* ignoring for this table deconvolution, which can improve an additional ~10% resolution.

Table added 20211007 because we are demo'ing the Olympus 150x/1.45NA objective lens (hopefully find money some day to buy it ... ideally Olympus would introduce X-line or X-Line-HR version),

widefield lambda = 500nm

d = 0.61 * 500 / 1.40 = 218nm  

d = 0.61 * 500 / 1.45 = 210nm    ... 150x/1.45NA lens

confocal, pinhole 1.0 Airy Unit,  lambda = 500nm

d = 0.51 * 500 / 1.40 = 182nm  

d = 0.51 * 500 / 1.45 = 178nm    ... 150x/1.45NA lens

confocal, pinhole ~0.666 Airy Unit

("confocal sweetest spot", re Jeff Reese "confocal sweet spot" range 0.6-0.7 A.U.} ,  lambda = 500nm

d = 0.475 * 500 / 1.40 = 170nm  

d = 0.475* 500 / 1.45 = 164nm    ... 150x/1.45NA lens

confocal, pinhole 0.5 Airy Unit,  lambda = 500nm

d = 0.44 * 500 / 1.40 = 157nm  

d = 0.44 * 500 / 1.45 = 151nm    ... 150x/1.45NA lens

confocal, pinhole 0.5 Airy Unit,  lambda = 440nm for Brilliant Violet BV421

d = 0.44 * 440 / 1.40 = 140nm  

d = 0.44 * 440 / 1.45 = 135nm    ... 150x/1.45NA lens

reviist widefield

widefield lambda = 500nm

d = 0.61 * 500 / 1.40 = 218nm  

d = 0.61 * 500 / 1.45 = 210nm    ... 150x/1.45NA lens

ORCA-FLASH4.0LT is 6.5x6.5 um pixel size, so:

6.5 um / 100 (mag) = 65nm 

6.5 um / 150 (mag) = 43nm

This might have benefit for super-resolution on widefield microscopy.

20210805 note

Preprint mentioned that the seven central detectors of AiryScan (and AiryScan2) have a diameter of 0.6 Airy Unit. Then clever math (Zeiss or this preprint - see also SVI.nl Huygens) is done with respect to all three rings. One consequence of 7 or all 32 detectors is noise adds. this suggests to me that for dim signal, one extremely good detector (i.e. avalanche photodiode or Leica STELLARIS confocal "S" SiPM) with 0.6 (or 0.666...) Airy Unit could outperform the 32-channel detector.

Prigent 20210802 bioRxiv - High-resolution reconstruction and deconvolution of array detector images
https://doi.org/10.1101/2021.08.02.454749
Each individual sub-detector has a diameter of 0.2 AU (Airy Unit). The inner hexagonal patch gathering the 7 central detectors has a diameter of 0.6 AU. The inner ring gathering the sub-detectors 8-19 has a diameter of 1 AU. The whole array of 32 sub-detectors has a diameter of 1.25 AU.

20220216W: Abberior Instruments (cofounded by Stefan Hell) introduced MATRIX detector with 20 "sub-detectors" (unclear if to 20 APDs or perhaps more likely to a SPAD). Same day early Feb 2022 introduced TIMEbox (fluorescence lifetime meets rainbow)

Lam F, ... Bolte S 2017 Super-resolution for everybody: An image processing workfl ow to obtain high-resolution images with a standard confocal microscope. Methods 115 (2017) 17–27. http://dx.doi.org/10.1016/j.ymeth.2016.11.003

We furthermore showed that the fixed biological tissue has an overall refractive index that is close to that of the optical system (1.518), rendering the tissue very transparent. {gm note: live cells R.I. ~1.40}

