Annals of Mathematical Sciences and Applications

Volume 5 (2020)

Number 2

Two-stage dimension reduction for noisy high-dimensional images and application to Cryogenic Electron Microscopy

Pages: 283 – 316



Szu-Chi Chung (Institute of Statistical Science, Academia Sinica, Tapei, Taiwan)

Shao-Hsuan Wang (Institute of Statistical Science, Acadmia Sinica, Taiwan)

Po-Yao Niu (Institute of Statistical Science, Academia Sinica, Tapei, Taiwan)

Su-Yun Huang (Institute of Statistical Science, Academia Sinica, Tapei, Taiwan)

Wei-Hau Chang (Institute of Chemistry, Acadmia Sinica, Taipei, Taiwan)

I-Ping Tu (Institute of Statistical Science, Academia Sinica, Tapei, Taiwan)


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.


generalized information criterion, image denoising and reconstruction, random matrix theory, rank selection, Stein’s unbiased estimate of risk

Winner of a Best Paper Award (Silver Medal) at the 2020 International Consortium of Chinese Mathematicians, December 2020.

I-Ping Tu was supported by Academia Sinica:AS-GCS-108-08 and MOST:106-2118-M-001-001-MY2.

The authors are grateful to AMSA chief editor Tze Leung Lai (Zhejiang University) for his editing work on this paper.

Received 30 April 2020

Accepted 22 May 2020

Published 13 October 2020