Corrigendum

IET Image ProcessingVolume 16, Issue 1 p. 285-288 CORRIGENDUMOpen Access Corrigendum This article corrects the following: A novel weighted total variation model for image denoising Meng-Meng Li, Bing-Zhao Li, Volume 15Issue 12IET Image Processing pages: 2749-2760 First Published online: May 16, 2021 First published: 04 October 2021 https://doi.org/10.1049/ipr2.12356AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat In this article the Section 3.3 needs revision, below is the modified section 3.3 3.3 Convergence analysis In this subsection, we mainly analyze the convergence property of solutions generated from Algorithm 1. The proof is referred to [34]. Suppose that ( u ∗ , h 1 ∗ , h 2 ∗ , r 1 ∗ , r 2 ∗ ) is a saddle point of problem (15), then it is unique and the sequence u k , h 1 k , h 2 k , r 1 k , r 2 k generated by algorithm 1 converges to it. Proof. Let ( u ∗ , h 1 ∗ , h 2 ∗ , r 1 ∗ , r 2 ∗ ) is a saddle point of problem (15), A = w 1 ∂ x , B = w 1 ∂ y , u k + 1 ∼ = u k + 1 − u ★ , h 1 k + 1 ∼ = h 1 k + 1 − h 1 ★ , h 2 k + 1 ∼ = h 2 k + 1 − h 2 ★ , r 1 k + 1 ∼ = r 1 k + 1 − r 1 ★ , r 2 k + 1 ∼ = r 2 k + 1 − r 2 ★ . Λ1 and Λ2 are subdifferential of ∂ | | h 1 | | 1 and ∂ | | h 2 | | 1 respectively. Denot Λ 1 k + 1 ∼ = Λ 1 k + 1 ∼ − Λ 1 ★ ∼ , Λ 2 k + 1 ∼ = Λ 2 k + 1 ∼ − Λ 2 ★ ∼ . u k + 1 , h 1 k + 1 , h 2 k + 1 , r 1 k + 1 , r 2 k + 1 and u ∗ , h 1 ∗ , h 2 ∗ , r 1 ★ , r 2 ★ satisfy the following conditions λ I + τ A ★ A + τ B ★ B u k + 1 − λ f − τ A ★ h 1 k + r 1 k / τ − τ B ★ h 2 k + r 2 k / τ = 0 , (25a) Λ 1 k + 1 + τ h 1 k + 1 − τ A u k + 1 + r 1 k = 0 , (25b) Λ 2 k + 1 + τ h 2 k + 1 − τ B u k + 1 + r 2 k = 0 , (25c) λ I + τ A ★ A + τ B ★ B u ∗ − λ f − τ A ★ h 1 ∗ + r 1 ∗ / τ − τ B ★ h 2 ∗ + r 2 ∗ / τ = 0 , (26a) Λ 1 ∗ + τ h 1 ∗ − τ A u ∗ + r 1 ∗ = 0 , (26b) Λ 2 ∗ + τ h 2 ∗ − τ B u ∗ + r 2 ∗ = 0 , (26c) Subtracted Eq. (26) from Eq. (25), it is derived as follows λ I + τ A ★ A + τ B ★ B u k + 1 ∼ − λ f − τ A ★ h 1 k + 1 ∼ + r 1 k + 1 ∼ / τ − τ B ★ h 2 k + 1 ∼ + r 2 k + 1 ∼ / τ = 0 , (27a) Λ 1 k + 1 ∼ + τ h 1 k + 1 ∼ − τ A u k + 1 ∼ + r 1 k + 1 ∼ = 0 , (27b) Λ 2 k + 1 ∼ + τ h 2 k + 1 ∼ − τ B u k + 1 ∼ + r 2 k + 1 ∼ = 0 . (27c) Taking the inter product of Eq. (27) by