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Goal

Learn the mathematics of single particle electron cryomicroscopy (cryoEM)

Why?

We can see living atoms with electrons. Electron microscopes use hundreds of thousands of volts to speed up single electrons to three quarters the speed of light. At such high speeds electrons have picometer wavelengths and can resolve the distances between atoms in biomolecules. Electron cryomicroscopy (cryoEM) won the 2017 Nobel prize in Chemistry, and pharmaceutical companies have invested in this technology for applications like structure based drug design.

After biochemical sample preparation of a purified biomolecule, two dimensional images are collected on electron microscopes. Images of single biomolecules are very noisy and computer algorithms average tens of thousands to millions of 2D images to reconstruct a 3D discrete scalar map that represents the Coulomb density (what the electron feels).

Historically cryoEM computational workflows have drawn from digital signal and image processing theory. With the recent popularization of cryoEM, many researchers have entered the field who do not have a strong physics/engineering background and treat algorithms as black boxes. Unfortunately, this can limit their intuition of what data processing strategies might work, or how to troubleshoot when confusing results are generated computationally. Furthermore, there is a growing trend of applied mathematicians, statisticians and computer scientists to approach cryoEM using the familiar methods of their discipline, providing novel breakthroughs. Therefore, a more transparent and pedagogical treatment of the mathematics of single particle electron cryomicroscopy would be welcomed by researchers who analyze cryoEM data and/or develop computational approaches to cryoEM data.

The mathematics of cryoEM spans several disciplines, from the physics of electron microscopes, through digital Fourier transformations, to Bayesian inference and beyond. This broad range of specialties makes the mathematics of cryoEM a challenge to master, and open to contributions from many fields.

In this repo I would like to open up the black box of cryoEM computation, and exhibit the mathematical objects contained inside. Those sufficiently curious and motivated can teach themselves more about cryoEM data processing by playing with simple models in interactive programming notebooks, made publicly available here.

NB: If there is a particular computational aspect of cryoEM analysis that you are curious about, feel free to request (via Issues in this repo) some learning materials. I will do my best to develop some code and data to play with for interactive learning.

Resources

Online Resources

  • Jensen, G. J. (2014). Getting Started in Cryo-EM. Retrieved from http://cryo-em-course.caltech.edu/videos
    • Recommended place to start for total beginners. Coursera style.
  • Jensen, G. J., & Vos, M. (2020). Electron Microscopy University. Retrieved from https://em-learning.com/
    • 70 hours of videos. From intro to very advanced. Theoretical white board lectures and videos of someone operating the microscope. Includes important concepts hard to find elsewhere (vacuum theory, heuristics of data collection / experimental design).
  • Sigworth, F., & Tagare, H. (2020). Cryo-EM Principles. Retrieved from https://cryoemprinciples.yale.edu/
    • Pedagogical step-by-step introduction to math
  • Shen, P., Iwasa, J., Thuesen, A., Thoms, J., & Mayson, P. (2020). CryoEM 101. Retrieved from https://cryoem101.org/
    • Visual animations help build intuition

CryoEM Textbooks

  • Frank, J. (1996). Electron Microscopy of Macromolecular Assemblies. In Three-Dimensional Electron Microscopy of Macromolecular Assemblies (pp. 12–53). Elsevier. http://doi.org/10.1016/B978-012265040-6/50002-3
  • Glaeser, R. (2007). Electron Crystallography of Biological Macromolecules. Oxford University Press.
    • Good historical overview and mathematical excursions. Narrative style. Works out analytical calculations. Before the major gains in resolution, and emphasis on 2D crystals.

