### Sampling

For a point-scanning confocal microscope (1.3 lens, 488 excitation) good sampling is around 50nm x 150nm. Laterally 70nm is fine, you shouldn't go beyond 100nm, axially it is best to stay at 150-200nm. If you sample around the 70nm x 150nm you will see that the restoration is capable to reduce noise considerably. As a result you might need less signal than you thought before. If practical considerations (bleaching, data size) don't allow these sampling densities, you'll just have to do the best you can. In our experience, unless you undersample dramatically, the restoration will always improve your image. For accurate values use our

See UnderSampling.

The aquisition zoom factor only affects the scanning parameters and as far as we know has no effect on the computation of the backprojected pinhole size.

If the pinhole is specified in terms of Airy units the backprojected radius is computed as (from the FAQ):
backprojected_RADIUS = number_of_Airy_disks * 0.61 * $$\frac{\lambda}{NA}$$
To compute the backprojected radius from the physical pinhole size see the tables in the Huygens Essential or Professional User Guide.

Oversampling is completely harmless. The more samples the better, though it could be argued that oversampling is not *necessary*. Of course there are practical reasons to limit sampling: object size, memory requirements, bleaching and so on.

If you have loaded an image in Huygens Pro, select it and select Edit-> Nyquist you get a report of the sampling situation taking into account all known parameters. Rule of thumb is for WF: 100nm lateral or better; around 200nm axially. 250nm would be fine too, but we distrust some z-motors with round numbers.

An undersampled image stack is a stack in which the Z or XY samping intervals are too large. Undersampling means that the sampling interval is too large to capture all information about the object generated by the microscope. In Huygens Essential the sampling values will be colored orange in case of undersampling. In case of severe undersampling the color will be red.

In Huygens Pro the optimal sampling density for the optical conditions under which an image is recorded can be computed by selecting its thumbnail and then Edit->Nyquist. When an image is recorded at this so called Nyquist frequency the digitized sequence contains *all* information carried in the signal. This makes it possible to reconstruct the image at any location, so not limited to the sampling positions. The Nyquist frequency is twice the highest spatial frequency (bandlimit) transmitted by the microscope.

There seems to be a discrepancy between the way Huygens defines stepsize and the way my microscope defines stepsize.

In the Huygens the stepsize (in any direction) is just the distance between samples, 'sampling distance'. Also, in all directions the numbering starts with 0, so the index for N samples runs from 0..N-1. This is the conventional way of indexing samples. Some microscope manufacturers use a special 1..N numbering scheme in the z-direction, but still 0..N-1 in the XY plane.

Yes, read our undersampling Is undersampling in Z-direction a serious problem?.

For a general discussion on correct sampling and aliasing see:
• Gonzalez, R.C. and R.E. Woods. (1992) Digital Image Processing. Addison-Wesley. ISBN 0-201-50803-6. p111 e.v.
For a discussion of microscopical bandpass characteristics see the papers below. Be sure to read both!
• Sheppard, C.J.R.The spatial frequency cut-off in three-dimensional imaging. (1986a). Optik 72 No. 4 131-133.
• Sheppard, C.J.R.The spatial frequency cut-off in three-dimensional imaging II. (1986b). Optik 74 No. 3, pp. 128-129.
For a practical discussion see also SVI's 'Deconvolution Recipes' manual or the Nyquist Calculator and Nyquist Rate.