u k + 1 ∼ , h 1 k + 1 ∼ and h 2 k + 1 ∼ respectively, we can obtain the following result λ | | u k + 1 | ∼ | 2 + τ | | A u k + 1 ∼ | | 2 + τ | | B u k + 1 ∼ | | 2 − τ ⟨ A ★ h 1 k ∼ , u k + 1 ∼ ⟩ − ⟨ A ★ r 1 k ∼ , u k + 1 ∼ ⟩ − τ ⟨ B ★ h 2 k ∼ , u k + 1 ∼ ⟩ − ⟨ B ★ r 2 k ∼ , u k + 1 ∼ ⟩ = 0 , (28a) ⟨ Λ 1 k + 1 ∼ , h 1 k + 1 ∼ ⟩ + τ | | h 1 k + 1 ∼ | | 2 − ⟨ A u k + 1 ∼ , h 1 k + 1 ∼ ⟩ + ⟨ r 1 k ∼ , h 1 k + 1 ∼ ⟩ = 0 , (28b) ⟨ Λ 2 k + 1 ∼ , h 2 k + 1 ∼ ⟩ + τ | | h 2 k + 1 ∼ | | 2 − ⟨ B u k + 1 ∼ , h 2 k + 1 ∼ ⟩ + ⟨ r 2 k ∼ , h 2 k + 1 ∼ ⟩ = 0 . (28b) Similarly, it is also obtained ⟨ r 1 k + 1 ∼ , h 1 k ∼ ⟩ = | | r 1 k ∼ | | 2 + τ ⟨ h 1 k + 1 ∼ , r 1 k ∼ ⟩ − ⟨ A u k + 1 ∼ , r 1 k ∼ ⟩ , (29a) ⟨ r 2 k + 1 ∼ , h 2 k ∼ ⟩ = | | r 2 k ∼ | | 2 + τ ⟨ h 2 k + 1 ∼ , r 2 k ∼ ⟩ − ⟨ B u k + 1 ∼ , r 1 k ∼ ⟩ . (29b) In fact, 1 2 τ | | r 1 k ∼ | | 2 − | | r 1 k + 1 ∼ | | 2 = ⟨ A u k + 1 ∼ − h 1 k + 1 ∼ , r 1 k ∼ ⟩ − τ 2 | | A u k + 1 ∼ − h 1 k + 1 ∼ | | 2 , (30a) 1 2 τ | | r 2 k ∼ | | 2 − | | r 2 k + 1 ∼ | | 2 = ⟨ B u k + 1 ∼ − h 2 k + 1 ∼ , r 2 k ∼ ⟩ − τ 2 | | B u k + 1 ∼ − h 2 k + 1 ∼ | | 2 . (30b) Summing Eq. (30a)-(30c) with Eq. (28), we have 1 2 τ r 1 k ∼ | | 2 − r 1 k + 1 | ∼ | 2 + r 2 k ∼ | | 2 − r 2 k + 1 ∼ | | 2 = A u k + 1 ∼ − h 1 k + 1 ∼ , r 1 k ∼ − τ 2 | | A u k + 1 ∼ − h 1 k + 1 ∼ | | 2 + B u k + 1 ∼ − h 2 k + 1 ∼ , r 2 k ∼ − τ 2 | | B u k + 1 ∼ − h 2 k + 1 ∼ | | 2 + λ | | u k + 1 ∼ | | 2 + τ | | A u k + 1 | ∼ | 2 + τ | | B u k + 1 ∼ | | 2 − τ A ★ h 1 k ∼ , u k + 1 ∼ − A ★ r 1 k ∼ , u k + 1 ∼ + B ★ r 2 k ∼ , u k + 1 ∼ − τ B ★ h 2 k ∼ , u k + 1 ∼ + Λ 1 k + 1 ∼ , h 1 k + 1 ∼ + τ | | h 1 k + 1 | ∼ | 2 − τ A u k + 1 ∼ , h 1 k + 1 ∼ + r 1 k ∼ , h 1 k + 1 ∼ + Λ 2 k + 1 ∼ , h 2 k + 1 ∼ + τ | | h 2 k + 1 ∼ | | 2 − τ B u k + 1 ∼ , h 2 k + 1 ∼ + r 2 k ∼ , h 2 k + 1 ∼ , (31)where τ A u k + 1 ∼ 2 + τ h 1 k + 1 ∼ 2 − τ 2 A u k + 1 ∼ − h 1 k + 1 ∼ 2 − τ A ★ h 1 k ∼ , u k + 1 ∼ − τ A u k + 1 ∼ , h 1 k + 1 ∼ = A u k + 1 ∼ − h 1 k + 1 ∼ 2 − τ A u k + 1 ∼ , h 1 k ∼ − h 1 k + 1 ∼ , (32) τ | | B u k + 1 ∼ | | 2 + τ | | h 2 k + 1 | ∼ | 2 − τ 2 | | B u k + 1 ∼ − h 2 k + 1 | ∼ | 2 − τ ⟨ B ★ h 2 k ∼ , u k + 1 ∼ ⟩ − τ ⟨ B u k + 1 ∼ , h 2 k + 1 ∼ ⟩ = τ 2 | | B u k + 1 ∼ − h 2 k + 1 | ∼ | 2 − τ ⟨ B u k + 1 ∼ , h 