Abstract

Based on truncated inverse filtering, a theory for deconvolution of complex fields is studied. The validity of the theory is verified by comparing with experimental data from digital holographic microscopy (DHM) using a high-NA system (NA=0.95). Comparison with standard intensity deconvolution reveals that only complex deconvolution deals correctly with coherent cross-talk. With improved image resolution, complex deconvolution is demonstrated to exceed the Rayleigh limit. Gain in resolution arises by accessing the objects complex field - containing the information encoded in the phase - and deconvolving it with the reconstructed complex transfer function (CTF). Synthetic (based on Debye theory modeled with experimental parameters of MO) and experimental amplitude point spread functions (APSF) are used for the CTF reconstruction and compared. Thus, the optical system used for microscopy is characterized quantitatively by its APSF. The role of noise is discussed in the context of complex field deconvolution. As further results, we demonstrate that complex deconvolution does not require any additional optics in the DHM setup while extending the limit of resolution with coherent illumination by a factor of at least 1.64.

© 2010 Optical Society of America

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    [CrossRef]
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2010 (1)

2009 (3)

2008 (1)

2007 (3)

G. Indebetouw, Y. Tada, J. Rosen, and G. Brooker, "Scanning holographic microscopy with resolution exceeding the Rayleigh limit of the objective by superposition of off-axis holograms," Appl. Opt. 46, 993-1000 (2007).
[CrossRef] [PubMed]

A. Marian, F. Charri`ere, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, "On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography," J. Microsc. 225, 156-169 (2007).
[CrossRef] [PubMed]

C. J. Sheppard, "Fundamentals of superresolution," Micron 38, 165-169 (2007).
[CrossRef]

2006 (5)

2005 (1)

B. Colicchio, O. Haeberl, C. Xu, A. Dieterlen, and G. Jung, "Improvement of the lls and map deconvolution algorithms by automatic determination of optimal regularization parameters and pre-filtering of original data," Opt. Commun. 244, 37-49 (2005).
[CrossRef]

2002 (1)

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2002).
[CrossRef] [PubMed]

2001 (1)

W. Wallace, L. H. Schaefer, and J. R. Swedlow, "A workingperson’s guide to deconvolution in light microscopy," Biotechniques 31, 1076 (2001).
[PubMed]

2000 (1)

M. G. L. Gustafsson, "Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy," J. Microsc. 198, 82-87 (2000).
[CrossRef] [PubMed]

1999 (3)

1997 (4)

D. Mendlovic, A. W. Lohmann, N. Konforti, I. Kiryuschev, and Z. Zalevsky, "One-dimensional superresolution optical system for temporally restricted objects," Appl. Opt. 36, 2353-2359 (1997).
[CrossRef] [PubMed]

M. Totzeck, and H. J. Tiziani, "Phase-singularities in 2d diffraction fields and interference microscopy," Opt. Commun. 138, 365-382 (1997).
[CrossRef]

C. J. Sheppard, and K. Larkin, "Vectorial pupil functions and vectorial transfer functions," Optik (Stuttg.) 107, 79-87 (1997).

V. Torczon, "On the convergence of pattern search algorithms," SIAM J. Optim. 7, 125 (1997).
[CrossRef]

1993 (1)

C. J. R. Sheppard, and M. Gu, "Imaging by a high aperture optical-system," J. Mod. Opt. 40, 1631-1651 (1993).
[CrossRef]

1987 (1)

1972 (1)

R. Gerchberg, and W. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttg.) 35, 227-246 (1972).

Aert, S. V.

Aguet, F.

Angell, D.

Brooker, G.

Charri`ere, F.

A. Marian, F. Charri`ere, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, "On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography," J. Microsc. 225, 156-169 (2007).
[CrossRef] [PubMed]

Chen, J.

Colicchio, B.

B. Colicchio, O. Haeberl, C. Xu, A. Dieterlen, and G. Jung, "Improvement of the lls and map deconvolution algorithms by automatic determination of optimal regularization parameters and pre-filtering of original data," Opt. Commun. 244, 37-49 (2005).
[CrossRef]

Colomb, T.

A. Marian, F. Charri`ere, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, "On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography," J. Microsc. 225, 156-169 (2007).
[CrossRef] [PubMed]

Conchello, J. A.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, "Three-dimensional imaging by deconvolution microscopy," Methods 19, 373-385 (1999).
[CrossRef] [PubMed]

Cooper, J.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, "Three-dimensional imaging by deconvolution microscopy," Methods 19, 373-385 (1999).
[CrossRef] [PubMed]

Cotte, Y.

