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A note on "Anisotropic total variation regularized L1-approximation and denoising/deblurring of 2D bar codes"

Dabrock, Nils; van Gennip, Yves

A note on "Anisotropic total variation regularized L1-approximation and denoising/deblurring of 2D bar codes" Thumbnail


Authors

Nils Dabrock

Yves van Gennip



Abstract

© 2018 American Institute of Mathematical Sciences. 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 F1 and F3 are faithful to are clean 2D bar codes. Lemma 4.2 stated that: if f ∈ BV (ℝ2; {0, 1}) is both the measured signal in F1 and a minimizer of F1 over BV (ℝ2), 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 = χΩ ∈ L1(ℝ) be the characteristic function of Ω. Define (Formula Presented) and let1 λ > 2/s. Define, for r ∈ ℝ, w(r):= min{1, max{−1, r/s}} and let, for (x, y) ∈ ℝ2, (Formula Presented) We will now show that v ∈ V(f) and hence, by [1, Theorem 3.2], F1 is faithful to f. It can be verified by direct computation that, for x ∈ ℝ2, |v(x)|∞ ≤ 1 and (Formula Presented). Furthermore, we have for all z ∈ ∂Ω that v(z) · n∂Ω(z) = |n∂Ω(z)|1, where n∂Ω is the outward normal vector to the boundary ∂Ω. By the definition of the anisotropic total variation in [1, Formula (1)] and [1, Appendix A, Corollary 3] we then find (Formula Presented) Since the second part of Theorem 6.4 was based directly on Lemma 4.2, that result is also incorrect (part 1 of Theorem 6.4 is unaffected). For a more general treatment of this topic by ND, including the abovementioned counterexample, we refer to [2].

Citation

Dabrock, N., & van Gennip, Y. (2018). A note on "Anisotropic total variation regularized L1-approximation and denoising/deblurring of 2D bar codes". Inverse Problems and Imaging, 12(2), 525-526. https://doi.org/10.3934/ipi.2018022

Journal Article Type Article
Acceptance Date Jan 15, 2018
Publication Date Feb 28, 2018
Deposit Date Feb 2, 2018
Publicly Available Date Mar 28, 2024
Journal Inverse Problems & Imaging
Electronic ISSN 1930-8337
Publisher American Institute of Mathematical Sciences (AIMS)
Peer Reviewed Peer Reviewed
Volume 12
Issue 2
Pages 525-526
DOI https://doi.org/10.3934/ipi.2018022
Keywords Anisotropic total variation, L1 approximation, 2D bar code, denoising, deblurring
Public URL https://nottingham-repository.worktribe.com/output/917034
Publisher URL http://aimsciences.org/article/doi/10.3934/ipi.2018022
Additional Information This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Inverse Problems & Imaging following peer review. The definitive publisher-authenticated version A note on "Anisotropic total variation regularized L1-approximation and denoising/deblurring of 2D bar codes" (2018) is available online at:http://aimsciences.org/article/doi/10.3934/ipi.2018022

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