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About the Optimal Optical Transfer Function
- f a Microscope
The Optical Microscope And Its Resolution
min
About the Optimal Optical Transfer Function of a Microscope Jrg - - PowerPoint PPT Presentation
About the Optimal Optical Transfer Function of a Microscope Jrg - - PowerPoint PPT Presentation
About the Optimal Optical Transfer Function of a Microscope Jrg Enderlein Georg August University Gttingen Germany Quantitative BioImaging 2019, January 9 11 Rennes, France The Optical Microscope And Its Resolution
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The Optical Transfer Function
sin ' sin M n Abbe‘s sine condition: detection and imaging as a superposition
- f plane waves
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The Optical Transfer Function
imaging as a superposition of plane waves Fourier transform of electric field
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The Optical Transfer Function
intensity = absolute square of electric field in Fourier space: convolution Fourier transform of electric field
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The Optical Transfer Function
Optical Transfer Function Fourier transform of electric field
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The Optical Transfer Function
PSF = Fourier back- transform of OTF Optical Transfer Function
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The Point Spread Function
electric field intensity
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The Optical Transfer Function
electric field Fourier transform of electric field
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The Optical Transfer Function
convolution Fourier transform of electric field
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The Optical Transfer Function
convolution OTF
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The Optical Transfer Function
electric field of collected light
- r
electric field of focused excitation OTF WFM
- r
OTF CLSM (no pinhole)
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The Optical Transfer Function
2 max sin n k
4 max sin n k
min
2 2 sin 2 N.A. max y n k
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But what would be the optimal Optical Transfer Function for a given finite frequency support of a microscope? Answer: The OTF which yields the sharpest image.
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One-dimensional case
max max max
lim exp 2
q q q
dq iqx x
But we have also to require strict non-negativeness
- f PSF!
OTF PSF Maybe, a uniform amplitude across all available frequencies is optimal.
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One-dimensional case
max max
2
exp 2
q q
dq U x iqx
strictly non-negative OTF PSF
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How about non-uniform amplitude distribution?
1 a q q OTF E-Field PSF
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A heuristic argument for the Optimal Optical Transfer Function
2
dxU x x dxU x
does generally not converge What is a good measure of „sharpness“? So what about the following idea?
2 2 2
dxU x x dxU x
minimize
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A heuristic argument for the Optimal Optical Transfer Function
2 2 2
dxU x x dxU x
1 a q q
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Recipe For Finding Optimal OTF: Minkowski Sum Decomposition
Given the frequency support of OTF (B), search Minkowski decomposition (A) so that its auto-convolution recovers the OTF support, then fill Minkowksi decomposition with uniform frequency amplitude and calculate via auto- convolution optimal OTF. A A B
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OTFs of Ideal Wide-Field Microscope or Pinhole-less Laser-Scanning Microscope Are Optimal
electric field of collected light
- r
electric field of focused excitation OTF WFM
- r
OTF CLSM (no pinhole)
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Minkowski Sum = Geometric Convolution
A B
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Minkowski Sum = Geometric Convolution
A B
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Minkowski Sum = Geometric Convolution
A B C
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Minkowski Sum Decomposition
generally unsolved problem easy for convex shapes
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Example: Ideal Confocal Microscope Or Image Scanning Microscope
OTF of excitation OTF of detection through infinitely small pinhole OTF of LSCM with infinitely small pinhole Stokes shift neglected
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Example: Ideal Confocal Microscope
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Example: Ideal Confocal Microscope
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Conclusion
- Optimal OTF can be generated via an auto-
convolution of a frequency distribution with uniform constant amplitude
- General problem of how to find Minkowski
decomposition for non-convex shapes - may be complicated
- Knowing the optimal OTF is important for
deconvolution but also for physical PSF engineering
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