No, it is a function of the optical properties of your system. It all revolves around the Shannon theorem, that states that for a bandlimited system (all our microscopes) it is totally sufficient to sample at the Nyquist rate. Now suppose the maximum spatial frequency passing a microscope is one cycle per 100nm (1.3 NA oil, confocal, 488/520nm, sampling at 50nm to sample peaks and valleys of the 100nm periodic wave). If you have a periodic structure of lumps spaced 80nm apart then this structure is not imaged, apart from its average value, nothing of it. Can't restore it, no way. If there is omly one single object and you know it is a sphere then restoration could consist of determining its center of mass. The accuracy of that depends on the SNR, but you could easily reach 10nm. Job done! The regular restoration procedure could also do it for you, but obviously to get such an accuracy in determining the peak location of the object you would have to resample the data to a higher sampling interval of 10nm. (You could also play it a bit dirty by not deconvolving with the PSF, but with the known image of the object; out comes a single peak where the center of the object is).
A more interesting object is for instance a two-blob object with a spacing at the Nyquist rate. Now the most interesting parts of the object spectrum are cut off by the microscope. The problem now is that the transmitted piece is the same for a whole family of objects. The family which has a spectrum quite like it is even larger. The restoration algorithm must now choose among them, the first selection being to exclude all objects with negative values. The better the SNR, the better the restoration algorithm can exclude objects of which the spectrum is slightly dissimilar to the measured spectrum. For confocals the situation is worse because they already attenuate everything beyond say 60% of the band practically to zero (depending on the pinhole). So in practice there is little hope for resolving objects at the edge of the band.

See Nyquist Rate and Nyquist Calculator. The following formulas can be used to compute the Nyquist rate.
Widefield microscope:
Nyquist_lateral = \lambda$$/ ( 4 n sin(\alpha$$))
Nyquist_axial = \lambda$$/ ( 4 n (1 - cos(\alpha$$)))
with \lambda$$the wavelength, N$$ the refractive index of the medium (1.515 for immersion oil). \alpha$$is the half-aperture angle obtained with: \alpha = arcsin(NA/N)$$ with NA the Numerical aperture. Many calculators use the 'sin-1' or 'asin' symbol for the arcsin function.
Confocal microscope:
Assuming the excitation and emission wavelength are equal a confocal microscope doubles the bandwidth so halves the Nyquist sampling density.
Both Huygens Pro and Essential take the exact wavelength into account when computing the Nyquist rate. In case of multi photon excitation they also take the number of excitation photons into account. Both will color the background of X, Y, Z sampling density entry fields orange (moderate undersampling) or red (serious undersampling) when detecting undersampling. In Huygens Pro you can look up the Nyquist rate for a particular image by selecting it and Edit->Nyquist rate.

The Z-sampling is modfied by foreshortening due to differences in refractive index of the media and the immersion oil: the 'reverse fishtank' effect. In most cases the z-sampling as specified in the raw datafile is the nominal sampling distance, i.e. the distance the table or objective actually moved in Z without taking foreshortening due to refractive index mismatch into account. The mismatch induces spherical abberation and can have a profound effect on the PSF shape and effective aperture. The PSF generator takes all this into account.

After deconvolution the remaining geometric distortion can be corrected by multiplying the z-sampling distance by the ratio of the medium and immersion refractive index, a number in most of the cases < 1. See also [FishtankEffect|Fishtank Effect]

To find out the ideal sampling, you can always use the online Nyquist Calculator, entering the image parameters (including the number of photons: 2). You can also use the Huygens Software in your computer. To find out how large are your samples in relation with the ideal Nyquist Rate, do the following:

In Essential
• If you have an image for which you want to compute the Nyquist rate, open it and check its parameters (right click -> "Show parameters").
• Whenever you change the Microscopic Parameters of an image, the Nyquist rate is recalculated. Modify the image parameters to match the microscopic conditions for which you want to compute the Nyquist rate, then check the parameters again (right click -> "Show parameters").
In Professional
• Select the image, select Edit -> Nyquist rate in the main menu. You'll get a popup displaying the Nyquist rate.

If in 2-photon excitation a pinhole is not used it is the excitation distribution which determines the imaging properties of the microscope and therefore the Nyquist rate. The excitation intensity field is that of a widefield microscope, but since due to the 2-photon effect the effective excitation distribution is the square of the intensity distribution, the imaging properties are vastly different. The squaring operation makes the distribution more 'peaked', resulting in an improved resolution. It also causes the bandwidth (and with that the Nyquist rate) in x, y, z to be twice that of a widefield microscope at the same wavelength.

