Constrained Total Variation Based Three-Dimension Single Particle Reconstruction in Cryogenic Electron Microscopy

Abstract

The single particle reconstruction (SPR) in cryogenic electron microscopy is considered in this paper. This is an emerging technique for determining the three-dimensional (3D) structure of biological specimens from a limited number of the micrographs. Because the micrographs are modulated by contrast transfer functions and corrupted by heavy noise, the number of micrographs might be limited, in general it is a serious ill-posed problem to reconstruct the original particle. In this paper, we propose a constrained total variation (TV) model for single particle reconstruction. The TV norm is represented by the dual formulation that changes the SPR problem into a minimax one. The primal-dual method is applied to find the saddle point of the minimax problem, and the convergence condition is given. Numerical results show that the proposed model is very effective in reconstructing the particle.

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Notes

  1. 1.

    http://www.ebi.ac.uk/pdbe/entry/emdb/EMD-1252/.

  2. 2.

    \(\mathrm {RD} = \frac{\Vert \mathbf{v}^{(k+1)}-\mathbf{v}^{(k)}\Vert _2}{\max (\Vert \mathbf{v}^{(k)}\Vert _2,10^{-4})}\).

References

  1. 1.

    Abbas, S.A., Sun, Q., Foroosh, H.: An exact and fast computation of discrete fourier transform for polar and spherical grid. IEEE Trans. Signal Process. 65(8), 2033–2048 (2017)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Andersen, A.: Algebraic reconstruction in CT from limited views. IEEE Trans. Med. Imaging 8(1), 50–55 (1989)

    Article  Google Scholar 

  3. 3.

    Babacan, S., Molina, R., Katsaggelos, A.: Parameter estimation in TV image restoration using variational distribution approximation. IEEE Trans. Image Process. 17(3), 326–339 (2008)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Babacan, S., Molina, R., Katsaggelos, A.: Variational Bayesian blind deconvolution using a total variation prior. IEEE Trans. Image Process. 18, 12–26 (2009)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bertsekas, D.: Convex optimization theory. Athena Scientific, Belmont (2009)

    Google Scholar 

  6. 6.

    Bhamre, T., Zhang, T., Singer, A.: Denoising and covariance estimation of single particle cryo-EM images. J. Struct. Biol. 195(1), 72–81 (2016)

    Article  Google Scholar 

  7. 7.

    Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Czarnocki-Cieciura, M., Nowotny, M.: Introduction to high-resolution cryo-electron microscopy. Postepy Biochem. 62(4), 383–394 (2016)

    Google Scholar 

  10. 10.

    Doerr, A.: Single-particle electron cryomicroscopy. Nat. Methods 1, 30–30 (2014)

    Article  Google Scholar 

  11. 11.

    Fessler, J., Lee, S., Olafsson, V., Shi, H., Noll, D.: Toeplitz-based iterative image reconstruction for mri with correction for magnetic field inhomogeneity. IEEE Trans. Signal Process. 53(9), 3393–3402 (2005)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Fessler, J.A., Sutton, B.P.: Nonuniform fast Fourier transforms using min–max interpolation. IEEE Trans. Signal Process. 51(2), 560–574 (2003)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Frank, J.: Three-dimensional electron microscopy of macromolecular assemblies. Oxford University Press, Oxford (2006)

    Google Scholar 

  14. 14.

    Galatsanos, N., Katsaggelos, A.: Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation. IEEE Trans. Image Process. 1(3), 322–336 (1992)

    Article  Google Scholar 

  15. 15.

    Gallier, J.: Fundamentals of Linear Algebra and Optimization (2018). https://www.seas.upenn.edu/~cis515/linalg.pdf

  16. 16.

    Gilbert, P.: Iterative methods for the three-dimensional reconstruction of an object from projections. J. Theor. Biol. 36(1), 105–117 (1972)

    Article  Google Scholar 

  17. 17.

    Golub, G.H., Heath, M., Wahba, G.: Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21(2), 215–223 (1979)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Gordon, R., Bender, R., Herman, G.T.: Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography. J. Theor. Biol. 29(3), 477–481 (1970)

    Article  Google Scholar 

  19. 19.

    Goris, B., Roelandts, T., Batenburg, K.J., Heidari Mezerji, H., Bals, S.: Advanced reconstruction algorithms for electron tomography: from comparison to combination. Ultramicroscopy 127, 40–47 (2013)

    Article  Google Scholar 

  20. 20.

    Greengard, L., Lee, J.-Y.: Accelerating the nonuniform fast Fourier transform. SIAM Rev. 46(3), 443–454 (2004)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Hansen, P.C.: Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 34(4), 561–580 (1992)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Heel, M.V., Frank, J.: Use of multivariates statistics in analysing the images of biological macromolecules. Ultramicroscopy 6(1), 187–194 (1981)

    Article  Google Scholar 

  23. 23.

    Kucukelbir, A., Sigworth, F., Tagare, H.: A Bayesian adaptive basis algorithm for single particle reconstruction. J. Struct. Biol. 179(1), 56–67 (2012)

    Article  Google Scholar 

  24. 24.