2 k ∼ − h 2 k + 1 ∼ ⟩ . (33) Therefore 1 2 τ r 1 k ∼ | | 2 − r 1 k + 1 | ∼ | 2 + r 2 k ∼ | | 2 − r 2 k + 1 ∼ | | 2 = A u k + 1 ∼ − h 1 k | ∼ | 2 + τ 2 B u k + 1 ∼ − h 2 k + 1 ∼ | | 2 − τ ⟨ A u k + 1 ∼ , h 1 k ∼ − h 1 k + 1 ∼ ⟩ − τ ⟨ B u k + 1 ∼ , h 2 k ∼ − h 2 k + 1 ∼ ⟩ + λ | | u k + 1 | ∼ | 2 + ⟨ Λ 1 k + 1 ∼ , h 1 k + 1 ∼ ⟩ + ⟨ Λ 2 k + 1 ∼ , h 2 k + 1 ∼ ⟩ = τ 2 A u k + 1 ∼ − h 1 k ∼ | | 2 + τ 2 B u k + 1 ∼ − h 2 k ∼ | | 2 + λ | u k + 1 ∼ | | 2 + ⟨ Λ 1 k + 1 ∼ , h 1 k + 1 ∼ ⟩ + ⟨ Λ 2 k + 1 ∼ , h 2 k + 1 ∼ ⟩ + τ 2 h 1 k + 1 ∼ | | 2 − h 1 k ∼ | | 2 + h 2 k + 1 | ∼ | 2 − h 2 k ∼ | | 2 . (34) With the monotonicity of subdifferential, the following inequalities holds ⟨ Λ 1 k + 1 ∼ , h 1 k + 1 ∼ ⟩ ≥ 0 , ⟨ Λ 2 k + 1 ∼ , h 2 k + 1 ∼ ⟩ ≥ 0 . (35) Thus 1 2 τ | | r 1 k ∼ | | 2 − | | r 1 k + 1 ∼ | | 2 + | | r 2 k ∼ | | 2 − | | r 2 k + 1 ∼ | | 2 − τ 2 | | h 1 k + 1 ∼ | | 2 − | | h 1 k ∼ | | 2 + | | h 2 k + 1 | ∼ | 2 − | | h 2 k ∼ | | 2 ≥ λ | | u k + 1 ∼ | | 2 + τ 2 | | A u ( k + 1 ) ∼ − h 1 k ∼ ∥ 2 + τ 2 ∥ B u ( k + 1 ) ∼ − h 2 k ∼ | | 2 . (36) Sum Eq. (36) with k increasing from 0 to 1, it is derived that 1 2 τ | | r 1 0 ∼ | | 2 − | | r 1 l + 1 | ∼ | 2 + | | r 2 0 ∼ | | 2 − | | r 2 l + 1 ∼ | | 2 − τ 2 | | h 1 l + 1 ∼ | | 2 − | | h 1 0 | ∼ | 2 + | | h 2 l + 1 ∼ | | 2 − | | h 2 0 | ∼ | 2 ≥ λ ∑ k = 0 l | | u k + 1 | ∼ | 2 + τ 2 ∑ k = 0 l | | A u k + 1 ∼ − h 1 k ∼ | | 2 + τ 2 ∑ k = 0 l | | B u k + 1 ∼ − h 2 k | ∼ | 2 . (37) In Eq. (37), the left side is bounded and hence the right side is convergent. Then it is deduced that lim k → ∞ | | u k + 1 ∼ | | 2 = 0 , (38a) lim k → ∞ | | A u k + 1 ∼ − h 1 k ∼ | | 2 = 0 , (38b) lim k → ∞ | | B u k + 1 ∼ − h 2 k ∼ | | 2 = 0 . (38c) Further lim k → ∞ u k + 1 = u ∗ , (39a) lim k → ∞ h 1 k + 1 = h 1 ∗ , (39b) lim k → ∞ h 2 k + 1 = h 2 ∗ . (39c) In addition, let k tend to infinity in Eq. (25) and Eq. (26), we have lim k → ∞ r 1 k + 1 = r 1 ∗ , (40a) lim k → ∞ r 2 k + 1 = r 2 ∗ . (40b) REFERENCE Li M-M, Li B-Z.: A novel weighted total variation model for image denoising. IET Image Process. 15(12), 2749– 2760 (2021). https://doi.org/10.1049/ipr2.12259Wiley Online LibraryWeb of Science®Google Scholar Volume16, Issue1January 2022Pages 285-288 ReferencesRelatedInformation