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Abstract
Quantifying the resolution of a super-resolution image is vital for biologists trying to apply super-resolution microscopy in various research fields. Among the reported image resolution estimation methods, the one that calculates the full width at half maximum (FWHM) of line profile, called FWHM resolution, continues the traditional resolution criteria and has been popularly used by many researchers. However, quantifying the FWHM resolution of a super-resolution image is a time-consuming, labor-intensive, and error-prone process because this method typically involves a manual and careful selection of one or several of the smallest structures. In this paper, we investigate the influencing factors in FWHM resolution quantification systematically and present an ImageJ plug-in called LuckyProfiler for biologists so that they can have an easy and effective way of quantifying the FWHM resolution of super-resolution images.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Super-resolution localization microscopy (SRLM) is an important member in the family of super-resolution fluorescence microscopy techniques. Built from a relatively simple optical setup, SRLM can realize typically a spatial resolution of 30-50 nm, and thus provides great opportunities for investigating subcellular structures and functions [1]. Generally, to ensure a successful biological experiment and/or data interpretation, biologists need to confirm whether the spatial resolution obtained in a certain SRLM experiment matches the required resolution for their specific biological questions [2]. However, the traditional spatial resolution criteria based on the wavelength of light and the numerical aperture (NA) of objective lens are no longer capable of quantifying the spatial resolution of SRLM [3]. The reason is, in SRLM, the imaging quality of a reconstructed super-resolution image is not only dependent on the optical system itself, but also closely related to the sample properties (such as labeling densities, signal brightness, structure size, etc.) and the image processing algorithms [2,4,5]. In other words, when some experimental conditions change, the spatial resolution of SRLM would also change, even if there are no changes in the optical setup. Therefore, when using SRLM in studying a certain biological question, biologists are more concerned about the image resolution of final super-resolution images, rather than the traditional system resolution [6]. In this case, it is not surprised to see that quantifying the spatial resolution of a super-resolution image obtained from SRLM has become a hot topic in the past decade.
In SRLM, currently there are two main approaches for quantifying image resolution. The first approach analyzes the information in a localization table to present a convolved resolution [7–9] or Fourier Ring Correlation (FRC) resolution [10]. The convolved resolution is determined jointly upon localization precision and localization density [9]; however, this kind of prior knowledge is not easy to calculate accurately. The FRC resolution is determined using the distribution of localization points rather than the prior knowledge, and thus is being popular used in SRLM. The second approach for image resolution quantification analyzes a reconstructed super-resolution image to present Decorrelation resolution [2], Single FRC resolution [11], Full Width at Half Maximum (FWHM) resolution [12], or two-point resolution [13]. The Decorrelation resolution and the Single FRC resolution overcome the problem in the FRC resolution, where two sets of localization points under the same experimental conditions are necessary. But, these two newly-reported methods for image resolution measurement require more trial use before being widely accepted within the SRLM community. The FWHM resolution includes Single-line FWHM resolution (calculated from a single line profile) and Projected-line FWHM resolution (calculated from the projected profile of a long line) [14], while the two-point resolution is calculated from two adjacent line profiles [15]. The FWHM resolution and the two-point resolution are both traditional resolution measurement methods. The former is derived from Houston's criterion, and the latter is directly from Rayleigh's criterion [16]. These two traditional resolution methods can be categorized into line profile-based resolution method, and have been widely used in various conventional optical microscopy and super-resolution optical microscopy techniques for a long time. The reason for this is simple: the physical meanings of line profile-based resolution methods are clear and the results can be easily understood by biologists.
However, in practice, the line profile-based resolution methods require researchers to find out carefully the finest structures from a super-resolution image, which is time-consuming, labor-intensive and error-prone [17]. In 2014, to solve the problem of manual selection, researchers developed two point spread function (PSF) measurement tools called PSFj [18] and PSFtracker [19] to measure the line profile-based resolution automatically using fluorescent beads. However, these two tools are only suitable for measuring resolution from traditional fluorescence microscopy. For SRLM, we still have no such tools. Moreover, a reconstructed super-resolution image in SRLM usually contains discontinuous or narrow structures. Selecting such structures for calculating line profile-based resolution will result in resolution overestimation, that is, a smaller resolution value than the true one is obtained. Clearly, in SRLM, developing an automatic and effective way for quantifying line profile-based resolution remains an important but unmet goal for researchers, especially biologists who are not familiar with image processing.
