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The super-resolution optical fluctuation imaging (SOFI) technique enhances image spatial resolution by evaluating the independent stochastic intensity fluctuations of emitters. In principle, it eliminates any noise uncorrelated temporally, and provides unlimited spatial resolution since the calculation of the nth-order cumulant followed by a deconvolution results in an image with n-fold resolution improvement in three dimensions. But in practice, due to limited data length, the statistical uncertainty of cumulants will affect the continuity and homogeneity of SOFI image, which results in the fact that the high order SOFI (typically over 3rd order) cannot improve spatial resolution significantly. Since the variance characterizes the statistical uncertainty of cumulant, we deduce its theoretical expression based on a single dataset. In traditional SOFI techniques, due to lack of statistical analysis of cumulant, there is no noise constraint condition of cumulant in the Lucy-Richardson deconvolution to prevent the algorithm from causing noise amplification. In this paper, based on the cumulant variance formula, we calculate the cumulant standard deviation in each pixel of SOFI image and introduce the results into the Lucy-Richardson algorithm as a DAMPAR to suppress the noise generation in such pixels. The simulation and experimental results show that under the same data length, the deconvolution optimization based on cumulant standard deviation significantly improves the uniformity and continuity of SOFI image. On the other hand, under the premise of identical image quality, this optimization technique can also greatly shorten the image frames to less than half the original, thus promoting the development of super-resolution imaging of living cells.
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Keywords:
- super-resolution microscopy /
- cumulant /
- deconvolution
[1] Dertinger T, Colyer R, Iyer G, Weiss S, Enderlein J 2009 Proc. Nat. Acad. Sci. 106 22287
[2] Geissbuehler S, Dellagiacoma C, Lasser T 2011 Biomed. Opt. Express 2 408
[3] Dertinger T, Colyer R, Vogel R, Enderlein J, Weiss S 2010 Opt. Express 18 18875
[4] Geissbuehler S, Bocchio N L, Dellagiacoma C, Berclaz C, Leutenegger M, Lasser T 2012 Opt. Nanoscopy 1 1
[5] Stein S C, Huss A, Höhnel D, Gregor I, Enderlein J 2015 Opt. Express 23 16154
[6] Betzig E, Patterson G H, Sougrat R, Lindwasser O W, Olenych S, Bonifacino J S, Davidson M W, Lippincott-Schwartz J, Hess H F 2006 Science 313 1642
[7] Rust M J, Bates M, Zhuang X 2006 Nat. Methods 3 793
[8] Chen D N, Liu L, Yu B, Niu H B 2010 Acta Phys. Sin. 59 6948 (in Chinese) [陈丹妮, 刘磊, 于斌, 牛憨笨2010 59 6948]
[9] Li H, Yu B, Chen D N, Niu H B 2013 Acta Phys. Sin. 62 124201 (in Chinese) [李恒, 于斌, 陈丹妮, 牛憨笨2013 62 124201]
[10] Wang X, Chen D, Yu B, Niu H 2015 Appl. Opt. 54 6919
[11] Koppel D E 1974 Phys. Rev. A 10 1938
[12] Qian H 1990 Biophys. Chem. 38 49
[13] Zeng Z, Chen X, Wang H, Huang N, Shan C, Zhang H, Teng J, Xi P 2015 Sci. Rep. 5 1
[14] Wang X, Chen D, Yu B, Niu H 2016 Appl. Opt. 55 7911
[15] Kendall M G, Stuart A 1977 The Advanced Theory of Statistics (Vol. 1) (New York: MacMillan Publishing) pp57-96
[16] Rose C, Smith M D 2002 Mathematical Statistics with Mathematica (New York: Springer) pp31-80
[17] Vandenberg W, Duwé S, Leutenegger M, Moeyaert B, Krajnik B, Lasser T, Dedecker P 2016 Biomed. Opt. Express 7 467
[18] Mller J D 2004 Biophys. J. 86 3981
[19] Biggs D S C, Andrews M 1997 Appl. Opt. 36 1766
[20] Arganda-Carreras I, Fernández-González R, Muöoz-Barrutia A, Ortiz-De-Solorzano C 2010 Microsc. Res. Tech. 73 1019
[21] Lee T C, Kashyap R L, Chu C N 1994 Graph. Model. Im. Proc. 56 462
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[1] Dertinger T, Colyer R, Iyer G, Weiss S, Enderlein J 2009 Proc. Nat. Acad. Sci. 106 22287
[2] Geissbuehler S, Dellagiacoma C, Lasser T 2011 Biomed. Opt. Express 2 408
[3] Dertinger T, Colyer R, Vogel R, Enderlein J, Weiss S 2010 Opt. Express 18 18875
[4] Geissbuehler S, Bocchio N L, Dellagiacoma C, Berclaz C, Leutenegger M, Lasser T 2012 Opt. Nanoscopy 1 1
[5] Stein S C, Huss A, Höhnel D, Gregor I, Enderlein J 2015 Opt. Express 23 16154
[6] Betzig E, Patterson G H, Sougrat R, Lindwasser O W, Olenych S, Bonifacino J S, Davidson M W, Lippincott-Schwartz J, Hess H F 2006 Science 313 1642
[7] Rust M J, Bates M, Zhuang X 2006 Nat. Methods 3 793
[8] Chen D N, Liu L, Yu B, Niu H B 2010 Acta Phys. Sin. 59 6948 (in Chinese) [陈丹妮, 刘磊, 于斌, 牛憨笨2010 59 6948]
[9] Li H, Yu B, Chen D N, Niu H B 2013 Acta Phys. Sin. 62 124201 (in Chinese) [李恒, 于斌, 陈丹妮, 牛憨笨2013 62 124201]
[10] Wang X, Chen D, Yu B, Niu H 2015 Appl. Opt. 54 6919
[11] Koppel D E 1974 Phys. Rev. A 10 1938
[12] Qian H 1990 Biophys. Chem. 38 49
[13] Zeng Z, Chen X, Wang H, Huang N, Shan C, Zhang H, Teng J, Xi P 2015 Sci. Rep. 5 1
[14] Wang X, Chen D, Yu B, Niu H 2016 Appl. Opt. 55 7911
[15] Kendall M G, Stuart A 1977 The Advanced Theory of Statistics (Vol. 1) (New York: MacMillan Publishing) pp57-96
[16] Rose C, Smith M D 2002 Mathematical Statistics with Mathematica (New York: Springer) pp31-80
[17] Vandenberg W, Duwé S, Leutenegger M, Moeyaert B, Krajnik B, Lasser T, Dedecker P 2016 Biomed. Opt. Express 7 467
[18] Mller J D 2004 Biophys. J. 86 3981
[19] Biggs D S C, Andrews M 1997 Appl. Opt. 36 1766
[20] Arganda-Carreras I, Fernández-González R, Muöoz-Barrutia A, Ortiz-De-Solorzano C 2010 Microsc. Res. Tech. 73 1019
[21] Lee T C, Kashyap R L, Chu C N 1994 Graph. Model. Im. Proc. 56 462
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