Math

Fourier

  • Lighthill, M. J. (1958). An Introduction to Fourier Analysis and Generalised Functions. An Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press. http://doi.org/10.1017/CBO9781139171427.001
    • subtelties about delta functions. Helps for not getting confused between discrete and continous FT
  • Vembu, S. (1961). Fourier Transformation of the n-Dimensional Radial Delta Function. The Quarterly Journal of Mathematics, 12(1), 165–168. http://doi.org/10.1093/qmath/12.1.165
  • Briggs, W. L., & Hensen, V. E. (1995). The DFT: an owner’s manual for the discrete Fourier transform (1st ed.). SIAM.
    • very pedagogical textbook on DFTs. Starts with gentle introduction and goes to advanced level.
  • Goodman, J. W. (1996). Introduction to Fourier Optics (2nd ed.).
    • Classic optics text. Incudes end of chapter problems. See math chapter, Chapter 2: Analysis of Two-Dimensional Signals and Systems.
  • Bracewell, R. (2000). The Fourier Transform And Its Applications. The Fourier Transform and It’s Applications (3rd ed.). http://doi.org/10.1017/cbo9780511623813.014
  • Khuri, A. I. (2004). Applications of Dirac’s delta function in statistics. International Journal of Mathematical Education in Science and Technology, 35(2), 185–195. http://doi.org/10.1080/00207390310001638313
  • Hobbie, R. K., & Roth, B. J. (2007). Intermediate Physics for Medicine and Biology (4th ed.). New York, NY: Springer New York. http://doi.org/10.1007/978-0-387-49885-0
    • Discussion of convolution theorems and slice theorems. Includes very thoughtful end of chapter problems. Author maintains blog where he works out some problems in detail.
  • Osgood, B. (2007). Lecture Notes for EE 261: The Fourier Transform and its Applications. Lecture Notes for EE 261 - The Fourier Transform and its Applications.
  • Filedman, J. (2007). Discrete-Time Fourier Series and Fourier Transform. Retrieved from https://www.math.ubc.ca/~feldman/m267/dft.pdf
  • Gonzalez, R. C., & Woods, R. E. (2008). Digital Image Processing (2nd ed.).
    • Classic image analysis chapter. Chapter of Fourier filtering helps build intuition of how to use DFT/FFT in practice.
  • Lam, E. (2008). Some Properties of Fourier Transform 1.
  • Wang, Q., Ronneberger, O., & Burkhardt, H. (2008). Fourier Analysis in Polar and Spherical Coordinates. Technical Report, University of Freiburg, (Internal Report 1/08).
    • Advances, but details (not terse) mathamatical analysis of 2D and 3D transforms (Fourier, Zernike, Spherical Harmonics, etc).
  • Jeong, D. (2010). Appendix A: Fourier transforms. In Cosmology with high (z > 1) redshift galaxy surveys (pp. 238–253). CRC Press.
  • Bright, A., & Wang, R. (2010). Delta Functions Generated by Complex Exponentials. Retrieved from http://fourier.eng.hmc.edu/e102/lectures/ExponentialDelta.pdf
  • Baddour, N. (2011). Two-Dimensional Fourier Transforms in Polar Coordinates. In Advances in Imaging and Electron Physics (Vol. 165, pp. 1–45). Elsevier Inc. http://doi.org/10.1016/B978-0-12-385861-0.00001-4
  • Berendsen, H. J. C. (2012). Fourier transforms. In Simulating the Physical World (pp. 315–334). Cambridge: Cambridge University Press. http://doi.org/10.1017/CBO9780511815348.014
  • Henning, A. J., Huntley, J. M., & Giusca, C. L. (2015). Obtaining the Transfer Function of optical instruments using large calibrated reference objects. Optics Express, 23(13), 16617. http://doi.org/10.1364/OE.23.016617
  • Haber, H. E. (2018). A Gaussian integral with a purely imaginary argument. Retrieved from http://scipp.ucsc.edu/~haber/ph215/Gaussian.pdf
  • Baddour. (2019). Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules. Mathematics, 7(8), 698. http://doi.org/10.3390/math7080698
  • Yao, X., & Baddour, N. (2020). Discrete two dimensional Fourier transform in polar coordinates part II: numerical computation and approximation of the continuous transform. PeerJ Computer Science, 6, e257. http://doi.org/10.7717/peerj-cs.257

Poisson (stats, pmf, random variable)

Linear Algebra

  • Abdi, H. (2007). The Eigen-Decomposition: Eigenvalues and Eigenvectors. Encyclopedia of Measurements and Statistics, 1–10.
    • Helpful for math in Grob et al (2013)

Complex Random Variables

Physics (electon optics, lenses, detector)