In this paper, we analyzed the influencing factors in several methods for characterizing the line profile-based resolution. We verified that the Projected-line FWHM resolution, which calculates image resolution using the projection from a single and relatively-long line, is more effective in determining the image resolution of a reconstructed super-resolution image. Based on this finding, we developed an ImageJ plug-in called LuckyProfiler, which is capable of automatically determining the position of a very small structure from a reconstructed super-resolution image. From this structure, the calculated Projected FWHM resolution is almost the smallest, thus avoiding the overestimation problem in line profile-based resolution quantification.
2. Method
2.1 Experimental datasets
2.1.1 Cell culture and sample preparation
U2OS cells were first cultured in McCoy's 5A medium with 10% fetal bovine serum, 100 U/ml penicillin and 0.1 mg/ml streptomycin in an environment containing 5% CO2 at 37 °C. Cells were then seeded on glass dishes (MatTek P35G-1.5-14-C) and incubated for 24 hours before use. Next, cells attached to the glass dish were fixed in 4% paraformaldehyde solution for 15 minutes, and then washed 3 times with PBS for 5 minutes each time. Subsequently, the fixed cells were incubated for 10 minutes in permeabilization solution (0.2% Triton X-100 in PBS). Afterwards, the fixed cells were washed again with PBS for 5 minutes and blocked with 3% BSA in PBS solution for 1 hour. The fixed cells were then labelled with anti-α-tubulin monoclonal antibody (T5168, Sigma) for 1 hour at room temperature, and washed 3 times with PBS for 10 minutes each time. Finally, the fixed cells were labelled with Alexa Fluor 647-conjugated secondary antibody (A-21236, Invitrogen) for 1 hour at room temperature, and washed 5 times with PBS for 10 minutes each time.
2.1.2 Imaging and data processing
Before imaging, cells were mounted with imaging buffer consisted of 10% (w/v) glucose, 100 mM cysteamine, 0.5 mg/mL glucose oxidase, and 40 µg/mL catalase in Tris-HCl (pH 8.0). SRLM imaging was then performed on a custom-built setup that is based on an Olympus IX73 inverted microscope. A 640 nm excitation laser (∼5 kW/cm2) was used to excite Alexa Fluor 647, and a 405 nm laser was used to control the activation density of the fluorophores. The two lasers used in the experiments were purchased from LaserWave, China. The fluorescence from Alexa Fluor 647 was collected by an Olympus 60×/NA 1.42 oil immersion objective, transmitted through a dichroic mirror and an emission filter (FF01 680/42, Semrock), and finally imaged by a Hamamatsu Flash 4.0 V3 sCMOS camera. The exposure time was 10 ms. Finally, raw images were processed to obtain localization tables, which were further used to reconstruct super-resolution images. All image processing was performed by QC-STORM [20].
2.2 Simulation datasets
2.2.1 SRLM images of straight line structure
We simulated SRLM images with straight line structures and different structure sizes. The radius of the lines was increased from 10 nm to 100 nm. The PSF of the imaging system was approximated with a Gaussian function. Other parameters are listed in the following: localization precision is 30 nm, labeling density is 0.1 µm-2, frame number is 10000, and signal-noise-ratio (SNR) is 100.
2.2.2 SRLM images of two adjacent emitters
We simulated SRLM images from two adjacent emitters with different separation distances (ranging from 15 nm to 80 nm). According to the literature, the localization precision was set to be 30 nm [4], and was kept the same for both emitters. The fluorescence intensity ratio of the two emitters was increased from 0.3 to 1. The SNR was 50.
2.2.3 SRLM images of microtubules with a crossing structure
We used SuReSim [21] to simulate fluorescence images of microtubules with a crossing structure. The parameters were as follows: the radius of the microtubules was 12.5 nm, the localization precision was 10 nm, the labeling efficiency was 10%, the detection efficiency was 100%, and the pixel size of the reconstructed images was 10 nm.
2.3 LuckyProfiler
2.3.1 Working principle of LuckyProfiler
LuckyProfiler firstly selects a default number of relatively small structures in a reconstructed super-resolution image, and calculate the skeleton, boundary and angle of each selected structure (Fig. 1 (a)). Then, based on the normal directions and the centers of different positions in each structure (Fig. 1 (b)), all intensity distributions within a certain length of each structure are aligned and projected (Fig. 1 (c)). Finally, the projected profile of each structure is used to calculate its Projected-line FWHM resolution (Fig. 1 (d)). The smallest resolution value in these structures is reported as the image resolution of the reconstructed super-resolution image. Details of these steps are described below.
