近年來，使用低空間解析度高光譜影像(hyperspectral image)與高空間解析度多光譜影像(multispectral image)融合來獲取高空間解析度高光譜影像已被視為較為經濟的方法。使用耦合非負矩陣分解(CNMF)準則已經被報導可以獲得良好的融合成果。然而，該準則是一個病態的逆向問題(ill-posed inverse problem)。在本論文中，我們使用了正規化技術(regularization)提出一新演算法遠勝過 CNMF 演算法。我們引入兩種正規子(regularizer)，其一為廣泛使用的稀疏化正規子，使豐度圖(abundance map)趨向於稀疏化；其二為針對端元(endmember，即物質之光譜特徵)之間距離平方總和降低的正規子。

由於所制定的問題具有雙凸(bi-convexity)的特性，我們將其分解成兩個子問題，
並使用交替方向乘子法(ADMM)設計出一個凸優化導向耦合非負矩陣分解演算法
(CO-CNMF)，其中每個疊代步驟中都可以從凸優化理論求出封閉形式解(closed-form solution)。由於問題的尺度非常大，導致計算複雜度太高，我們藉由原閉式解中固有的矩陣結構進一步推導出替代的閉式解，因此大幅降低計算複雜度。

最後，我們以真實高光譜影像進行實驗，實驗內容分成三部分:首先，分析演算法中的參數選擇；其次，與現存的融合演算法進行比較，以驗證所提出之融合演算法的優良性能；最後，我們亦針對不完美圖像配準(imperfect co-registration)情況下所造成此演算法的性能損失。

In recent years, fusing a low-spatial-resolution hyperspectral image with a highspatial-resolution multispectral image has been thought of as an economical approach for obtaining high-spatial-resolution hyperspectral image. A fusion criterion, termed
coupled nonnegative matrix factorization (CNMF) has been reported to be effective in yielding promising fusion performance. However, the CNMF criterion amounts to an ill-posed inverse problem. In this thesis, we propose a new data fusion algorithm by suitable regularization that significantly outperforms the unregularized
CNMF algorithm. Besides utilizing the sparsity-promoting regularizer, which promotes the sparsity of the abundance map, we also incorporate the sum-of-squared endmember distances demoting regularizer.

Owing to the bi-convexity of the formulated optimization problem, we can decouple it into two convex subproblems. Each subproblem is then solved by a carefully designed alternating direction method of multiplers (ADMM), leading to a convex-optimization based CNMF (CO-CNMF) fusion algorithm, where each ADMM iterate is equipped with a closed-form solution. Since the problem size is very large, leading to high computational complexity of the proposed CO-CNMF algorithm, we futher obtain alternative expressions by exploiting some inherent matrix structure in those closed-form solutions, which greatly reduce the computational complexity.

Finally, we present some experiments using real hyperspectral data, which can be divided into three parts. The first part is to analyze how we choose parameters in our proposed algorithm. Second, we demonstrate its superior performance to some state-of-the-art fusion algorithms. Third, by some experimental results, we discuss its performance loss due to imperfect co-registration.

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