Resolution improvement by closing the pinhole aperture
We then wanted to test if we could increase resolution of the confocal microscope by closing the detection pinhole. We compared resolution at the coverslip and in a depth of 60 lm with the detector pinhole set to 0.6 AU (Fig. 1C, Table 1) and after deconvolution of the data. Our choice of the 0.6 AU pinhole size was based on several tests on our biological data. We acquired the same type of biological sample with different pinhole sizes, from 1 AU to 0.4 AU and observed that 0.6 AU is the threshold where we discard enough diffraction signal without photo-bleaching and with a good contrast. Since the result depends largely on the quality and the photo-stability of the biological sample the optimal pinhole value has to be evaluated for each biological sample. 
For the AF1+-medium {R.I. 1.518}, lateral resolutions of 106 nm ± 4 nm (coverslip) and 120 nm ± 7 nm (depth) and axial resolutions of 164 nm ± 8 nm (coverslip) and 192 nm ± 17 nm (depth) were measured.
Closing the pinhole indeed increased lateral and axial resolution 1.3–1.4-fold. These results are in good agreement with data
measured by Cox and Sheppard [28], who observed a 1.4-fold increase in resolution after closing the pinhole aperture to 0.5
using a Leica TSC SP2 confocal microscope. 
By optimizing sample preparation, image acquisition parameters and performing deconvolution, our workflow allowed us to
obtain a considerable gain in lateral and axial resolution throughout the sample thickness.
We used AF1 (Citifluor, UK), a commercially available mounting medium with a refractive index of 1.463 and AF1+, a modified AF1 solution harbouring a refractive index of 1.518. The refractive index increase of AF1+ solution was obtained by adding 83% (w/w) of Methyl-Phenylsulfoxid (Sigma-Aldrich, #261696) to AF1-solution. Refractive indices were verified at 21 C using a refractometer (Mettler Toledo, Switzerland).
* [17] C. Fouquet, J.-F. Gilles, N. Heck, M. Dos Santos, R. Schwartzmann, V. Cannaya, M.-P. Morel, R.S. Davidson, A. Trembleau, S. Bolte, PLoS ONE (2015), http://dx.doi.org/10.1371/journal.pone.0121096

20230224F update

SPLIT-PIN software enabling confocal and super-resolution imaging with a virtually closed pinhole
Elisabetta Di Franco, Angelita Costantino, Elena Cerutti, Morgana D’Amico, Anna P. Privitera, Paolo Bianchini, Giuseppe Vicidomini, Massimo Gulisano, Alberto Diaspro & Luca Lanzanò 
Scientific Reports volume 13, Article number: 2741 (2023) 
https://www.nature.com/articles/s41598-023-29951-9
In point-scanning microscopy, optical sectioning is achieved using a small aperture placed in front of the detector, i.e. the detection pinhole, which rejects the out-of-focus background. The maximum level of optical sectioning is theoretically obtained for the minimum size of the pinhole aperture, but this is normally prevented by the dramatic reduction of the detected signal when the pinhole is closed, leading to a compromise between axial resolution and signal-to-noise ratio. We have recently demonstrated that, instead of closing the pinhole, one can reach a similar level of optical sectioning by tuning the pinhole size in a confocal microscope and by analyzing the resulting image series. The method, consisting in the application of the separation of photons by lifetime tuning (SPLIT) algorithm to series of images acquired with tunable pinhole size, is called SPLIT-pinhole (SPLIT-PIN). Here, we share and describe a SPLIT-PIN software for the processing of series of images acquired at tunable pinhole size, which generates images with reduced out-of-focus background. The software can be used on series of at least two images acquired on available commercial microscopes equipped with a tunable pinhole, including confocal and stimulated emission depletion (STED) microscopes. We demonstrate applicability on different types of imaging modalities: (1) confocal imaging of DNA in a non-adherent cell line; (2) removal of out-of-focus background in super-resolved STED microscopy; (3) imaging of live intestinal organoids stained with a membrane dye.

A user-friendly version of the Matlab (The MathWorks) code is available at https://github.com/llanzano/SPLITPIN. A step-by-step description of this user-friendly version is available as Supplementary Text.

20240512S  statistical resolution measure of fluorescence microscopy with finite photons

• GM: please note that biomedical research funders provide money to do research on biological objects, not on sinusoidal objects (though occasionally cells like muscle have more-or-less sinusoidal features in them). They do compare infinitely thin planar specimen (single plane) and 30 um thick volumetric object (latter 5000 photons at imaging plane, 500 photons outside as background).
• - this paper compares 0.5 and 1.0 Airy unit pinhole size, not 0.6-0.7AU range (Jeff Reece) or 0.666 AU (GM). the paper also fails to mention spatial deconvolution (re: svi.nl Huygens, Microvolution.com, Leica Lightning [confocal] and Thunder [widefield], etc.). My rule of thumb for deconvolution is it can improve spatial resolution (XY and Z) by ~10% for each imaging mode, and can de-haze in a qualitative sense = eliminate or more practically reduce background. ... End of their discussion claims "deterministic post-processing methodd cannot increase Fisher information. Therefore, post-processing methods, including image reconstruction algorithms, can either keep IbR constant or worsen IbR." ... GM disagrees -- for 3D datasets that have sufficient photon counts from the objects, I expect 10% improvement in spatial resolution if pixel size and Z-step size are optimized [their text mentions tiny pixel size may be helpful in widefield.
• I suggest the Spring 2024 Hamamatsu ORCA-Quest2 "quantitative CMOS camera" - photon resolving [1 - 200 photon range; could be extended with multiple exposures; 4.6x4.6um pixel size], optimal objective lens (perhaps Olympus 150x 1.45NA, to get the extra mag the authors discuss for widfield small pixels) (4.6um / 150 = 0.00892 um/pixel), optimal Z-step size, non-bleaching fluorophore!!! may help to have modest average intensity pulsed illumination, ex. 1 microsecond on light to excite each fluorophore , 9 microseconds off pulses [give time for fluorophores 'shelved' in excited triplet state to go back to ground singlet state), optimal mounting medium, spatial deconvolution (example products mentioned above). ... Supplemental Note 2 ends with a caveat with respect to "if other prior information" - for deconvolution, this would be calculated or measured point spread function (PSF). Their text: "Last step of inequality comes from fact that any Fisher information is greater or equal to 0. If the processing is deterministic, then Fisher information in the processed data �� would always be smaller or equal to raw data ��. In another word, deterministic image post-processing cannot increase Fisher information of any parameter. Note that this statement is under condition that image processing is deterministic. If other prior information could be accessed, this relationship would not stand.  "

