1. Introduction

Structured illumination microscopy (SIM) is the method of choice for high-resolution live cell imaging, because of its large field of view, low laser intensities and thus low phototoxicity. Standard 3D- (three-beam) SIM has a lateral and axial resolution of about 110 and 360 nm respectively [1–3]. For a single focus slice, commercial systems can achieve 10 high resolution frames per second [4].

In SIM, a fine sinusoidal shaped illumination pattern excites the fluorophores in the specimen. The illumination pattern encodes high spatial frequencies outside the passband of the optical transfer function (OTF) into a low frequency signal so that they can be transferred by the imaging system and contribute to the final image. Several raw images with a phase shifted excitation pattern have to be acquired to enable a decoding of the original information. The procedure has to be repeated in several directions, since the encoding is, like the illumination pattern, not isotropic. 3D-SIM uses 5 phases and 3 directions per focus slice [5].

The decoding is done by a reconstruction algorithm that assumes a static specimen. Any deviation from this assumption, such as a movement, leads to a reconstruction artefact. However, small structures, such as cell components (e.g. actin filaments) can move quickly in living cells and produce these artefacts [6]. The larger the movement, the worse the artefact. It is impossible to generally distinguish between real high-resolution features in an unknown specimen and an artefact [1,7–9].

Appendix 5.2 gives an overview of the velocities of various cellular components and the typical acquisition time of a fastSIM system [10–12]. It can be seen that some cell processes happen too fast to be imaged with SIM. Hence, research on increasing the frame rate and thus the maximum allowed velocity is still necessary [13,14].

Shao et al. claim that motion artefacts emerge if the specimen moves over a distance larger than the resolution of the microscope during acquisition [15]. This limit is certainly right in classical wide-field microscopy, where a movement smears the image only, which is unnoticeable as long as the smear is smaller than the resolution. However, the reconstruction process of SIM consists of several complex operations, so that artefacts may arise already at much lower velocities. Finding a verified maximum velocity for artefact-free SIM imaging enables the possibility to decide whether a biological process can be imaged with high quality and confidence or not. Ströhl et al. investigated the resolution capability of 2D- (two-beam) SIM on a moving double-slit. However, they analyze a two-dimensional image only. In addition, it is unclear whether the results of a slit can be applied on objects of different shape [13]. Thus, we use single beads as test objects instead, to gain a more general insight.

We previously reported a method to detect and locate motion in 3D-SIM data [16]. The algorithm applies the frame-difference method on two independent wide-field images, which are created out of the conventional 3D-SIM raw data in a standard z-series configuration, without the need for extra images. A probability theory based analysis distinguishes between movement and noise using a threshold. Yet, the threshold has been set empirically and needed further investigation to fully automatize the motion artefact detection algorithm. Here we present a thorough signal-to-noise (SNR) analysis to obviate the need for a user-defined empirical threshold.

2. Theory and simulations

2.1. Test object and imaging parameters

To link an objects velocity quantitatively to an artefact, we designed a simulated 3D test object that consists of 25 individual beads in the center z-slice (shown in Appendix 5.3). Studying single beads is sufficient, because fluorescent microscopy is incoherent and SIM image reconstruction is essentially linear. Other shapes can be understood as a linear combination of single beads. Each simulated bead moves horizontally with its own constant velocity v through the specimen during the entire simulated acquisition procedure. One bead does not move yielding an ideal SIM reconstruction for comparison. Only lateral movements are investigated, because an axial movement only stretches or compresses the PSF along z, with negligible lateral artefacts [15]. The simulation uses the experimental parameters of a fastSIM system (see Appendix 5.4). The deconvolved wide-field and SIM images are shown in Appendix 5.5. The reconstructed SIM-image of a bead in slice z with velocity v is called B^(v,z).

2.2. Shortcomings of previous algorithm

The shortcomings of the previous motion detection algorithm [16] is seen by considering two consecutive acquired wide-field images I_1,2(r⃗). They are normalized to 1, so that the later introduced factor N represent the maximum number of photons. The object has moved between the two acquisitions by a shift d⃗:

 

Fig. 1 Ultimate z-score of simulated moving beads acquired with fastSIM. The (ultimate) z-score of a bead depends on the velocity v and the number of photons N. The previous algorithm detects motion artefacts by thresholding the z-score at 6.5 (black line).

