Fast Steerable Principal Component Analysis

Abstract

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 n images of size L × L pixels, the computational complexity of our algorithm is O(nL3 + L4), while existing algorithms take O(nL4). 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.

Two-Dimensional Tomography from Noisy Projection Tilt Series Taken at Unknown View Angles with Non-Uniform Distributi

“…1. With the estimated coefficients a, we use the analytical expression of the inverse Fourier transform of ψ k,q c to reconstruct the image in the real domain as detailed in [17]. The computational complexity is O(N L log L) for non-uniform FFT (5), O(N L 2 ) moment features generation (8), and O(L 6 ) for each iteration in Alg.…”

“…where J k is the k th order Bessel function of the first kind, R k,q is the q th root of J k , and N k,q = (c √ π |J k+1 (R k,q )|) −1 is a normalization factor. Assuming that the object is also well concentrated in real domain, with the radius of support R, the Fourier transformed function can be well approximated by the truncated FB expansion [16,17] according to the sampling criterion, R k,q+1 ≤ 2πcR,…”

Single-Particle Cryo-Electron Microscopy: Mathematical Theory, Computational Challenges, and Opportunities

“…Luckily, there is a simple way to account for all in-plane rotations without rotating the images explicitly. to O(N L 3 + L 4 ): the first term is the cost of computing the sample covariance over N images, and the second term is the cost of the eigendecomposition over all blocks [100].…”

Factorization of the translation kernel for fast rigid image alignment

“…This justifies our naming (21) the Fourier-Bessel coefficients of the image. A similar basis is used by Zhao et al [45,44], who make different assumptions regarding the support of the images. The above remark, (23), and x ≤ 1 show that the coefficients a(k; q) are exponentially small, and thus numerically negligible, for |q| > K + O(K 1/3 ).…”