Multi-Target Detection With an Arbitrary Spacing Distribution

Lan, Ti-Yen; Bendory, Tamir; Boumal, Nicolas; Singer, Amit

doi:10.1109/tsp.2020.2975943

Cited by 18 publications

(26 citation statements)

References 34 publications

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“…The estimation problem described in Section I-A is a simplified version of the cryo-EM reconstruction problem: the tomographic projection operator is omitted and we observe the same 2-D image multiple times with random in-plane rotations. This image recovery problem is an instance of the multi-target detection statistical model, in which a set of signals appear multiple times at unknown locations in a noisy measurement [3], [4], [13]. Here, we extend previous works by taking in-plane rotations into account, which forms an important step towards the analysis of the full cryo-EM problem.…”

Section: B Motivationmentioning

confidence: 84%

“…We have demonstrated the reconstruction of a target image from its noiseless invariants, and showed that our algorithmic approach is robust to noise. In future work, we intend to extend the framework to include the recovery of the target image from the observation M , to mitigate the separation condition [13], and to allow the recovery of multiple images simultaneously, in a similar fashion to [4], [6]. From a theoretical perspective, we wish to complement the empirical results of this work by proving that indeed a generic image f is determined uniquely by its rotationally-averaged third-order autocorrelation.…”

Section: Discussionmentioning

confidence: 96%

“…For simplicity, we assume that images in the measurement are separated by at least one image diameter, see (5). Recent work on a related problem suggests that this separation restriction can be alleviated [13].…”

Section: A Problem Statementmentioning

confidence: 99%

“…From a computational perspective, this approach is highly efficient as it requires only one pass over the measurement to calculate the autocorrelations; this is in contrast to the likelihoodbased techniques that may become prohibitive as the data size increases. For a more detailed discussion, see [13].…”

Section: Main Contributionmentioning

confidence: 99%

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Image Recovery from Rotational And Translational Invariants

Marshall

Lan

Bendory

et al. 2020

ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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We introduce a framework for recovering an image from its rotationally and translationally invariant features based on autocorrelation analysis. This work is an instance of the multi-target detection statistical model, which is mainly used to study the mathematical and computational properties of singleparticle reconstruction using cryo-electron microscopy (cryo-EM) at low signal-to-noise ratios. We demonstrate with synthetic numerical experiments that an image can be reconstructed from rotationally and translationally invariant features and show that the reconstruction is robust to noise. These results constitute an important step towards the goal of structure determination of small biomolecules using cryo-EM.

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Section: B Motivationmentioning

confidence: 84%

Section: Discussionmentioning

confidence: 96%

Section: A Problem Statementmentioning

confidence: 99%

Section: Main Contributionmentioning

confidence: 99%

See 2 more Smart Citations

Image Recovery from Rotational And Translational Invariants

Marshall

Lan

Bendory

et al. 2020

ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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“…In addition, autocorrelation analysis provides a flexible framework to extend the multi-target detection model by relating the expected autocorrelations of the data with the signals, and all parameters necessary to describe the generative model. For instance, a follow-up paper relaxes the separation condition (1.2) and allows an arbitrary spacing of targets, as long as the signal occurrences do not overlap [29]. In that case, the autocorrelations of the data are functions of the signal and the unknown target distribution.…”

Section: Discussionmentioning

confidence: 99%

Multi-target detection with application to cryo-electron microscopy

et al. 2019

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We consider the multi-target detection problem of recovering a set of signals that appear multiple times at unknown locations in a noisy measurement. In the low noise regime, one can estimate the signals by first detecting occurrences, then clustering and averaging them. In the high noise regime, however, neither detection nor clustering can be performed reliably, so that strategies along these lines are destined to fail. Notwithstanding, using autocorrelation analysis, we show that the impossibility to detect and cluster signal occurrences in the presence of high noise does not necessarily preclude signal estimation. Specifically, to estimate the signals, we derive simple relations between the autocorrelations of the observation and those of the signals. These autocorrelations can be estimated accurately at any noise level given a sufficiently long measurement. To recover the signals from the observed autocorrelations, we solve a set of polynomial equations through nonlinear least-squares. We provide analysis regarding well-posedness of the task, and demonstrate numerically the effectiveness of the method in a variety of settings.The main goal of this work is to provide theoretical and numerical support for a recently proposed framework to image 3-D structures of biological macromolecules using cryo-electron microscopy in extreme noise levels. arXiv:1903.06022v2 [cs.IT] 3 Jun 2019In words: the starting positions of any two occurrences must be separated by at least 2L − 1 positions, so that their end points are necessarily separated by at least L − 1 signal-free (but still noisy) entries in the data. Furthermore, we require that the last signal occurrence in y is also followed by at least L − 1 signal-free entries. This property ensures that correlating y with versions of itself shifted by at most L−1 entries does not involve correlating distinct signal occurrences. Once s is determined, for each position i such that s[i] = 1, one of the signals x k is selected independently at random, and accordingly we set s k [i] = 1. As a result, the only properties of s 1 , . . . , s K that affect the autocorrelations of y (for shifts up to L − 1) are the total number of occurrences of the distinct signals: their individual and relative locations do not intervene. We detail this in Section 3.The Poisson model. If the separation condition is violated, more knowledge about the location distribution is necessary to disentangle the autocorrelations of y. To that effect, we analyze a Poisson generative model. Specifically, for each position i, the number s[i] of signal occurrences starting at that position is drawn independently from a Poisson distribution with parameter γ/L, that is, s[i] i.i.d.

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Likelihood Maximization and Moment Matching in Low SNR Gaussian Mixture Models

Katsevich

Bandeira

2022

Comm Pure Appl Math

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We derive an asymptotic expansion for the log-likelihood of Gaussian mixture models (GMMs) with equal covariance matrices in the low signal-to-noise regime. The expansion reveals an intimate connection between two types of algorithms for parameter estimation: the method of moments and likelihood optimizing algorithms such as Expectation-Maximization (EM). We show that likelihood optimization in the low SNR regime reduces to a sequence of least squares optimization problems that match the moments of the estimate to the ground truth moments one by one. This connection is a stepping stone towards the analysis of EM and maximum likelihood estimation in a wide range of models. A motivating application for the study of low SNR mixture models is cryo-electron microscopy data, which can be modeled as a GMM with algebraic constraints imposed on the mixture centers. We discuss the application of our expansion to algebraically constrained GMMs, among other example models of interest.

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Multi-Target Detection With an Arbitrary Spacing Distribution

Cited by 18 publications

References 34 publications

Image Recovery from Rotational And Translational Invariants

Image Recovery from Rotational And Translational Invariants

Multi-target detection with application to cryo-electron microscopy

Likelihood Maximization and Moment Matching in Low SNR Gaussian Mixture Models

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