Fast Steerable Principal Component Analysis

Cryo-electron microscopy nowadays often requires the analysis of hundreds of thousands of 2-D images as large as a few hundred pixels in each direction. Here, we introduce an algorithm that efficiently and accurately performs principal component analysis (PCA) for a large set of 2-D images, and, for each image, the set of its uniform rotations in the plane and their reflections. For a dataset consisting of <i>n</i> images of size <i>L</i> × <i>L</i> pixels, the computational complexity of our algorithm is <i>O</i>(<i>nL</i>³ + <i>L</i>⁴), while existing algorithms take <i>O</i>(<i>nL</i>⁴). The new algorithm computes the expansion coefficients of the images in a Fourier-Bessel basis efficiently using the nonuniform fast Fourier transform. We compare the accuracy and efficiency of the new algorithm with traditional PCA and existing algorithms for steerable PCA.