Quantification of model bias underlying the phenomenon of Einstein from Noise

Abstract

Arising from cryogenic electron microscopy image analysis, "Einstein from noise" is a phenomenon of significant statistical interest because spurious patterns could easily emerge by averaging a large number of white-noise images aligned to a reference image through rotation and translation. While this phenomenon is often attributed to model bias, quantitative studies on such a bias are lacking. Here, we introduce a simple framework under which an image of p pixels is treated as a vector of dimension p and a white-noise image is a random vector uniformly sampled from the (p − 1)-dimensional unit sphere. Moreover, we adopt the cross correlation of two images which is a similarity measure based on the dot product of image pixels. This framework geometrically explains how the bias results from averaging a properly chosen set of white-noise images that are most highly cross-correlated with the reference image. We quantify the bias in terms of three parameters: the number of white-noise images (n), the image dimension (p), and the size of the selection set (m). Under the conditions that n, p and m are all large and (ln n) 2 /p and m/n are both small, we show that the bias is approximately 2γ 1+2γ where γ = m p ln n m .

Uncertainty Quantification in Dynamic Image Reconstruction with Applications to Cryo-EM

“…From the perspectives of statistical analysis, the "Einstein from noise" can be formulated as a problem of model bias (also see Supplementary Material S1). To capture the essence of model bias in such image alignment, Wang et al (2021) treat an image of p pixels as vector of dimension p and a white-noise image as a random vector uniformly distributed on the (p − 1)dimensional unit sphere. Instead of delving into the technical details of rotating 1000 images, Wang et al (2021) generate millions of random vectors to compensate the increased samples and simplify the pixel correlated format through the rotating process in image alignment.…”

“…To capture the essence of model bias in such image alignment, Wang et al (2021) treat an image of p pixels as vector of dimension p and a white-noise image as a random vector uniformly distributed on the (p − 1)dimensional unit sphere. Instead of delving into the technical details of rotating 1000 images, Wang et al (2021) generate millions of random vectors to compensate the increased samples and simplify the pixel correlated format through the rotating process in image alignment. The cross correlation (CC) of two images is defined as the inner product of the corresponding vectors, which is a similarity measure widely used in image processing.…”

“…The results are summarized in S1. Letting ρ n,p,m = r X m / X m be the cross-correlation of r and X m , Wang et al (2021) have carried out an elaborate asymptotic analysis of ρ 2 n,p,m to explain the results in the simulation study, as also summarized in Supplementary Material S1.…”