Super-resolution lasers
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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. ***** We are looking into setting up a super-resolution system using our existing laser combiner and Nikon TI TIRF arm setup to bring the light to the sample. The system is run with Micro-Manager, so QuickPalm seems like a good place to start. We currently have 50W 405/75 mW 488/ 50mW 561/ 40mW 640 diode lasers on the system. I have no experience with super resolution at all yet but I have been told that this is not enough laser power for QuickPalm or STORM. Some funds from the university have suddenly appeared and I need to respond rapidly so I would like to know if people have suggestions as to specifically what lasers I should be looking at to add to the system and what they are likely to cost. Any hints as to other problems I am not expecting that will need $$$ to solve would be helpful as well. Thanks- Dave Dr. David Knecht Professor of Molecular and Cell Biology Core Microscopy Facility Director University of Connecticut Storrs, CT 06269 860-486-2200 |
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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 David, those powers could potentially be made to work at a pinch, if you were prepared to limit yourself to a very small field of view, but in general around 200-500mW, or possibly even higher is where you want to be. The 405 probably already has enough power as is. Are you planning on doing PALM (fluorescent proteins) or STORM(organic dyes)? In my opinion, the first laser to replace would be the 640, as this is where you will get the most mileage. A good 200mW laser at 640nm should go for somewhere in the vicinity of 7-8K. If you are prepared to go longer, reasonable but decidedly no-frills 671nm lasers at ~ 1W can be had from Viasho for about 2K (although there are some caveats with using this wavelength). A good 561/568nm laser would probably be the next step - although these are typically more expensive (12-15k). You will also want to replace the TIRF coupler (probably with something home built) as the stock one spreads the laser light out over too large a field of view to be useful (ideally you want to just illuminate an area the size of your camera FOV, potentially even smaller). Depending on how your current focusing is implemented, you might want to consider a piezo-focusser and/or some form of 'perfect focus' scheme. I've got a custom system based on a Nikon Ti running down here in New Haven - if you want to come down for a look, drop me an email at [hidden email]. cheers,David On Friday, 16 January 2015 4:20 PM, "Knecht, David" <[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. ***** We are looking into setting up a super-resolution system using our existing laser combiner and Nikon TI TIRF arm setup to bring the light to the sample. The system is run with Micro-Manager, so QuickPalm seems like a good place to start. We currently have 50W 405/75 mW 488/ 50mW 561/ 40mW 640 diode lasers on the system. I have no experience with super resolution at all yet but I have been told that this is not enough laser power for QuickPalm or STORM. Some funds from the university have suddenly appeared and I need to respond rapidly so I would like to know if people have suggestions as to specifically what lasers I should be looking at to add to the system and what they are likely to cost. Any hints as to other problems I am not expecting that will need $$$ to solve would be helpful as well. Thanks- Dave Dr. David Knecht Professor of Molecular and Cell Biology Core Microscopy Facility Director University of Connecticut Storrs, CT 06269 860-486-2200 |
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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 all, Nyquist again: I gather that the actual Nyquist criterion says that pixel size must be smaller than 1/2 the physical resolution. In the literature, I also find the factor 1/2.3 and I wonder where the 2.3 comes from. Is this just one interpretation of <1/2 or is this the result of some calculation of which I could not find the source? (I am aware that if the structure of interest is oriented diagonally to the pixel pattern, an additional factor of 1.41 comes into play, see discussion on this list in April 2012 or chapter 4 in the Handbook, so that it could be argued the factor should rather be <1/2.8 or 1/3.2, but my question is about the origin of the 2.3). Steffen -- ------------------------------------------------------------ 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 Germany |
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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. ***** Dear Steffen Pixels are square which means that in diagonal pixel distance is higher. The correction factor should then be sqrt(2)/4, i.e., not 50 but 35% of the resolution. But this is not 1/2.3. I presume that 1/2.3 is for a confocal. Bear in mind thought that this does not take in consideration statistics and optical aberrations which should decrease resolution and the need of having such a small pixel size. Best, nm > On 19 Jan 2015, at 11:32, 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. > ***** > > Hi all, > > Nyquist again: I gather that the actual Nyquist criterion says that > pixel size must be smaller than 1/2 the physical resolution. In the > literature, I also find the factor 1/2.3 and I wonder where the 2.3 > comes from. Is this just one interpretation of <1/2 or is this the > result of some calculation of which I could not find the source? > > (I am aware that if the structure of interest is oriented diagonally to > the pixel pattern, an additional factor of 1.41 comes into play, see > discussion on this list in April 2012 or chapter 4 in the Handbook, so > that it could be argued the factor should rather be <1/2.8 or 1/3.2, but > my question is about the origin of the 2.3). > > Steffen > > > > -- > ------------------------------------------------------------ > 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 > Germany |
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I interpret this as the difference between Rayleigh and FWHM. One is the point where two structures can just be resolved, the other is the point where they definitively cannot.