  • Shaw, R. (1978). Evaluating the efficient of imaging processes. Reports on Progress in Physics, 41(7), 1103–1155. http://doi.org/10.1088/0034-4885/41/7/003
  • Rabbani, M., Van Metter, R., & Shaw, R. (1987). Detective quantum efficiency of imaging systems with amplifying and scattering mechanisms. Journal of the Optical Society of America A, 4(5), 895. http://doi.org/10.1364/JOSAA.4.000895
  • Snyder, D. L., White, R. L., & Hammoud, A. M. (1993). Image recovery from data acquired with a charge-coupled-device camera. Journal of the Optical Society of America A, 10(5), 1014. http://doi.org/10.1364/JOSAA.10.001014
  • Vliet, L. J. Van, Boddeke, F. R., Sudar, D., & Young, I. T. (1998). Image Detectors for Digital Image Microscopy. In Digital Image Analysis of Microbes; Imaging, Morphometry, Fluorometry and Motility Techniques and Applications (pp. 37–64).
    • readout noise as Gaussian noise from electronics. semi techincal explanation of how CCDs work. pedagogical outline of sources of noise: 1. Readout noise: on-chip electronics, pre-amplifier and KTC noise (Gaussian additive) 2. Quantization noise (uniform, additive) 3. Photon (ie shot) (Poisson) 4. Thermal (ie dark current and hot pixels) (Poisson). See Table 3 for their distribution, dependencies, SNR, and other remakrs. Detailed discussion of gain, but likely specific to CCD cameras.
  • Meyer, R. R., & Kirkland, A. (1998). The effects of electron and photon scattering on signal and noise transfer properties of scintillators in CCD cameras used for electron detection. Ultramicroscopy, 75(1), 23–33. http://doi.org/10.1016/S0304-3991(98)00051-5
  • Faruqi, A. R., Henderson, R., & Tlustos, L. (2005). Noiseless direct detection of electrons in Medipix2 for electron microscopy. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 546(1–2), 160–163. http://doi.org/10.1016/j.nima.2005.03.022
    • Experimental characterization of Medipix2. Dose as low as < 0.01 e/pixel. Counting stats "essentially Poissonian". Discussions of MTF.
  • Faruqi, A. R., Henderson, R., Pryddetch, M., Allport, P., & Evans, A. (2005). Direct single electron detection with a CMOS detector for electron microscopy. Nuclear Instruments and Methods in Physics Research, Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 546(1–2), 170–175. http://doi.org/10.1016/j.nima.2005.03.023
    • Experimental characterization of MAPS detector. Single electon events detected and pixel values of nearby pixels studied: profile of sorrounding pixels, SNR.
  • Evans, D. A., Allport, P. P., Casse, G., Faruqi, A. R., Gallop, B., Henderson, R., … Waltham, N. (2005). CMOS active pixel sensors for ionising radiation. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 546(1–2), 281–285. http://doi.org/10.1016/j.nima.2005.03.108
    • Plots of "Electron Transfer Curve", ie deposited energy per electron event.
  • Kirkland, E. J. (2006). Image Simulation in Transmission Electron Microscopy, 1–14.
  • Glaeser, R. (2007). Electron Crystallography of Biological Macromolecules. Oxford University Press.
  • McMullan, G., Cattermole, D. M., Chen, S., Henderson, R., Llopart, X., Summerfield, C., … Faruqi, A. R. (2007). Electron imaging with Medipix2 hybrid pixel detector. Ultramicroscopy, 107(4–5), 401–413. http://doi.org/10.1016/j.ultramic.2006.10.005
    • Experimental characterizatoin of the Medipix2, originally a high energy particle physics detector, and repositioned to detection electrons in TEM. They derive a simple and useful analytical result of the DQE (at zero spatial frequency) for counting detectors. It perhaps could be modified for other intermadiate spatial frequencies. They investigate the effect of threshold energy (to call a count) on the DQE with Monte Carlo simulations, and suggest a threshold of half the incident energy (e.g. 100 keV for a 200 keV scope). Dong MC simulations allows them to investigate how many pixels are being miscounted through backscattering, or would not be observed experimentally (the number of zero count events).
  • Turchetta, R. (2007). CMOS monolithic active pixel sensors (MAPS) for scientific applications: Some notes about radiation hardness. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 583(1), 131–133. http://doi.org/10.1016/j.nima.2007.08.226
    • review the state-of-art of MAPS as imaging sensors. historical perspective. consumer and scientific applications. references to electronics literature.
  • Mooney, P. (2009). A Noise-Tolerant Method for Measuring MTF from Found-Object Edges in a TEM. Microscopy and Microanalysis, 15(S2), 234–235. http://doi.org/10.1017/S1431927609098407
  • Baxter, W. T., Grassucci, R. A., Gao, H., & Frank, J. (2009). Determination of signal-to-noise ratios and spectral SNRs in cryo-EM low-dose imaging of molecules. Journal of Structural Biology, 166(2), 126–132. http://doi.org/10.1016/j.jsb.2009.02.012
  • McMullan, G., Chen, S., Henderson, R., & Faruqi, A. R. (2009). Detective quantum efficiency of electron area detectors in electron microscopy. Ultramicroscopy, 109(9), 1126–1143. http://doi.org/10.1016/j.ultramic.2009.04.002
    • Rigerous and pedagogical (and not too long) overview of how to measure modulation transfer function (MTF), noise power spectrum (NPS), detective quantum efficiency (DQE) from images (knife method). Analytical results for edge spread function, line spread function given which are derived/presented in more detail in references (textbooks, etc); see equations 10-13. special care is taken for DQE(w=0), because there is much noise in the NPS at low spatial frequency and the MTF can drop off quickly at low spatial frequency. Methods to measure the NPS from the difference of digitized frames is presented. All this theory is used to compare film, CCD, Medipix2 (hybrid pixel detector) and a MAPS (monolithic active pixel sensor) detector. For detectors (such as MAPS) that can distinguish individual electron events, one can calculate the DQE from the probability density of the energy (Landau plot), although this density is hard to estimate. The authors do Monte Carlo simulations at two levels of theory (“continuous slowing down approximation”, and another they call “Full Monte Carlo” that treats both elastic and inelastic collisions as stochastic), and give references to a previous study and a textbook. Using this theory they simulate electron trajectories, deposited energy, and the resulting image in film, and MAPS. They are then able connect detector performance (DQE, etc) to physical/causal explanations, and suggest how to make the ideal detector. History has vindicated their calls for how to make a detector of choice.
