Fwd: Re: Nyquist and the factor 2.3

[Steffen Dietzel]

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Am 19.01.2015 um 14:50 schrieb Zdenek Svindrych:
> Dear all,
> Steffen's argument is simply not right, sorry to say that.

Don't be sorry on that one - I am happy to stand corrected on these
issues. I am not entirely convinced though. I can see from your image
how extended (linear) structures may be separated in real space at pixel
spacing of 1/2 the resolution. But after playing around a bit in
Photoshop with a picture of two PSFs at Rayleigh distance (i.e. point
like structures) and a grid with a pixel size of about 1/2 the
resolution, I am not so sure they can always be separated.

Never mind the other problems one has in real life in actually achieving
the theoretical resolution that you and others mentioned, such as noise,
spherical aberrations.

Concerning my original question and considering the answers so far, I
take it that the 2.3 is really some empirical factor that seems
reasonable and turned out to lead to good results (and is not the result
of some calculation, although it may have a base in the comparison of
FWHM and Rayleigh length, 0.61/0.51=1.19; 1.19*2=2.39).

Thanks, everybody, for replying.

Steffen

> It is without
> doubt that in the fourier-transformed image there is more headroom in the
> diagonal directions, so higher frequencies can be encoded with the same
> sampling. In real space it's illustrated here:
> https://drive.google.com/file/d/0B5vWyBYrDvcJckZNbE5lRVIyaUk/view?usp=
> sharing
> The Nyquist is, however, somewhat puzzling. The following notes may not
> clarify the issue much:
> (1) There is a hard limit of the OTF, it's the 'lambda/2NA' criterion
> (properly adjusted for confocal/SIM/STED/...). But usually the magnitude of
> the OTF rolls off quickly and you are left with nothing but noise even at
> frequencies well below the hard limit.
> (2) Review what Nyquist says: any band-limited signal sampled at frequency
> at least twice the bandwidth can be restored exactly. But his sampling was
> very different, he considered sampling of 1D (e.g. electrical) signal at
> discrete time points (I call it 'sampling with delta-functions'). In
> widefield microscopy each pixel integrates all pixels hitting the area of
> that pixel ('sampling with box functions'). This sampling attenuates the
> highest frequencies (compare fourier-transforms of a delta-function to that
> of a box function), i.e. the frequiencies that are so weak and precious in
> microscopy...
> (3) Most of the time our photon budget is limited and the associated poisson
> noise is critical for the final resolution. Maybe the simple OTF concept is
> not appropriate and should be replaced by something like 'Stochastic
> Transfer Function' (Somekh et al).
> (4) Also there are other effects, such as limited modulation transfer
> function of camera chips, that further attenuate the highest frequencies,
> calling for finer sampling.
> Bottom line? I think 2.3 x the_ultimate_frequency_limit is sufficient
> sampling.
> Best, zdenek

--
------------------------------------------------------------
Steffen Dietzel, PD Dr. rer. nat
Ludwig-Maximilians-Universität München
Walter-Brendel-Zentrum für experimentelle Medizin (WBex)
Head of light microscopy

Marchioninistr. 27
D-81377 München

Phone: [phone]
Fax-to-email: +49/89/2180-9976509
skype: steffendietzel
e-mail: [hidden email]

[Brian Northan]

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Hi Steffen

But after playing around a bit in
> Photoshop with a picture of two PSFs at Rayleigh distance (i.e. point
> like structures) and a grid with a pixel size of about 1/2 the
> resolution, I am not so sure they can always be separated.
>

It would be interesting to set up a simulation where two PSFs are placed at
different distances from each other, resampled at various rates and then
apply a deconvolution followed by autothresholding (Otsu or perhaps
something else).  Afterwards see if you get two objects back or just one.

As you mention, for practical purposes it would be important to add in the
effect of noise and spherical aberrations.

