影像填補是一種影像處理的技術，主要的目的是利用得到的資訊去填補待填補的區域。影像填補的問題已經有許多不同的處理方法，主要分為三個類別：以擴散為基礎的方法、以補丁為基礎的方法、和深度學習法。本文採用稀疏表示與字典學習的架構，這樣的架構屬於以補丁為基礎的方法，主要的優勢是能較好地填補紋理的部分，相對於深度學習法利用許多不同的影像取得填補所需的資訊，本文所採用的方法則是利用同一張影像中待填補區以外的區域所得到的資訊進行填補，所以稱為單一影像填補方法。在整個方法中，字典扮演著最關鍵的角色，然而訓練一本字典通常需要相當長的時間，這是字典學習的最大缺點。為了改善這個問題，本文提出了一個基於分治法的演算法，主要的想法是經由解決許多較小問題，從而得到本來問題的解答，藉由這個演算法可以更有效率地訓練一本字典。最後，我們進行多個數值實驗以驗證該演算法的性能。

Image inpainting is a technique that fills in the pixels in a missing data region. There are many different approaches in the literature for dealing with image inpainting problems. These approaches are categorized into three classes: diffusion-based methods, patch-based methods, and deep learning methods. In this thesis, we study image inpainting based on the dictionary learning and sparse representation framework, which is a patch-based method. Such an approach generally has better image texture results. Compared to deep learning methods, which acquire information from multiple images, our framework, called single-image inpainting, trains a dictionary with the information from the same image. However, training an efficient dictionary usually takes a lot of time. Therefore, we propose an algorithm based on the divide-and-conquer concept for learning a dictionary, which is computationally
more efficient. We verify the algorithm’s performance with several test problems.

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