Cuche, E.

Cui, X. Q.

Debailleul, M.

den Dekker, A. J.

Depeursinge, C.

Dieterlen, A.

B. Colicchio, O. Haeberl, C. Xu, A. Dieterlen, and G. Jung, "Improvement of the lls and map deconvolution algorithms by automatic determination of optimal regularization parameters and pre-filtering of original data," Opt. Commun. 244, 37-49 (2005).
[CrossRef]

Dyck, D. V.

Ferreira, C.

Garcia, J.

García, J.

Geissbühler, S.

Georges, V.

Gerchberg, R.

R. Gerchberg, and W. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttg.) 35, 227-246 (1972).

Gu, M.

C. J. R. Sheppard, and M. Gu, "Imaging by a high aperture optical-system," J. Mod. Opt. 40, 1631-1651 (1993).
[CrossRef]

Guo, H.

Gustafsson, M. G. L.

M. G. L. Gustafsson, "Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy," J. Microsc. 198, 82-87 (2000).
[CrossRef] [PubMed]

Haeberl, O.

B. Colicchio, O. Haeberl, C. Xu, A. Dieterlen, and G. Jung, "Improvement of the lls and map deconvolution algorithms by automatic determination of optimal regularization parameters and pre-filtering of original data," Opt. Commun. 244, 37-49 (2005).
[CrossRef]

Haeberle, O.

Heng, X.

Indebetouw, G.

Jung, G.

B. Colicchio, O. Haeberl, C. Xu, A. Dieterlen, and G. Jung, "Improvement of the lls and map deconvolution algorithms by automatic determination of optimal regularization parameters and pre-filtering of original data," Opt. Commun. 244, 37-49 (2005).
[CrossRef]

Karpova, T.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, "Three-dimensional imaging by deconvolution microscopy," Methods 19, 373-385 (1999).
[CrossRef] [PubMed]

Kiryuschev, I.

Knapp, D. W.

Konforti, N.

Kuei, C. P.

Kühn, J.

N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, "Suppression of the zero-order term in off-axis digital holography through nonlinear filtering," Appl. Opt. 48, H186-H195 (2009).
[CrossRef] [PubMed]

A. Marian, F. Charri`ere, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, "On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography," J. Microsc. 225, 156-169 (2007).
[CrossRef] [PubMed]

Larkin, K.

C. J. Sheppard, and K. Larkin, "Vectorial pupil functions and vectorial transfer functions," Optik (Stuttg.) 107, 79-87 (1997).

Lasser, T.

Lauer, V.

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2002).
[CrossRef] [PubMed]

Leitgeb, R. A.

Leith, E. N.

Leutenegger, M.

Liang, Z.

Lohmann, A. W.

Marian, A.

A. Marian, F. Charri`ere, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, "On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography," J. Microsc. 225, 156-169 (2007).
[CrossRef] [PubMed]

Märki, I.

Marquet, P.

A. Marian, F. Charri`ere, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, "On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography," J. Microsc. 225, 156-169 (2007).
[CrossRef] [PubMed]

E. Cuche, P. Marquet, and C. Depeursinge, "Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms," Appl. Opt. 38, 6994-7001 (1999).
[CrossRef]

Martinez, P. G.

McDowell, E. J.

McNally, J. G.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, "Three-dimensional imaging by deconvolution microscopy," Methods 19, 373-385 (1999).
[CrossRef] [PubMed]

Mendlovic, D.

Mico, V.

Montfort, F.

A. Marian, F. Charri`ere, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, "On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography," J. Microsc. 225, 156-169 (2007).
[CrossRef] [PubMed]

Morin, R.

Nehorai, A.

P. Sarder, and A. Nehorai, "Deconvolution methods for 3-d fluorescence microscopy images," IEEE Signal Process. Mag. 23, 32-45 (2006).
[CrossRef]

Pavillon, N.

Psaltis, D.

Rao, R.

Rosen, J.

Sarder, P.

P. Sarder, and A. Nehorai, "Deconvolution methods for 3-d fluorescence microscopy images," IEEE Signal Process. Mag. 23, 32-45 (2006).
[CrossRef]

Saxton, W.

