regularization parameter increases, extending previous results to a broader class of functions.

Contribution

The study extends monotonicity results of jump sets and amplitudes in TV denoising to functions in $L^ Infty$, using convexity and competitor constructions, beyond the traditional bounded variation setting.

Findings

Jump set size decreases with regularization

Jump amplitudes are nonincreasing as regularization grows

Results apply to broader class of functions than previous work

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.