二階張量多維線性主成分分析在低溫電子顯微鏡影像的應用

Abstract

在本論文中，我們將介绍多重線性的主要成分分析(MPCA)，並推導出它在統計上的一些性質，之後將它運用在低溫電子顯微鏡的影像分析上。 由於此影像具有較低的雜訊比(SNR)，因此將相類似的圖像做平均是必要的。 K-means是將樣本點做分類最常用的演算法。 然而，我們發現此方法對具有低雜訊比特性的低溫電子顯微鏡影像的分析是不理想的，因此這裡我們將採用self-updating process (SUP) 演算法 (Chen and Shiu, 2007) 運用此演算配合多重線性的主要成分分析來重建物體的結構。

In statistics, dimension reduction is a process of reducing the number of random variables under consideration, and can be divided into feature selection and feature extraction. Principal component analysis (PCA) belongs to the latter category. Traditional linear techniques for dimensionality reduction like PCA reshapes image matrices into vectors. It leads to vectors in a very high-dimensional space and thus easily suff ers from the curse of dimensionality. Multilinear principal component analysis (MPCA) has the potential to serve the similar purpose for analyzing tensor structure data. MPCA aims to preserve the natural data structure, based on 2D matrices rather than 1D vectors, and searches for low-dimensional multilinear projections. It can decrease the dimensionality in a more stable and efficient way than traditional PCA. MPCA and other tensor decomposition methods have been shown to have good performance in both real data analysis and simulations (Ye, 2005; Lu, Plataniotis and Venetsanopoulos, 2008; Kolda and Bader, 2009; Li, Kim and Altman, 2010). However, there is not much statistical theoretic study of it. In this thesis, we place the MPCA in a statistical framework and investigate its statistical properties, including asymptotic distributions for principal components, associated projections and explained variances. We also apply it to electron microscopy images analysis. Due to the nature of low signal to noise ratio (SNR) of electron microscopy images, an averaging process for similar images is needed for denoising. The k-means algorithm is probably the most commonly used algorithm for clustering. However, we find it not ideal for low SNR electron microscopy images. The k-means algorithm needs quite some manual tuning and care in order to get reasonable clustering results. Here we adopt a self-updating process (SUP) clustering algorithm (Chen and Shiu, 2007) on the MPCA-extracted core tensors to recover the hidden cluster structure.