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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 .

This paper introduces a new approach, which combines empirical Bayes modeling with recent advances in Markov chain Monte Carlo filters for hidden Markov models, to address long-standing challenging problems in reconstruction of 3D images, with uncertainty quantification, from noisy 2D pixels in cryogenic electron microscopy and other applications such as brain network development in infants.