Importantly, the 3D shape of the band-pass area is very different: while the widefield area has a wedge at the center causing the large widefield blur cones, the 2-photon bandpass area has no such defects.

See Pinhole And Bandwidth.

Old information
In principle, the Nyquist rate is independent of the pinhole size. This is due to the choice to relate the Nyquist rate to the theoretical bandwidth of the system: the spatial frequency beyond which *nothing* is transferred by the microscope. It would be a different story if we would have used a criterion based on attenuation of spatial frequencies below a certain factor. (Larger pinhole sizes attenuate higher frequencies more, but still are not zero). Although a practical approach (because about last third of the band has so low intensities that they can be considered zero most of the times) this involves an arbitrary choice, so therefore we base the Nyquist rate on the well defined theoretical bandwidth.

A problem occurs with extremely large pinholes like those used in two-photon systems. In these cases the optical properties are practically identical to a widefield system whereas due to the presence of the pinhole the theoretical bandlimit is still that of a confocal microscope. In the two-photon case it is best to set the microscope type to 'widefield' when doing deconvolution with the Huygens Software, since this will result in the same optical properties but with a more practical Nyquist rate. Note however that with single photon confocals even a very large pinhole will still have a noticeable effect on the blur contributions of far off focus regions, thus improving resolution along the optical axis.

The Confocal Microscope provides more information than the widefield system; in theory the confocal Sampling Density should be twice the widefield density. This holds for in all 3 dimensions, so you'd get 8 times more voxels.

A 'typical' widefield setup (1.3 effective NA lens, 500nm emission) is sampled well at 100 nm laterally, 300 nm axially. However, due to good SNR ratio's deconvolution can often gain a lot in Z, so you might as well go for 200 nm. With a 100x lens and a CCD with 6.7 micron cells you get 67 nm laterally. (This is assuming that there is no extra magnification; otherwise the total magnification must be used when calculating the pixel size). If this bloats your data too much you can try binning to increase the lateral size to 134 nm, but you will already start seeing some 'staircasing' effects on thin filaments in the deconvolved image (Aliasing Artifacts).

In the same typical confocal case a nice sampling rate would be 50 nm in Z, 150 nm axially. In case of bleaching problems you can stretch this up to 75 nm lateral, and after that increase the Z-sampling.

If the Parameter Editor or the Parameter wizard starts coloring the sampling fields orange, then you start to undersample; red means severely undersampled. See Nyquist Rate.

This greatly depends on the optical paramaters, in particular on the NA and microscope type. Together with the wavelenghts and refractive indexes these determine the so called Nyquist sample distance, the maximum sampling distance at which all image information is captured. For a plot of the dependancy of the Nyquist rate on the NA and microscope type see the User guides, or go to Nyquist Rate. To calculate this figures online see the Nyquist Calculator.
• Confocal microscopes: While sampling at the Nyquist rate is a very good idea, it is in many practical situations hard to attain. In these cases larger sampling distances may be used. For confocal images with a Airy disk sized pinhole the lateral sampling distance may be up to about 1.6 of the Nyquist distance. When much larger pinholes are used, up to 2x.
When very small pinholes are used to optimize resolution, or any other case where high resolution is required, we do not not recommend undersampling since this would defeat the purpose, and since it might limit the deconvolution result.
• Two photon microscopes: Two photon microscopes without a detection pinhole (e.g. non descanned systems) follow the rules of a small pinhole confocal microscope relative to the longer excitation wavelength. The addition of a detection pinhole increases the theoretical bandwidth, but since usually large pinholes are used the practical bandwidth increase is small.
• Spinning (Nipkow) disk microscopes: Regarding sampling criteria spinning disk microscopes behave like confocal microscopes with, depending on the disk, a fairly large pinhole. For a typical system we do not recommend more than 1.6 times undersampling.
• Widefield: Widefield data are more sensitive to undersampling; stay below 1.3. In case of low numerical apertures like 0.4 we recommend not to undersample in the axial direction.
• STED : The remarkable property of STED microscopes is that they do not have a band limit in the strict sense. That means that any sampling rate is a compromise between practical considerations and the to be reached resolution. Since lateral STED HIW resolution can achieve 50nm in good conditions we recommend sampling around 25nm. In difficult condition this can be increased, but that will limit deconvolution.
• Multi channel data: The sampling rate should be derived from the highest resolution channel.
The actual and the ideal sampling distances of an image can be seen in Huygens Essential right-clicking on the image thumbnail and choosing 'Show parameters'.

