A note on "Anisotropic total variation regularized $L^1$-approximation and denoising/deblurring of 2D bar codes"
Nils Dabrock, Yves van Gennip · Inverse Problems and Imaging · 2018
This note addresses an error in [1].In this short note, we address an error in [1, Lemma 4.2 and Theorem 6.4] which was pointed out to YvG by ND in April 2016.In this note we assume familiarity with the notation from [1].That paper erroneously argues that the only binary signals F 1 and F 3 are faithful to are clean 2D bar codes.Lemma 4.2 stated that: if f ∈ BV (R 2 ; {0, 1}) is both the measured signal in F 1 and a minimizer of F 1 over BV (R 2 ), then f ∈ B. This statement is false, as can be seen from the following counterexample, which was provided in a private communication by ND.Let h ∈ (1/ √ 2, 1) and let Ω := B(0, 1) ∩ [-h, h] 2 be a truncated circle.Let f = χ Ω ∈ L 1 (R) be the characteristic function of Ω. Define s := √ 1 -h 2 and let 1 λ > 2 s .Define, for r ∈ R, w(r) := min{1, max{-1, r/s}} and let, for (x, y) ∈ R 2 ,