We present an efficient and accurate algorithm for principal component analysis (PCA) of a large set of two-dimensional images and, for each image, the set of its uniform rotations in the plane and its reflection. The algorithm starts by expanding each image, originally given on a Cartesian grid, in the Fourier–Bessel basis for the disk. Because the images are essentially band limited in the Fourier domain, we use a sampling criterion to truncate the Fourier–Bessel expansion such that the maximum amount of information is preserved without the effect of aliasing. The constructed covariance matrix is invariant to rotation and reflection and has a special block diagonal structure. PCA is efficiently done for each block separately. This Fourier–Bessel-based PCA detects more meaningful eigenimages and has improved denoising capability compared to traditional PCA for a finite number of noisy images.
© 2013 Optical Society of America
Original Manuscript: October 24, 2012
Revised Manuscript: February 15, 2013
Manuscript Accepted: March 18, 2013
Published: April 15, 2013
Zhizhen Zhao and Amit Singer, "Fourier–Bessel rotational invariant eigenimages," J. Opt. Soc. Am. A 30, 871-877 (2013)