此博士論文之主題為非負盲蔽訊號源分離 (non-negative blind source separation (nBSS))。nBSS 已經在科學和工程上找到許多成功的應用，例如：生物醫學成像、基因表達數據分析、及用於遙感探測 (remote sensing) 之超光譜影像分析。不同於傳統的 nBSS 算法，如非負獨立成份分析 (nICA) 或是非負矩陣分解 (NMF)，我們從單形幾何 (simplex geometry) 的觀點來探討 nBSS 問題。藉此，我們不需要訊號源之統計獨立假設、以及純像素 (完全由單一訊號源構成) 之存在假設。
　　混合矩陣用來描述這些非負訊號源是如何混合的，而其行向量可藉由包住所有像素的最小體積單形 (simplex) 之頂點 (亦被稱為端元 (endmembers)) 來估計──此即為著名的克雷格 nBSS 準則。從實證經驗得知克雷格準則有能力分離重度混合之訊號源，然而從理論的觀點來看，為何此準則能具有此能力尚待證實。在我們採用此強大的準則來設計高效率且高效能的 nBSS 演算法之前，我們將先建立一套分析框架，藉此量化定義了訊號源混合程度 (亦即數據之純度)，然後證明只要此數據純度高於一個特定的小臨界值，那麼克雷格準則確實可以完美地辨識出端元 (在無雜訊的情況下)。我們的理論結果也以數值模擬結果加以驗證。
　　考慮到現存的克雷格單形辨識 (Craig-simplex-identification (CSI))演算法都遭遇了高計算複雜度的問題，而其主因是在冗長的數值優化過程中需要大量的單形體積之計算，上述的分析結果促使我們去設計一個超快速用於 nBSS 的 CSI 演算法，不需要計算任何單形體積。具體來說，藉由利用一個凸幾何事實 (亦即一個 N 個頂點的最簡單之單形 (simplest simplex) 可被 N 個相應的超平面 (hyperplane) 唯一定義)，我們藉由估計 N 個超平面來重建克雷格單形，其中每個超平面則是由 N-1 個仿射獨立的數據像素所估計出。無需數值優化的方式，我們提出的演算法只需簡單線性的運算來搜尋這 N(N-1) 個數據像素，說明了其高計算效率。我們除了提供端元辨識分析來做為其效能保證之外，亦藉由提供仿真、真實超光譜遙感探測 (hyperspectral remote sensing (HRS)) 成像數據實驗來展示其 (相較於目前最先進的 CSI 演算法) 之優越的效能 (無論是計算效率或是估計精確度)。
　　最後，我們基於一個源自於訊息理論之最小描述長度 (minimum description length (MDL)) 準則，提出估計訊號源數目 N 之模型階數選擇 (model-order selection (MOS)) 方法，此準則避免了依賴數據的參數調整 (例如：特徵值之閾值的調整)。現存基於 MDL 的框架多藉由高斯競爭模型 (Gaussian competing model) 來描述 nBSS 數據，然而此模型可能太過於簡化以至於無法適當描述 nBSS 數據；不同於此，我們基於標準化之後的 nBSS 數據通常呈現出單形結構之此一事實，而考慮了更為全面的數據建模。具體來說，我們採用了一個經過線性變換之狄利克雷分佈 (Dirichlet distribution) 來捕捉這些鑲嵌於無雜訊數據裡的單形結構，再連同高斯雜訊建模一起產生了高斯─狄利克雷褶積 (convolution) 競爭模型。接著，藉由建立具隨機性之最大似然 (maximum-likelihood (ML)) 估計子和單形幾何兩者之間的鏈結，我們推導出了此高斯─狄利克雷機率密度之 ML 參數估計。最後，使用蒙地卡羅積分 (Monte Carlo integration) 有效率地計算對應的描述長度。我們藉由大量的模擬研究來驗證我們的 nBSS-MDL 準則、以及用四組真實的生物醫學、HRS 成像數據來證實其性能和適用性；在我們所有的四組案例研究裡，所提供的 nBSS-MDL 準則都能一致地偵測到正確的訊號源個數。

Non-negative blind source separation (nBSS), the focus of this dissertation, has found many successful applications in science and engineering, such as biomedical imaging, gene expression data analysis, and hyperspectral imaging in remote sensing. In contrast to conventional nBSS methods, including non-negative independent component analysis (nICA) and non-negative matrix factorization (NMF), we consider the nBSS problem from the perspective of simplex geometry without requiring sources' statistical independence and existence of pure pixel (fully contributed by a single source).

The columns of the mixing matrix, describing how the non-negative sources are mixed, can be estimated by the vertices (also referred to as endmembers) of the minimum-volume simplex that encloses all pixel vectors---the well-known Craig's nBSS criterion. Empirical experience has suggested that Craig's criterion is capable of unmixing heavily mixed sources, but it was not clear why this is true from a theoretical viewpoint. Before we adopt this powerful criterion for devising a highly efficient and effective nBSS algorithm,
we develop an analysis framework wherein the source mixing level (or data purity level) is quantitatively defined, and prove that Craig's criterion indeed can yield perfect endmember identifiability (in the noiseless scenario) as long as this quantity is greater than a {certain} small threshold. Our theoretical results are substantiated by numerical simulation results.

Considering that existing Craig-simplex-identification (CSI) algorithms suffer from high computational complexity due to heavy simplex volume computations, our identifiability analysis results motivated us to devise a super fast CSI algorithm for nBSS without involving any simplex volume computations. Specifically, by exploiting a convex geometry fact that a simplest simplex of N vertices can be defined by N associated hyperplanes, we reconstruct Craig's simplex from N hyperplane estimates, where each hyperplane is estimated from N-1 affinely independent data pixels. Without resorting to numerical optimization, the proposed algorithm searches for the N(N-1) data pixels via simple linear algebraic computations, accounting for its computational efficiency. Besides an endmember identifiability analysis for its performance support, synthetic/real hyperspectral remote sensing (HRS) imaging data experiments are also provided to demonstrate its superior efficacy over state-of-the-art CSI algorithms in both computational efficiency and estimation accuracy.

Finally, model-order selection (MOS), determining the number of sources N, is done based on an information theoretic-oriented minimum description length (MDL) criterion that {avoids data-dependent parameter tuning} (e.g., eigenvalue threshold). Instead of describing nBSS data via Gaussian competing models (which may be too simplified to advisably describe nBSS data) as in existing MDL-based frameworks, we consider more comprehensive modeling based on the fact that (standardized) nBSS data often can be configured as a simplex. Specifically, we employ a (linearly transformed) Dirichlet distribution to capture the simplex structure embedded in the noiseless counterpart of data, which, together with a Gaussian noise modeling, gives rise to Gaussian-Dirichlet convolution competing models. Then, maximum-likelihood (ML) estimates of the Gaussian-Dirichlet density are derived by building up a link between stochastic ML estimator and simplex geometry. Consequently, the corresponding description lengths are efficiently calculated by Monte Carlo integration. We validate our nBSS-MDL criterion through extensive simulations and experiments on real-world biomedical and HRS imaging datasets, to demonstrate its performance/applicability, and it consistently detects the true number of sources in all of our four case studies.

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