    Li, M., Xu, G., Sorzano, C.O., Sun, F., Bajaj, C.L.: Single-particle reconstruction using L2-gradient flow. J. Struct. Biol. 176(3), 259–267 (2011)

    Article  Google Scholar 

  25. 25.

    Mallat, S.: A Wavelet Tour of Signal Processing, 2nd edn. Cademic Press, San Diego (1999)

    Google Scholar 

  26. 26.

    Morozov, V.: Methods for Solving Incorrectly Posed Problems. Springer, New York (1984)

    Google Scholar 

  27. 27.

    Oliveira, J.P., Bioucas-Dias, J.M., Figueiredo, M.A.T.: Adaptive total variation image deblurring: a majorization–minimization approach. Signal Process. 89(9), 1683–1693 (2009)

    Article  Google Scholar 

  28. 28.

    Penczek, P.A.: Image restoration in cryo-electron microscopy. In: Jensen, G.J. (ed.) Cryo-EM, Part B: 3-D Reconstruction, volume 482 of Methods in Enzymology, pp. 35–72. Academic Press, London (2010)

    Google Scholar 

  29. 29.

    Penczek, P.A., Grassucci, R.A., Frank, J.: The ribosome at improved resolution: new techniques for merging and orientation refinement in 3D cryo-electron microscopy of biological particles. Ultramicroscopy 53, 251–270 (1994)

    Article  Google Scholar 

  30. 30.

    Pryor, A., Yang, Y., Rana, A., Gallagher-Jones, M., Zhou, J., Lo, Y.H., Melinte, G., Chiu, W., Rodriguez, J.A., Miao, J.: GENFIRE: a generalized Fourier iterative reconstruction algorithm for high-resolution 3D imaging. Sci. Rep. 7, 10409 (2017)

    Article  Google Scholar 

  31. 31.

    Radermacher, M., Wagenknecht, T., Verschoor, A., Frank, J.: Three-dimensional reconstruction from a single-exposure, random conical tilt series applied to the 50S ribosomal subunit of Escherichia coli. J. Microsc. 146(2), 113–136 (1987)

    Article  Google Scholar 

  32. 32.

    Radon, J.: On the determination of functions from their integral values along certain manifolds. IEEE Trans. Med. Imaging 5(4), 170–176 (1986)

    Article  Google Scholar 

  33. 33.

    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Scheres, S.H.: A Bayesian view on cryo-EM structure determination. J. Mol. Biol. 415(2–3), 406–418 (2012)

    Article  Google Scholar 

  35. 35.

    Scheres, S.H.: RELION: implementation of a Bayesian approach to cryo-EM structure determination. J. Struct. Biol. 180(3), 519–530 (2012)

    Article  Google Scholar 

  36. 36.

    Singer, A., Zhao, Z., Shkolnisky, Y., Hadani, R.: Viewing angle classification of cryo-electron microscopy images using eigenvectors. SIAM J. Imaging Sci. 4(2), 723–759 (2011)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Wade, R.H.: A brief look at imaging and contrast transfer. Ultramicroscopy 46(1–4), 145–156 (1992)

    Article  Google Scholar 

  38. 38.

    Wang, L., Shkolnisky, Y., Singer, A.: A Fourier-Based Approach for Iterative 3D Reconstruction from Cryo-EM Images (2013)

  39. 39.

    Wang, L., Singer, A., Wen, Z.: Orientation determination of cryo-EM images using least unsquared deviations. SIAM J. Imaging Sci. 6(4), 2450–2483 (2013)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Yang, Y., Chen, C.C., Scott, M.C., Ophus, C., Xu, R., Pryor, A., Wu, L., Sun, F., Theis, W., Zhou, J., Eisenbach, M., Kent, P.R., Sabirianov, R.F., Zeng, H., Ercius, P., Miao, J.: Deciphering chemical order/disorder and material properties at the single-atom level. Nature 542, 75–79 (2017)

    Article  Google Scholar 

  41. 41.

    Zanetti, G., Riches, J.D., Fuller, S.D., Briggs, J.A.G.: Contrast transfer function correction applied to cryo-electron tomography and sub-tomogram averaging. J. Struct. Biol. 168(2), 305–312 (2009)

    Article  Google Scholar 

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Correspondence to You-Wei Wen.

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This work is supported by NSFC Grant Nos. 11871210 and 11971215, the Construct Program of the Key Discipline in Hunan Province, the SRF of Hunan Provincial Education Department (No. 17A128), the Hunan Province Graduate Research and Innovation Project (No. CX20190336). The work of Tieyong Zeng was supported by the NSFC under Grant 11671002, in part by the CUHK Start-Up, and in part by the CUHK DAG under Grant Nos. 4053342, 4053405, RGC 14300219, RGC 14302920, and NSFC/RGC N_CUHK 415/19.

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Pan, H., Wen, YW. & Zeng, T. Constrained Total Variation Based Three-Dimension Single Particle Reconstruction in Cryogenic Electron Microscopy. J Sci Comput 85, 37 (2020). https://doi.org/10.1007/s10915-020-01344-4

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Keywords

  • Cryogenic electron microscopy (cryo-EM)
  • Single particle reconstruction
  • Micrographs
  • Total variation
  • Primal-dual