Fig. 1. Schematic diagram of the working principle of LuckyProfiler. (a) Illustration on skeleton, boundary and structural radius (R1 and R2). (b) Single Line Profiles along the normal directions. (c) Projected line profiles from two different projection lengths. Here Lproj 2 is longer than Lproj 1. (d) Illustration showing that a projected line profile is convolution of a rectangular function (structural width) with a Gaussian function (the image resolution).
Step 1: Calculate structural skeleton. We use the Fast Marching method [22] to obtain the skeleton of a default number of selected structures in a super-resolution image, use the Laplacian operator to obtain the boundary of the structures, and use gradient to calculate the normal vector directions of each structure.
Step 2: Obtain projected line profile. For each structure, we calculate the distance (or called radius, R) between the center of the skeleton to the boundary, along the normal direction. Here we use the characteristics of skeleton to calculate the radius R. Specifically, a line is drawn from the center of the structure (that is, skeleton) to its boundary in both sides, according to the normal direction at each skeleton position. When both sides of this line reaches the boundary, the radius of the line is considered to be R. It is then easy to obtain the intensity values of the corresponding pixels based on the skeleton (structure center), angle (how to draw a line profile) and boundary (where to stop), and from the line at each skeleton position. Moreover, according to the projection length obtained in Step 3, the line profiles at the adjacent positions are superimposed to obtain the intensity distribution (projected line profile) with the center of the structure as the projection center.
Step 3: Determine the optimal projection length. We randomly select several positions (M) in the skeleton map. Then, we use Gaussian fitting to calculate the Projected-line FWHM resolution of these positions. For each position, we increase the projection length gradually until a stable value in the Projected-line FWHM resolution is obtained. For each position, we also calculate the average radius (Rave) within the optimal projection length (Lproj).
Step 4: Calculate the image resolution. We sort the Rave obtained in Step 3, and determine N candidate positions (P) with smallest structural radii. Then, we use Gaussian fitting to calculate the Projected-line FWHM resolution from these positions. The optimal projection length determined in Step 3 is used in this step, and the smallest Projected-line FWHM in these N positions is reported as the image resolution of this reconstructed super-resolution image.
Step 5: Calculate the deconvolved image resolution. This step is designed for minimizing the influences of structure size on image resolution. It is suggested that deconvolution should be done when the structure size is larger than 0.3 times the image resolution [18]. In this case, according to the literature [23], we should apply deconvolution to the projected line profile. Here we use the least squares method (LSM) to calculate the image resolution for structures with large size. In principle, a reconstructed image can be regarded as a convolution of a rectangular function (structural width) with a Gaussian function (image resolution) (see Fig. 1(d) and Eq. (2)), where parameter r and $\vec{\theta }$ need to be solved. Here, the structure is modeled as a skeleton with a specific width (Eq. (3)), which is represented as Sigmoid equation (Eq. (4)-5). Note that Sigmoid function is used to approximate rectangular function, because the former is a nonlinear and continuous derivation function, and thus is easier in programming. The PSF of the optical system is approximately represented as Gaussian function (Eq. (6)). After solving these equations (using global optimal solution obtained from multiple initial values), we obtain the standard deviation (δ) of the Gaussian function, and the image resolution quantified by the projected-line profile FWHM resolution, and express the FWHM resolution as FWHW = 2.35×δ.
We note that LuckyProfiler is suitable for measuring the FWHM resolution of line structure, ring structure and other samples with obvious structural features, because LuckyProfiler needs structural features to perform image projection. For samples without linear structures, LuckyProfiler calculates the image resolution by quantifying the projection of the main structure. For example, for the nuclear pore complex (NPC) structures that are composed of 8 nuclear pores, we need to deal with three different cases: 1) If the image resolution reaches 40 nm, each nuclear pore in the image should be visible, and LuckyProfiler calculates the projected-line FWHM resolution in each nuclear pore along the normal direction; 2) If the image resolution is between 40 and 100 nm, the NPC presents a ring-like structure in the image, and LuckyProfiler calculates the projected-line FWHM resolution using projected structures along the ring structure; 3) If the image resolution is greater than 100 nm, the NPC in the image looks like a bead, and LuckyProfiler calculates the image resolution using the projection of the bead. Moreover, for other samples that may not have enough structure length for projection, LuckyProfiler uses the longest projection length available to calculate the Projected-line FWHM resolution.