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The previous algorithm detects motion artefacts by thresholding the z-score at 6.5 (black line). However, the shape of the artefact does not depend on the beads brightness, which appears just as a scaling factor. Thus, slow (negligible artefact) but bright beads are marked wrongly as motion (dark blue), while fast (notable artefact) but dim beads are not detected (light blue).

Thus, the SNR dependency of the z-score has to be removed (section 2.3). In addition, the empirical thresholding of the z-score is not equal to a threshold at a maximum allowed artefact or a certain maximum velocity. Thus the maximum acceptable artefact needs to be defined and linked to the underlying object velocity with the measurable z-score (section 2.4 and 2.5).

2.3. Normalized z-score

To remove the influence of the objects brightness on the z-score for fast velocities (Eq. (6)), the z-score is divided by $\sqrt{N}$ yielding to the ’normalized z-score’ z_norm(r⃗). In practice, the number of photons is calculated by the raw images and the camera gain for each pixel. The normalized z-score is visualized later in section 2.5.

2.4. Artefacts in SIM - distortion and shearing

There are two types of artefacts. First, a distortion of the bead in a single reconstructed slice, leading to misinterpretations of the data (Fig. 2 top). Such a distorted image is of little practical use. Second, a misplacement of the center of the bead between neighboring (well imaged) z-slices, which is moving with the bead itself in each focus position (Fig. 2 bottom). This leads to an overall sheared appearance of the moving object, which is usually considered much less problematic than a distortion, because the image in each z-plane is artefact free and useable [16].

 

Fig. 2 Lateral (top) and axial (bottom) cross-sections of the images of simulated beads at different velocities (left: v = 0; middle: v = shearing threshold velocity [STV]; right: v = distortion threshold velocity [DTV]). Lateral: left: ideal image (similarity ς = 1); middle: negligible distortion (ς > 0.89); right: distorted image (ς = 0.89). Axial: left: ideal image; middle: sheared (or tilted) artefacts; right: heavy shearing. (scalebar = 100 nm)

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2.4.1. Distortions

Measurement

In order to compare different artefacts, they need to be quantified. This is done by comparing the ideal image without artefacts A^ideal(x, y) with the image with artefacts A(x, y) by a normalized cross-correlation (NCC) Γ. We use ’*’ as operator for the NCC and ’〈 〉_xy’ to indicate the 2D-mean over x and y. The simulated data fulfils the requirements of a NCC (equal scale, no rotation and global distortion) [19].

Tolerance limit

Experimental data always carries noise and some (minor) aberrations, so that every image is distorted in some way. Unfortunately, there does not exist a global definition or measure, where an arising image artefact goes beyond experimental imperfections and the image represents the specimen incorrectly. Thus, we suggest a limit starting at the Sparrow criteria: if two (static) beads are so close to each other, that their image has no intensity dip in-between them, they are considered as being the image of one bead as shown in (Appendix 5.7) [20]. This is a strong but globally accepted distortion artefact.

Similarity-threshold

We define the limit of acceptable distortion artefacts with the similarity value obtained from a single bead correlated with two beads at the Sparrow distance. Unfortunately their similarity ς^(z) depends on their focus position z. However, the axial position of the specimen is unknown in experimental data, so that a global threshold is necessary. In order to have a strict criteria, the maximum of ς^(z) is used and is called ’similarity-threshold’. The similarity-threshold is 0.89 and is actually the value in-focus (see Appendix 5.7). Thus, every measured distortion with a similarity value above this similarity-threshold is less problematic than two Sparrow beads in focus, which are interpreted as being one.

Distortion threshold velocity - DTV

The similarity value ς^(v,z) of a SIM imaged bead B^(v,z) of the test object is calculated with the NCC (A^ideal = B^(v=0,z), A = B^(v,z)) and shown in Fig. 3. It can be seen that, the similarity value decreases with increasing velocity and defocus. In-focus, the bead is imaged well up to a velocity of 252 nm/s, where the similarity value falls below the similarity threshold (red line). However, out-of-focus beads distort already at much slower velocities of 233 nm/s and 89.6 nm/s for the first two out-of-focus slices (compare Appendix 5.5). Finally, occurring distortions are linked to the underlying velocities.

 

Fig. 3 The similarity value ς^(v,z), comparing a moving to a static SIM reconstructed single noise-free bead, decreases with increasing velocity and defocus.

