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.