2.3.2 ImageJ plug-in
LuckyProfiler was compiled with Java 1.8, and is compatible with ImageJ 1.53s or later. The plug-in was tested on Windows 10, and was confirmed to work well with ImageJ, Fiji and µManager. In LuckyProfiler, OpenCV library is used for image processing, and apache-commons-math3 library is used for fitting.
3. Result and discussion
3.1 Comparing different line profile-based image resolution methods
As mentioned in the Introduction section, image resolution quantified from the line profile-based methods include mainly Single-line FWHM resolution, Projected FWHM resolution and two-point resolution. We used these methods to analyze the image resolution of a typical super-resolution image from microtubules (Fig. 2(a)). We found that the resolution values are 50 nm for the Single-line FWHM resolution (Fig. 2(b)), 80 nm for the two-point resolution (Fig. 2(c)) and 81 nm for the Projected FWHM resolution (Fig. 2(d)), respectively. Obviously, in this super-resolution image, the calculated Projected-line FWHM resolution and the two-point resolution are significantly larger than the Single-line FWHM resolution.
Fig. 2. Comparing three image resolution calculation methods based on line profile. (a) A reconstructed super-resolution image of microtubules from U2OS cells. The image marked by the while square in the lower-left of the image was enlarged into the sub-image showed in the upper-right, and the structure marked by the line in this sub-image was furthered analyzed to present the distribution in the upper-right. The calculated Single-line FWHM resolution is also shown in the distribution. (b) Single-line FWHM resolution of the structure marked by the green line in (a). (c) Two-point resolution of the structure marked by the blue line in (a). (d) Projected-line FWHM resolution of the structure marked by the yellow rectangle in (a). Scale bar: 2 µm (a), 100 nm (the inset in (a)).
Theoretically, the Single-line FWHM resolution (derived from Houston's criterion) and the two-point resolution (derived from Rayleigh's criterion) obtained from the same sample should have only a slight difference. Therefore, we took a closer look at the super-resolution image in Fig. 2, considered carefully the measurement processes for these two kinds of image resolution, and figured out three possible reasons for the significant difference. Firstly, researchers need to choose a most suitable structure for calculating the Single-line FWHM resolution; otherwise, the image resolution may be wrongly estimated. For example, selecting a small structure from fluorescent contaminants or insufficient labeling density (see the inset of Fig. 2(a)) may result in an overestimated image resolution. Secondly, according to the Rayleigh criterion, when calculating the two-point resolution, the fluorescence intensities of the two adjacent structures need to be equal, and the central dip from the intensity distribution needs to be equal to 26%. Obviously, the structures used to quantify the image resolution in Fig. 2(c) didn't satisfy these conditions. Note that it is very difficult to find structures that meet the conditions described in Rayleigh's criterion. Finally, the finding that the Projected-line FWHM resolution is significantly larger than the Single-line FWHM resolution may be related to the broadening of the intensity distribution caused by the curvature of the structure used. In theory, researchers could pick up a straight line structure to avoid such a broadening effect. However, in actual experiments, the complexity of biological structures and the manual selection of biological structures make it difficult to guarantee the straightness of the structures.
3.2 Influencing factors in quantifying two-point resolution
As mentioned in Section 3.1, two-point resolution should be characterized using two adjacent structures with equal intensity and suitable central dip. To investigate the influencing factors on the measurement of two-point resolution in SRLM, we simulated two adjacent emitters with the same localization precision (30 nm). Here we used two adjacent emitters instead of two adjacent structures, because discontinuous structures and uneven intensity will further complicate the problem. To help further analysis, we firstly introduced the concept of central dip, and defined it as the intensity ratio between the valley gap (in the center) and the lower peak (Fig. 3(a)). Then, to quantify the effect of intensity ratios, we simulated an image with two adjacent emitters, where the ratio of their intensities was between 0.3-1 and the distance between them was 15∼80 nm. Note that the central dip is positively correlated with the distance. If we still require a central dip of 26.4% (Rayleigh’s criterion) to calculate the two-point resolution, the distance would be 30 nm for an intensity ratio of 1, and 35 nm for an intensity ratio of 0.4, respectively (Fig. 3 (b)). These findings point out the necessity of considering the relationship between intensity ratio and central dip in quantifying two-point resolution.
Fig. 3. The relationship between intensity ratio and central dip. (a) Illustration on the concept of central dip. (b) The relationship among central dip, separation distance and intensity ratio. (c) Overlaid image and line profiles of two adjacent emitters with 30 nm separation and two different intensity ratio (1 and 0.4). (d) The relationship between central dip and intensity ratio. Scale bar: 30 nm (a) and (c).
