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Denoising of piecewise constant signal based on total variation

  • S.I. : NCAA 2021
  • Published:
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Abstract

Total variation denoising (TVD) algorithm is suitable for the restoration of piecewise constant signals (PCS). The amplitude accuracy of discontinuities affects the overall denoising effect. Improving the amplitude accuracy of discontinuities is the difficulty of PCS denoising. Recent papers have presented TVD-based algorithms to improve the denoising precision. After applying the forward–backward splitting (FBS) method, the iteration of the denoising is comprised by the traditional TVD. This paper proposes a new upper bound function to enhance the accuracy of the traditional TVD method. Then, the enhanced TVD is used to update the TVD-based algorithm to improve the denoising efficiency. The experimental results demonstrate that the proposed method has superior performance compared to other methods in PCS denoising.

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Abbreviations

N :

The number of sampling points of the signal

t, r :

The tth and rth iterations, \(t\in \mathbb {N^*}\), and \(r\in \mathbb {N^*}\)

\(\varvec{y}\) :

Noisy signal, \(\varvec{y}\in \mathbb {R}^N\)

\(\varvec{x}\) :

Denoised signal, \(\varvec{x}\in \mathbb {R}^N\)

\(\varvec{x}_{r}\) :

TVD result of the rth iteration, \(\varvec{x}_r\in \mathbb {R}^N\)

\(\hat{\varvec{x}}_{r}\) :

Tolerance error TVD result of the rth iteration, \(\hat{\varvec{x}}_r\in \mathbb {R}^N\)

\(\Vert \varvec{x}\Vert _1\) :

The \(\ell _1\) norm of \(\varvec{x}\)

\(\Vert \varvec{x}\Vert _2\) :

The \(\ell _2\) norm of \(\varvec{x}\)

\(\varvec{D}\) :

First-order difference matrix

x(n):

The nth component of \(\varvec{x}\) and \(n=1,2,\ldots ,N\)

\(\eta\) :

Penalty, \(\eta :\mathbb {R}^N\rightarrow \mathbb {R}\)

\(\lambda\) :

Regularization parameter

\(G_{\lambda \eta (\cdot )}\) :

Objective function, \(G_{\lambda \eta (\cdot )}: \mathbb {R}^N\rightarrow \mathbb {R}\)

\(\varvec{x}_{\lambda \eta (\cdot )}^*\) :

Minimizer of \(G_{\lambda \eta (\cdot )}\), \(\varvec{x}_{\lambda \eta (\cdot )}^*\in \mathbb {R}^N\)

\(\mathrm{TVD}(\varvec{y};\lambda )\) :

Result of TVD for the noisy signal \(\varvec{y}\)

\(\mathrm{eTVD}(\varvec{y};\lambda )\) :

Result of the enhanced TVD for the noisy signal \(\varvec{y}\)

\(j_t, h_r, H_r\) :

Upper bound functions, \(j_t:\mathbb {R}\rightarrow \mathbb {R}\), \(h_r:\mathbb {R}\rightarrow \mathbb {R}\), and \(H_r:\mathbb {R}^N\rightarrow \mathbb {R}\)

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Acknowledgements

This work was supported by the Hubei Province Natural Science Foundation under Grant 2020CFA031, the Inner Mongolia Natural Science Foundation under Grant 2019BS06004, the National Natural Science Foundation of China under Grants 61773354, 61903345 and 62003318, and the Fundamental Research Funds for the Central Universities under Grant 137-162301202638.

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Authors and Affiliations

  1. School of Automation, China University of Geosciences, No. 388 Lumo Road, Wuhan, 430074, Hubei, China

    Donghao Lv, Weihua Cao, Wenkai Hu, Chao Gan & Min Wu

  2. School of Information Engineering, Inner Mongolia University of Science and Technology, No. 7 Arding Street, Baotou, 014010, Inner Mongolia, China

    Donghao Lv

  3. Hubei Key Laboratory of Advanced Control and Intelligent Automation for Complex Systems, China University of Geosciences, No. 388 Lumo Road, Wuhan, 430074, Hubei, China

    Donghao Lv, Weihua Cao, Wenkai Hu, Chao Gan & Min Wu

  4. Engineering Research Center of Intelligent Technology for Geo-Exploration, Ministry of Education, No. 388 Lumo Road, Wuhan, 430074, Hubei, China

    Donghao Lv, Weihua Cao, Wenkai Hu, Chao Gan & Min Wu

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  1. Donghao Lv
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Correspondence to Weihua Cao.

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Lv, D., Cao, W., Hu, W. et al. Denoising of piecewise constant signal based on total variation. Neural Comput & Applic 34, 16341–16349 (2022). https://doi.org/10.1007/s00521-022-06937-8

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