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Journal of Mathematical Imaging and Vision

, Volume 61, Issue 7, pp 1037–1050 | Cite as

Optimal Multivariate Gaussian Fitting with Applications to PSF Modeling in Two-Photon Microscopy Imaging

  • Emilie ChouzenouxEmail author
  • Tim Tsz-Kit Lau
  • Claire Lefort
  • Jean-Christophe Pesquet
Article
  • 242 Downloads

Abstract

Fitting Gaussian functions to empirical data is a crucial task in a variety of scientific applications, especially in image processing. However, most of the existing approaches for performing such fitting are restricted to two dimensions and they cannot be easily extended to higher dimensions. Moreover, they are usually based on alternating minimization schemes which benefit from few theoretical guarantees in the underlying nonconvex setting. In this paper, we provide a novel variational formulation of the multivariate Gaussian fitting problem, which is applicable to any dimension and accounts for possible nonzero background and noise in the input data. The block multiconvexity of our objective function leads us to propose a proximal alternating method to minimize it in order to estimate the Gaussian shape parameters. The resulting FIGARO algorithm is shown to converge to a critical point under mild assumptions. The algorithm shows a good robustness when tested on synthetic datasets. To demonstrate the versatility of FIGARO, we also illustrate its excellent performance in the fitting of the point spread functions of experimental raw data from a two-photon fluorescence microscope.

Keywords

Gaussian fitting Kullback–Leibler divergence Alternating minimization Proximal methods PSF identification Two-photon fluorescence microscopy 

Notes

Acknowledgements

This work was supported by the CNRS under grant MI-AAP Interne2018-SupRéMA.

