有效的高光譜解混（hyperspectral unmixing）對於從高光譜場景的給定圖像（hyperspectral image）中識別端元（endmember，即物質之光譜特徵）及其相對豐度圖（abundance maps）是不可缺少的。近年來，在具有不可忽略的端元變異性（endmember variability）和異常值效應（outlier effects）下研究高光譜解混引起了廣泛的關注。迄今為止一些最先進的高光譜解混工作僅考慮端元變異性，或者僅考慮異常值效應，尚未有能夠同時考慮端元變異性和異常值效應的現存高光譜解混演算法。在本文中，我們提出了一種新的高光譜解混演算法，稱為VOIMU演算法，它對於同時具有不可忽略的端元變異性和異常值效應的高光譜影像能夠強健的進行解混。由Thouvenin等人提出的擾動線性混合模型（perturbed linear mixing model）所定義的一個非凸最小化問題，考慮端元變異性於此模型，以及將 p quasi-norm (0 <p <1) 應用於數據擬合上來隱式處理異常值效應，再加上兩個能使端元變異性及端元光譜特徵更符合高光譜影像特性的正規子（regularizer）。接著我們將問題重新規劃為一個多凸（multi-convex）問題，通過塊座標下降（block coordinate descent）方法求解，此方法可以被視為是分塊連續上界最小化（block successive upper bound minimization）方法的特例，經推導得出VOIMU演算法可以保證收斂至鞍點。再者，儘管異常值並沒有在上述問題之物理模型，也沒有在演算法操作中檢測，但一些潛在的有趣信息依舊能顯示出異常值像素。最後，我們利用實際數據提供了一些仿真與實驗結果，證明所提出的VOIMU演算法的有效性和實際應用性。此論文主要貢獻為祁忠勇教授及林家祥教授包含題目定義、數學推導及英文撰寫等。

Effective hyperspectral unmixing (HU) is essential to the estimation of the underlying materials’ signatures (endmember signatures) and their spatial distributions (abundance maps) from a given image (data) of a hyperspectral scene. Recently, investigating HU under the non-negligible endmember variability (EV) and outlier effects (OE) has drawn extensive attention. Some state-of-the-art works either consider EV or consider OE, but none of them considers both EV and OE simultaneously. In this thesis, we propose a novel HU algorithm, referred to as the variability/outlierinsensitive multi-convex unmixing (VOIMU) algorithm, which is robust against both EV and OE. Considering two suitable regularizers, a nonconvex minimization problem is formulated for which the perturbed linear mixing model (PLMM) proposed by Thouvenin et al., is used for modeling EV, while OE is implicitly handled by applying a p quasi-norm to the data fitting with 0 < p < 1. Then we reformulate it into a multi-convex problem which is then solved by the block coordinate descent (BCD) method, with convergence guarantee by casting it into a block successive upper bound minimization (BSUM) framework. The proposed VOIMU algorithm can yield a stationary-point solution with convergence guarantee, together with some intriguing information of potential outlier pixels though outliers are neither physically modeled
in the above problem nor detected in the algorithm operation. Finally, we provide some simulation results and experimental results using real data to demonstrate the efficacy and practical applicability of the proposed VOIMU algorithm. Note that the main contributions of this thesis are made by Prof. Chong-Yung Chi and Prof. Chia-Hsiang Lin, including problem formulation, theoretical derivations, and English writing.

Chinese Abstract - ii
Abstract - iii
Acknowledgments - v
List of Figures - viii
List of Tables - xiii
List of Notations - 1
1 Introduction - 3
2 Signal Model and Problem Formulation - 7
3 The Proposed VOIMU Algorithm - 11
3.1 Closed-form Solution for Solving (3.5) - 14
3.2 Algorithm for Solving (3.6) - 15
3.3 Closed-form Solution for Solving (3.7) - 19
3.4 Algorithm for Solving (3.8) - 20
4 Simulation Results - 22
4.1 Generation of Synthetic Data - 22
4.2 Performance Measures - 24
4.3 Performance Comparison - 25
5 Experimental Results - 32
5.1 Subscene of Moffett Field - 32
5.2 Sub-image of Cuprite: Desert Varnish - 35
5.3 Further Experiment Results: Cuprite Mining Site - 37
6 Conclusions - 67
A Proof of Example 1 - 69
Bibliography - 75

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