雖然盲訊號源分離(blind source separation)技術已被廣泛研究，但多數現存方法需假定訊號源是統計獨立或不相關的。然而在許多生物醫學應用中，因訊號源體現了活體細胞中蛋白質和分子的交互作用，故訊號源是彼此相關的，導致傳統的獨立成分分析法(independent component analysis)無法適用。有鑑於此我們專注於確定性質(deterministic properties)的相關性訊號源盲分離技術。我們主要研究兩種類型的訊號源分離：在本論文的第一部分，處理非負訊號源的分離問題，這常出現於以下生物醫學成像方法中，包含動態對比增強的磁共振成像(dynamic contrast enhanced-magnetic resonance imaging)，X-射線成像、超音波成像(ultrasonic imaging)和螢光顯微鏡成像(fluorescence microscope imaging)，或者出現於頻譜分析中，例如核磁共振頻譜(Nuclear magnetic resonance spectrum)和紅外光譜等分析方法。藉由降低兩非負訊號間的相關係數，一個非負最小關聯分析法(non-negative least-correlated component analysis)被提出來，以設計訊號源分離矩陣。我們證明在兩個訊號源和兩個混和訊號的情形下，可以獲得一個封閉型式解(closed-form solution)。為分離超過二個以上的訊號源，一個多訊號聯合相關函數被提出來。基於最小化非負訊號間的聯合相關函數，我們提出了一個疊代式體積最大化(iterative volume maximization)方法，其中僅需解決線性規畫(linear programming)問題就能分離出非負訊號源。同時該法的訊號源可辨識性(source identifiability)也被進一步分析與證明。模擬數據和真實生醫數據的實驗結果顯示了所提出的方法在性能上優於現存的基準方法。在本論文的第二部分，我們研究了生物醫學上常見的指數訊號源分析(exponential signal analysis)。指數信號盲抽取的問題出現在許多應用中，包括超音波傳感器陣列處理(ultrasonic sensor array processing)，血液流速成像和螢光細胞成像。依據不同應用，訊號源的特定性質可被利用以作為源分離時的限制。基於此想法，我們提出``多重根-多重信號分類”(multiple rooting-multiple signal classification)演算法，以整合訊號源的已知特定性質，進而提高指數訊號源的分離性能。此外對於螢光衰減信號，一個子空間距離數據分割(subspace distance data segmentation)法被提出來以辨認具有相同訊號特徵的興趣區域(region of interest)。藉由主成分分析法(principal component analysis)處理興趣區域中的所有影像像素後，多重根-多重信號分類法可用以獲得更加準確的衰減常數(decay constant)估測。對這些被提出的方法我們以模擬數據進行性能評估以證實其優於現存的基準方法。

Although significant efforts have been made in developing blind source separation (BSS) techniques, most of the existing methods rely on the foundational assumption that the sources are statistically independent or uncorrelated. However, in many biomedical applications the source signals, which represent interactions of specific proteins and molecules in living cells, may be mutually correlated, leading to the conventional independent component analysis (ICA) not applicable. In view of this, we focus on the blind extraction of correlated signals with specific deterministic properties. Two classes of BSS problems are studied. In the first part of the thesis, the separation of non-negative sources is considered, which could appear in biomedical imaging modalities including dynamic contrast enhanced magnetic resonance imaging (DCE-MRI), X-ray imaging, ultrasonic imaging, and fluorescence microscope imaging, or appear in spectrum signals including nuclear magnetic resonance (NMR) spectrum and infrared (IR) spectrum. By minimizing the correlation coefficient of two non-negative sources, a non-negative least-correlated component analysis ($n$LCA) method is proposed to design the unmixing matrix. We show that a closed-form solution is available for unmixing two mixtures of two sources. For extracting more than two sources, a joint correlation function of multiple signals is proposed to determine the unmixing matrix. Based on minimizing the joint correlation function among the estimated non-negative sources, we propose an iterative volume maximization (IVM) principle which involves solving linear programming problem only for non-negative source extraction. The source identifiability is further discussed and analyzed. Both simulation data and real biomedical data were used to demonstrate its superior performance of the proposed nLCA method over some existing benchmark algorithms. In the second part of the thesis, the exponential signal analysis for biomedical applications is studied. The exponential signal extraction problem arises in many applications including ultrasonic sensor array processing, blood flow imaging, and fluorescence cellular imaging. Depending on the applications, the sources have specific properties that can be used as constraints for source separation. Based on this idea, we propose a multiple rooting technique for multiple signal classification (MR-MUSIC) algorithm, which can integrate the prior information of signals for improving the source extraction performance. Moreover, for fluorescence decay signals, a subspace distance data segmentation (SDDS) is proposed to identify the region of interest (ROI) with the same characteristics. By using principal component analysis (PCA) on all pixel data in the ROI and MR-MUSIC algorithm, an accurate estimation of image signatures, i.e., the decay constants, can be obtained. These proposed methods were evaluated with simulation data to demonstrate their superior performance over several existing benchmark methods.

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