Local Hölder regularity of minimizers for nonlocal denoising problems

Abstract

We study the regularity of solutions to a nonlocal version of the image denoising model and we show that, in two dimensions, minimizers have the same Hölder regularity as the original image. More precisely, if the datum is (locally) β-Hölder continuous for some β ∈ (1 − s, 1], where s ∈ (0, 1) is a parameter related to the nonlocal operator, we prove that the solution is also β-Hölder continuous.

Convergence of level sets in fractional Laplacian regularization

“…Let us also mention that in the recent work [27], the authors study a denoising scheme with the W s,1 seminorm as regularizer, proving preservation of Hölder continuity and hence recreating the well-known result of [12] for total variation denoising. In the fractional case, this seminorm satisifies a coarea formula (first proved in [39], see also [8, Thm.…”

“…The barrier constructed in [31, Lemma 2] corresponds to a = −1 and b = 1, and in their case a parameter τ appears multiplying the right hand side of (26), which in our case is always τ = 1. Then, we just notice that multiplying w by a constant changes the offset a in (26) but not the terms with w by homogeneity, while in (27) it just affects the constant C B .…”

Convergence of level sets in fractional Laplacian regularization

“…Let us also mention that in the recent work [28], the authors study a denoising scheme with the W s,1 seminorm as regularizer, proving preservation of Hölder continuity and hence recreating the well-known result of [13] for total variation denoising. In the fractional case, this seminorm satisfies a coarea formula (first proved in [41], see also [9, theorem 2.2.2]) in terms of the fractional perimeter, which makes a purely geometric point of view applicable and leads one to expect that results along the lines of those in [14,23,24] also hold.…”

“…Remark 5. Note that since there is a fixed universal lower bound for R and also the tolerance η needs to be accommodated in (28), a significant distance from x to ∂ D and ∂ Ω is needed. This will be the case when we apply this lemma below, because the domains D and Ω will be rescaled versions of fixed ones, with freedom to choose the rescaling factor.…”