To further investigate this relationship, we simulated two emitters with fixed distance (30 nm, also considered as the ground truth) and different ratios. Gaussian fitting was used to analysis the profile. When the intensity ratio is 1, we found that the fluorescence intensity distribution of the two emitters can still be separated from the summed fluorescence distribution, although the noises from the sample or the camera may distort the profile. When the intensity ratio is < 0.4, the summed fluorescence intensity distribution mimics the intensity distribution from the emitter with stronger signal (Fig. 3(c)). We further observed that, when the intensity ratio is less than 0.4, the central dip is almost zero. Starting from a ratio of 0.4, the central dip increases linearly with the increase of the ratio (Fig. 3(d)). Ultimately, when the ratio satisfies the Rayleigh’s criterion (that is, an intensity ratio of 1), the central dip reaches 26.4%. These results verified again that, when calculating two-point resolution, we should choose carefully a suitable structure based on the fluorescence intensity ratio and the central dip.
3.3 Influencing factors in quantifying single-line FWHM resolution
Single-line FWHM resolution has been popularly used to quantify the image resolution in various super-resolution microscopy techniques, including but not limited to SRLM. However, Single-line FWHM resolution is easily overestimated or underestimated. The overestimation is mainly due to the selection of a structure that is originated from incomplete or inappropriate sampling, and thus can be minimized by using the projected distribution of a continuous structure. The underestimation is mainly due to the structural broadening, which is originated from non-negligible structure size, and structural bending if the Projected-line FWHM resolution is used. If the structural broadening is not handled carefully, the calculated Single-line FWHM resolution will be larger than the true image resolution, resulting in an underestimated image resolution.
Early study in fluorescence microscopy has already pointed out that, when the structure size is smaller than 0.3 times the image resolution, the projected profile is dominated by PSF [18]. In this case, Gaussian fitting is sufficient to estimate approximately the image resolution. For SRLM, the image resolution is usually around 30-50 nm. This means, the structure size should be less than 15 nm, if we want to use Gaussian fitting in image resolution calculation. However, the structure size would be usually larger than 30 nm if conventional immunofluorescence labelling is used [21,23]. In this case, we need to use deconvolution analysis to extract the image resolution properly (details can be seen in Section 2.3.1). Here, to quantify the influence factors of structure size on Single-line FWHM resolution, we simulated straight lines with different radius. Typical images with 10 nm, 30 nm and 50 nm radius are shown in Fig. 4(a). Clearly, when the structure size increases, the experimental line profile is gradually flattened, and Gaussian fitting becomes unsuitable for measuring Single-line FWHM resolution, as compared with Deconvolution analysis (Fig. 4(b)). In the simulation, the image resolution values (that is, Single-line FWHM resolution) calculated by Deconvolution analysis are all around 30 nm, while the image resolution values calculated by Gaussian fitting range from 30 nm to 170 nm (Fig. 4(c)). That is to say, to overcome the underestimation issue in image resolution calculation, deconvolution analysis would be necessary in many super-resolution images from SRLM.
Fig. 4. The dependence of structure size on image resolution. (a) Simulated super-resolution images with different structure sizes. The radii of the lines are shown on the top of each line. Scale bar: 200 nm. (b) The corresponding experimental line profiles (Experiment), Gaussian fitted line profiles (Gaussian) and deconvolution PSF (deconvolution) in (a). (c) The relationship between structure size (radius) and image resolution calculated by Gaussian fitting and Deconvolution.
3.4 Influencing factors in quantifying projected-line FWHM resolution
In SRLM, a reconstructed super-resolution image often includes discontinuous and narrow structures due to various reasons in sample preparation (for example, labeling efficiency, photobleaching, insufficient sampling, etc) and molecule localization. Quantifying the Single-line FWHM resolution with these structures may result in a smaller image resolution value than the actual one, thus leading to resolution overestimation. Quantifying the Projected-line FWHM resolution is a good way to avoid this overestimation. However, there is no research on how to optimize the projection length (Lproj) for a certain structure. Here, we used a simulated image with cross structure (Fig. 5(a)) to study the relationship between projection length and Projected-line FWHM resolution. We found that, when the projection length increases from 100 nm to 2000nm, the projected profile gradually increases, and eventually becomes stable (Fig. 5(b)). Then, we analyzed the projected profile via both Gaussian fitting and Deconvolution. The image resolution value and coefficient of variation (C.V.) shows that the Projected-line FWHM resolution calculated by Gaussian fitting becomes stable (∼10% C. V.) at 61 nm, when Lproj is larger than 500 nm (Fig. 5(c)). For the Projected-line FWHM resolution calculated by Deconvolution, the values are stabilized at 48 nm, when Lproj is larger than 2000nm (Fig. 5(d)). That is to say, when the Projection-line FWHM resolution is used for quantifying image resolution, it is necessary to optimize the projection length for a stable image resolution value.
