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研究生:洪悅慈
研究生(外文):Hung, Yueh-Tzu
論文名稱:總變差流正則化之數值研究
論文名稱(外文):Numerical Study of Regularization for the Total Variation Flow
指導教授:薛名成
指導教授(外文):Shiue, Ming-Cheng
口試委員:謝世峰吳金典
口試委員(外文):Shieh, Shih-FengWu, Chin-Tien
口試日期:2023-07-14
學位類別:碩士
校院名稱:國立陽明交通大學
系所名稱:應用數學系所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2023
畢業學年度:111
語文別:英文
論文頁數:26
中文關鍵詞:總變差流正則化硬閾值法軟閾值法
外文關鍵詞:Total Variation FlowRegularizationHard ThresholdingSoft Thresholding
相關次數:
  • 被引用被引用:0
  • 點閱點閱:75
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  • 下載下載:6
  • 收藏至我的研究室書目清單書目收藏:0
本論文探討了Rudin等人於1992年引入的Rudin-Osher-Fatemi(ROF)模型的正則化方法。 ROF模型旨在通過最小化總變差範數來為圖像降噪。它將問題表述為由參數$\lambda$控制數據保真度和總變差(Total Variation)之間的權衡。

為了解決總變差範數最小化中出現的奇異性問題,現存文獻中探討了許多不同的正則化方法。硬閾值正則化去除噪聲同時保留邊緣和細節,而軟閾值正則化則減小梯度值以獲得更平滑的圖像。收斂分析對於評估迭代算法的效率至關重要,先前的研究提供了有限差分方案和迭代方法的證明。

本論文研究了硬閾值和軟閾值正則化技術在解決總變差流問題中的應用與表現,另外也解釋了相關的控制方程和計算過程。我們透過了有限元方法和預處理技術等數值方法解決正則化逆問題。最後我們進行了一個數值實驗來驗證所提出的方法,並突出顯示兩種正則化方法的性能差異和權衡。

總結來說,本論文討論了關於總變差流的正則化技術的見解與參數選擇的挑戰,並提出了正則化方法的未來研究方向。
This thesis explores regularization techniques for the Rudin-Osher-Fatemi (ROF) model introduced by Rudin et al. in 1992. The ROF model aims to minimize the total variation norm to denoise images. It formulates the problem as a trade-off between data fidelity and total variation, controlled by the parameter $\lambda$.

Different regularization approaches have been developed to address the singularity issue in Total Variation (TV)-norm minimization. Hard threshold regularization removes noise while preserving edges and details, while soft threshold regularization reduces the gradient values for a smoother image. Convergence analysis is crucial for evaluating the efficiency of iterative algorithms, with previous studies providing proofs for finite difference schemes and iterative methods.

In this thesis, we investigates the use of hard and soft threshold regularization techniques for solving the TV flow. The governing equations and their calculations are explained. Numerical methods, including the finite element method and preconditioning techniques, are examined for solving regularized inverse problems. Numerical examples are presented to validate the proposed methods and highlight performance differences and trade-offs.

In conclusion, this thesis provides insights into regularization techniques for TV flow. Challenges in parameter selection and potential degradation of image features are discussed, and future research directions in regularization methods are suggested.
誌謝 -i
中文摘要 -ii
Abstract -iii
Content -vi
List of Figures -vii
List of Tables - viii
1 Introduction -1
2 Regularization Type -5
2.1 Hard Threshold -6
2.2 Soft Threshold -7
3 Numerical Method -10
3.1 Finite element method -11
3.1.1 Preconditioning -13
3.2 Numerical example -15
3.2.1 Example 1 -15
3.2.2 Example 2 -21

4 Conclusions -25
Bibliography -26
[1] Hong, Q., Lai, M.J. and Wang, J., 2021. The convergence of a numerical method
for total variation flow. Journal of Algorithms and Computational Technology,
15, p.17483026211011323.
[2] Tai, X.C., Winther, R., Zhang, X. and Zheng, W., 2022. A uniform preconditioner for a Newton algorithm for total-variation minimization and minimumsurface problems. arXiv preprint arXiv:2208.01390.
[3] Rudin, L.I., Osher, S. and Fatemi, E., 1992. Nonlinear total variation based noise
removal algorithms. Physica D: nonlinear phenomena, 60(1-4), pp.259-268.
[4] Tong, Q., Liang, G. and Bi, J., 2022. Calibrating the adaptive learning rate to
improve convergence of ADAM. Neurocomputing, 481, pp.333-356.
[5] Bartels, S., Nochetto, R.H. and Salgado, A.J., 2014. Discrete total variation flows without regularization. SIAM Journal on Numerical Analysis, 52(1),
pp.363-385.
[6] Dobson, D.C. and Vogel, C.R., 1997. Convergence of an iterative method for
total variation denoising. SIAM Journal on Numerical Analysis, 34(5), pp.1779-
1791.
[7] HECHT, Frédéric. New development in FreeFem++. Journal of numerical mathematics, 2012, vol. 20, no 3-4, p. 251-266.
[8] Steidl, G., Weickert, J., Brox, T., Mrázek, P. and Welk, M., 2004. On the
equivalence of soft wavelet shrinkage, total variation diffusion, total variation
regularization, and SIDEs. SIAM Journal on Numerical Analysis, 42(2), pp.686-
713.
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