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Nonconvex optimization for 3-dimensional point source localization using a rotating point spread function. (English) Zbl 1423.94010

Summary: We consider the high-resolution imaging problem of 3-dimensional (3D) point source image recovery from 2-dimensional data using a method based on point spread function (PSF) engineering. The method involves a new technique, recently proposed by Prasad, based on the use of a rotating PSF with a single lobe to obtain depth from defocus. The amount of rotation of the PSF encodes the depth position of the point source. Applications include high-resolution single molecule localization microscopy as well as the problem addressed in this paper on localization of space debris using a space-based telescope. The localization problem is discretized on a cubical lattice where the coordinates of nonzero entries represent the 3D locations and the values of these entries the fluxes of the point sources. Finding the locations and fluxes of the point sources is a large-scale sparse 3D inverse problem. A new non-convex regularization method with a data-fitting term based on Kullback-Leibler (KL) divergence is proposed for 3D localization for the Poisson noise model. In addition, we propose a new scheme of estimation of the source fluxes from the KL data-fitting term. Numerical experiments illustrate the efficiency and stability of the algorithms that are trained on a random subset of image data before being applied to other images. Our 3D localization algorithms can readily be applied to other kinds of depth-encoding PSFs as well.

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
65K05 Numerical mathematical programming methods
65F22 Ill-posedness and regularization problems in numerical linear algebra
90C26 Nonconvex programming, global optimization

Software:

