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Hirn M, Little A. Wavelet invariants for statistically robust multi-reference alignment. INFORMATION AND INFERENCE : A JOURNAL OF THE IMA 2021; 10:1287-1351. [PMID: 35070296 PMCID: PMC8782248 DOI: 10.1093/imaiai/iaaa016] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
We propose a nonlinear, wavelet-based signal representation that is translation invariant and robust to both additive noise and random dilations. Motivated by the multi-reference alignment problem and generalizations thereof, we analyze the statistical properties of this representation given a large number of independent corruptions of a target signal. We prove the nonlinear wavelet-based representation uniquely defines the power spectrum but allows for an unbiasing procedure that cannot be directly applied to the power spectrum. After unbiasing the representation to remove the effects of the additive noise and random dilations, we recover an approximation of the power spectrum by solving a convex optimization problem, and thus reduce to a phase retrieval problem. Extensive numerical experiments demonstrate the statistical robustness of this approximation procedure.
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Affiliation(s)
- Matthew Hirn
- Department of Computational Mathematics, Science and Engineering, Department of Mathematics and Center for Quantum Computing, Science and Engineering, Michigan State University, East Lansing, MI 48824
| | - Anna Little
- Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI 48824
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A Fast Image Alignment Approach for 2D Classification of Cryo-EM Images Using Spectral Clustering. Curr Issues Mol Biol 2021; 43:1652-1668. [PMID: 34698131 PMCID: PMC8928942 DOI: 10.3390/cimb43030117] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/29/2021] [Revised: 10/14/2021] [Accepted: 10/14/2021] [Indexed: 01/22/2023] Open
Abstract
Three-dimensional (3D) reconstruction in single-particle cryo-electron microscopy (cryo-EM) is a significant technique for recovering the 3D structure of proteins or other biological macromolecules from their two-dimensional (2D) noisy projection images taken from unknown random directions. Class averaging in single-particle cryo-EM is an important procedure for producing high-quality initial 3D structures, where image alignment is a fundamental step. In this paper, an efficient image alignment algorithm using 2D interpolation in the frequency domain of images is proposed to improve the estimation accuracy of alignment parameters of rotation angles and translational shifts between the two projection images, which can obtain subpixel and subangle accuracy. The proposed algorithm firstly uses the Fourier transform of two projection images to calculate a discrete cross-correlation matrix and then performs the 2D interpolation around the maximum value in the cross-correlation matrix. The alignment parameters are directly determined according to the position of the maximum value in the cross-correlation matrix after interpolation. Furthermore, the proposed image alignment algorithm and a spectral clustering algorithm are used to compute class averages for single-particle 3D reconstruction. The proposed image alignment algorithm is firstly tested on a Lena image and two cryo-EM datasets. Results show that the proposed image alignment algorithm can estimate the alignment parameters accurately and efficiently. The proposed method is also used to reconstruct preliminary 3D structures from a simulated cryo-EM dataset and a real cryo-EM dataset and to compare them with RELION. Experimental results show that the proposed method can obtain more high-quality class averages than RELION and can obtain higher reconstruction resolution than RELION even without iteration.
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Ma C, Bendory T, Boumal N, Sigworth F, Singer A. Heterogeneous multireference alignment for images with application to 2-D classification in single particle reconstruction. IEEE TRANSACTIONS ON IMAGE PROCESSING : A PUBLICATION OF THE IEEE SIGNAL PROCESSING SOCIETY 2019; 29:10.1109/TIP.2019.2945686. [PMID: 31613760 PMCID: PMC11367667 DOI: 10.1109/tip.2019.2945686] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
Motivated by the task of 2-D classification in single particle reconstruction by cryo-electron microscopy (cryo-EM), we consider the problem of heterogeneous multireference alignment of images. In this problem, the goal is to estimate a (typically small) set of target images from a (typically large) collection of observations. Each observation is a rotated, noisy version of one of the target images. For each individual observation, neither the rotation nor which target image has been rotated are known. As the noise level in cryo-EM data is high, clustering the observations and estimating individual rotations is challenging. We propose a framework to estimate the target images directly from the observations, completely bypassing the need to cluster or register the images. The framework consists of two steps. First, we estimate rotation-invariant features of the images, such as the bispectrum. These features can be estimated to any desired accuracy, at any noise level, provided sufficiently many observations are collected. Then, we estimate the images from the invariant features. Numerical experiments on synthetic cryo-EM datasets demonstrate the effectiveness of the method. Ultimately, we outline future developments required to apply this method to experimental data.