Fig. 5. The dependence of projection length on Projection-line FWHM resolution. (a) Illustration of a typical super-resolution image with cross structure. Scale bar: 500 nm. (b) Line profiles of the yellow structure in (a) with different projection lengths shown on the right. (c) The dependence of Projection-line FWHM resolution (calculated by Gaussian-fitting) and coefficient of variation (C. V.) on projection length. (d) The dependence of Projection-line FWHM resolution (calculated by deconvolution) and coefficient of variation (C. V.) on projection length.
After a close look at Fig. 5(c) and Fig. 5(d), we noticed that the image resolution calculated by deconvolution has a larger C.V. value, as compared with Gaussian fitting. This difference is easier to observe when Lproj is smaller than 500 nm. This finding shows that, although the deconvolution method may provide better image resolution calculation in SRLM, a relatively long profile is required; otherwise, the image resolution obtained from the deconvolution method would be noisy or unstable. Therefore, as described in Section 2.3, in LuckyProfiler, we firstly use the Gaussian fitting method to identify a position with the smallest image resolution, and then apply the deconvolution method to the projected line profile in that position (with a sufficient projection length) to improve the precision in the image resolution calculation. After considering different application needs, we allow users to choose either the Gaussian fitting method or the deconvolution method in the LuckyProfiler plugin (see Fig. 6).
Fig. 6. The user interface of and results from LuckyProfiler. (a) The user interface. (b) A simulated image for testing the plug-in. Each alphabet in this image has a structure radius, as shown on the top of each alphabet. (c) The Projection-line FWHM resolution of the chosen region (“L”) in (b). Scale bar: 500 nm.
3.5 User interface and features of the ImageJ plug-in — LuckyProfiler
We took advantage of the findings in the previous sections, and developed an ImageJ plug-in called LuckyProfiler for image resolution quantification. The user interface of the LuckyProfiler plug-in is presented in Fig. 6(a). The plug-in can automatically calculate the image resolution of a final SR image or an ROI. The only parameter we need to know is the pixel size for super-resolution image reconstruction. To test the capability of the LuckyProfiler, we generated a simulated dataset consisting of the word “Lucky” (Fig. 6(b)). “Lucky” consists of five letters, and the structural radii of the letters are set to 0 nm, 10 nm, 20 nm, 30 nm and 40 nm, respectively (Fig. 6(b)). In the simulation, the localization accuracy is 30 nm, the localization density is 0.01 µm-2, and the number of frames is 10000. A Gaussian function is used to render the localizations, and the SNR is set to 50. Results show that the LuckyProfiler can automatically find the best position for quantifying image resolution, that is, the letter “L” among the five letters. The image resolution obtained by LuckyProfiler is 31 nm (Deconvolution) or 34 nm (Gaussian fitting), which is close to the ground truth (Fig. 6(c)). These results show that the plug-in can automatically find the position suitable for quantifying the image resolution, and then present an accurate estimation on the image resolution.
3.6 Optimizing the projection length in LuckyProfiler
The LuckyProfiler plug-in calculates the Projected-line FWHM resolution rather than the Single-line FWHM resolution to avoid image resolution overestimation. Estimating image resolution from samples with sparser structures will require a larger projection length. And in our LuckyProfiler plugin, according to the structural characteristics and structural direction, we automatically determine a proper projection length to ensure a robust image resolution. The working principle of projection length optimization was described in Section 2.3. Here, we use a super-resolution image of microtubules (Fig. 7(a)) to show the effectiveness of LuckyProfiler in determining the optimal projection length (Lproj).
Fig. 7. Optimizing the projection length using LuckyProfiler. (a) A super-resolution image of microtubules. The yellow regions represent the projection of the structure. Scale bar: 500 nm. (b) The dependence of Projected-line FWHM resolution on projection length. (c) The dependence of coefficient of variance (C. V.) on projection length. (d) The distribution of the optimal projection lengths. (e) The dependence of optimal projection length on sampling size. (f) The relationship between coefficient of variance and sampling size.
