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研究生： 曾柏逸
Po-I Tseng
論文名稱： 卷積稀疏字典學習及其在影像超解析度的應用
Convolutional Sparse Dictionary Learning with Application to Single Image Super-Resolution
指導教授： 楊肅煜
Suh-Yuh Yang
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2020
畢業學年度： 108
語文別： 英文
論文頁數： 69
中文關鍵詞： 稀疏表示 、卷積稀疏表示 、字典學習 、影像超解析度 、迭代優化
外文關鍵詞： sparse representation, convolutional sparse representation, dictionary learning, image super-resolution, iterative refinement
相關次數： 點閱：18  下載：0 
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    • 影像超解析度是影像處理領域裡的一個重要的議題，它主要的目的是希望經由特定演算法將一張或結合多張的低解析度影像生成一張高解析度影像。本文的目標是改進 Gu 等人 [9] 所發展的求解單張影像超解析度方法，我們提出一種以迭代優化為基礎提高影像解析度的預測校正方法。首先，我們概括回顧傳統稀疏表示與字典學習的問題及如何運用交錯方向乘子法求解上述問題。然後考慮卷積稀疏表示與字典學習的問題，整合交錯方向乘子法、 Fourier 轉換、Hadamard 乘積和 Sherman-Morrison 公式等技術來加速求解上述卷積稀疏字典學習問題。接著，我們運用卷積稀疏字典學習理論到數學影像處理問題上，提出一種單張影像提高影像解析度的預測校正迭代程序，這也是本文最主要的貢獻。最後，數值模擬實驗驗證了此方法的有效性及效能。

    Image super-resolution is an essential issue of image processing, and its primary goal is to use single or multiple low-resolution images to generate a high-resolution image through specific algorithms. The objective of this thesis is to improve the method developed by Gu et al. [9] for solving single image super-resolution problems. We propose a novel iterative refinement-based prediction-correction approach. First, we give a general review of the traditional sparse representation and dictionary learning problems and solve the problems using the alternating direction method of multipliers (ADMM).
    Second, we consider the convolutional sparse representation and dictionary learning problems,
    and then integrate all the techniques of ADMM, Fourier transform, Hadamard product,
    and Sherman-Morrison formula to speed up the computations for solving the problems.
    Then we apply the convolutional sparse dictionary learning theory to mathematical image processing.
    We propose a prediction-correction iterative approach
    for the image processing of single image super-resolution
    that is the main contribution of the thesis.
    Finally, numerical experiments are performed to validate the effectiveness and efficiency
    of the newly proposed approach.

    1 Introduction 1 2 Sparse representation and dictionary learning 4 2.1 Original sparse representation problem 4 2.1.1 Two dual `0 minimization problems 4 2.1.2 General sparse representation problem 7 2.2 Alternating direction method of multipliers (ADMM) 7 2.2.1 Derivation of the ADMM: augmented Lagrangian 8 2.2.2 Scaled form of the augmented Lagrangian 8 2.2.3 ADMM for `1-norm sparse representation problem 9 2.2.4 Solving sparse representation problem 9 2.3 Sparse dictionary learning problem 13 2.3.1 Methods for solving subproblems of coefficient Z 15 2.3.2 Methods for solving subproblems of dictionary D 16 2.4 Convergence and stopping criterion of ADMM 17 3 Convolutional sparse representation and dictionary learning 18 3.1 Convolutional sparse representation problem 18 3.1.1 ADMM for convolutional sparse representation 20 3.1.2 Sherman-Morrison formula 23 3.2 Convolutional sparse dictionary learning problem 25 4 Application to single image super-resolution 28 4.1 Single image super-resolution 28 4.2 Residuals of training images 31 4.3 A prediction-correction approach 33 5 Numerical experiments 35 5.1 Grayscale images 38 5.2 Color images 38 5.3 Promotion to higher resolution 51 6 Summary and conclusion 58 References 59

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