https://www.nature.com/articles/s41467-024-48155-x

Li Y, Huang F 2024 A statistical resolution measure of fluorescence microscopy with finite photons. Nature Communications 15: 3760 ("NComm") (open access).

Abstract

First discovered by Ernest Abbe in 1873, the resolution limit of a far-field microscope is considered determined by the numerical aperture and wavelength of light, approximately �� / 2��A. With the advent of modern fluorescence microscopy and nanoscopy methods over the last century, this definition is insufficient to fully describe a microscope’s resolving power. To determine the practical resolution limit of a fluorescence microscope, photon noise remains one essential factor yet to be incorporated in a statistics-based theoretical framework. We proposed an information density measure quantifying the theoretical resolving power of a fluorescence microscope in the condition of finite photons. The developed approach not only allows us to quantify the practical resolution limit of various fluorescence and super-resolution microscopy modalities but also offers the potential to predict the achievable resolution of a microscopy design under different photon levels.

**

Parts of text (bold text highlighted by GM -- see text for equations/symbols that did not paste here):

In 1873, Abbe published his work stating that microscopy resolution solely depends on the numerical aperture and wavelength of light1,2, a statement later verified theoretically3,4. This limit generally suffices for traditional microscopes, which collect transmission, reflection, or scattered light as signals5. The signal-to-noise ratio (SNR) can be optimized by adjusting the illumination power. However, in fluorescence microscopy, photons—the sole source of molecular information generated by individual fluorescent probes—are limited due to the photobleaching and photochemical environment of the fluorophores6,7.

The discrete nature of light results in inherent photon counting noise, which follows a Poisson distribution. Consequently, the SNR diminishes as the number of detected photons decreases. This reduction in SNR at low photon levels complicates the distinction between actual structural differences and random noise fluctuations, thereby hindering the ability of microscopy techniques to reach their theoretical resolution limits8,9,10,11,12,13,14,15,16,17.

Here, we propose a theoretical measure for quantifying the resolving power of microscopes, accounting for numerical aperture, emission wavelength, and photon statistics. Our approach considers the Fisher information of a sinusoidal grating’s phase estimation per area, defined as information density, to measure the imaging system’s resolving power. Based on an adjustable criterion of the information density threshold, we define an information-based resolution (IbR). This measure is applied to evaluate and distinguish the significant practical resolution differences across various conventional and super-resolution imaging modalities, including wide-field microscopy, confocal microscopy23,24, two-beam structured illumination microscopy (SIM)25,26, and image scanning microscopy (ISM)27,28. We expect IbR to be a useful measure in estimating the noise-considered resolution to guide and validate the design of newly developed or proposed imaging modalities.

...

The above results can also be demonstrated from the view of photon emission requirement. To achieve a specific practical resolution (IbR), the minimum numbers of photons required for different imaging modalities drastically defer (Fig. 2f). To resolve an object in a planar specimen with a frequency as low as 0.2������, a confocal system requires 100 photons/μm2, whereas other systems need fewer than 40 photons/μm2—a more than twofold difference. At a frequency of ������, ISM requires 175 photons/μm2, in contrast to other modalities that need at least 350 photons/μm2. 

...

In the case of volumetric specimens, wide-field and SIM experience significant performance declines due to the increased background from out-of-focus planes in thick samples. Conversely, confocal and ISM, using pinhole for background rejection, maintain information density similar to that of planar specimens. For instance, when imaging an object at a frequency of 1.85������, as the sample volume thickness increases from 0 to 30 μm, the information density of ISM decreases only slightly from 33 rad−2·μm−2 to 27 rad−2·μm−2. In comparison, the information density of SIM drops drastically from 58 rad−2·μm−2 to 14 rad−2·μm−2, a more than four-fold difference. This superior background resistance of ISM and confocal can be attributed to their optical sectioning capabilities, due to the use of pinholes9,29,38,39.