  • Rullgård, H., Öfverstedt, L.-G., Masich, S., Daneholt, B., & Öktem, O. (2011). Simulation of transmission electron microscope images of biological specimens. Journal of Microscopy, 243(3), 234–256. http://doi.org/10.1111/j.1365-2818.2011.03497.x
    • Early forward model of image formation. Includes: Poisson shot noise, inverse Gaussian (or Wald) for the Landau distribution for the amount of energy an electron deposits.
  • Ghadimi, R., Daberkow, I., Kofler, C., Sparlinek, P., & Tietz, H. (2011). Characterization of 16 MegaPixel CMOS Detector for TEM by Evaluating Single Events of Primary Electrons. Microscopy and Microanalysis, 17(S2), 1208–1209. http://doi.org/10.1017/S143192761100691X
  • Ruskin, R. S., Yu, Z., & Grigorieff, N. (2013). Quantitative characterization of electron detectors for transmission electron microscopy. Journal of Structural Biology, 184(3), 385–393. http://doi.org/10.1016/j.jsb.2013.10.016
    • Details on how to experimentally measure the DQE for different detectors (CMOS, CCD, film). Mathematical details about modulation transfer function, noise poser spectrum, signal to noise ratio
  • Vulović, M., Ravelli, R. B. G., van Vliet, L. J., Koster, A. J., Lazić, I., Lücken, U., Rullgård, H., Öktem, O., Rieger, B. (2013). Image formation modeling in cryo-electron microscopy. Journal of Structural Biology, 183(1), 19–32. http://doi.org/10.1016/j.jsb.2013.05.008
    • detailed motivation of an image formation (forward model). Authors include employees of the FEI company, which makes electron microscopes.
  • Grob, P., Bean, D., Typke, D., Li, X., Nogales, E., & Glaeser, R. M. (2013). Ranking TEM cameras by their response to electron shot noise. Ultramicroscopy, 133, 1–7. http://doi.org/10.1016/j.ultramic.2013.01.003
    • Details mathematical analysis of detector noise, based on minimal assumptions (poisson stats of the number of general random variable events for the count registered, with mean and std given). See supplementary info for detailed proofs.
  • Jensen, G. J. (2014). Getting Started in Cryo-EM. Retrieved from http://cryo-em-course.caltech.edu/videos
  • Vulović, M., Voortman, L. M., Van Vliet, L. J., & Rieger, B. (2014). When to use the projection assumption and the weak-phase object approximation in phase contrast cryo-EM. Ultramicroscopy, 136, 61–66. http://doi.org/10.1016/j.ultramic.2013.08.002
  • Öktem, O. (2015). Mathematics of Electron Tomography. In Handbook of Mathematical Methods in Imaging (pp. 937–1031). New York, NY: Springer New York. http://doi.org/10.1007/978-1-4939-0790-8_43
  • Kirkland, E. J. (2016). Computation in electron microscopy. Acta Crystallographica Section A Foundations and Advances, 72(1), 1–27. http://doi.org/10.1107/S205327331501757X
  • McMullan, G., Faruqi, A. R., & Henderson, R. (2016). Direct Electron Detectors. In Methods in Enzymology (1st ed., Vol. 579, pp. 1–17). Elsevier Inc. http://doi.org/10.1016/bs.mie.2016.05.056
    • Excellent historical overview of improvements in electron detectors, especially the 1990s and 2000s. Good place to start for further delving into detector literature, especially in the journals Ultramicroscopy and Nuclear Instruments & Methods in Physics Research Section A—Accelerators, Spectrometers, Detectors and Associated Equipment.
  • Maigné, A., & Wolf, M. (2018). Low-dose electron energy-loss spectroscopy using electron counting direct detectors. Microscopy, 67(suppl_1), i86–i97. http://doi.org/10.1093/jmicro/dfx088
    • Characterization of a Gatan K2 for EELS on a biological protein embedded in amorphous ice. Promising applications for time resolved spectra measurments on cryoEM samples.
  • Booth, C. (2019). Detection Technologies for Cryo-Electron Microscopy.
  • Datta, A., Ban, Y., Ding, M., Chee, S. W., Shi, J., & Loh, N. D. (2019). ReCoDe: A Data Reduction and Compression Description for High Throughput Time-Resolved Electron Microscopy.
  • Kirkland, E. J. (2020). Advanced Computing in Electron Microscopy. Cham: Springer International Publishing. http://doi.org/10.1007/978-3-030-33260-0
    • Superb textbook! Very pedagical treatment of physics and computation. Emphasis on TEM and SEM as well as materials science and biological cryoEM. Computer programs are available online accompanying the text. Derivation and analysis of: wave equation for fast electrons, multislice (thick specimens). Important details about appropriate discretizations (rules of thumb and the reason why) and other matters surrounding numerical computations.
  • Nakane, T., Kotecha, A., Sente, A., McMullan, G., Masiulis, S., Brown, P. M. G. E., … Scheres, S. H. W. (2020). Single-particle cryo-EM at atomic resolution. BioRxiv, 2020.05.22.110189. http://doi.org/10.1101/2020.05.22.110189
    • electron-event representation data format for electron counting
  • Saxton, W. O. (2020). Advances in Imaging and Electron Physics: Computer Techniques for Image Processing in Electron Microscopy. (M. Hÿtch & P. W. Hawkes, Eds.).
    • Multichapter reprints from Advances in Electronics and Electron Physics, Supplement 10, 1978. Although much of the numerical implementation details are dated, the modelling choices and their motivations can be discerned.
  • Sigworth, F., & Tagare, H. (2020). Cryo-EM Principles. Retrieved from https://cryoemprinciples.yale.edu/