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Schnitzbauer J, Wang Y, Zhao S, Bakalar M, Nuwal T, Chen B, Huang B. Correlation analysis framework for localization-based superresolution microscopy. Proc Natl Acad Sci U S A 2018; 115:3219-3224. [PMID: 29531072 PMCID: PMC5879654 DOI: 10.1073/pnas.1711314115] [Citation(s) in RCA: 42] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Superresolution images reconstructed from single-molecule localizations can reveal cellular structures close to the macromolecular scale and are now being used routinely in many biomedical research applications. However, because of their coordinate-based representation, a widely applicable and unified analysis platform that can extract a quantitative description and biophysical parameters from these images is yet to be established. Here, we propose a conceptual framework for correlation analysis of coordinate-based superresolution images using distance histograms. We demonstrate the application of this concept in multiple scenarios, including image alignment, tracking of diffusing molecules, as well as for quantification of colocalization, showing its superior performance over existing approaches.
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Affiliation(s)
- Joerg Schnitzbauer
- Department of Pharmaceutical Chemistry, University of California, San Francisco, CA 94143
| | - Yina Wang
- Department of Pharmaceutical Chemistry, University of California, San Francisco, CA 94143
| | - Shijie Zhao
- School of Life Sciences, Peking University, Beijing 100871, China
| | - Matthew Bakalar
- UC Berkeley-UCSF Joint Graduate Group in Bioengineering, University of California, Berkeley, CA 94720
| | - Tulip Nuwal
- Department of Pharmaceutical Chemistry, University of California, San Francisco, CA 94143
| | - Baohui Chen
- Department of Pharmaceutical Chemistry, University of California, San Francisco, CA 94143
| | - Bo Huang
- Department of Pharmaceutical Chemistry, University of California, San Francisco, CA 94143;
- Department of Biochemistry and Biophysics, University of California, San Francisco, CA 94143
- Chan Zuckerberg Biohub, San Francisco, CA 94158
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Bendory T, Boumal N, Ma C, Zhao Z, Singer A. Bispectrum Inversion with Application to Multireference Alignment. IEEE TRANSACTIONS ON SIGNAL PROCESSING : A PUBLICATION OF THE IEEE SIGNAL PROCESSING SOCIETY 2018; 66:1037-1050. [PMID: 29805244 PMCID: PMC5966049 DOI: 10.1109/tsp.2017.2775591] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We consider the problem of estimating a signal from noisy circularly-translated versions of itself, called multireference alignment (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.
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Affiliation(s)
- Tamir Bendory
- Princeton University in PACM and the Mathematics Department
| | - Nicolas Boumal
- Princeton University in PACM and the Mathematics Department
| | - Chao Ma
- Princeton University in PACM and the Mathematics Department
| | - Zhizhen Zhao
- University of Illinois at Urbana-Champaign in the Department of Electrical and Computer Engineering
| | - Amit Singer
- Princeton University in PACM and the Mathematics Department
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Bhamre T, Zhao Z, Singer A. MAHALANOBIS DISTANCE FOR CLASS AVERAGING OF CRYO-EM IMAGES. PROCEEDINGS. IEEE INTERNATIONAL SYMPOSIUM ON BIOMEDICAL IMAGING 2017; 2017:654-658. [PMID: 29081898 DOI: 10.1109/isbi.2017.7950605] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Abstract
Single particle reconstruction (SPR) from cryo-electron microscopy (EM) is a technique in which the 3D structure of a molecule needs to be determined from its contrast transfer function (CTF) affected, noisy 2D projection images taken at unknown viewing directions. One of the main challenges in cryo-EM is the typically low signal to noise ratio (SNR) of the acquired images. 2D classification of images, followed by class averaging, improves the SNR of the resulting averages, and is used for selecting particles from micrographs and for inspecting the particle images. We introduce a new affinity measure, akin to the Mahalanobis distance, to compare cryo-EM images belonging to different defocus groups. The new similarity measure is employed to detect similar images, thereby leading to an improved algorithm for class averaging. We evaluate the performance of the proposed class averaging procedure on synthetic datasets, obtaining state of the art classification.
