2SDR: Two Stage Dimension Reduction to Denoise Cryo-EM Images

Principal component analysis (PCA) is arguably the most widely used dimension reduction method for vector type data. When applied to a set of images, PCA demands that the images be vectorized. This demand consequentially introduced a weakness in the application: heavy computation due to solving the eigenvalue problem of a huge covariance matrix. In this paper, we propose a two stage dimension reduction (2SDR) method for images based on a statistical model with two layers of noise structures. 2SDR first applies multi-linear PCA (MPCA) to extract core scores from the images as well as to screen the first layer of noise, and then applies PCA on these scores to further reduce the second layer of noise. MPCA has computation advantages that it avoids image vectorization and applies the Kronecker product on column and row eigenvectors to model the image bases. In contrast, PCA can diagonalize the covariance matrix that its projected scores are guaranteed to be uncorrelated. Combining MPCA and PCA, 2SDR has two benefits that it inherits the computation advantage of MPCA and its projection scores are uncorrelated as those of PCA. Testing with two cryo-electron microscopy (cryo-EM) benchmark experimental datasets shows that 2SDR performs better than MPCA and PCA alone in terms of the computation efficiency and denoising performance. We further propose a rank selection method for 2SDR and prove that this method has the consistency property under some regular conditions.