Abstract
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.