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As discussed, it is impossible to use a threshold that depends on the unknown focus-position of the object. If the global limit was one of the rather low out-of-focus threshold velocities, essentially distortion-free in-focus objects would be wrongly marked as motion. Since this is unacceptable, the in-focus threshold velocity of 252 nm/s is used as the ’distortion threshold velocity’ (DTV). As a consequence, out-of-focus artefacts are not marked if their underlying velocity is smaller than the DTV. This might be acceptable, considering that an out-of-focus image is much dimmer than in-focus, due to the section effect of SIM (see Appendix 5.5 and 5.7). In addition, strong artefacts are still detected.

2.4.2. Shearing

According to Shao et al., the motion of an object does not lead to visible artefacts in wide-field detection, if the change of its position during the acquisition is smaller than the final resolution. This approach can be applied on SIM, because it is an wide-field technique [15]. The final resolution is estimated by the lateral full width at half maximum (FWHM) of the SIM PSF (FWHM_xy). Since SIM has an extended PSF along z (depth of focus), a bead appears bright on 4 z-slices (FWHM_z = 279 nm, z-sampling of 110 nm) [15]. The fastSIM needs 1.1 seconds to acquire all raw-images in 4 focus positions and has a lateral FWHM_xy of 103 nm. Thus, the corresponding ’shearing threshold velocity’ (STV) is 94 nm/s.

2.4.3. Comparison of distortion and shearing

An ordinary shear occurs in SIM at slower velocities (STV = 94 nm/s) than a distortion (DTV = 252 nm/s). The STV can be calculated easily by acquisition time and resolution and gives a very good estimation if a structure of known velocity can be imaged without artefacts. Nevertheless, the distortion is the more problematic artefact (see subsection 2.4) and is thus considered further. Because, the velocities are not measurable directly either, they need to be linked further to the measurable normalized z-score.

2.5. Thresholding the normalized z-score

Figure 4 shows the ultimate normalized z-score ($z_{\text{norm}}^{\text{ult}}$) over the bead velocity v for various maximum number of photons N. It can be seen that $z_{\text{norm}}^{\text{ult}}$ is photon-independent for fast movements, enabling a conversion from velocities to normalized z-scores. Thus, the STV is converted into the ’shearing threshold normalized z-score’ (STZ) of 0.085. The DTV is converted into the ’distortion threshold normalized z-score’ (DTZ) of 0.52. Finally we conclude, that our fastSIM system can image the polymerization of microtubules before cell division in 3D but not their depolymerization afterwards, which will contain distortion artefacts [11,12].

 

Fig. 4 The (ultimate) normalized z-score $z_{\text{norm}}^{\text{ult}}$ of a bead depends on its velocity only (compare Fig. 1). However, the intended conversion between the artefact creating velocity and the measurable normalized z-scores (orange and red) is disturbed by the substantial noise floor $n_{\text{norm}}^{\text{floor}}(N)$. The noise floor can be mitigated by increasing the number of photons.

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In order to convert the measurable normalized z-score into the underlying velocity and thus to an artefact, $z_{\text{norm}}^{\text{ult}}$ must not perish in the noise floor $n_{\text{norm}}^{\text{floor}}$ at the thresholds STZ and DTZ. The more photons, the lower the noise floor (see Eq. (9)). Thus a minimum number of photons is required, which is $N_{\mathit{min}}^{\mathit{STZ}}\approx 4800$ for a shearing and $N_{\mathit{min}}^{\mathit{DTZ}}\approx 128$ for a distortion (see Appendix 5.8). The minimum number of photons for shearing is much higher than for distortions, because shearing artefacts occur at slower velocities and thus at lower signals $z_{\text{norm}}^{\text{ult}}$. The classification parameters are independent of exposure time and used SIM setup (shown in Appendix 5.9).

Figure 5 shows the false color map for artefact classification, which depends on the number of photons N and z_norm. Distorted, sheared and correctly (static) imaged specimen information are colored in red, orange and green respectively. A shear is only marked in a warning orange, because it does not necessarily destroy the benefits of SIM. Areas, where there are neither enough photons nor the signal $z_{\text{norm}}^{\text{ult}}$ is high enough are marked as black, turning into yellow if the signal rises suspiciously above the noise floor without reaching a sufficient SNR to classify it.

