Bo Zhang,1
Josiane Zerubia,2
and Jean-Christophe Olivo-Marin1
1B. Zhang (bzhang@pasteur.fr) and J.-C. Olivo-Marin (jcolivo@pasteur.fr) are with the Unité d'Analyse d'Images Quantitative, Institut Pasteur, 25-28 rue du Docteur Roux, 75724 Paris Cedex 15, France. B. Zhang is also with the Ecole Nationale Supérieure des Télécommunications, 46 rue Barrault, 75013 Paris, France.
2J. Zerubia (josiane.zerubia@inria.fr) is with the Ariana Project (INRIA∕I3S), Institut National de Recherche en Informatique et en Automatique, 2004 route des lucioles-BP 93, 06902 Sophia-Antipolis Cedex, France.
Bo Zhang, Josiane Zerubia, and Jean-Christophe Olivo-Marin, "Gaussian approximations of fluorescence microscope point-spread function models," Appl. Opt. 46, 1819-1829 (2007)
We comprehensively study the least-squares Gaussian approximations of the diffraction-limited 2D–3D paraxial–nonparaxial point-spread functions (PSFs)
of the wide field fluorescence microscope (WFFM), the laser scanning confocal microscope
(LSCM), and the disk scanning confocal microscope (DSCM). The PSFs are expressed using the Debye integral. Under an
constraint imposing peak matching, optimal and near-optimal Gaussian parameters are derived for the PSFs. With an
constraint imposing energy conservation, an optimal Gaussian parameter is derived for the 2D paraxial WFFM PSF. We found that (1) the 2D approximations are all very accurate; (2) no accurate Gaussian approximation exists for 3D WFFM PSFs; and (3) with typical pinhole sizes, the 3D approximations are accurate for the DSCM and nearly perfect for the LSCM. All the Gaussian parameters derived in this study are in explicit analytical form, allowing their direct use in practical applications.
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The top part of the table shows the RSE% and the bottom part shows the PRE%. In parentheses, the minimal and maximal errors are shown (Min.Err.%, Max.Err.%). The NA varies from 0.2 to 0.7 in the paraxial cases and from 0.8 to 1.4 in the nonparaxial cases.
The top part of the table shows the RSE%, and the bottom part shows the PRE%. In parentheses, the minimal and maximal errors are shown (Min.Err.%, Max.Err.%). The NA varies from 0.2 to 0.7 in the paraxial cases and from 0.8 to 1.4 in the nonparaxial cases. D is the pinhole diameter, and its unit is the Airy Unit (1 AU = 1.22λex∕NA). D = 0 corresponds to the vanishing pinhole situation.
The top part of the table shows the RSE%, and the bottom part shows the PRE%. In parentheses, the minimal and maximal errors are shown (Min.Err.%, Max.Err.%). The NA varies from 0.2 to 0.7 in the paraxial cases and from 0.8 to 1.4 in the nonparaxial cases. d is set to . D is the pinhole diameter, and its unit is the Airy Unit (1 AU = 1.22λex∕NA). D = 0 corresponds to the vanishing pinhole situation.
Table 7
Approximation Errorsa on the 2D Paraxial WFFM PSF (L1Constraint)
RSE%
PRE%(σ̂ρ*)
(1.1, 1.3)
(0.0, 1.7)
In parentheses the minimal and maximal errors are shown (Min.Err.%, Max.Err.%). The NA varies from 0.2 to 0.7.
The top part of the table shows the RSE% and the bottom part shows the PRE%. In parentheses, the minimal and maximal errors are shown (Min.Err.%, Max.Err.%). The NA varies from 0.2 to 0.7 in the paraxial cases and from 0.8 to 1.4 in the nonparaxial cases.
The top part of the table shows the RSE%, and the bottom part shows the PRE%. In parentheses, the minimal and maximal errors are shown (Min.Err.%, Max.Err.%). The NA varies from 0.2 to 0.7 in the paraxial cases and from 0.8 to 1.4 in the nonparaxial cases. D is the pinhole diameter, and its unit is the Airy Unit (1 AU = 1.22λex∕NA). D = 0 corresponds to the vanishing pinhole situation.
The top part of the table shows the RSE%, and the bottom part shows the PRE%. In parentheses, the minimal and maximal errors are shown (Min.Err.%, Max.Err.%). The NA varies from 0.2 to 0.7 in the paraxial cases and from 0.8 to 1.4 in the nonparaxial cases. d is set to . D is the pinhole diameter, and its unit is the Airy Unit (1 AU = 1.22λex∕NA). D = 0 corresponds to the vanishing pinhole situation.
Table 7
Approximation Errorsa on the 2D Paraxial WFFM PSF (L1Constraint)
RSE%
PRE%(σ̂ρ*)
(1.1, 1.3)
(0.0, 1.7)
In parentheses the minimal and maximal errors are shown (Min.Err.%, Max.Err.%). The NA varies from 0.2 to 0.7.