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Kim JS, Afsari B, Chirikjian GS. Cross-Validation of Data Compatibility Between Small Angle X-ray Scattering and Cryo-Electron Microscopy. J Comput Biol 2017; 24:13-30. [PMID: 27710115 PMCID: PMC5220572 DOI: 10.1089/cmb.2016.0139] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/24/2022] Open
Abstract
Cryo-electron microscopy (EM) and small angle X-ray scattering (SAXS) are two different data acquisition modalities often used to glean information about the structure of large biomolecular complexes in their native states. A SAXS experiment is generally considered fast and easy but unveils the structure at very low resolution, whereas a cryo-EM experiment needs more extensive preparation and postacquisition computation to yield a three-dimensional (3D) density map at higher resolution. In certain applications, we may need to verify whether the data acquired in the SAXS and cryo-EM experiments correspond to the same structure (e.g., before reconstructing the 3D density map in EM). In this article, a simple and fast method is proposed to verify the compatibility of the SAXS and EM experimental data. The method is based on averaging the two-dimensional correlation of EM images and the Abel transform of the SAXS data. Orientational preferences are known to exist in cryo-EM experiments, and we also consider these effects on our method. The results are verified on simulations of conformational states of large biomolecular complexes.
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Affiliation(s)
- Jin Seob Kim
- Department of Mechanical Engineering, Johns Hopkins University, Baltimore, Maryland
| | - Bijan Afsari
- Center for Imaging Science, Johns Hopkins University, Baltimore, Maryland
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Kye M, Lim YB. Reciprocal Self-Assembly of Peptide-DNA Conjugates into a Programmable Sub-10-nm Supramolecular Deoxyribonucleoprotein. Angew Chem Int Ed Engl 2016; 55:12003-7. [DOI: 10.1002/anie.201605696] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/13/2016] [Revised: 07/22/2016] [Indexed: 02/03/2023]
Affiliation(s)
- Mahnseok Kye
- Department of Materials Science and Engineering; Yonsei University; 50 Yonsei-ro Seoul 03722 Korea
| | - Yong-beom Lim
- Department of Materials Science and Engineering; Yonsei University; 50 Yonsei-ro Seoul 03722 Korea
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Kye M, Lim YB. Reciprocal Self-Assembly of Peptide-DNA Conjugates into a Programmable Sub-10-nm Supramolecular Deoxyribonucleoprotein. Angew Chem Int Ed Engl 2016. [DOI: 10.1002/ange.201605696] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/01/2023]
Affiliation(s)
- Mahnseok Kye
- Department of Materials Science and Engineering; Yonsei University; 50 Yonsei-ro Seoul 03722 Korea
| | - Yong-beom Lim
- Department of Materials Science and Engineering; Yonsei University; 50 Yonsei-ro Seoul 03722 Korea
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Zhao Z, Singer A. Rotationally invariant image representation for viewing direction classification in cryo-EM. J Struct Biol 2014; 186:153-66. [PMID: 24631969 DOI: 10.1016/j.jsb.2014.03.003] [Citation(s) in RCA: 37] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/25/2013] [Revised: 03/01/2014] [Accepted: 03/02/2014] [Indexed: 10/25/2022]
Abstract
We introduce a new rotationally invariant viewing angle classification method for identifying, among a large number of cryo-EM projection images, similar views without prior knowledge of the molecule. Our rotationally invariant features are based on the bispectrum. Each image is denoised and compressed using steerable principal component analysis (PCA) such that rotating an image is equivalent to phase shifting the expansion coefficients. Thus we are able to extend the theory of bispectrum of 1D periodic signals to 2D images. The randomized PCA algorithm is then used to efficiently reduce the dimensionality of the bispectrum coefficients, enabling fast computation of the similarity between any pair of images. The nearest neighbors provide an initial classification of similar viewing angles. In this way, rotational alignment is only performed for images with their nearest neighbors. The initial nearest neighbor classification and alignment are further improved by a new classification method called vector diffusion maps. Our pipeline for viewing angle classification and alignment is experimentally shown to be faster and more accurate than reference-free alignment with rotationally invariant K-means clustering, MSA/MRA 2D classification, and their modern approximations.