 

Fig. 5 Motion is classified by the normalized z-score z_norm and the number of photons N.

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3. Experimental results

We demonstrate the improvements by reprocessing the previously published [16] data of an autofluorescent plant cells (Epipremnum aureum) with the new algorithm. Figure 6 shows the result of the old method (top row, motion encircled in red) and the new one (bottom row). To capture the motion in one single image, a maximum projection along z was performed. The specimen is focused on the free chloroplast (left column) and the fixed stoma cell (right column).

 

Fig. 6 Motion encircled in red using the z-score (old method, a and b) and the classification by the normalized z-score (new method, c and d; colour map as in Fig. 5) applied on a biological sample, which shows two relevant structures: one free round chloroplast at the top and a large fixed stoma cell at the bottom. The specimen is focused on the free chloroplast (a and c) and the fixed stoma cell (b and d). The classification shows that the image of the chloroplast has just a shear (c) and no motion artefact(a), which corresponds to the SIM image [16].

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The two main differences are: 1) The empirical thresholding of the old method detects motion, while the new method classifies the occurring artefacts and distinguishes between shearing and distortion. 2) The new method is brightness independent, which can be seen best at the free moving chloroplast (left column).

While the z-score (top row) marks the free chloroplast as a motion, it is classified as a shear by the normalized z-score (bottom row). Thus the SIM-image of each focus-slice is good, only the 3D representation is wrong (see Fig. 2). This matches the experimental results very well [16]. In addition, the new algorithm is much more sensitive to dim object features like individual beads (left column) and also matches the experimental results better than the old method [16].

4. Conclusion

We investigated successfully fundamental shortcomings of our previous motion detection. We showed why the z-score, as a measure of motion, depends on an objects brightness and how to eliminate this by introducing the normalized z-score. Simulating perfect and aberrated beads and comparing them via a normalized cross-correlation, allows to judge artefacts quantitatively. Motion of a bead leads primarily to a 3D shear rather than 2D failures (distortions) in the reconstruction. Distortions emerge mainly from out-of-focus slices which are dimmed by the sectioning effect of SIM. Thus, the wide-field threshold can be applied on SIM. Finally, our investigation links the calculated motion signal to the corresponding velocity and arising artefacts, allowing us to classify occurring artefacts. Reconstruction artefacts are not an issue in 3D-SIM if the investigated biological structure has a movement velocity below the wide-field limit.

5. Appendices

5.1. Outlook

We want to extend the motion detection for 2D-SIM methodsTable 1, such as TIRF-SIM [21] or standard two-beam SIM [10,11]. Because neither of the two techniques acquires a z-stack, they need to acquire one additional phase step to get two independent wide-field images. However, the here studied signal-to-noise analysis can then be transferred.

5.2. Velocity of various cells

Table 1. Velocity of various cells and their components [12]. Important in-vivo processes happen too fast for the fastSIM setup, because the fluorophores cross the SIM PSF (FWHM = 103 nm) before all raw images are acquired [11]. 2D-SIM needs less images and does not perform a z-scan, so that faster processes can be imaged.

View Table

5.3. Test object

The object contains 25 beads in the center z-slice (Fig. 7 left). Each bead moves with its own constant velocity (right). In practice, the objects velocity and direction changes. This effect is neglected, to keep the number of dimensions reasonable for the following evaluation. This is justified by the idea that a homogenous movement leads to a stronger artefact, than any arbitrary movement of smaller maximum velocity.

 

Fig. 7 Left: Test object with 25 horizontally moving beads in the center z-slice. Right: Individual velocity of the beads. The top left bead is static and is used as the ideal object and image.

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However, the direction of the linear movement has an influence on the occurring motion artefact. A movement along the excitation stripe produces less artefacts than a movement perpendicular to them. The reason is, that a bead moving along a stripe changes indeed its position, however not its phase on the excitation pattern. It is this phase mismatch which gives rise to artefacts in the SIM reconstruction [22]. Thus, a bead moving along a stripe has a better reconstruction, than a bead moving perpendicular to it.

In order to estimate the velocity thresholds, we run the simulation several times, each with a different moving direction of the beads in the test pattern (see error bars in Fig. 3).

