Posted by
Daniel White-2 on
URL: http://confocal-microscopy-list.275.s1.nabble.com/In-search-of-old-article-lens-frequency-band-limit-doesnt-really-exist-tp7585504.html
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Hi Sergey,
I've been thinking about this lots recently,
while trying to describe what happens in contrast restoration (by
deconvolution) microscope, and in 3D SIM - because that's what I do
You can see an imageJ/Fiji javascript linked below, that shows a noise free
image blurred by a perfectly known point spread function can be deconvolved
and recover features much smaller than Abbes limit of wavelength / 2NA.
This is because the system is linear and determined and can be simply
inverted. 2+3=5, and 5-3=2.
The OTF doesnt have a sharp edge where it goes to zero.... its just gets
really small after 2NA.
https://raw.githubusercontent.com/chalkie666/imagejMacros/master/DeconvolutionDemos/Convolution_Deconvolution_Demo.jsHowever, the real resolution limit is the noise and other errors in any
measurement of any real image, and in a real measured point spread function
(OTF, contrast transfer function, modulation transfer function, whatever
you want to call it)
Why?:
When the noise or other error is larger than the OTF at a certain spatial
frequency, that is the effective resolution limit. Cookie cutter, sharp
band limit is a first approximation only.
This is what Abbe effectively observed, and modeled as ray optics (which is
wrong in the same way that Newtonian mechanics is wrong - quantum mechanics
is right). The contrast transfer function (OTF) is so small after about
2NA, that you cant see any finer features with your eyes, or with a camera,
because contrast is too low or lost in noise.
If we could do something about photon shot noise at high signals maybe we
could see more resolution from a given lens. Structured illumination
microscopy and STED work around this limit - 3D SIM turns Abbe's equation
into
d = lambda / 4NA
by frequency mixing modulation. Abbe could not have guessed this is the
real, and easy to access resolution limit, by using only ray optics, but
any signal processing engineer knows this is true.
If an image is really noisy, then the effective resolution is less than
Abbe's limit, because the noise is larger then the contrast transfer
function (OTF) already at lower spatial frequencies than 2NA.
In short -
No Contrast = No Resolution
Noise and other errors kill contrast.
Noise magnitude vs the OTF magnitude is the real resolution limit, and NA
is a part of that, but not the only part.
We should be slightly less obsessed with theoretical band limits (cookie
cutters have sharp edges - OTFs dont - they just get really small), and
worry more about noise, and the fact that the OTF kills contrast of medium
and small features very strongly, and we need to fix that systematic error,
restore that contrast, by a method such as iterative constrained
convolution that can handle the noise. Also true for confocal.
cheers
Dan
Date: Wed, 3 Aug 2016 09:35:17 +0100
From: Mark Cannell <
[hidden email]>
Subject: Re: In search of old article
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Abbe ?
Cheers
On 2/08/2016, at 10:03 pm, Sergey Tauger <
[hidden email]> wrote:
frequency response is bandlimited to a know frequency? I browsed through
articles cited by Hopkins in 50s and Sheppard in 60s, but I still cannot
get who was the first.
>
> Best,
> Sergey
Mark B. Cannell Ph.D. FRSNZ FISHR
Professor of Cardiac Cell Biology
School of Physiology & Pharmacology
Faculty of Biomedical Sciences
University of Bristol
Bristol
BS8 1TD UK
[hidden email]
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