Bispectrum Inversion With Application to Multireference Alignment

Abstract

We consider the problem of estimating a signal from noisy circularly-translated versions of itself, called (MRA). One natural approach to MRA could be to estimate the shifts of the observations first, and infer the signal by aligning and averaging the data. In contrast, we consider a method based on estimating the signal directly, using features of the signal that are invariant under translations. Specifically, we estimate the power spectrum and the bispectrum of the signal from the observations. Under mild assumptions, these invariant features contain enough information to infer the signal. In particular, the bispectrum can be used to estimate the Fourier phases. To this end, we propose and analyze a few algorithms. Our main methods consist of non-convex optimization over the smooth manifold of phases. Empirically, in the absence of noise, these non-convex algorithms appear to converge to the target signal with random initialization. The algorithms are also robust to noise. We then suggest three additional methods. These methods are based on frequency marching, semidefinite relaxation and integer programming. The first two methods provably recover the phases exactly in the absence of noise. In the high noise level regime, the invariant features approach for MRA results in stable estimation if the number of measurements scales like the cube of the noise variance, which is the information-theoretic rate. Additionally, it requires only one pass over the data which is important at low signal-to-noise ratio when the number of observations must be large.

Multi-Target Detection With an Arbitrary Spacing Distribution

“…On the other hand, autocorrelation analysis requires shorter computation time by summarizing the data as autocorrelation statistics with one pass over the data. We note that the distinct computational speeds for autocorrelation analysis and the EM algorithm is also observed in the related problem of multi-reference alignment [8], [14]. The difference in run time and the similar reconstruction quality of the two methods at low SNR make autocorrelation analysis the preferred approach for large datasets.…”

Toward single particle reconstruction without particle picking: Breaking the detection limit

“…Below, we start with p = 2. The PSWFs are eigenfunctions of the truncated Fourier transform and are given in polar coordinates by 5 ψ k,q (r, ϕ) = 1 √ 8π α k,q R k,q (r)e ιkϕ , r ≤ 1, 0, r > 1, (31) where the range of k, q is determined by Eq. (8) in [24], the R k,q are a family of real, one-dimensional functions and the α k,q are scaling factors which will be dened in the next section.…”

“…Thus, we proceed as follows: for each position i in the micrograph I, we extract the corresponding patch of size (2P − 1) × (2P − 1), expand it in the discretized PSWFs as in (36), and collect the a k,q as per (34) to constitute the second-order autocorrelation of the micrograph. 5 A dierent normalization is used in [24].…”

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

“…For the cryo-EM setup, assuming perfect particle picking and no CTF, it was shown that the sample complexity depends on the distribution of rotations: for uniform distribution the sample complexity scales as 1/SNR 3 (that is, the structure is determined by the third moment), while for generic non-uniform distribution the second moment suffices, and thus the sample complexity scales as 1/SNR 2 [11], [77]. Similar results were derived for simpler setups, for instance, when a discrete signal is acted upon by cyclic shifts (as shown in Figure 7) [22], [3], [11].…”