References

  1. 1.
    Anthony, S.M., Granick, S.: Image analysis with rapid and accurate two-dimensional Gaussian fitting. Langmuir 25(14), 8152–8160 (2009).  https://doi.org/10.1021/la900393v CrossRefGoogle Scholar
  2. 2.
    Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kurdyka-Łojasiewicz inequality. Math. Oper. Res. 35(2), 438–457 (2010).  https://doi.org/10.1287/moor.1100.0449 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Attouch, H., Bolte, J., Svaiter, B.F.: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. Math. Program. 137(1), 91–129 (2013).  https://doi.org/10.1007/s10107-011-0484-9 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Basseville, M., Cardoso, J.F.: On entropies, divergences, and mean values. In: Proceedings of 1995 IEEE international symposium on information theory, pp. 330– (1995).  https://doi.org/10.1109/ISIT.1995.550317
  5. 5.
    Bauschke, H., Combettes, P.L., Noll, D.: Joint minimization with alternating Bregman proximity operators. Pacific J. Optim. 2(3), 401–424 (2006)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, Berlin (2017)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)zbMATHGoogle Scholar
  8. 8.
    Bolte, J., Combettes, P.L., Pesquet, J.C.: Alternating proximal algorithm for blind image recovery. In: Proceedings IEEE international conference image process. (ICIP 2010), pp. 1673–1676. Hong-Kong, China (2010)Google Scholar
  9. 9.
    Bolte, J., Daniilidis, A., Lewis, A., Shiota, M.: Clarke subgradients of stratifiable functions. SIAM J. Optim. 18(2), 556–572 (2007).  https://doi.org/10.1137/060670080 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bolte, J., Sabach, S., Teboulle, M.: Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math. Program. 146(1), 459–494 (2014).  https://doi.org/10.1007/s10107-013-0701-9 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Briceno-Arias, L.M., Chierchia, G., Chouzenoux, E., Pesquet, J.C.: A random block-coordinate Douglas-Rachford splitting method with low computational complexity for binary logistic regression. Comput. Optim. Appl. 72(3), 707–726 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Burger, M., Sawatzky, A., Steidl, G.: First Order Algorithms in Variational Image Processing, pp. 345–407. Springer, Cham (2016)zbMATHGoogle Scholar
  13. 13.
    Caruana, R., Searle, R., Heller, T., Shupack, S.: Fast algorithm for the resolution of spectra. Anal. Chem. 58(6), 1162–1167 (1986)CrossRefGoogle Scholar
  14. 14.
    Chan, R.H., Chan, T.F., Shen, L., Shen, Z.: Wavelet deblurring algorithms for spatially varying blur from high-resolution image reconstruction. Linear Algebra Appl. 366, 139–155 (2003). Special issue on Structured Matrices: Analysis, Algorithms and ApplicationsMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chen, Y.C., Furenlid, L.R., Wilson, D.W., Barrett, H.H.: Calibration of scintillation cameras and pinhole SPECT imaging systems, pp. 195–202. Springer, Berlin (2005). 12Google Scholar
  16. 16.
    Cherni, A., Chouzenoux, E., Delsuc, M.A.: PALMA, an improved algorithm for DOSY signal processing. Analyst 142(5), 772–779 (2017)CrossRefGoogle Scholar
  17. 17.
    Chouzenoux, E., Lamassé, L., Chaux, C., Jaouen, A., Vanzetta, I., Debarbieux, F.: Approche variationnelle pour la déconvolution rapide de données 3d en microscopie biphotonique. In: Actes du 25e colloque GRETSI (2015)Google Scholar
  18. 18.
    Chouzenoux, E., Pesquet, J.C., Repetti, A.: A block coordinate variable metric forward-backward algorithm. J. Glob. Optim. 66(3), 457–485 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Bauschke, H.H., Burachik, R.S., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 185–212. Springer, New York (2011)CrossRefGoogle Scholar
  20. 20.
    Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the Lambert W function. Adv. Comput. Math. 5(1), 329–359 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    van den Dries, L., Miller, C.: Geometric categories and o-minimal structures. Duke Math. J. 84(2), 497–540 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Friesen, W.I., Michaelian, K.H.: Deconvolution and curve-fitting in the analysis of complex spectra: the CH stretching region in infrared spectra of coal. Appl. Spectrosc. 45(1), 50–56 (1991)CrossRefGoogle Scholar
  23. 23.
    Galatsanos, N.P., Katsaggelos, A.K.: 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)CrossRefGoogle Scholar
  24. 24.
    Guo, H.: A simple algorithm for fitting a Gaussian function [DSP tips and tricks]. IEEE Signal Proc. Mag. 28(5), 134–137 (2011).  https://doi.org/10.1109/MSP.2011.941846 CrossRefGoogle Scholar
  25. 25.