Firstly, we calculated the Projected-line FWHM resolution for different projection lengths of a microtubule structure. We found that the Projected-line FWHM resolution gradually stabilizes with the increase of projection length (from 10 nm to 3000 nm). And, when the projection length is larger than 200 nm, the Projected-line FWHM resolution values calculated by both Gaussian fitting and deconvolution become stable (Fig. 7(b)). We took a close look at the results in Fig. 7(b) and realized that, we will easily obtain an overestimated image resolution if we calculate the Single-line FWHM Resolution, or the Projected-line FWHM resolution, using an insufficient projection length (less than 200 nm in this image). Moreover, we found that the image resolution values calculated by Gaussian fitting are almost all higher than those by Deconvolution, indicating the broadening effect from non-ignorable structural size. To further determine an optimal projection length, we calculated the dependence of coefficient of variation (variance divided by mean) on the projection length (see Fig. 6(c)). We found that, when the projection length is greater than 1.3 µm (about 20-25 times the image resolution of this image), the coefficient of variation becomes stable.
Then, we randomly selected a large number of positions, and obtained the optimal projection length at each position using the Projected-line FWHM resolution as a guide. The results are shown in Fig. 7(d). The average value of the optimal projection lengths at all positions is 1.3 µm, which is consistent with the results in Fig. 7(b) and Fig. 7(c). We also found that, when the sampling size increases from 10 to 950, the optimal projection lengths will stabilize at 1.3 µm (Fig. 7(e)), but the coefficient of variation of the optimal projection length becomes relatively stable only when the sampling positions exceed 100 (Fig. 7(f)). Therefore, in LuckyProfiler, we set the default sampling number to 100. In other words, LuckyProfiler will randomly select 100 sampling positions on a super-resolution image to calculate the optimal projection length. This setting ensures that the projection length used to calculate the image resolution is good enough to avoid image resolution overestimation. Note that this sampling size was found to be good for different sample types (data not shown).
We also note that, for structures that are relatively short and thus cannot support best projection length (like nuclear pore complex), the length of the projection should be optimized to be as large as possible. And, if there has no possibility to perform projection in some special structures, we can use the traditional method, that is, manually select a small ROI and calculate the Single-line FWHM resolution.
3.7 Calculating the projected-line FWHM resolution using LuckyProfiler
We noted previously the radius of a structure on a super-resolution image is positively correlated with the Projected-line FWHM resolution. Therefore, the structure with the smallest radius means the best position for image resolution calculation. Actually, as shown by the yellow data points in Fig. 8(a), when the averaged structure radius (Rave) increases, a larger Projected-line FWHM resolution can be found. We further investigated the dependence of image resolution on sampling size. To ensure the accuracy in image resolution quantification, we used LuckyProfiler to identify N candidate positions with smallest averaged structure radii, then calculated the Projected-line FWHM of these positions. The smallest FWHM resolution value was selected as the image resolution. As shown in Fig. 8(b), the FWHM resolution value changes from 49 nm to 45 nm when the sampling percentage (the number of sampling size divided by the population) increases from 1% to 4%, and is kept stable when the sampling percentage is larger than 4%. Here we obtained the minimal image resolution value from the selected samples which have small Rave values. The results in Fig. 8(b) indicate that we can estimate properly the image resolution of a super-resolution image using a sampling percentage of 4%. Note that, in this super-resolution image, the total sampling size is 2603, and a sampling percentage of 4% means a sampling size of 100, which is consistent with the results in Fig. 7(f). Under this sampling size, LuckyProfiler is capable of selecting a good position (Fig. 8(c)) for calculating image resolution. And, the image resolution calculated by Gaussian fitting and deconvolution are 45 nm and 36 nm, respectively (Fig. 8(d)).
Fig. 8. Calculating image resolution with LuckyProfiler. (a) The dependence of Projected-line FWHM resolution on averaged structure radius. The blue scattered points represent experimental results. The scattered points were divided into 8 regions based on their values, and averaged to present the data points in yellow. (b) The dependence of Projected-line FWHM resolution on sampling size. (c) The best location (indicated by the white rectangle area) determined by LuckyProfiler for image resolution calculation. Scale bar: 500 nm. (d) The projected line profile of the white rectangle area in (c). Gaussian fitting and deconvolution analysis were used to analyze the profile, and the calculated image resolution (FWHM) values are shown on the right side.