Guy -----Original Message----- From: Confocal Microscopy List [mailto:[hidden email]] On Behalf Of Nuno Moreno Sent: Monday, 19 January 2015 11:13 PM 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. ***** Dear Steffen Pixels are square which means that in diagonal pixel distance is higher. The correction factor should then be sqrt(2)/4, i.e., not 50 but 35% of the resolution. But this is not 1/2.3. I presume that 1/2.3 is for a confocal. Bear in mind thought that this does not take in consideration statistics and optical aberrations which should decrease resolution and the need of having such a small pixel size. Best, nm > On 19 Jan 2015, at 11:32, 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. > ***** > > Hi all, > > Nyquist again: I gather that the actual Nyquist criterion says that > pixel size must be smaller than 1/2 the physical resolution. In the > literature, I also find the factor 1/2.3 and I wonder where the 2.3 > comes from. Is this just one interpretation of <1/2 or is this the > result of some calculation of which I could not find the source? > > (I am aware that if the structure of interest is oriented diagonally > to the pixel pattern, an additional factor of 1.41 comes into play, > see discussion on this list in April 2012 or chapter 4 in the > Handbook, so that it could be argued the factor should rather be > <1/2.8 or 1/3.2, but my question is about the origin of the 2.3). > > Steffen > > > > -- > ------------------------------------------------------------ > 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 > Germany |
 
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Christian Soeller |
Re: Nyquist and the factor 2.3
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In reply to this post by Steffen Dietzel
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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, First, I guess we may be seeing a long thread as Nyquist always seems to resonate with microscopists. I have heard about and read the 'diagonal argument' and I think it is flawed, at least as presented. My reason for that is as follows: IF a 2D signal is bandwidth limited at a frequency f, i.e. if all Fourier components outside a circle with radius f are truly zero, then it can be mathematically rigorously shown that sampling it on a 2D grid with spacing < 1/2f is sufficient to fully reconstruct the original signal. This makes no reference to the "orientation of features etc" and argues that the diagonal argument cannot be strictly speaking correct. (There are some issues with a signal of limited spatial extent being incompatible with finite bandwidth). It may well be that the 'diagonal argument' leads to a result that is sort of "the right result" but I think does that by incorrect reasoning for reasons as above. At the very least I would like to see how those who think the diagonal argument is ok deal with the rigorous Fourier result which can be found in the literature. One issue is the term "The Resolution". The frequency response of a microscope (OTF) rolls of in a characteristic way and it is not always clear how this can be properly captured by one number ("The Resolution"). Choices are, for example, the FWHM of the lateral PSF, some XX dB rolloff of the frequency response (and then using the inverse of that) etc. For example, the FWHM "resolution" r_fwhm does not mean that the frequency response outside the circle with radius 1/r_fwhm is zero. Therefore sampling with r_fwhm/2 is generally not enough. In that sense a factor of >2 may be thought of as a safety factor to take care of the fact that "The Resolution" may be underestimating the true finest detail where a frequency response is still distinguishable from 0. In that sense before thinking about the "right factor" it may be more important to clarify how "resolution" may be properly defined (and measured). If you look at the Nyquist calculators for deconvolution these generally recommend what might appear to be quite fine sampling. Happy to hear other's perspective. Cheers, Christian On Monday, 19 January 2015 at 11:32 AM, Steffen Dietzel 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 all, > > Nyquist again: I gather that the actual Nyquist criterion says that > pixel size must be smaller than 1/2 the physical resolution. In the > literature, I also find the factor 1/2.3 and I wonder where the 2.3 > comes from. Is this just one interpretation of <1/2 or is this the > result of some calculation of which I could not find the source? > > (I am aware that if the structure of interest is oriented diagonally to > the pixel pattern, an additional factor of 1.41 comes into play, see > discussion on this list in April 2012 or chapter 4 in the Handbook, so > that it could be argued the factor should rather be <1/2.8 or 1/3.2, but > my question is about the origin of the 2.3). > > Steffen > > > -- > ------------------------------------------------------------ > 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 > Germany > > > |
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Zdenek Svindrych-2 |
Re: Nyquist and the factor 2.3
