Resolution of diffraction-limited imaging systems using the point spread function

Characterizing the resolution of a diffraction-limited imaging system, such as a microscope, can be done in different ways. In this article, I propose using the point spread function (PSF) calculated in OpticStudio to obtain an objective measure of the resolution of these imaging systems. Two methods, which overlap the PSFs of two field points on the image (detector) plane are introduced. The first method uses the multi-configuration editor, and the second the image simulation tool. Both methods are compared, and their pros and cons are discussed.

Authored By David Nguyen with the help of Berta Bernard, and Chris Normanshire

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Introduction

Performance of imaging systems relates to their resolution, but the definition of resolution varies. In super-resolution microscopy, Fourier Ring Correlation [1] is used to evaluate the resolution. In diffraction-limited microscopes the resolution is estimated with the Rayleigh, or Sparrow criterion [2]. In practice, the resolution of those systems can also be measured with microbeads, chosen significantly smaller than the expected resolution, assuming one of the above criterions. These microbeads act as point emitters forming a PSF, which size gives an estimate of the resolution in the image, once again, this size varies depending on its definition. In this article, I use the PSF in OpticStudio to evaluate the resolution of diffraction-limited imaging systems more objectively.

Method 1: Multi-Configuration Editor (coherent imaging)

Microscope design

Throughout this article, I use a microscope design based on a TL4X-SAP objective lens (4X, 0.2 NA), and a TTL200 tube lens, as depicted in Figure 1. Both lenses available as black boxes from THORLABS website.

Figure 1 - Microscope design

Figure 1 - Microscope design composed of black box elements from THORLABS. The magnification is 4X, and numerical aperture (NA) is 0.2.

I use a Real Image Height field definition and specify five fields with equal areas over a square of X, and Y half-width of 6.656 mm, corresponding to 1.664 mm in the object plane. The field definition models a scientific  CMOS (sCMOS) detector with 2048 by 2048 pixels, and a physical size of 13.312 by 13.312 mm2, in the image plane. These detectors are commonly used in microscopy, and can be found in products such as the Orca-Flash4.0 V3 (Hamamatsu), or Zyla 4.2 plus (Andor) cameras. I also use the wavelength F, d, C (Visible) preset of OpticStudio. The optimization is performed using a criterion: RMS wavefront centroid with four rings, and six arms, as well as default air boundary constraints (0 to 1000 mm). Additionally, the system was constrained for telecentricity, and -4X magnification (in this microscope design, the image is reversed).

I have chosen this microscope design because it is relatively easy to setup and has real-life applications. For example, telecentricity is often required in machine vision applications (as discussed here). I have kept the optimization simple, and this article is not meant to describe microscope design in general. However, the results, and conclusions of this article apply to most imaging systems with conjugated object, and image plane, in which most contribution to diffraction occurs from the exit pupil, meaning that the Huygens point spread function (PSF) is a good representation of the system performance.

Multi-Configuration setup

To address the resolution of the microscope design, I create two point sources in my object plane, which I gradually separate with a distance close to the Rayleigh criterion, and I observe how their PSFs overlap in the image plane. In the microscope design, the Rayleigh criterion yRayleigh is calculated as

Rayleigh criterion

where λPrimary is the primary wavelength, 0.588 um, and NA is the numerical aperture of the objective lens, 0.2 (I do not disucuss about condenser NA in this article). While the Rayleigh criterion can serve as a measure of the system resolution, it assumes a perfectly circular, unaberrated aperture stop, and incoherent illumination (more details about the Rayleigh criterion can be found here). Additionally, the Rayleigh criterion is a subjective metric to establish the discernability of two PSFs, which actually depends on the observer, and the kind of information which needs to be retrieved from the microscope images, as we shall see in the remainder of this section.

I start by removing all but the on-axis field (Field 1), and I convert it to Object Height, as depicted in Figure 2.

Field setup

Figure 2 - Field setup for the Multi-Configuration method to analyse the microscope resolution. Only the on-axis field is kept, and it has been converted to Object Height.

Then, I create two configurations with a single YFIE operand, and I specify a value of 1.8e-3 mm in the second configuration, as shown in Figure 3.

Multi-Configuration Editor

Figure 3 - Multi-Configuration setup for the PSF overlap analysis. The two point sources are separated by 1.8 um in the object plane.

Finally, I use a Huygens PSF, and Huygens PSF Cross Section to analyse the overlap of the two PSFs in the image plane. Those two analyses can perform a coherent sum of the individual PSFs across the two configurations (see the Help File for more details). The analyses settings are displayed in Figure 4, and the peculiar Multi-Configuration setting is showed with a red box and arrow (this option is not available with the FFT PSF).

Huygens PSF settings

Figure 4 - Huygens PSF settings. By checking all configurations from the menu bar, a coherent sum is performed of the individual PSFs.

I focus the resolution analysis on the On-axis Field, but the same analysis can be conducted in every part of the field.

The results of the Huygens PSF are shown in Figure 5.