On Tue, Jan 20, 2015 at 4:00 AM, Steffen Dietzel <[hidden email]>
wrote:

> *****
> To join, leave or search the confocal microscopy listserv, go to:
> http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
> Post images on http://www.imgur.com and include the link in your posting.
> *****
>
> Am 19.01.2015 um 14:50 schrieb Zdenek Svindrych:
> > Dear all,
> > Steffen's argument is simply not right, sorry to say that.
>
> Don't be sorry on that one - I am happy to stand corrected on these
> issues. I am not entirely convinced though. I can see from your image
> how extended (linear) structures may be separated in real space at pixel
> spacing of 1/2 the resolution. But after playing around a bit in
> Photoshop with a picture of two PSFs at Rayleigh distance (i.e. point
> like structures) and a grid with a pixel size of about 1/2 the
> resolution, I am not so sure they can always be separated.
>
> Never mind the other problems one has in real life in actually achieving
> the theoretical resolution that you and others mentioned, such as noise,
> spherical aberrations.
>
> Concerning my original question and considering the answers so far, I
> take it that the 2.3 is really some empirical factor that seems
> reasonable and turned out to lead to good results (and is not the result
> of some calculation, although it may have a base in the comparison of
> FWHM and Rayleigh length, 0.61/0.51=1.19; 1.19*2=2.39).
>
> Thanks, everybody, for replying.
>
> Steffen
>
> > It is without
> > doubt that in the fourier-transformed image there is more headroom in the
> > diagonal directions, so higher frequencies can be encoded with the same
> > sampling. In real space it's illustrated here:
> > https://drive.google.com/file/d/0B5vWyBYrDvcJckZNbE5lRVIyaUk/view?usp=
> > sharing
> > The Nyquist is, however, somewhat puzzling. The following notes may not
> > clarify the issue much:
> > (1) There is a hard limit of the OTF, it's the 'lambda/2NA' criterion
> > (properly adjusted for confocal/SIM/STED/...). But usually the magnitude
> of
> > the OTF rolls off quickly and you are left with nothing but noise even at
> > frequencies well below the hard limit.
> > (2) Review what Nyquist says: any band-limited signal sampled at
> frequency
> > at least twice the bandwidth can be restored exactly. But his sampling
> was
> > very different, he considered sampling of 1D (e.g. electrical) signal at
> > discrete time points (I call it 'sampling with delta-functions'). In
> > widefield microscopy each pixel integrates all pixels hitting the area of
> > that pixel ('sampling with box functions'). This sampling attenuates the
> > highest frequencies (compare fourier-transforms of a delta-function to
> that
> > of a box function), i.e. the frequiencies that are so weak and precious
> in
> > microscopy...
> > (3) Most of the time our photon budget is limited and the associated
> poisson
> > noise is critical for the final resolution. Maybe the simple OTF concept
> is
> > not appropriate and should be replaced by something like 'Stochastic
> > Transfer Function' (Somekh et al).
> > (4) Also there are other effects, such as limited modulation transfer
> > function of camera chips, that further attenuate the highest frequencies,
> > calling for finer sampling.
> > Bottom line? I think 2.3 x the_ultimate_frequency_limit is sufficient
> > sampling.
> > Best, zdenek
>
>
> --
> ------------------------------------------------------------
> Steffen Dietzel, PD Dr. rer. nat
> Ludwig-Maximilians-Universität München
> Walter-Brendel-Zentrum für experimentelle Medizin (WBex)
> Head of light microscopy
>
> Marchioninistr. 27
> D-81377 München
>
> Phone: [phone]
> Fax-to-email: +49/89/2180-9976509
> skype: steffendietzel
> e-mail: [hidden email]
>

[Remko Dijkstra]

*****
To join, leave or search the confocal microscopy listserv, go to:
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*****

**Vendor response**

Dear all,

Most microscope systems (widefield, spinning-disk, confocal etc.) are
indeed bandwidth limited, i.e. they have a hard practical bandwidth limit.
The max. resolution is related to the bandwidth of the system, but also
depends on how much the higher spatial frequencies within the band are
attenuated.
Instead of defining the Nyquist rate based on physical resolution (which
also depends on your definition of resolution), it makes much more sense
to define the ideal Nyquist rate purely on the system bandwidth, which
is microscope dependent: http://www.svi.nl/NyquistRate

