基於非高斯性及最佳正交近似轉換方法之獨立成分分析研究

Abstract

人的大腦可以幫助我們處理在吵雜環境中的多個混和聲音源，讓我們彷彿可以只聽到想聽的特定聲音，不論是播放的音樂，抑或是正與你交談的對話內容。這類型的問題即為雞尾酒會問題(cocktail-party problem)。針對這類型的問題，常見的方法是使用獨立成分分析(Independent Component Analysis ; ICA)，用來將多個混和的獨立訊號源重新進行線性組合(linear combination)，藉此還原得原始的獨立訊號源，亦即前述中提及的特定聲音源。 獨立成分分析目標是將混和訊號還原至統計獨立(statistically independent)，而統計獨立的必要條件之一是統計不相關(statistically uncorrelated)，亦即在向量空間上的相互正交。利用最佳正交近似轉換(Johnson Transformation)可將一個矩陣轉換為與其本身最相近的標準正交矩陣，即是將訊號轉換成零相關性的訊號，獨立成分分析再進一步將這些零相關訊號轉換成獨立的非高斯性訊號。 本論文探討獨立成分分析中廣為使用的快速獨立成分分析(Fast Independent Component Analysis;FastICA)演算法，針對其非高斯性(Non-Gaussianity;NG)及白化(whitening)的使用加以探討，提出結合使用最佳正交近似轉換與非高斯性排序之FastICA演算法，並針對每一輪迭代後應保持其正交性，提出迭代使用NG排序及最佳正交近似轉換之FastICA演算法，讓每一輪迭代都能從標準正交訊號轉換為獨立訊號。 最後，使用公開的腦電圖(Electroencephalography, EEG)資料集、常用的影像處理資料集及動物聲音資料集進行比較，並對獨立成分分析演算法在不同類型資料集上的結果進行討論。

The brain can deal with several mixed sounds and focus on certain signal no matter it is the sound of music or conversations. Such a problem is known as the cocktail-party problem. The most common computerized method to isolate the original signal is called Independent Component Analysis (ICA). ICA is to linearly recombine the mixed signals to recover the original independent signals. One of the necessary conditions for signals to be statistically independent to one another is for the signals to be uncorrelated to one another, that is, the signals expressing as vectors are orthogonal to one another in the vector space. Johnson’s Transformation can be used to transform a matrix into an orthonormal matrix with columns closest to the columns in the original matrix. Therefore, this research proposes to use the Johnson’s Transformation to transform signals into uncorrelated signals and then transform the uncorrelated signals into independent non-Gaussian signals by the FastICA algorithm. The FastICA algorithm is a well-known algorithm consisting of signal whitening and Non-Gaussianity (NG) maximization. To incorporate Johnson’s Transformation with the FastICA algorithm, this research proposes to obtain uncorrelated signals by the Johnson’s Transformation first and rank the signal’s NG before each round of the NG iteration. Hence, the signals are recovered by iterative NG-ranked Johnson-Transformed signals. To demonstrate and validate the proposed ICA algorithm, the Electroencephalography (EEG) dataset, image dataset and animal sound dataset from open databases are tested and results are discussed.