Huygens PSF overlap under Rayleigh criterion

Figure 5 - Results of the Huygens PSF, and PSF Cross Section overlap with an object plane Y-field separation of 1.8 um (Rayleigh criterion) in multiple configurations. The two point sources are hardly distinguishable by eye in this microscope design.

As one can see, the two field points are severly overlapped in the image plane, and their respective PSFs are nearly undistinguishable. Two reasons can explain this result. First, by performing a coherent sum of the PSFs, the incoherent illumination assumption of the Rayleigh criterion is violated, and causes a degradation of the resolution. Second, the OPD Fan shows aberrations in the order of 0.25 waves, and this microscope sits at the edge of the diffraction limit, meaning it is sufficiently diffraction limited to allow for analyses such as the Huygens PSF, but it still present some geometric aberrations, which alters the diffraction-limited performance of the system. In my experience, microscope designs, which maximize both field of view, and resolution tend to fall in that category of near-diffraction-limit systems, and are often difficult to characterize based solely on the Rayleigh criterion.

From the Rayleigh criterion, I can increase the separation distance of my fields, and re-evaluate the results. I have done it in Figure 6 with a separation of 2.3 um in object plane.

Distinct overlap

Figure 6 - Results of the Huygens PSF, and PSF Cross Section overlap with an object plane Y-field separation of 2.3 um in multiple configurations. By increasing the separation distance between the field points, the PSFs start to separate in the image plane, and one can observe two distinct peaks.

With a greater field separation, the resulting PSFs become distinguishable. The peak separation in the Huygens PSF Cross Section is almost 10 um, which is in agreement with the system magnification (4X). When I say "distinguishable", it is a qualitative assessement of what I see in Figure 6. However, this criterion can be made more objective if one defines how the peaks should be separated in terms of post-processing. For example, a criterion could be "I want to be able to apply a threshold at 80% and detect two separate spots", in which case, one can use OpticStudio to optimize the peak interdistance to correspond to 80% of the maximum relative irradiance (this is out of the scope of this article).

Lastly, we can also account for the physical pixel size of our detector to get an image as seen from the microscope. The PSFs have a full-width at half maximum of approximately 12 um, and our hypothetic detector has a physical pixel size of 6.5 um, clearly violating Nyquist-Shanon's sampling theorem, which is yet another limitation of the microscope design. Figure 7 shows the Huygens PSF results when the image sampling is changed to 32 by 32 pixels with an Image Delta (the physical pixel size) of 6.5 um.

Pixelated overlap

Figure 7 - PSFs overlap when accounting for the physical pixel size of the detector. Too few pixels compose the PSFs overlap and further degrade the resolution of the microscope.

As one can see, the inadequate physical pixel size further degrades the resolution of the microscope, and while the two peaks were distinguishable in Figure 6, they are now overlapping again in Figure 7. In this case, the microscope resolution is said to be pixel-limited, and is given by, at least, twice the pixel size scaled by the magnification, meaning 3.25 um (two times 6.5 divided by 4). The result of a 3.25 um separation distance between the fields in object plane is depicted in Figure 8.

Distinct pixelated overlap

Figure 8 - PSFs overlap when accounting for the physical pixel size of the detector. A separation of 3.25 um allows to separate the close fields again. This distance corresponds to twice the pixel size divided by the magnification, a consequence of Nyquist-Shanon's sampling theorem.

By accounting for the detector pixel size, a greater separation is needed to avoid aliasing of the PSF, and ensuring it is represented by at least 2 pixels. The fields separation of 3.25 um, is quite different from the 1.8 um Rayleigh criterion, and shows just how ambiguous the definition of resolution can be, and yet I have not considered tolerancing of the microscope in this article, which would further reduce this metric.

The method presented in this section considered the coherent sum of two nearby PSFs, and how these could be distinguished from one another. While this method is suited to coherent imaging systems, it is generally more conservative for incoherent imaging systems, and could lead to over-performing designs, which are inherently more costly. For example, in fluorescence microscopy, the resolution is measured with fluorescent microbeads. The light emitted from fluorescent microbeads is often considered to be incoherent. In this case, the performance of the microscope is expected to be more in line with the Rayleigh criterion.

In the next section, I show how to use the Image Simulation feature of OpticStudio to investigate the resolution of the microscope design under the incoherent illumination assumption by summing PSFs from closely located field points incoherently.

Method 2: Image Simulation (incoherent imaging)

For this method, I can keep the original microscope design with its 5 field points, and I convert those to Object Height (Figure 2). Before I delve into the description of the Image Simulation method, I'd like to inform the reader that I do not cover the details of this feature of OpticStudio, and I assume that you are familiar with it, and are able to set it up properly (for more details about Image Simulation, I refer the reader to this knowledgebase article).