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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. ***** Dear all, Steffen's argument is simply not right, sorry to say that. 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 -- Zdenek Svindrych, Ph.D. W.M. Keck Center for Cellular Imaging (PLSB 003) University of Virginia, Charlottesville, USA http://www.kcci.virginia.edu/workshop/index.php ---------- Původní zpráva ---------- Od: Christian Soeller <[hidden email]> Komu: [hidden email] Datum: 19. 1. 2015 7:47:31 Předmět: 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. ***** Hi Steffen, First, I guess we may be seeing a long thread as Nyquist always seems to resonate with microscopists. I have heard about and read the 'diagonal argument' and I think it is flawed, at least as presented. My reason for that is as follows: IF a 2D signal is bandwidth limited at a frequency f, i.e. if all Fourier components outside a circle with radius f are truly zero, then it can be mathematically rigorously shown that sampling it on a 2D grid with spacing < 1/2f is sufficient to fully reconstruct the original signal. This makes no reference to the "orientation of features etc" and argues that the diagonal argument cannot be strictly speaking correct. (There are some issues with a signal of limited spatial extent being incompatible with finite bandwidth). It may well be that the 'diagonal argument' leads to a result that is sort of "the right result" but I think does that by incorrect reasoning for reasons as above. At the very least I would like to see how those who think the diagonal argument is ok deal with the rigorous Fourier result which can be found in the literature. One issue is the term "The Resolution". The frequency response of a microscope (OTF) rolls of in a characteristic way and it is not always clear how this can be properly captured by one number ("The Resolution"). Choices are, for example, the FWHM of the lateral PSF, some XX dB rolloff of the frequency response (and then using the inverse of that) etc. For example, the FWHM "resolution" r_fwhm does not mean that the frequency response outside the circle with radius 1/r_fwhm is zero. Therefore sampling with r_ fwhm/2 is generally not enough. In that sense a factor of >2 may be thought of as a safety factor to take care of the fact that "The Resolution" may be underestimating the true finest detail where a frequency response is still distinguishable from 0. In that sense before thinking about the "right factor" it may be more important to clarify how "resolution" may be properly defined (and measured). If you look at the Nyquist calculators for deconvolution these generally recommend what might appear to be quite fine sampling. Happy to hear other's perspective. Cheers, Christian On Monday, 19 January 2015 at 11:32 AM, Steffen Dietzel 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 all, > > Nyquist again: I gather that the actual Nyquist criterion says that > pixel size must be smaller than 1/2 the physical resolution. In the > literature, I also find the factor 1/2.3 and I wonder where the 2.3 > comes from. Is this just one interpretation of <1/2 or is this the > result of some calculation of which I could not find the source? > > (I am aware that if the structure of interest is oriented diagonally to > the pixel pattern, an additional factor of 1.41 comes into play, see > discussion on this list in April 2012 or chapter 4 in the Handbook, so > that it could be argued the factor should rather be <1/2.8 or 1/3.2, but > my question is about the origin of the 2.3). > > Steffen > > > -- > ------------------------------------------------------------ > 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 > Germany > > >" |
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David Baddeley |
Re: Nyquist and the factor 2.3
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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 Steffan, I've always worked under the assumption that the 2.3 (sometimes 2.35) was an empirical factor to take the possibility of under-estimating the cutoff frequency into account. It should be noted that this factor is also often used when choosing a sampling frequency in electronics, audio, and radio applications, so it's not specific to some resolution metric. If I had to hazard a guess, it looks awfully like the factor that you multiply the std. deviation of a Gaussian by to get the FWHM, it could equally be related to the the typical frequency response of anti-aliasing filters. For true Nyquist sampling in microscopy you should theoretically use the band limit, rather than the resolution (ie the less than half the Abbe resolution formula for widefield, and half that again for confocal) - resulting in confocal pixel sizes on the order of 45 nm regardless of pinhole size. This does not seem necessary in practice, and might well be overkill. Does anyone know of a rigorous analysis of how low the OTF must be for it to effectively be considered as zero? cheers,David On Monday, 19 January 2015 8:52 AM, Zdenek Svindrych <[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. ***** Dear all, Steffen's argument is simply not right, sorry to say that. 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 -- Zdenek Svindrych, Ph.D. W.M. Keck Center for Cellular Imaging (PLSB 003) University of Virginia, Charlottesville, USA http://www.kcci.virginia.edu/workshop/index.php ---------- Původní zpráva ---------- Od: Christian Soeller <[hidden email]> Komu: [hidden email] Datum: 19. 1. 