R. Gerchberg, and W. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttg.) 35, 227-246 (1972).

Schaefer, L. H.

W. Wallace, L. H. Schaefer, and J. R. Swedlow, "A workingperson’s guide to deconvolution in light microscopy," Biotechniques 31, 1076 (2001).
[PubMed]

Seelamantula, C. S.

Shaffer, E.

Shemer, A.

Sheppard, C. J.

C. J. Sheppard, "Fundamentals of superresolution," Micron 38, 165-169 (2007).
[CrossRef]

C. J. Sheppard, and K. Larkin, "Vectorial pupil functions and vectorial transfer functions," Optik (Stuttg.) 107, 79-87 (1997).

Sheppard, C. J. R.

C. J. R. Sheppard, and M. Gu, "Imaging by a high aperture optical-system," J. Mod. Opt. 40, 1631-1651 (1993).
[CrossRef]

Simon, B.

Swedlow, J. R.

W. Wallace, L. H. Schaefer, and J. R. Swedlow, "A workingperson’s guide to deconvolution in light microscopy," Biotechniques 31, 1076 (2001).
[PubMed]

Tada, Y.

Tiziani, H. J.

M. Totzeck, and H. J. Tiziani, "Phase-singularities in 2d diffraction fields and interference microscopy," Opt. Commun. 138, 365-382 (1997).
[CrossRef]

Torczon, V.

V. Torczon, "On the convergence of pattern search algorithms," SIAM J. Optim. 7, 125 (1997).
[CrossRef]

Totzeck, M.

M. Totzeck, and H. J. Tiziani, "Phase-singularities in 2d diffraction fields and interference microscopy," Opt. Commun. 138, 365-382 (1997).
[CrossRef]

Toy, M. F.

Unser, M.

Wallace, W.

W. Wallace, L. H. Schaefer, and J. R. Swedlow, "A workingperson’s guide to deconvolution in light microscopy," Biotechniques 31, 1076 (2001).
[PubMed]

Wu, J. G.

Xu, C.

B. Colicchio, O. Haeberl, C. Xu, A. Dieterlen, and G. Jung, "Improvement of the lls and map deconvolution algorithms by automatic determination of optimal regularization parameters and pre-filtering of original data," Opt. Commun. 244, 37-49 (2005).
[CrossRef]

Yang, C. H.

Yaqoob, Z.

Zalevsky, Z.

Zhuang, S.

Appl. Opt. (5)

Biotechniques (1)

W. Wallace, L. H. Schaefer, and J. R. Swedlow, "A workingperson’s guide to deconvolution in light microscopy," Biotechniques 31, 1076 (2001).
[PubMed]

IEEE Signal Process. Mag. (1)

P. Sarder, and A. Nehorai, "Deconvolution methods for 3-d fluorescence microscopy images," IEEE Signal Process. Mag. 23, 32-45 (2006).
[CrossRef]

J. Microsc. (3)

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2002).
[CrossRef] [PubMed]

M. G. L. Gustafsson, "Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy," J. Microsc. 198, 82-87 (2000).
[CrossRef] [PubMed]

A. Marian, F. Charri`ere, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, "On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography," J. Microsc. 225, 156-169 (2007).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

C. J. R. Sheppard, and M. Gu, "Imaging by a high aperture optical-system," J. Mod. Opt. 40, 1631-1651 (1993).
[CrossRef]

J. Opt. Soc. Am. A (1)

Methods (1)

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, "Three-dimensional imaging by deconvolution microscopy," Methods 19, 373-385 (1999).
[CrossRef] [PubMed]

Micron (1)

C. J. Sheppard, "Fundamentals of superresolution," Micron 38, 165-169 (2007).
[CrossRef]

Opt. Commun. (2)

B. Colicchio, O. Haeberl, C. Xu, A. Dieterlen, and G. Jung, "Improvement of the lls and map deconvolution algorithms by automatic determination of optimal regularization parameters and pre-filtering of original data," Opt. Commun. 244, 37-49 (2005).
[CrossRef]

M. Totzeck, and H. J. Tiziani, "Phase-singularities in 2d diffraction fields and interference microscopy," Opt. Commun. 138, 365-382 (1997).
[CrossRef]

Opt. Express (5)

Opt. Lett. (3)

Optik (Stuttg.) (2)

C. J. Sheppard, and K. Larkin, "Vectorial pupil functions and vectorial transfer functions," Optik (Stuttg.) 107, 79-87 (1997).