5.4. Simulation parameters

The SIM data is simulated with the imaging parameters of our fastSIM setup: 512 × 512 × 20 voxels; lateral sampling = 40 nm; axial sampling 110 nm; exposure time: 10 ms; readout time: 10 ms; phase stepping time: 0.434 ms; refocusing time: 20 ms. The acquisition order is: 1) phase step; 2) grating rotation; 3) z-scan [10,11].

5.5. Deconvolved wide-field and SIM images of the test object

Figure 8 and 9 show the deconvolved wide-field (left) and SIM (right) image of the test object in focus and 110 nm out-of-focus respectively. For higher visibility the beads are magnified and allocated to their original position. Motion artefacts are visible clearly and become more significant with increasing velocity (compare Fig. 7 right) and defocus. The better sectioning of SIM compared to wide-field can also be seen.

 

Fig. 8 Deconvolved wide-field (left) and SIM (right) image of the test object in focus (scalebar = 1 μm).

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Fig. 9 Deconvolved wide-field (left) and SIM (right) image of the test object 110 nm out of focus (scalebar = 1 μm).

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5.6. Assignment of a ultimate z-score to a bead

To assess the effect of motion of a single bead, we need one single meaningful value. Thus, we consider the maximum of all calculated z-scores in a square pixel (px) area surrounding the bead (81x81) px² = (3.24x3.24) μm². The z-scores are obtained for each illumination direction independently and then a maximum is performed over them to obtain the ’ultimate z-score’ representing the entire bead. The ultimate z-score is thus a maximum over 81 × 81 × 3 = 19 683 individual z-scores per beads. The size of the area is chosen such that even the fastest bead does not leave it in the simulated acquisition.

The z-score of static beads and dark areas follows the distribution |𝒢{0, 1}|. The noise floor is estimated as the maximum value out of (above mentioned) 19 683 draws out of the distribution |𝒢{0, 1}|. Repeating this entire statistics several times leads to an average noise floor of 4.2 with a standard deviation of 0.3:

(12)$$n^{\text{floor}}\approx 4.2\pm 0.3$$

5.7. Sparrow Criteria as the maximum allowed distortion

Figure 10 shows a simulated SIM image of a single bead (top row) and two beads at the Sparrow limit (bottom row, bead distance 101 nm) for in- and out-of-focus (oof) slices. Interestingly the two beads in-focus cannot be separated and look like one bead (high similarity), while they can be identified as two beads (low similarity) at a defocus of 110 nm. This illustrates that the similarity of a single bead and two beads at the Sparrow limit depends on the focus position (see Fig. 11).

 

Fig. 10 Top: simulated SIM image of a single bead for in- and out-of-focus (oof) slices. Bottom: two beads at the Sparrow limit (distance 101 nm). (scale bar = 100 nm; all images in each column are normalized to a maximum absolute value of 1 for higher visibility)

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Fig. 11 Black: similarity of a single bead and two beads at the Sparrow limit of the same focus for different focus positions. Green: absolute intensity of a SIM PSF.

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The black curve in Fig. 11 shows the focus-dependency of the similarity of a single bead and two beads at the Sparrow limit of the same focus position. Because the precise axial position of a structure in an object stays unknown, the defined similarity threshold must be a constant value. In order to use a strict threshold, the lokal maxima in focus (ς^(z=0) = 0.89) is used. Several slightly larger similarity values can be found at higher defocus positions (190 nm and 306 nm). However, since the SIM PSF has a section effect, there is less and less light in these planes as the green curve shows (23% and 14% respectively). The simulation is noise-free, because noise is irrelevant for the similarity threshold.

5.8. Minimum number of photons

Considering normal noise in the roughly 60 million pixels of the investigated image ([x; y; z; directions] = [1000; 1000; 20; 3]) we chose a SNR of 5.64, which leads to one false positive assignment according the Gaussian error function (erf) [16]:

(13)$$\frac{1}{1-\text{erf}\left(\frac{\text{SNR}}{\sqrt{2}}\right)}=1000\cdot 1000\cdot 20\cdot 3\to \text{SNR}=5.64$$

SNR is defined as signal over noise. The signal is given by the measurement z_norm itself minus the offset given by the known photon number dependent noise floor $\left(n_{\text{floor}}^{\text{ult}}(N)\right)$. The noise is the standard deviation $\left(\sigma \left(n_{\text{norm}}^{\text{floor}}\right)\right)$.