2015 7:47:31 Předmět: 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. ***** Hi Steffen, First, I guess we may be seeing a long thread as Nyquist always seems to resonate with microscopists. I have heard about and read the 'diagonal argument' and I think it is flawed, at least as presented. My reason for that is as follows: IF a 2D signal is bandwidth limited at a frequency f, i.e. if all Fourier components outside a circle with radius f are truly zero, then it can be mathematically rigorously shown that sampling it on a 2D grid with spacing < 1/2f is sufficient to fully reconstruct the original signal. This makes no reference to the "orientation of features etc" and argues that the diagonal argument cannot be strictly speaking correct. (There are some issues with a signal of limited spatial extent being incompatible with finite bandwidth). It may well be that the 'diagonal argument' leads to a result that is sort of "the right result" but I think does that by incorrect reasoning for reasons as above. At the very least I would like to see how those who think the diagonal argument is ok deal with the rigorous Fourier result which can be found in the literature. One issue is the term "The Resolution". The frequency response of a microscope (OTF) rolls of in a characteristic way and it is not always clear how this can be properly captured by one number ("The Resolution"). Choices are, for example, the FWHM of the lateral PSF, some XX dB rolloff of the frequency response (and then using the inverse of that) etc. For example, the FWHM "resolution" r_fwhm does not mean that the frequency response outside the circle with radius 1/r_fwhm is zero. Therefore sampling with r_ fwhm/2 is generally not enough. In that sense a factor of >2 may be thought of as a safety factor to take care of the fact that "The Resolution" may be underestimating the true finest detail where a frequency response is still distinguishable from 0. In that sense before thinking about the "right factor" it may be more important to clarify how "resolution" may be properly defined (and measured). If you look at the Nyquist calculators for deconvolution these generally recommend what might appear to be quite fine sampling. Happy to hear other's perspective. Cheers, Christian On Monday, 19 January 2015 at 11:32 AM, Steffen Dietzel 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 all, > > Nyquist again: I gather that the actual Nyquist criterion says that > pixel size must be smaller than 1/2 the physical resolution. In the > literature, I also find the factor 1/2.3 and I wonder where the 2.3 > comes from. Is this just one interpretation of <1/2 or is this the > result of some calculation of which I could not find the source? > > (I am aware that if the structure of interest is oriented diagonally to > the pixel pattern, an additional factor of 1.41 comes into play, see > discussion on this list in April 2012 or chapter 4 in the Handbook, so > that it could be argued the factor should rather be <1/2.8 or 1/3.2, but > my question is about the origin of the 2.3). > > Steffen > > > -- > ------------------------------------------------------------ > 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 > Germany > > >" |
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In reply to this post by Steffen Dietzel
It's been a while since I cracked those textbooks, but...
As I remember long-past classes in signal processing, the extra 0.3 is a fairly heuristic buffer to ensure that you have sufficient high frequency information to recover data at the band limit. For an in-focus widefield microscope the modulation transfer function (MTF) decreases in a near-linear fashion, with average intensities (zero frequencies) coming through intact, and frequencies near the cutoff having very low values. Frequencies just below the sampling limit are, however, _just_ detectable, and that may be impossible in the presence of noise. Sampling somewhat _above_ the Nyquist limit prevents the sampling limit (where frequencies are barely, theoretically, detectable) from utterly compounding the MTF reduction, and provides some chance of accurately capturing (and potentially restoring, for example through deconvolution) those high frequencies. And in practice 2.3 has been a reasonable factor for this across multiple application domains. Kevin Ryan Media Cybernetics, Inc. -----Original Message----- From: Confocal Microscopy List [mailto:[hidden email]] On Behalf Of Steffen Dietzel Sent: Monday, January 19, 2015 6:33 AM To: [hidden email] Subject: 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. ***** Hi all, Nyquist again: I gather that the actual Nyquist criterion says that pixel size must be smaller than 1/2 the physical resolution. In the literature, I also find the factor 1/2.3 and I wonder where the 2.3 comes from. Is this just one interpretation of <1/2 or is this the result of some calculation of which I could not find the source? (I am aware that if the structure of interest is oriented diagonally to the pixel pattern, an additional factor of 1.41 comes into play, see discussion on this list in April 2012 or chapter 4 in the Handbook, so that it could be argued the factor should rather be <1/2.8 or 1/3.2, but my question is about the origin of the 2.3). Steffen -- ------------------------------------------------------------ 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 Germany |
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A side note on Nyquist sampling: sampling _exactly_ 2x the highest frequency in the observed object means you _can_ detect that frequency, but not that you _will_. 2x sampling means you can pick up the peak and trough of the high frequency, but if you are 90 degrees out of phase your sampling will read all zero crossings instead. Hence the >2x from Nyquist.