CryoEM algorithms / data processing

Overview of algorithms

  • Cong, Y., & Ludtke, S. J. (2010). Single Particle Analysis at High Resolution. In Methods in Enzymology (1st ed., Vol. 482, pp. 211–235). Elsevier Inc. http://doi.org/10.1016/S0076-6879(10)82009-9
  • Scheres, S. H. W. (2010). Classification of Structural Heterogeneity by Maximum-Likelihood Methods. In Methods in Enzymology (1st ed., Vol. 482, pp. 295–320). Elsevier Inc. http://doi.org/10.1016/S0076-6879(10)82012-9
  • Jensen, G. J. (2014). Getting Started in Cryo-EM. Retrieved from http://cryo-em-course.caltech.edu/videos
  • Sigworth, F. J. (2016). Principles of cryo-EM single-particle image processing. Microscopy (Oxford, England), 65(1), 57–67. http://doi.org/10.1093/jmicro/dfv370
  • Singer, A. (2018). Mathematics for cryo-electron microscopy.
  • Sigworth, F., & Tagare, H. (2020). Cryo-EM Principles. Retrieved from https://cryoemprinciples.yale.edu/
  • Singer, A., & Sigworth, F. J. (2020). Computational Methods for Single-Particle Cryo-EM, 1–40.
  • Bendory, T., Bartesaghi, A., & Singer, A. (2020). Single-Particle Cryo-Electron Microscopy: Mathematical Theory, Computational Challenges, and Opportunities. IEEE Signal Processing Magazine, 37(2), 58–76. http://doi.org/10.1109/MSP.2019.2957822