(14)$$\text{SNR}:=\frac{z_{\text{norm}}-n_{\text{floor}}^{\text{ult}}(N)}{\sigma \left(n_{\text{floor}}^{\text{ult}}(N)\right)}$$

Thus it is possible to estimate the value $z_{\text{norm}}^{(5.64)}$ the measurement z_norm has to exceed, so that the SNR is at least 5.64 for a given number of photons N and a thresholding can be performed with sufficient confidence:

(15)$$\text{with}\hspace{0.17em}\text{Eq}.(9)\hspace{0.17em}\text{and}\hspace{0.17em}\text{Eq}.(13):5.64=\frac{z_{\text{norm}}^{(5.64)}-\frac{4.2}{\sqrt{N_{\text{min}}}}}{\frac{0.3}{\sqrt{N_{\text{min}}}}}$$

(16)$$z_{\text{norm}}^{(5.64)}(N)=\frac{5.89}{\sqrt{N_{\text{min}}}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{or}:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{N}_{\text{min}}={\left(\frac{5.89}{z_{\text{norm}}^{(5.64)}(N)}\right)}^2$$

Conversely, it is possible to calculate how many photons N_min are necessary to threshold at a specific normalized z-score. The minimum number of photons to classify a shearing $\left(\text{STZ}=0.085=z_{\text{norm}}^{(5.64)}\right)$ and a distortion $\left(\text{DTZ}=0.52=z_{\text{norm}}^{(5.64)}\right)$ are:

(17)$$N_{\text{min}}^{\text{STZ}}\approx 4800$$

(18)$$N_{\text{min}}^{\text{DTZ}}\approx 128$$

5.9. Transfer to SIM with different acquisition parameters

The experimental parameters of the (simulated) fastSIM are fixed except the exposure time, which needs to be modified with the used fluorophores and the excitation intensity the specimen can withstand. Reducing the exposure time reduces motion and thus artefacts. Nevertheless, the artefact classification parameters (STZ, DTZ, $N_{\mathit{min}}^{\mathit{STZ}}$ and $N_{\mathit{min}}^{\mathit{DTZ}}$) in the fastSIM system are independent of the exposure time, as shown in Appendix 5.10.

There exist two different possible acquisition orders. The z-scan can be done before (Elyra S.1, Zeiss, Germany) or after (fastSIM) the rotation of the grating. The motion detection evaluates the raw data of all focus positions for one grating direction [16]. Thus, the acquisition order influences only the time which is needed to acquire the necessary data. For this reason the choice of acquisition order and exposure time lead to the same consequences, meaning that both influence the maximum allowed velocities but not the motion detection (see Appendix 5.10).

5.10. Classification parameters for different exposure times

The shorter the exposure time the less motion occurs and the faster a bead can be and is still images correctly. Thus, the distortion threshold velocity (DTV) for our fastSIM system increases with decreasing exposure time (see Fig. 12).

 

Fig. 12 The distortion threshold velocity (DTV) decreases with an increasing exposure time. The maximum permissible exposure time can be found by estimating the occurring velocity of a biological process.

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An artefact and the (normalized) z-score depend on the distance d an object moves during acquisition and not on the velocity itself (see Eq. 1). Thus, the distortion threshold normalized z-score (DTZ) is independent of the exposure time (Fig. 13). Thus, a change in the exposure time does not change the artefact classification parameters (STZ, DTZ, $N_{\mathit{min}}^{\mathit{STZ}}$ and $N_{\mathit{min}}^{\mathit{DTZ}}$) in the fastSIM-system.

 

Fig. 13 The ultimate normalized z-score of a bead increases with its velocity (which is normalized to the distortion threshold velocity (DTV) of each exposure time (see Fig. 12)). Fortunately, the measurable normalized ultimate z-score ($z_{\text{norm}}^{\text{ult}}$) is independent of the exposure time so that the distortion threshold normalized z-score (DTZ) is thus a global threshold for all fastSIM systems.

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Funding

Federal Ministry of Education and Research, Germany (BMBF) (FKZ 13N13140); Deutsche Forschungsgemeinschaft (DFG) (TRR 166; Project Number: 279219770).

Acknowledgments

The authors thank Christian Karras for helpful discussion about structured illumination microscopy and Linda Klaus for explaining details of cell biology.

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