And if you have a sampling only _slightly_ higher than the Nyquist criterion, your highest frequency recovery may alias as a lower (beat) frequency at the difference between your sampling and input frequency. Again, better to exceed Nyquist by a comfortable margin, and 2.3x has been found heuristically workable in multiple domains. Kevin Ryan Media Cybernetics, Inc. -----Original Message----- From: Confocal Microscopy List [mailto:[hidden email]] On Behalf Of Kevin Ryan Sent: Monday, January 19, 2015 10:00 AM To: [hidden email] Subject: Re: Nyquist and the factor 2.3 It's been a while since I cracked those textbooks, but... As I remember long-past classes in signal processing, the extra 0.3 is a fairly heuristic buffer to ensure that you have sufficient high frequency information to recover data at the band limit. For an in-focus widefield microscope the modulation transfer function (MTF) decreases in a near-linear fashion, with average intensities (zero frequencies) coming through intact, and frequencies near the cutoff having very low values. Frequencies just below the sampling limit are, however, _just_ detectable, and that may be impossible in the presence of noise. Sampling somewhat _above_ the Nyquist limit prevents the sampling limit (where frequencies are barely, theoretically, detectable) from utterly compounding the MTF reduction, and provides some chance of accurately capturing (and potentially restoring, for example through deconvolution) those high frequencies. And in practice 2.3 has been a reasonable factor for this across multiple application domains. Kevin Ryan Media Cybernetics, Inc. -----Original Message----- From: Confocal Microscopy List [mailto:[hidden email]] On Behalf Of Steffen Dietzel Sent: Monday, January 19, 2015 6:33 AM To: [hidden email] Subject: 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. ***** Hi all, Nyquist again: I gather that the actual Nyquist criterion says that pixel size must be smaller than 1/2 the physical resolution. In the literature, I also find the factor 1/2.3 and I wonder where the 2.3 comes from. Is this just one interpretation of <1/2 or is this the result of some calculation of which I could not find the source? (I am aware that if the structure of interest is oriented diagonally to the pixel pattern, an additional factor of 1.41 comes into play, see discussion on this list in April 2012 or chapter 4 in the Handbook, so that it could be argued the factor should rather be <1/2.8 or 1/3.2, but my question is about the origin of the 2.3). Steffen -- ------------------------------------------------------------ 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 Germany |
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Kyle Michael Douglass |
Re: Nyquist and the factor 2.3
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In reply to this post by Steffen Dietzel
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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, On 01/19/2015 12:32 PM, Steffen Dietzel wrote: > > Nyquist again: I gather that the actual Nyquist criterion says that > pixel size must be smaller than 1/2 the physical resolution. In the > literature, I also find the factor 1/2.3 and I wonder where the 2.3 > comes from. Is this just one interpretation of <1/2 or is this the > result of some calculation of which I could not find the source? > There's an explanation of the factor 2.3 (more precisely 2.355) in this book: Steve B. Howell, "Handbook of CCD Astronomy," 2nd ed., Cambridge University Press (2006) Look at pages 130--133, Section 5.9: Pixel sampling. He states that 2.3 is the FWHM in pixels of the Gaussian approximation to the PSF. Cheers, Kyle -- Kyle M. Douglass, PhD Post-doctoral researcher The Laboratory of Experimental Biophysics EPFL, Lausanne, Switzerland http://kmdouglass.github.io http://leb.epfl.ch |
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