Influence of pinhole size on confocal microscope (and see figure 4 in article online or PDF)

In confocal imaging, shrinking pinhole size affects the resolving power in two opposite ways—improving it by broadening effective OTF, while worsening it by decreasing photon detection due to photon rejections of the pinhole24,37,39,40 (Supplementary Note 4, Supplementary Fig. 11). A too large pinhole diameter, such as 2 AU, yields an extended OTF akin to that of a wide-field system. (6, 7). Yet, for a 30 μm thick volumetric specimen, a confocal system with a larger pinhole provides a superior IbR compared to wide field, thanks to its ability to reduce out-of-focus background. For instance, for a volumetric specimen of 30 μm with a signal photon density of 5000 photons/μm2 and background photon density 500 photons/μm3, confocal systems with pinhole diameter of 2 AU (����=1.22������) result in an IbR of 313 nm versus 384 nm for the wide-field system. As the pinhole diameter shrinks, the confocal system acquires better resolving power: decreasing the pinhole diameter to 1 AU and 0.5 AU improves IbR to 300 nm and 285 nm, respectively (Fig. 4a). However, excessively small pinholes, such as 0.1 AU, significantly deteriorate resolution, leading to an IbR of 833 nm due to photon loss.

The tradeoff between improvement and deterioration of the confocal system’s resolving power, often balanced by pinhole size from experience, can now be quantified through information density ����. Our simulation suggests an ideal confocal pinhole diameter between 0.5 AU and 1 AU—in agreement with the common practice in confocal systems12,29. In the case of 30 μm thick volumetric specimen, for frequency below 1.3������, a 1 AU confocal pinhole diameters yields higher ���� than a 0.5 AU diameter. Above 1.3������, a 0.5 AU diameter is more effective (Figure. S7). To seek an optimal resolving power for an object at specific frequencies, Fig. 4b demonstrates the optimal pinhole diameter to achieve the largest ���� given four sinusoidal grating objects of different frequencies. In the case of a 30 μm thick volumetric specimen, for objects of frequency 0.5������, ������, 1.5������, and 2������, the optimal pinhole diameters are 1.2, 0.9, 0.75, 0.5 AU, respectively. The selection of the optimal confocal pinhole diameter is influenced by the balance between photon collection efficiency and the effective Optical Transfer Function (OTF) enhancement. This balance is not uniform across all frequencies, which leads to varying optimal pinhole sizes depending on the specific spatial frequencies of the sinusoidal grating being imaged (Fig. 4b). Generally, a small pinhole size suits objects of high frequencies, while a large pinhole size suits objects of low frequencies.

Confocal is well acknowledged for its background reduction capability. Another important, often overlooked advantage is its extension of the effective OTF of the imaging system, which enhances resolution beyond that of wide-field system28,41. This can be reflected by our simulation in the planar specimen case: at a signal photon density of 5000 photons/μm2, confocal systems with pinhole diameters of 1 AU and 0.5 AU achieve an IbR of 308 nm and 303 nm respectively, outperforming the wide-field system’s 400 nm (Supplementary Fig. 9). Across a broad frequency range [0,1.5������], a confocal system with a 1 AU pinhole diameter approaches maximum information density. For frequency above 1.5������, a 0.5 AU pinhole diameter in confocal microscopy is near optimal for information density (Supplementary Fig. 9).

...

Influence of pixel size on resolving power in wide-field microscope (and see fig 6 online) (reduce Id is bad; increase Id is good)

•     * GM note: tiny pixel size requires either a camera with tiny pixels, or extra magnification (150x objective lens; 2x or other "optovar" in microscope, 2x or other mag on camera adapter) - any additional lenses may decrease photon number or introduce abberations; Evident Scientific (ex-Olympus) offers a UAPON 150XOTIRF plan-apochromatic 150x / 1.45NA objective lens (with correction collar - if you have this lens, set the collar correctly!) Very good transmission 380-760nm (%T curve in data sheet) - likely better to use 150x lens and 1x additional mag (optovar, camera adapter) than 100x lens 1.5x additional mag - but quality also depends on how well the lens and microscope are maintained, coverglass thickness (high NA lenses are designed for exactly 170um coverglass, specimen directly at the coverglass); perfect matching [1.518] of refractive index of mounting medium - coverglass - immersion medium (unless doing TIRF, in which case prioritize mounting medium R.I. for TIRF -- and have a TIRF illumination syestem).
• GM note 2: spatial deconvolution (svi.nl Huygens etc) should be especially useful with tiny pixel size -- if algorithm(s) have been optimized for this and you acquire enough photons from the features of interest. The deconvolution algorims also do "some amount" of denoising. 