SPIRAL; PDCO
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] H. Babcock, Y. M. Sigal, and X. Zhuang, {\it A high-density 3D localization algorithm for stochastic optical reconstruction microscopy}, Opt. Nanoscopy, 1 (2012), 6.
[2] A. Barsic, G. Grover, and R. Piestun, {\it Three-dimensional super-resolution and localization of dense clusters of single molecules}, Sci. Rep., 4 (2014), 5388.
[3] U. J. Birk, {\it Super-resolution Microscopy: A Practical Guide}, Wiley, Weinheim, Germany, 2017.
[4] T. Blumensath and M. E. Davies, {\it Iterative thresholding for sparse approximations}, J. Fourier Anal. Appl., 14 (2008), pp. 629-654. · Zbl 1175.94060
[5] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, {\it Distributed optimization and statistical learning via the alternating direction method of multipliers}, Found. Trends Mach. Learn., 3 (2011), pp. 1-122. · Zbl 1229.90122
[6] S. S. Chen, D. L. Donoho, and M. A. Saunders, {\it Atomic decomposition by basis pursuit}, SIAM Rev., 43 (2001), pp. 129-159. · Zbl 0979.94010
[7] O. Daigle and S. Blais-Ouellette, {\it Photon counting with an EMCCD}, in Sensors, Cameras, and Systems for Industrial/Scientific Applications XI, Proc. SPIE 7536, SPIE, Bellingham, WA, 2010, .
[8] C. R. Englert, J. T. Bays, K. D. Marr, C. M. Brown, A. C. Nicholas, and T. T. Finne, {\it Optical orbital debris spotter}, Acta Astronaut., 104 (2014), pp. 99-105.
[9] S. Foucart and H. Rauhut, {\it A Mathematical Introduction to Compressive Sensing}, Vol. 1, Birkhäuser, Basel, 2013. · Zbl 1315.94002
[10] J. W. Goodman, {\it Introduction to Fourier Optics}, 4th ed., Freeman, New York, 2017.
[11] D. Hampf, P. Wagner, and W. Riede, {\it Optical technologies for observation of low Earth orbit objects}, in Proceedings of the 66th International Astronautical Congress, Vol. 3, The International Astronautical Federation, Paris, 2015,
[12] Z. T. Harmany, R. F. Marcia, and R. M. Willett, {\it This is SPIRAL-TAP: Sparse Poisson intensity reconstruction algorithms–theory and practice}, IEEE Trans. Image Process., 21 (2012), pp. 1084-1096. · Zbl 1372.94381
[13] B. Huang, W. Wang, M. Bates, and X. Zhuang, {\it Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy}, Science, 319 (2008), pp. 810-813.
[14] M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, {\it Three-dimensional sub–\textup100 nm resolution fluorescence microscopy of thick samples}, Nature Methods, 5 (2008), pp. 527-529.
[15] R. Kumar and S. Prasad, {\it PSF rotation with changing defocus and applications to 3D imaging for space situational awareness}, in Proceedings of the 2013 AMOS Technical Conference, Maui, HI, 2013.
[16] T. Le, R. Chartrand, and T. J. Asaki, {\it A variational approach to reconstructing images corrupted by Poisson noise}, J. Math. Imaging Vision, 27 (2007), pp. 257-263.
[17] M. D. Lew, S. F. Lee, M. Badieirostami, and W. Moerner, {\it Corkscrew point spread function for far-field three-dimensional nanoscale localization of pointlike objects}, Opt. Lett., 36 (2011), pp. 202-204.
[18] M. D. Lew, A. R. von Diezmann, and W. Moerner, {\it Easy-DHPSF open-source software for three-dimensional localization of single molecules with precision beyond the optical diffraction limit}, Protocol Exchange, 2013 (2013), 026.
[19] J. Min, S. J. Holden, L. Carlini, M. Unser, S. Manley, and J. C. Ye, 3{\it D high-density localization microscopy using hybrid astigmatic/biplane imaging and sparse image reconstruction}, Biomed. Opt. Express, 5 (2014), pp. 3935-3948.
[20] J. Min, C. Vonesch, H. Kirshner, L. Carlini, N. Olivier, S. Holden, S. Manley, J. C. Ye, and M. Unser, {\it FALCON: Fast and unbiased reconstruction of high-density super-resolution microscopy data}, Sci. Rep., 4 (2014), 4577.
[21] B. K. Natarajan, {\it Sparse approximate solutions to linear systems}, SIAM J. Comput., 24 (1995), pp. 227-234. · Zbl 0827.68054
[22] M. Nikolova, M. K. Ng, and C. Tam, {\it Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction}, IEEE Trans. Image Process., 19 (2010), pp. 3073-3088. · Zbl 1371.94277
[23] M. Nikolova, M. K. Ng, and C.-P. Tam, {\it On \(ℓ_1\) data fitting and concave regularization for image recovery}, SIAM J. Sci. Comput., 35 (2013), pp. A397-A430. · Zbl 1267.65028
[24] M. Nikolova, M. K. Ng, S. Zhang, and W.-K. Ching, {\it Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization}, SIAM J. Imaging Sci., 1 (2008), pp. 2-25. · Zbl 1207.94017
[25] J. Nocedal and S. Wright, {\it Numerical Optimization}, Springer, New York, 2006. · Zbl 1104.65059
[26] P. Ochs, A. Dosovitskiy, T. Brox, and T. Pock, {\it On iteratively reweighted algorithms for nonsmooth nonconvex optimization in computer vision}, SIAM J. Imaging Sci., 8 (2015), pp. 331-372. · Zbl 1326.65078
[27] Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, {\it Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition}, in 1993 Conference Record of the Twenty-Seventh Asilomar Conference on Signals, Systems and Computers, 1993, IEEE Computer Society, Los Alamitos, CA, 1993, pp. 40-44.
[28] S. R. P. Pavani and R. Piestun, {\it High-efficiency rotating point spread functions}, Opt. Express, 16 (2008), pp. 3484-3489.
[29] S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. Moerner, {\it Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function}, Proc. Natl. Acad. Sci. USA, 106 (2009), pp. 2995-2999.
[30] P. Prabhat, S. Ram, E. S. Ward, and R. J. Ober, {\it Simultaneous imaging of different focal planes in fluorescence microscopy for the study of cellular dynamics in three dimensions}, IEEE Trans. Nanobiosci., 3 (2004), pp. 237-242.
[31] S. Prasad, {\it Innovations in space-object shape recovery and 3D space debris localization}, in AFOSR-SSA Workshop, Maui, 2017, .
[32] S. Prasad, {\it Rotating point spread function via pupil-phase engineering}, Opt. Lett., 38 (2013), pp. 585-587.
[33] Y. Shechtman, S. J. Sahl, A. S. Backer, and W. Moerner, {\it Optimal point spread function design for 3D imaging}, Phys. Rev. Lett., 113 (2014), 133902.
[34] Y. Shechtman, L. E. Weiss, A. S. Backer, S. J. Sahl, and W. Moerner, {\it Precise three-dimensional scan-free multiple-particle tracking over large axial ranges with tetrapod point spread functions}, Nano Lett., 15 (2015), pp. 4194-4199.
[35] G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, et al., {\it Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure}, Proc. Natl. Acad. Sci. USA, 106 (2009), pp. 3125-3130.
[36] B. Shuang, W. Wang, H. Shen, L. J. Tauzin, C. Flatebo, J. Chen, N. A. Moringo, L. D. Bishop, K. F. Kelly, and C. F. Landes, {\it Generalized recovery algorithm for 3D super-resolution microscopy using rotating point spread functions}, Sci. Rep., 6 (2016), 30826.
[37] E. Soubies, L. Blanc-Féraud, and G. Aubert, {\it A continuous exact \(ℓ_0\) penalty (CEL0) for least squares regularized problem}, SIAM J. Imaging Sci., 8 (2015), pp. 1607-1639. · Zbl 1325.65086
[38] T. Teuber, G. Steidl, and R. H. Chan, {\it Minimization and parameter estimation for seminorm regularization models with I-divergence constraints}, Inverse Problems, 29 (2013), 035007. · Zbl 1267.65046
[39] A. von Diezmann, Y. Shechtman, and W. Moerner, {\it Three-dimensional localization of single molecules for super-resolution imaging and single-particle tracking}, Chem. Rev., 117 (2017), pp. 7244-7275.
[40] P. Wagner, D. Hampf, F. Sproll, T. Hasenohr, L. Humbert, J. Rodmann, and W. Riede, {\it Detection and laser ranging of orbital objects using optical methods}, in Remote Sensing System Engineering VI, Proc. SPIE 9977, SPIE, Bellingham, WA, 2016, 99770D.
[41] J. Xiao, M. K. Ng, and Y. Yang, {\it On the convergence of nonconvex minimization methods for image recovery}, IEEE Trans. Image Process., 24 (2015), pp. 1587-1598. · Zbl 1408.94726
[42] L. Zhu, W. Zhang, D. Elnatan, and B. Huang, {\it Faster STORM using compressed sensing}, Nature Methods, 9 (2012), 721.
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