We also tested whether LuckyProfiler is applicable to more complex structures, for example, nuclear pore complexes (NPCs) which are good reference standards for quantifying the image resolution. We obtained a super-resolution image of Nup96, a well-known protein in NPCs, from the corresponding author of a published paper [24], and calculated the image resolution using LuckyProfiler. The resolution value (30 nm from both the Gaussian fitting method and the deconvolution method) and the corresponding position determined by LuckyProfiler are shown in Fig. 9(a). To verify the effectiveness of this image resolution, we also estimated two-point resolution from two manually selected positions (see Fig. 9(a), the enlarged area). From the cross-section of two adjacent Nup96 structures, we obtained a two-point resolution value of 31 nm and 33 nm for position 1 and position 2, respectively. Therefore, the image resolution value calculated by LuckyProfiler is almost the same as the two-point resolution. This finding verifies that LuckyProfiler is not only applicable to microtubules (uniform and tube-like structures that have been widely used in many papers related to resolution estimation), but also other complex structures (NPCs). However, the optimal projection length was found to depend on sample types. For example, the optimal projection length by LuckyProfiler for the NPCs is around 120 nm, which is different with that of microtubules (around 1 µm).
Fig. 9. Characterizing the image resolution in NPCs with LuckyProfiler. (a) A super-resolution image of Nup96. The best position identified by LuckyProfiler for calculating image resolution is marked by the green circle, and enlarged in the lower-left corner. A manually selected position for two-point resolution measurement is marked by the yellow line, and enlarged in the bottom. Position 1 and 2 in these enlarged area were further analyzed to present the results in (c). (b) Image resolution estimated by LuckyProfiler. (c) Two-point resolution estimated at position 1 (left) and position 2 (right). Scale bar: 500 nm.
Finally, it is worthy to note that, as explained in section 2.3.1, LuckyProfiler calculates image resolution using the line profile information, thus the quality of the super-resolution images are important for guaranteeing the measurement precision.
4. Conclusion
In this paper, three common line profile-based methods for calculating the resolution of a super-resolution image were analyzed and compared, and the influencing factors that should be paid attention to in each method were discussed. After analyzing simulated dataset and experimental images, we found that: 1) if the two-point resolution method is used to quantify the image resolution, the measured resolution value may deviate significantly from the actual value due to uneven signal intensities in the two adjacent structures; 2) The resolution measured by the Single-line FWHM resolution method may lead to overestimation due to incomplete or inappropriate sampling; 3) The Projected-line FWHM resolution method provides the best result in quantify the image resolution. Basing on these findings, we developed an ImageJ plug-in called LuckyProfiler to calculate the image resolution. LuckyProfiler is capable of finding out the best location automatically from a super-resolution image. LuckyProfiler uses the Projected-line FWHM resolution method to determine the resolution of the reconstructed image from the selected location, thus presenting an effectively way for quantifying image resolution.
Researchers usually consider that FRC resolution is a good way for quantifying image resolution in SRLM. However, several reports pointed out that the calculated FRC value can be affected by “bad localizations” from fluorescent background, camera noise, or repeated fluorescence from the same fluorophore [25–27]. And, unfortunately, these “bad localizations” are common in biological experiments and cannot be easily eliminated. Furthermore, another report pointed out that comparison of FRC resolution should be performed on similar sample structures and labeling density [28].
However, unlike FRC resolution or others that are based on Fourier transform, the FWHM resolution studies in this paper is based on line profile analysis from a super-resolution image. The physical principle of FWHM resolution is clear and easy to understand, and the FWHM resolution method is still widely used by many researchers. Therefore, we think it is still valuable to investigate the FWHM resolution method, especially for biologists who have a long history in using this method. We believe this study not only provides useful information for quantifying image resolution, but also a useful tool, the LuckyProfiler plug-in, for biologists to use SRLM easier and more effective.
Funding
National Natural Science Foundation of China (81827901); Hainan Province Science and Technology Special Fund (ZDYF2022SHFZ126); Start-up Fund from Hainan University (KYQD(ZR)-20077); Fundamental Research Funds for the Central Universities (2018KFYXKJC039); Wuhan National Laboratory for Optoelectronics.
Acknowledgments
We thank the Optical Bioimaging Core Facility of WNLO-HUST for technical support.
Disclosures
The authors declare no conflicts of interests.
Data availability
The plug-in presented in this paper is available in Ref. [29].
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