R. Gerchberg, and W. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttg.) 35, 227-246 (1972).

SIAM J. Optim. (1)

V. Torczon, "On the convergence of pattern search algorithms," SIAM J. Optim. 7, 125 (1997).
[CrossRef]

Other (8)

D. E. Goldberg, Genetic Algorithms in Search, Optimization & Machine Learning (Addison-Wesley, 1989).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, 1987), 6th ed.

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).

Y. Cotte, and C. Depeursinge, "Measurement of the complex amplitude point spread function by a diffracting circular aperture," in "Focus on Microscopy," (2009), Advanced linear and non-linear imaging, pp. TU-AF2-PAR-D.

Z. Zalevsky, and D. Mendlovic, Optical superresolution, vol. 91 (Springer, 2004).

C. Depeursinge, P. Jourdain, B. Rappaz, P. Magistretti, T. Colomb, and P. Marquet, "Cell biology explored with digital holographic microscopy," Biomed. Opt. p. BMD58 (2008).

C. Vonesch, "Fast and automated wavelet-regularized image restoration in fluorescence microscopy," Ph.D. thesis, EPFL, LIB Laboratoire d’imagerie biomédicale (2009).

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Figures (7)

Fig. 1.
Fig. 1.

SEM image of pair of nano-holes drilled by FIB in aluminum film at 100 000× magnification. The images show nominal center-to-center pitches η of 600nm (a), 500nm (b), 400nm (c) and 300nm (d) with according scale bars.

Fig. 2.
Fig. 2.

Experimental and synthetic transfer functions in focal plane at λ =532nm and NA=0.95. The experimental amplitude CTF ∣cexp ∣ (a) and phase CTF arg[cexp ] (c) are imaged from a single nano-metric aperture. According to Eq. (22), (b) shows the fitted synthetic amplitude CTF ∣csyn ∣ and (d) its phase part arg[csyn ].

Fig. 3.
Fig. 3.

Experimental transfer functions in focal plane at λ =532nm and NA=0.95 of the test target (cf. Fig. 1). The Γ-masked amplitude spectrum ∣∣ (a) and phase spectrum arg[] (b) are illustrated for η=400nm. (c) compares ∣∣ cross-sections in ky for kx = 0 with the experimental CTF of a single nano-metric hole. (d) shows the same comparison of the experimental OTF and ∣∣.

Fig. 4.
Fig. 4.

Comparisons of unresolved and super-resolved profiles of two nano-holes of test target (cf. Fig. 1) with center-to-center distances η=600nm in (a), η=500nm in (b), η=400nm in (c), and η=300nm in (d). The raw data images I are reconstructed in the focal plane at λ =532nm and NA=0.95 (dmin,coh =460nm). The ‘raw’ profile shows the central y cross-section of the resolution limited raw data I (cf. ‘rw’ insert). The ‘deconvolution complex’ profile shows the corresponding section of ∣o2 (cf. ‘cd’ insert) resulting from complex deconvolution by the experimental CTF. Additionally, ‘deconvolution intensity’ compares the profile of oi resulting from intensity deconvolution by the experimental OTF.

Fig. 5.
Fig. 5.

Influence of kmax (dmin ) on complex deconvolution results according to Eq. (4). (a–c) statistics for η=400nm for deconvolution of Uexp with cexp ‘experimental’, for deconvolution of Uexp with csyn ‘experimental-synthetic’, for deconvolution of Usyn with csyn synthetic (no noise)’, and deconvolution of Unoise with cnoise ‘synthetic (SNR=35)’. (d) statistics of p-t-p in dependence of dmin for all targets η. The grey bars indicate error margin of 25nm.

Fig. 6.
Fig. 6.

Experimental transfer functions in focal plane for λ =532nm and NA=0.95 of test target (cf. Fig. 1) after deconvolution. The amplitude spectrum ∣O∣ (a) and phase spectrum arg[O] (b) are illustrated for η=400nm after division by CTF. (c) compares ∣O∣ cross-section in ky for kx = 0 with for the η=400nm case. (d) shows the same comparison for ∣Oi ∣ and ∣∣.

Fig. 7.
Fig. 7.