    Hagen, N., Dereniak, E.L.: Gaussian profile estimation in two dimensions. Appl. Opt. 47(36), 6842–6851 (2008).  https://doi.org/10.1364/AO.47.006842 CrossRefGoogle Scholar
  26. 26.
    Hagen, N., Kupinski, M., Dereniak, E.L.: Gaussian profile estimation in one dimension. Appl. Opt. 46(22), 5374–5383 (2007).  https://doi.org/10.1364/AO.46.005374 CrossRefGoogle Scholar
  27. 27.
    Helmchen, F., Denk, W.: Deep tissue two-photon microscopy. Nat. Methods 2, 12 (2005)CrossRefGoogle Scholar
  28. 28.
    Henriques, R., Lelek, M., Fornasiero, E.F., Valtorta, F., Zimmer, C., Mhlanga, M.M.: QuickPALM: 3D real-time photoactivation nanoscopy image processing in ImageJ. Nat. Methods 7(5), 339–340 (2010)CrossRefGoogle Scholar
  29. 29.
    Kazovsky, L.G.: Beam position estimation by means of detector arrays. Opt. Quantum Electron. 13, 201–208 (1981)CrossRefGoogle Scholar
  30. 30.
    Kincaid, D., Cheney, E.: Numerical Analysis: Mathematics of Scientific Computing, 3th edn.Pure and applied undergraduate texts. American Mathematical Society, Providence (2002)Google Scholar
  31. 31.
    Kirshner, H., Ahuet, F., Sage, D., Unser, M.: 3-D PSF fitting for fluorescence microscopy: implementation and localization application. J. Microsc. 249(1), 13–25 (2013).  https://doi.org/10.1111/j.1365-2818.2012.03675.x CrossRefGoogle Scholar
  32. 32.
    Landman, D.A., Roussel-Dupré, R., Tanigawa, G.: On the statistical uncertainties associated with line profile fitting. Astrophys. J. 261, 732–735 (1982)CrossRefGoogle Scholar
  33. 33.
    Lapin, M., Hein, M., Schiele, B.: Analysis and optimization of loss functions for multiclass, top-k, and multilabel classification. IEEE Trans. Pattern Anal. Mach. Intell. 40(7), 1533–1554 (2018)CrossRefGoogle Scholar
  34. 34.
    Marim, M., Zhang, B., Olivo-Marin, J.C., Zimmer, C.: Improving single particle localization with an empirically calibrated Gaussian kernel. In: 5th IEEE international symposium biomedical imaging: From Nano to Macro (ISBI 2008), pp. 1003–1006. Paris, France (2008).  https://doi.org/10.1109/ISBI.2008.4541168
  35. 35.
    Rockafellar, R., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998).  https://doi.org/10.1007/978-3-642-02431-3 CrossRefzbMATHGoogle Scholar
  36. 36.
    Roonizi, E.K.: A new algorithm for fitting a Gaussian function riding on the polynomial background. IEEE Signal Process. Lett. 20(11), 1062–1065 (2013).  https://doi.org/10.1109/LSP.2013.2280577 CrossRefGoogle Scholar
  37. 37.
    Sarder, P., Nehorai, A.: Estimating locations of quantum-dot-encoded microparticles from ultra-high density 3-D microarrays. IEEE Trans. NanoBiosci. 7(4), 284–297 (2008)CrossRefGoogle Scholar
  38. 38.
    Tal, E., Oron, D., Silverberg, Y.: Improved depth resolution in video-rate line-scanning multiphoton microscopy using temporal focusing. Opt. Lett. 30, 1686–1688 (2005)CrossRefGoogle Scholar
  39. 39.
    Thompson, R.E., Larson, D.R., Webb, W.W.: Precise nanometer localization analysis for individual fluorescent probes. Biophys. J. 82, 2775–2783 (2002)CrossRefGoogle Scholar
  40. 40.
    Tseng, P.: Convergence of a block coordinate descent method for nondifferentiable minimization. J. Optim. Theory Appl. 109(3), 475–494 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Wolter, S., Löschberger, A., Holm, T., Aufmkolk, S., Dabauvalle, M.C., van de Linde, S., Sauer, M.: rapidSTORM: accurate, fast open-source software for localization microscopy. Nat. Methods 9, 1040–1041 (2012)CrossRefGoogle Scholar
  42. 42.
    Xu, Y., Yin, W.: A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion. SIAM J. Imag. Sci. 6(3), 1758–1789 (2013).  https://doi.org/10.1137/120887795 MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Zhang, B., Zerubia, J., Olivo-Marin, J.C.: Gaussian approximations of fluorescence microscope point-spread function models. Appl. Opt. 46(10), 1819–1829 (2007).  https://doi.org/10.1364/AO.46.001819 CrossRefGoogle Scholar
  44. 44.
    Zhu, X., Zhang, D.: Efficient parallel Levenberg-Marquardt model fitting towards real-time automated parametric imaging microscopy. PLOS ONE 8(10), 1–9 (2013).  https://doi.org/10.1371/journal.pone.0076665 Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Center for Visual Computing, CentraleSupélec, INRIA SaclayUniversité Paris-SaclayGif-sur-YvetteFrance
  2. 2.Laboratoire d’Informatique Gaspard Monge, UMR CNRS 8049Université Paris-Est Marne-la-ValléeChamps-sur-MarneFrance
  3. 3.Department of StatisticsNorthwestern UniversityEvanstonUSA
  4. 4.XLIM Research Institute, UMR CNRS 7252Université de LimogesLimogesFrance

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