在本論文中，我們著重在高光譜分解(hyperspectral unmixing)中的一個重要的步驟－豐度估計(abundance estimation)，所謂的豐度是指每個端元(endmember，即物質之光譜特徵)在一像素(pixel)中的含量比例。在豐度為非負且總合為一(non-negative and sum-to-one)之限制下，解豐度估計的問題是一完全約束最小二乘法(fully constrained least squares, FCLS)問題。然而高光譜影像含有的光譜波段很多，使得傳統最佳化做法的運算複雜度太高，因此為加速高光譜分解的步驟，維度縮減(dimension reduction)扮演重要的角色；另一方面，即使FCLS問題有嚴格凸(strictly convex)的特性，但由於影像的總像素個數可能非常多，使得解此問題會導致高運算複雜度。本論文是利用豐度估計之問題的幾何性質，提出一快速的豐度估計演算法。此論文有兩個主要貢獻：首先，我們將維度縮減問題重構成一矩陣分解問題，然後以眾所週知的Cholesky分解技術解決此問題。其次，我們對於豐度估計提出一個閉合式(closed-form)近似解。因大部分的高光譜像素會落於端元所形成的單體(simplex) (即端元之凸包(convex hull))內，對於此單體內的像素，所提出之閉合式近似解可被證明為豐度估計問題的全局最優解(global optimum)。此外，不論影像的像素數目多寡，所提出之閉合式近似解中許多變數只需計算一次，所以此近似解的運算效率很高。最後，我們以電腦模擬和真實數據實驗，將我們提出的演算法與現存的豐度估計演算法作比較，以驗證所提出之演算法的優良效能和實用性。

This thesis considers a widely studied problem in hyperspectral unmixing---the abundance estimation of hyperspectral images. Abundances are the proportions of different endmembers (the spectral signatures of materials) present in an imaging pixel. Conventionally, these are estimated by solving least-squares problems under the sum-to-one constraint and the non-negativity constraint, known as the fully constrained least squares (FCLS) problem. However, those traditional optimization methods may yield high computational complexity, since the number of spectral bands in hyperspectral images is high (usually several hundreds). Hence, dimension reduction of the data plays a pivotal role in speeding up the unmixing stage. On the other hand, the total number of pixels in the considered image may be very large. Therefore, solving the FCLS problem becomes computationally complex, although it is strictly convex. In this thesis, based on some geometric properties of the abundance estimation problem, we propose a fast abundance estimation algorithm. The main contributions are twofold:
Firstly, we formulate the dimension reduction problem into a matrix factorization problem and solve it by a modified version of the widely known Cholesky factorization technique.
Secondly, we propose a closed-form approximation as the abundance estimates. Due to the simplex structure of the hyperspectral data, most pixels lie within the simplex with vertices being the endmembers, i.e., the convex hull of the endmembers. For those pixels in the simplex, it can be proved that the proposed closed-form abundance approximation exactly yields the global optimum solution to the abundance estimation problem. In addition, the computation of the derived closed-form formula is highly efficient, since some quantities in this formula need to be computed only once, regardless of the number of pixels. Finally, the superior computational efficiency and estimation accuracy of the proposed algorithm over state-of-the-art algorithms are verified through simulations and real data experiments.

Chinese Abstract ii
Abstract iii
Acknowledgments v
List of Figures viii
List of Tables x
List of Notations 1
1 Introduction to Hyperspectral Imaging 2
2 Signal Model and Problem Formulation 6
2.1 Linear Mixing Model and Assumptions . . . . . . . . . . . . . . . . . 6
2.2 Abundance Estimation Problem . . . . . . . . . . . . . . . . . . . . . 8
2.3 A Literature Review of A State-of-the-Art Algorithm: FCLS . . . . . 9
3 Dimension Reduction 10
3.1 Convex Geometry Preliminary . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Method I: Gram-Schmidt Procedure . . . . . . . . . . . . . . . . . . . 12
3.3 Method II: Cholesky Factorization . . . . . . . . . . . . . . . . . . . . 13
4 Convex Geometry-Based Abundance Estimation Algorithm 17
4.1 Geometrical Meaning of Abundance Estimation Problem . . . . . . . 17
4.2 Solution to Problem (4.4) . . . . . . . . . . . . . . . . . . . . . . . . 21
5 Computer Simulations 24
5.1 Monte Carlo Simulations with Dirichlet
Distributed Abundances . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.2 Simulations with Spatially Correlated and
Sparse Abundances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6 Real Data Experiments 32
6.1 Remote Sensing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.2 Food Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7 Conclusions 43
A Proof of Generalized Pythagorean Lemma 44
B Proof of Theorem 4.2.1 46
Bibliography 48

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