For confocal systems the pinhole does not effect the bandwidth of the
system, but the pinhole does have a direct effect on the spatial
resolution: http://www.svi.nl/PinholeAndResolution
Although the confocal microscope is able to transmit twice as fine
details as a widefield, it attenuates these very strongly. Beyond say
70% of the highest frequency practically nothing is transmitted,
especially for not-ideal pinhole sizes. Therefore, if you use the
theoretical Nyquist sampling rate (based on the bandwidth of the system)
you are very much in the clear, but if you would use the 1/2.3
resolution definition, you would be undersampling in the case of
confocal-based systems.

The calculations of our free Nyquist Calculator
(http://www.svi.nl/NyquistCalculator) and the recently released free
Nyquist Calculator app (Android) are based on the systems bandwidth. You
can find the free Android Nyquist calculator app in the Google Play
store:
https://play.google.com/store/apps/details?id=com.svi.nyquistcalculator.app

As some of you already pointed out: STED is one of those special
techniques that doesn't have a bandwidth limit, which means it also does
not have a theoretical Nyquist rate.
However, we can still estimate a 'practical' Nyquist rate for STED based
on the attenuation of  high spatial frequencies. This attenuation is
dependent on the STED depletion intensity, STED wavelength and
back-projected pinhole size.

We will integrate a STED option very soon in both the Android app and
the online Nyquist calculator, so that you can also easily calculate the
ideal (practical) sampling rate for your STED images soon.

The Nyquist calculator app is in active development so any further ideas
and feedback are welcome.

Kind regards from SVI,

Remko

Try out the FREE Android SVI Nyquist App - http://www.svi.nl/NyquistApp
***********************************************************
Remko Dijkstra, MSc
Imaging Specialist/Account Manager
Scientific Volume Imaging bv
Tel: + 31 35 642 1626
www.svi.nl
***********************************************************
For support matters contact: [hidden email]

On 01/20/2015 02:08 PM, Brian Northan wrote:

> *****
> To join, leave or search the confocal microscopy listserv, go to:
> http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
> Post images on http://www.imgur.com and include the link in your posting.
> *****
>
> Hi Steffen
>
> But after playing around a bit in
>> Photoshop with a picture of two PSFs at Rayleigh distance (i.e. point
>> like structures) and a grid with a pixel size of about 1/2 the
>> resolution, I am not so sure they can always be separated.
>>
> It would be interesting to set up a simulation where two PSFs are placed at
> different distances from each other, resampled at various rates and then
> apply a deconvolution followed by autothresholding (Otsu or perhaps
> something else).  Afterwards see if you get two objects back or just one.
>
> As you mention, for practical purposes it would be important to add in the
> effect of noise and spherical aberrations.
>
> On Tue, Jan 20, 2015 at 4:00 AM, Steffen Dietzel <[hidden email]>
> wrote:
>
>> *****
>> To join, leave or search the confocal microscopy listserv, go to:
>> http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
>> Post images on http://www.imgur.com and include the link in your posting.
>> *****
>>
>> Am 19.01.2015 um 14:50 schrieb Zdenek Svindrych:
>>> Dear all,
>>> Steffen's argument is simply not right, sorry to say that.
>> Don't be sorry on that one - I am happy to stand corrected on these
>> issues. I am not entirely convinced though. I can see from your image
>> how extended (linear) structures may be separated in real space at pixel
>> spacing of 1/2 the resolution. But after playing around a bit in
>> Photoshop with a picture of two PSFs at Rayleigh distance (i.e. point
>> like structures) and a grid with a pixel size of about 1/2 the
>> resolution, I am not so sure they can always be separated.
>>
>> Never mind the other problems one has in real life in actually achieving
>> the theoretical resolution that you and others mentioned, such as noise,
>> spherical aberrations.
>>
>> Concerning my original question and considering the answers so far, I
>> take it that the 2.3 is really some empirical factor that seems
>> reasonable and turned out to lead to good results (and is not the result
>> of some calculation, although it may have a base in the comparison of
>> FWHM and Rayleigh length, 0.61/0.51=1.19; 1.19*2=2.39).
>>
>> Thanks, everybody, for replying.
>>
>> Steffen
>>
>>> It is without
>>> doubt that in the fourier-transformed image there is more headroom in the
>>> diagonal directions, so higher frequencies can be encoded with the same
>>> sampling. In real space it's illustrated here:
>>> https://drive.google.com/file/d/0B5vWyBYrDvcJckZNbE5lRVIyaUk/view?usp=
>>> sharing
>>> The Nyquist is, however, somewhat puzzling. The following notes may not
>>> clarify the issue much:
>>> (1) There is a hard limit of the OTF, it's the 'lambda/2NA' criterion
>>> (properly adjusted for confocal/SIM/STED/...). But usually the magnitude
>> of
>>> the OTF rolls off quickly and you are left with nothing but noise even at
>>> frequencies well below the hard limit.
>>> (2) Review what Nyquist says: any band-limited signal sampled at
>> frequency
>>> at least twice the bandwidth can be restored exactly. But his sampling
>> was
>>> very different, he considered sampling of 1D (e.g. electrical) signal at
>>> discrete time points (I call it 'sampling with delta-functions'). In
>>> widefield microscopy each pixel integrates all pixels hitting the area of
>>> that pixel ('sampling with box functions'). This sampling attenuates the
>>> highest frequencies (compare fourier-transforms of a delta-function to
>> that
>>> of a box function), i.e. the frequiencies that are so weak and precious
>> in
>>> microscopy...
>>> (3) Most of the time our photon budget is limited and the associated
>> poisson
>>> noise is critical for the final resolution. Maybe the simple OTF concept
>> is
>>> not appropriate and should be replaced by something like 'Stochastic
>>> Transfer Function' (Somekh et al).
>>> (4) Also there are other effects, such as limited modulation transfer
>>> function of camera chips, that further attenuate the highest frequencies,
>>> calling for finer sampling.
>>> Bottom line? I think 2.3 x the_ultimate_frequency_limit is sufficient
>>> sampling.
>>> Best, zdenek
>>
>> --
>> ------------------------------------------------------------
>> Steffen Dietzel, PD Dr. rer. nat
>> Ludwig-Maximilians-Universität München
>> Walter-Brendel-Zentrum für experimentelle Medizin (WBex)
>> Head of light microscopy
>>
>> Marchioninistr. 27
>> D-81377 München
>>
>> Phone: [phone]
>> Fax-to-email: +49/89/2180-9976509
>> skype: steffendietzel
>> e-mail: [hidden email]
>>

[Lutz Schaefer]

*****
To join, leave or search the confocal microscopy listserv, go to:
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*****

Dear Nyquist posters,
it is interesting to find this otherwise well understood topic repeatedly
here on the list. As you all agree, sampling a bit more than twice the
optical bandwidth is the currently preferred approach. Let me give you a
slightly different angle. When the recorded image is subject to
deconvolution, it should have been sampled twice according to its expected
resolution after reconstruction. The problem with too coarse sampling in
linear inverse filtering is well understood, it causes Gibb's oscillations
around discontinuous edges, which is prevented in positivity constrained
iterative algorithms at the cost of loosing detail. Now, many may say,
deconvolution does not reconstruct beyond the bounds of the OTF, which is
not entirely true, but considering the low SNR around these frequencies a
practical thought, at least for the lateral resolution component. Its is
entirely different for the axial part within a positivity constrained
algorithm. Going further, in presence of depth variant spherical aberration,
we do not want to sample at twice the lowest instrumental axial frequency,
when we attempt to restore our image using a depth variant algorithm (e.g.
C. Preza et.al). Aside from finer sampling of such aberrated image material,
one could think of an "upsampling" restoration algorithm that would
internally work on a finer grid as was suggested some time ago from Walter
Carrington. I have not seen such approach practically working though.
Information that is lost is lost...

