  Principal component analysis (PCA) is arguably the most widely used
dimension-reduction method for vector-type data. When applied to a sample of
images, PCA requires vectorization of the image data, which in turn entails
solving an eigenvalue problem for the sample covariance matrix. We propose
herein a two-stage dimension reduction (2SDR) method for image reconstruction
from high-dimensional noisy image data. The first stage treats the image as a
matrix, which is a tensor of order 2, and uses multilinear principal component
analysis (MPCA) for matrix rank reduction and image denoising. The second stage
vectorizes the reduced-rank matrix and achieves further dimension and noise
reduction. Simulation studies demonstrate excellent performance of 2SDR, for
which we also develop an asymptotic theory that establishes consistency of its
rank selection. Applications to cryo-EM (cryogenic electronic microscopy),
which has revolutionized structural biology, organic and medical chemistry,
cellular and molecular physiology in the past decade, are also provided and
illustrated with benchmark cryo-EM datasets. Connections to other
contemporaneous developments in image reconstruction and high-dimensional
statistical inference are also discussed.