The first step to set up the Image Simulation is to provide it with an input image file. Since Image Simulation essentially performs a convolution of the microscope PSF with this input image, I want to have a Kronecker delta in my input image to model my closely-separated fields. In other words, my input image is a completely black (zero-valued pixels) background, with a single white (maximum-valued) pixel at the position of each field. The size of the pixels is made as small as possible while keeping an image size which leads to decent computation time (in the order of minutes). I choose to represent approximately four times the area of the Huygens PSF presented in Figure 5-8, that is 400 by 400 um2, which makes for an intrinsic guard band. This area is scaled with the magnification to correspond to a field size of 100 by 100 um2 (remember, the magnification is 4X). I also choose my image size to be 1600 by 1600 pixels, hence my pixel dimension is 0.0625 by 0.0625 um2, well below the Rayleigh criterion of 1.8 um. I create six white pixels in my image which correspond to three cases of two nearby fields. The three cases are approximately the same as investigated in the first method: 1.8 (29), 2.3 (37), and 3.25 (52) um (pixels), respectively. A zoom in the region of the input image containing the white pixels is shown in Figure 9, and the image is available from the article attachements (PointSources.BMP).

Input image

Figure 9 - Zoom in the input image showing the six white pixels corresponding to three pairs of field separated by 1.8 (29), 2.3 (37), and 3.25 (52) um (pixels), respectively. The vertical separation is 6.25 (100) um (pixels), and avoids overlap between the field pairs. The upper pixel in the middle is located at coordinate (800, 800) in the image.

The Source Bitmap Settings of Image Simulation are shown in Figure 10.

Source bitmap settings

Figure 10 - Image Simulation settings for the Source Bitmap. The field height is 100 um (0.1 mm in lens units), and the input image is centered on the On-axis Field.

The Field Height is 100 um. I also focus the analysis on the On-axis Field for this method, but the same analysis can also be conducted at different field positions, and I use the combination of wavelengths 1 to 3, currently defined in the System Explorer.

The PSF grid settings are shown in Figure 11.

PSF Grid Settings

Figure 11 - Left: Convolution Grid Settings of the Image Simulation feature. It uses a single PSF on-axis due to the retricted field where the analysis is conducted. The sampling settings are chosen the same as for the Huygens PSF method. Aberrations are set to Diffraction. Right: resulting PSF Grid as shown by the Image Simulation feature.

I use the same sampling settings as the Huygens PSF method, and a single PSF on-axis. I am doing so because the Field Height is 100 um, which corresponds to a fraction of the field of view where I don't expect my PSF to change much across the field. The results of Image Simulation are shown in Figure 12 with default Detector and Display Settings (all zero), these reuslts can take several minutes to be computed.

Image simulation results with default detector settings

Figure 12 - Image Simulation results with default Detector and Display Settings (all zero). The first row depicts the intensity profile across the center-line going through both PSFs, and the second row is the Image Simulation output for (A) 3.25, (B) 2.3, and (C) 1.8 um separation between the fields. The image is reversed due to the optical property of the microscope (negative magnification).

From Figure 12 (C), one can observe that a separation of 1.8 um, the Rayleigh criterion, makes the two PSFs distinguishable. There is a small dip of approximately 15% in intensity, which can be used to post-process the image with a threshold for example. The greater the distance between the fields, the better the separation. When compared to the Huygens PSF method, which uses the coherent sum of PSFs across configurations, the results are better in terms of resolvability of the PSFs. However, we have not considered the physical size of the detector yet. The Image Simulation settings can be adjusted like in Figure 13 to account for the detector characteristics.

Detector settings

Figure 13 - Detector and Display Settings when considering the physical size of the microscope detector.

The result of accounting for the detector physical properties is shown in Figure 14, these reuslts can take several minutes to be computed.

Pixelated Image Simulation due to detector physical properties

Figure 14 - Image Simulation results accounting for the physical size of the microscope detector. Even a separation of 3.25 um in the object plane does not separate the PSFs. The vertical separation of 6.25 um makes the field pairs distinguishable.

The separation between closely located fields is not possible anymore, and one can observe that strictly using Nyquist-Shanon's sampling theorem to determine the pixel-limited resolution is often not sufficient. I am probably lucky to have found the results of Figure 8, and those might suffer from aliasing. The vertical separation between the field pairs is 6.25 um in the object plane (Figure 9), and makes for a clean separation between those pairs. Therefore, I assume the resolution is in between 3.25, and 6.25 um. A further investigation shows that a separation of 5.125 um gives a visual, qualitative, separation of the source points, as shown in Figure 15.

Pixelated Image Simulation due to detector physical properties

Figure 15 - Image Simulation results for a pair of points separated by 5.125 um in the object plane. Two brighter pixels seem to be qualitatively distinguishable.

Once again, this is an arbitrary criterion, and it would be better to define it in terms of post-processing needs, but I hope I made it clear that various aspects of the design contribute to the resolution, and there needs to be a clear definition to this concept to ensure proper assessement of the design performance, which will also help testing down the line. I emphasizes once again, that I did not discuss tolerancing, which will further reduce the performance of actual microscopes.

References

[1] N. Banterle, K. HuyBui, E. A.Lemke, and M. Beck, "Fourier ring correlation as a resolution criterion for super-resolution microscopy," J. Struct. Biol 183 (3), pp 363-367 (2013)

[2] J. S. Silfies, and S. A. Schwartz, "The Diffraction Barrier in Optical Microscopy," Retrieved from: https://www.microscopyu.com/techniques/super-resolution/the-diffraction-barrier-in-optical-microscopy on November 12, 2019.

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