To compute the ideal Nyquist rate of an image in Huygens Professional, select the image and choose Edit -> Nyquist rate.

Widefield and confocal microscopes differ in the amount of information they are able to extract from a specimen. One way to express this is to look at the finest details or highest spatial frequency the microscopes transmits: for a confocal microscope this is nearly twice as much as in an equivalent widefield microscope, in all directions.

According to the Nyquist theorem a signal should be sampled at twice its highest bandwidth so confocal microscopes need twice the sampling density of widefield microscopes. Although the confocal microscope is able to transmit twice as fine details as the widefield microscope, it attenuates these very strongly. Beyond say 60% of the highest frequency practically nothing is transmitted, especially for not-ideal pinhole cases. Therefore, while sampling according to Nyquist rate remains the safest solution, in the case of confocal imaging it is defensible to reduce the sampling rate to about 60% of the theoretical rate, for example in typical condition one sample per 50/0.6 = 80nm.

In the widefield case, high spatial frequencies are also attenuated as the band limit is approached, but to a much lesser degree than in the confocal case. Therefore we do not recommend to stay below 1.3 of the Nyquist rate. In case of low numerical apertures like 0.4 we recommend not to undersample widefield images in the axial direction.

A practical example:
Assuming a 1.3 N.A. objective lens and 488nm excitation, 520nm emission you need to sample around 50 x 50 x 165 nm to get an optimally sampled confocal 3D image; 100 x 100 x 330 nm in the 3D widefield case. As mentioned above, widefield images are more sensitive to undersampling, i.e. a violation of the sampling rule has a more dramatic effect on widefield images than on confocal images.

For more details on sampling densities for other microscope type, see the FAQ What is the maximal voxel size at which Huygens can still do a good job?.You may also want to have a look at Sampling Density and Nyquist Rate.

If your CCD camera has the option of binning, the binning can be used to increase the signal at the cost of resolution. With binning the intensity of several pixels is integrated to form a super pixel. For instance, 4 x 4 pixels are summed to form a bigger and brighter signal at the cost of resolution. When binning is applied the effective voxel size increases.
Example: If the actual detector element size is 6 micron and a 2 x 2 binning function is active, then the effective pixel size is 12 x 12 micron. As a consequence the magnification has to be increased by a factor of 2 relative to the non-binning mode in order to satisfy the sampling criteria.

Indeed, lowering the NA to allow for larger samples is the correct way. The drawback is that the axial resolution of the image will decrease with the square of the reduction of the NA. The sampling density as function of the NA can be found in the Huygens User Guides in the chapter Establishing Image Parameters.

### Particular instruments

See Biorad MRC_500_600_1024 and Biorad_Radiance. Old information: from communication with Brad Amos we learned that the factor for the 1024 is 53 and for the Radiance is 60. We had feed back from a customer who reported "Using Brad Amos' value, the deconvolution is working well".

For the most used Yokokawa disk spacing is 250 micron, so with an 100x lens backprojected about 2.5 micron This you can check by stopping the disk.
The pinhole diameter is probably 25-50 micron, resulting in a backprojected radius of .125 - .25. The latter is about an Airy disk.

The Perkin Elmer Ultra View is a Yokagawa disc and the Yokagawa disc is always a Nipkow disc. You must choose the Huygens option for Nipkow disc microscopes.