Any suggestions?

Regards
Lutz

__________________________________
L u t z   S c h a e f e r
Sen. Scientist
Mathematical modeling / Computational microscopy
Advanced Imaging Methodology Consultation
16-715 Doon Village Rd.
Kitchener, ON, N2P 2A2, Canada
Phone/Fax: [phone]
Email: [hidden email]
___________________________________

-----Original Message-----
From: Remko Dijkstra
Sent: Tuesday, January 20, 2015 8:35 AM
To: [hidden email]
Subject: Re: Nyquist and the factor 2.3

*****
To join, leave or search the confocal microscopy listserv, go to:
http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
Post images on http://www.imgur.com and include the link in your posting.
*****

**Vendor response**

Dear all,

Most microscope systems (widefield, spinning-disk, confocal etc.) are
indeed bandwidth limited, i.e. they have a hard practical bandwidth limit.
The max. resolution is related to the bandwidth of the system, but also
depends on how much the higher spatial frequencies within the band are
attenuated.
Instead of defining the Nyquist rate based on physical resolution (which
also depends on your definition of resolution), it makes much more sense
to define the ideal Nyquist rate purely on the system bandwidth, which
is microscope dependent: http://www.svi.nl/NyquistRate

For confocal systems the pinhole does not effect the bandwidth of the
system, but the pinhole does have a direct effect on the spatial
resolution: http://www.svi.nl/PinholeAndResolution
Although the confocal microscope is able to transmit twice as fine
details as a widefield, it attenuates these very strongly. Beyond say
70% of the highest frequency practically nothing is transmitted,
especially for not-ideal pinhole sizes. Therefore, if you use the
theoretical Nyquist sampling rate (based on the bandwidth of the system)
you are very much in the clear, but if you would use the 1/2.3
resolution definition, you would be undersampling in the case of
confocal-based systems.

The calculations of our free Nyquist Calculator
(http://www.svi.nl/NyquistCalculator) and the recently released free
Nyquist Calculator app (Android) are based on the systems bandwidth. You
can find the free Android Nyquist calculator app in the Google Play
store:
https://play.google.com/store/apps/details?id=com.svi.nyquistcalculator.app

As some of you already pointed out: STED is one of those special
techniques that doesn't have a bandwidth limit, which means it also does
not have a theoretical Nyquist rate.
However, we can still estimate a 'practical' Nyquist rate for STED based
on the attenuation of  high spatial frequencies. This attenuation is
dependent on the STED depletion intensity, STED wavelength and
back-projected pinhole size.

We will integrate a STED option very soon in both the Android app and
the online Nyquist calculator, so that you can also easily calculate the
ideal (practical) sampling rate for your STED images soon.

The Nyquist calculator app is in active development so any further ideas
and feedback are welcome.

Kind regards from SVI,

Remko

Try out the FREE Android SVI Nyquist App - http://www.svi.nl/NyquistApp
***********************************************************
Remko Dijkstra, MSc
Imaging Specialist/Account Manager
Scientific Volume Imaging bv
Tel: + 31 35 642 1626
www.svi.nl
***********************************************************
For support matters contact: [hidden email]

On 01/20/2015 02:08 PM, Brian Northan wrote:

> *****
> To join, leave or search the confocal microscopy listserv, go to:
> http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
> Post images on http://www.imgur.com and include the link in your posting.
> *****
>
> Hi Steffen
>
> But after playing around a bit in
>> Photoshop with a picture of two PSFs at Rayleigh distance (i.e. point
>> like structures) and a grid with a pixel size of about 1/2 the
>> resolution, I am not so sure they can always be separated.
>>
> It would be interesting to set up a simulation where two PSFs are placed
> at
> different distances from each other, resampled at various rates and then
> apply a deconvolution followed by autothresholding (Otsu or perhaps
> something else).  Afterwards see if you get two objects back or just one.
>
> As you mention, for practical purposes it would be important to add in the
> effect of noise and spherical aberrations.
>
> On Tue, Jan 20, 2015 at 4:00 AM, Steffen Dietzel <[hidden email]>
> wrote:
>
>> *****
>> To join, leave or search the confocal microscopy listserv, go to:
>> http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
>> Post images on http://www.imgur.com and include the link in your posting.
>> *****
>>
>> Am 19.01.2015 um 14:50 schrieb Zdenek Svindrych:
>>> Dear all,
>>> Steffen's argument is simply not right, sorry to say that.
>> Don't be sorry on that one - I am happy to stand corrected on these
>> issues. I am not entirely convinced though. I can see from your image
>> how extended (linear) structures may be separated in real space at pixel
>> spacing of 1/2 the resolution. But after playing around a bit in
>> Photoshop with a picture of two PSFs at Rayleigh distance (i.e. point
>> like structures) and a grid with a pixel size of about 1/2 the
>> resolution, I am not so sure they can always be separated.
>>
>> Never mind the other problems one has in real life in actually achieving
>> the theoretical resolution that you and others mentioned, such as noise,
>> spherical aberrations.
>>
>> Concerning my original question and considering the answers so far, I
>> take it that the 2.3 is really some empirical factor that seems
>> reasonable and turned out to lead to good results (and is not the result
>> of some calculation, although it may have a base in the comparison of
>> FWHM and Rayleigh length, 0.61/0.51=1.19; 1.19*2=2.39).
>>
>> Thanks, everybody, for replying.
>>
>> Steffen
>>
>>> It is without
>>> doubt that in the fourier-transformed image there is more headroom in
>>> the
>>> diagonal directions, so higher frequencies can be encoded with the same
>>> sampling. In real space it's illustrated here:
>>> https://drive.google.com/file/d/0B5vWyBYrDvcJckZNbE5lRVIyaUk/view?usp=
>>> sharing
>>> The Nyquist is, however, somewhat puzzling. The following notes may not
>>> clarify the issue much:
>>> (1) There is a hard limit of the OTF, it's the 'lambda/2NA' criterion
>>> (properly adjusted for confocal/SIM/STED/...). But usually the magnitude
>> of
>>> the OTF rolls off quickly and you are left with nothing but noise even
>>> at
>>> frequencies well below the hard limit.
>>> (2) Review what Nyquist says: any band-limited signal sampled at
>> frequency
>>> at least twice the bandwidth can be restored exactly. But his sampling
>> was
>>> very different, he considered sampling of 1D (e.g. electrical) signal at
>>> discrete time points (I call it 'sampling with delta-functions'). In
>>> widefield microscopy each pixel integrates all pixels hitting the area
>>> of
>>> that pixel ('sampling with box functions'). This sampling attenuates the
>>> highest frequencies (compare fourier-transforms of a delta-function to
>> that
>>> of a box function), i.e. the frequiencies that are so weak and precious
>> in
>>> microscopy...
>>> (3) Most of the time our photon budget is limited and the associated
>> poisson
>>> noise is critical for the final resolution. Maybe the simple OTF concept
>> is
>>> not appropriate and should be replaced by something like 'Stochastic
>>> Transfer Function' (Somekh et al).
>>> (4) Also there are other effects, such as limited modulation transfer
>>> function of camera chips, that further attenuate the highest
>>> frequencies,
>>> calling for finer sampling.
>>> Bottom line? I think 2.3 x the_ultimate_frequency_limit is sufficient
>>> sampling.
>>> Best, zdenek
>>
>> --
>> ------------------------------------------------------------
>> Steffen Dietzel, PD Dr. rer. nat
>> Ludwig-Maximilians-Universität München
>> Walter-Brendel-Zentrum für experimentelle Medizin (WBex)
>> Head of light microscopy
>>
>> Marchioninistr. 27
>> D-81377 München
>>
>> Phone: [phone]
>> Fax-to-email: +49/89/2180-9976509
>> skype: steffendietzel
>> e-mail: [hidden email]
>>