Expectation-Maximization

  • Sigworth, F. J. (1998). A Maximum-Likelihood Approach to Single-Particle Image Refinement. Journal of Structural Biology, 122(3), 328–339. http://doi.org/10.1006/jsbi.1998.4014
    • First application of expecation-maximization in cryoEM, although different language (maximum-liklihood) used by the author. Synthetic data analyzed in successfull attempt to overcome template bias in 2D class averages.
  • Do, C. B., & Batzoglou, S. (2008). What is the expectation maximization algorithm? Nature Biotechnology, 26(8), 897–899. http://doi.org/10.1038/nbt1406
  • Sigworth, F. J., Doerschuk, P. C., Carazo, J.-M., & Scheres, S. H. W. (2010). An Introduction to Maximum-Likelihood Methods in Cryo-EM. In Methods in Enzymology (1st ed., Vol. 482, pp. 263–294). Elsevier Inc. http://doi.org/10.1016/S0076-6879(10)82011-7
  • Tagare, H. D., Barthel, A., & Sigworth, F. J. (2010). An adaptive Expectation–Maximization algorithm with GPU implementation for electron cryomicroscopy. Journal of Structural Biology, 171(3), 256–265. http://doi.org/10.1016/j.jsb.2010.06.004
  • Scheres, S. H. W. (2010). Classification of Structural Heterogeneity by Maximum-Likelihood Methods. In Methods in Enzymology (1st ed., Vol. 482, pp. 295–320). Elsevier Inc. http://doi.org/10.1016/S0076-6879(10)82012-9
  • Nelson, P. C. (2019). Chapter 12 : Single Particle Reconstruction in Cryo-electron Microscopy. In Physical Models of Living Systems (pp. 305–325).
    • Superb! Very good place to start. Very pedagogical treamtment and motivation of the EM algorithm, in real space under Gaussian stats. 1D case (shifts) discussed first to motivate 2D case (shits and rotation). Care taken to explain what the motation means (subscripts, what is a vector, what is a scalar, etc.). Notebooks used to make the textbook are available on this repo by permission from the authors.

Resolution (FSC, SSNR, etc)

  • Penczek, P. A. (2010). Resolution Measures in Molecular Electron Microscopy. In Methods in Enzymology (1st ed., Vol. 482, pp. 73–100). Elsevier Inc. http://doi.org/10.1016/S0076-6879(10)82003-8
  • Kucukelbir, A., Sigworth, F. J., & Tagare, H. D. (2014). Quantifying the local resolution of cryo-EM density maps. Nature Methods, 11(1), 63–65. http://doi.org/10.1038/nmeth.2727
  • Penczek, P. A. (2020). Reliable cryo-EM resolution estimation with modified Fourier shell correlation. IUCrJ, 7(6), 1–14. http://doi.org/10.1107/s2052252520011574
    • modified FSC with real space masking done at end. Fourier shell still taken of each half map, but then goes back to real space and applies mask before correlation. Have to do an inverse FT for each shell, but can be parallelized.
  • van Heel, M., & Schatz, M. (2020). Information: to Harvest, to Have and to Hold, 1–43.

Ewald Sphere

  • Leong, P. A., Yu, X., Zhou, Z. H., & Jensen, G. J. (2010). Correcting for the Ewald Sphere in High-Resolution Single-Particle Reconstructions. In Methods in Enzymology (1st ed., Vol. 482, pp. 369–380). Elsevier Inc. http://doi.org/10.1016/S0076-6879(10)82015-4

Variational Autoencoders

  • Doersch, C. (2016). Tutorial on Variational Autoencoders, 1–23.
  • Bepler, T., Zhong, E. D., Kelley, K., Brignole, E., & Berger, B. (2019). Explicitly disentangling image content from translation and rotation with spatial-VAE, (NeurIPS 2019).
  • Zhong, E. D., Bepler, T., Davis, J. H., & Berger, B. (2019). Reconstructing continuous distributions of 3D protein structure from cryo-EM images, 1–20.
  • Miolane, N., Poitevin, F., Holmes, S., & Li, Y. T. (2019). Estimation of orientation and camera parameters from cryo-electron microscopy images with variational autoencoders and generative adversarial networks. ArXiv.
    • Ribosome (one simulated dataset in same 2D pose, three emprical datasets from differetne 2D classes) analyzed with a VAE-GAN architecture. In latent space, defocus and rotations are disentagled (in unsupervised way).

Reconstruction

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