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.