Although the pixel size of the digital image detector is often considered irrelevant to the conventional resolution limit, it has an impact on IbR. In scenarios with negligible sensor noise (e.g., readout noise), reducing pixel size can significantly increase information density-����. We observed this trend even when pixel size got smaller than that required by the Nyquist sampling theorem44,45. In wide-field microscopy, increasing the pixel size from 0.125 μm (0.2 AU) (Nyquist sampling pixel size) to 0.2 μm (0.33 AU) can reduce the ���� value from 20 rad−2·μm−2 to 12 rad−2·μm−2, roughly two-fold difference. This result underscores the importance of meeting Nyquist sampling pixel size requirement (Fig. 6). In addition, we investigated IbR in situations of applying a pixel size even smaller than that required by Nyquist sampling. We found that further reducing pixel size to 0.04 μm (0.07 AU) enhanced ���� by a quarter compared to the Nyquist sampling pixel size of 0.125 μm (0.2 AU). The presented findings suggest that grouping pixels—akin to using larger pixel sizes in a microscope—compromises the system’s effective resolution under photon-limited conditions, given an ideal scenario of zero camera readout noise. While microscope system essentially performs a low pass filter resulting in an diffraction limited image, pixelization (binning pixels) performs another layer of low pass filter on the image. The final image captured by the camera is thus a result of the image being filtered through these two sequential low-pass filters. The low-pass filter effect of pixelization is weaker compared to the OTF of the microscope system (Supplementary Note 3). Reducing the pixel size could improve the frequency transmission rate and increase the information density. Such improvement is obvious when the frequency is close to the diffraction limit boundary, while less pronounced when frequency is close to DC (Supplementary Fig. 15).

Discussion (entire section)

IbR is designed to establish a noise-considered theoretical resolution limit predicting the performance of imaging modalities with finite photon counts. The concept of IbR relies on the ideal image formation of a periodic object. In our calculation, we assumed the fluorescence response is linear, meaning the emission intensity is proportional to the illumination power. Thus, our IbR is not directly applicable to some of the super-resolution imaging modalities, such as single-molecule localization microscopy (SMLM)46,47,48,49 and stimulated emission depletion (STED) microscopy50,51, which rely on nonlinear fluorescence response. For example, SMLM requires the stochastic “blinking” of individual emitters. Thus, the resolution limit of SMLM relies on the exact on-off time sequences of imaged single molecules, which is challenging to summarize for IbR. The resolution of STED depends on the power of the depletion laser and its PSF. In an ideal situation where the depletion PSF has a perfect donut shape and infinite power, the resolution of STED can reach the molecule’s size51. IbR can potentially provide a method for assessing its practical resolution when providing the properties of the non-linear behavior of the probe and its physical model during depletion. In addition, another limitation of IbR is that it only quantifies the lateral resolution in a 2D structure with either planar or volumetric specimens.

While current resolution criteria are mostly defined as the smallest distance at which two closely spaced point objects remain distinguishable. In Abbe’s 1873 study, he concluded the resolution expression by examining the visibility of periodic grating structures, not point objects2. From a frequency perspective, evaluating resolution with grating structure is more appropriate. A sinusoidal wave structure, for instance, has only one frequency component pair besides the DC component. When these non-DC components surpass the diffraction limit, the structure vanishes, leading to a complete loss of resolvability. On the other hand, the spatial frequency spectrum of two-point objects extends infinitely. Consequently, even when the distance between two points gets closer beyond the diffraction limit (λ/2NA), there will not be a definitive distance at which they become unresolvable, since the remaining frequency components will still traverse the diffraction barrier (Supplementary Fig. 3).

In modern microscopy, raw data often undergo post-processing to form the image for visualization. It raises the question of whether such post-processing can increase the information and thus the resolution (e.g., IbR). To this end, we provided a theoretical derivation (Supplementary Note 2) showing that deterministic data post-processing methods cannot increase Fisher information. Therefore, post-processing methods, including image reconstruction algorithms, can either keep IbR constant or worsen IbR.

IbR provides a new measure of quantifying the practical resolving power of microscopy imaging modalities considering finite photons. The noise-considered resolution measure offers a theoretical and statistical reference for fluorescence microscope imaging modalities in photon-limited conditions. We believe IbR will become a new concept to provide theoretical guidance for advancement of novel microscopy methods.