XY images in focal plane of test target with sub-resolution pitch η=400nm [cf. insert (a) imaged by SEM]. Insert (b) shows the unresolved test target’s raw image I at λ =532nm and NA=0.95. Insert (c) shows oi resulting from intensity deconvolution. Insert (d) shows ∣o2 resulting from complex deconvolution by the synthetic CTF and insert (e) the according result for deconvolution by the experimental CTF.

Tables (2)

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Table 1. Results of fit of experimental data from optical system at λ =532nm and NA=0.95

Tables Icon

Table 2. Results of peak-to-peak (p-t-p) distance measurements of the test target at λ =532nm and NA=0.95. The standard precision is based on the lateral sampling of 56nm, the complex deconvolution is determined in Fig. 5(d)

Equations (29)

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v c , incoh = 2 v c , coh ,
d min = α λ NA ,
k = k = 2 π ν = 2 π d ,
k c , coh = 2 π d min , coh .
I ( x 2 , y 2 ) = h ( x 1 + M x 2 , y 1 + My 2 ) 2 o i ( x 1 , y 1 ) d x 1 d y 1 ,
J ( k x , k y ) = C ( k x , k y ) O i ( k x , k y ) ,
and I ( x 2 , y 2 ) = J ( k x , k y ) exp [ i 2 π ( k x M x 2 + k y M y 2 ) ] d k x d k y ,
C ( k x , k y ) = h ( x 1 , y 1 ) 2 exp [ i 2 π ( k x x 1 + k y y 1 ) ] d x 1 d y 1 .
o i ( x 1 , y 1 ) = O i ( k x , k y ) exp [ i 2 π ( k x x 1 + k y y 1 ) ] d k x d k y = 1 { J ˜ ( k x , k y ) C ( k x , k y ) } .
J ˜ ( k x , k y ) = J ( k x , k y ) Γ k max ( k x , k y ) where Γ k max ( k x , k y ) = { 1 k x 2 + k y 2 k s f s k s k x 2 + k y 2 k max 0 k x 2 + k y 2 > k max
U ( x 2 , y 2 ) = h ( x 1 + M x 2 , y 1 + M y 2 ) o ( x 1 , y 1 ) d x 1 d y 1 ,
G ( k x , k y ) = c ( k x , k y ) O ( k x , k y ) ,
and U ( x 2 , y 2 ) = G ( k x , k y ) exp [ i 2 π ( k x M x 2 + k y M y 2 ) ] d k x d k y ,
c ( k x , k y ) = h ( x 1 , y 1 ) exp [ i 2 π ( k x x 1 + k y y 1 ) ] d x 1 d y 1 ,
o ( x 1 , y 1 ) = O ( k x , k y ) exp [ i 2 π ( k x x 1 + k y y 1 ) ] d k x d k y = 1 { G ˜ ( k x , k y ) c ( k x , k y ) } .
G ˜ ( k x , k y ) = G ( k x , k y ) Γ k max ( k x , k y ) .
U ( x , y ) = a n A ( x , y ) exp [ i Φ ( x , y ) ] ,
NA = n m λ N δ x m px ,
{ x 1 = f sin θ cos ϕ , y 1 = f sin θ sin ϕ , z 1 = f cos θ , which satisfies f 2 = x 1 2 + y 1 2 + z 1 2 ,
{ x 2 = r 2 cos Ψ , y 2 = r 2 sin Ψ , z 2 , which satisfies r 2 2 = x 2 2 + y 2 2 .
U δ ( r 2 , Ψ , z 2 ) = i λ 0 2 π 0 α P ( θ , ϕ ) exp [ i k r 2 sin θ cos ( ϕ Ψ ) i k z 2 cos θ i k Φ ( θ , ϕ ) ] sin θ d θ d ϕ ,
P ( θ , ϕ ) = cos θ .
h ( z 2 ) = Ω U δ ( r 2 , Ψ , z 2 ) d r 2 d Ψ .
f ( A n , m ) = Σ k x , k y arg [ c exp ] arg [ c syn ( A n , m ) ] 2 ,
U syn = h syn ( x , y + η 2 ) + h syn ( x , y η 2 ) .
d min cd = min [ λ 2 ( 1 NA ± Δ ϕ π ) ] .
I ( x , y ) = U ( x , y ) 2 ,
g ( x ) = a 1 exp [ ( x μ 1 ) 2 2 b 1 2 ] + a 2 exp [ ( x μ 2 ) 2 2 b 2 2 ] .
FWHM = 2 ln 2 ( b 1 + b 2 ) .

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