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Affiliation(s)
- Zhizhen Zhao
- Courant Institute of Mathematical Sciences, New York University, Warren Weaver Hall, 251 Mercer Street, New York, NY 10012, USA.
| | - Amit Singer
- Department of Mathematics and PACM, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA.
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Sikora M, Adam D, Korczyk PM, Prodi-Schwab A, Szymczak P, Cieplak M. Geometrical and electrical properties of indium tin oxide clusters in ink dispersions. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2012; 28:1523-1530. [PMID: 22136161 DOI: 10.1021/la203886b] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
The analysis of the TEM images of indium tin oxide (ITO) clusters in ink solutions deposited from ink dispersions reveals that their geometry arises from a diffusion limited cluster aggregation (DLCA) process. We model films of ITO clusters as built through deposition of DLCA clusters made of primary spherical nanoparticles of 13 nm in diameter. The deposition is then followed by a further compactification process that imitates sintering. We determine the conductivity of the sintered films by mapping the problem to that of the resistor network in which the contact regions between the touching spheres provide the dominant electric resistance. For a given volume fraction, conductivity of the sintered films is shown to be larger than that for the randomly packed spheres. However, the larger a typical radius of gyration of the clusters the smaller the enhancement. We also provide numerical tests for the routines used in the interpretation of the TEM images.
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Affiliation(s)
- Mateusz Sikora
- Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland
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Park W, Madden DR, Rockmore DN, Chirikjian GS. Deblurring of Class-Averaged Images in Single-Particle Electron Microscopy. INVERSE PROBLEMS 2010; 26:3500521-35005229. [PMID: 20221416 PMCID: PMC2835172 DOI: 10.1088/0266-5611/26/3/035002] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/28/2023]
Abstract
This paper proposes a method for deblurring of class-averaged images in single-particle electron microscopy (EM). Since EM images of biological samples are very noisy, the images which are nominally identical projection images are often grouped, aligned and averaged in order to cancel or reduce the background noise. However, the noise in the individual EM images generates errors in the alignment process, which creates an inherent limit on the accuracy of the resulting class averages. This inaccurate class average due to the alignment errors can be viewed as the result of a convolution of an underlying clear image with a blurring function. In this work, we develop a deconvolution method that gives an estimate for the underlying clear image from a blurred class-averaged image using precomputed statistics of misalignment. Since this convolution is over the group of rigid body motions of the plane, SE(2), we use the Fourier transform for SE(2) in order to convert the convolution into a matrix multiplication in the corresponding Fourier space. For practical implementation we use a Hermite-function-based image modeling technique, because Hermite expansions enable lossless Cartesian-polar coordinate conversion using the Laguerre-Fourier expansions, and Hermite expansion and Laguerre-Fourier expansion retain their structures under the Fourier transform. Based on these mathematical properties, we can obtain the deconvolution of the blurred class average using simple matrix multiplication. Tests of the proposed deconvolution method using synthetic and experimental EM images confirm the performance of our method.
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Affiliation(s)
- Wooram Park
- Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD, USA
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