## Abstract

Whole Slide Imaging (WSI) systems are high-throughput automated microscopes for digital pathology applications. We present a method for testing and monitoring the optical quality of WSI-systems using a measurement of the through-focus Optical Transfer Function (OTF) obtained from the edge response of a custom made resolution target, composed of sagittal and tangential edges. This enables quantitative analysis of a number of primary aberrations. The curvature of the best focus as a function of spatial frequency is indicative for spherical aberration, the argument of the OTF quantifies for coma, and the best focus as a function of field position for sagittal and tangential edges allows assessment of astigmatism and field curvature. The statistical error in the determined aberrations is typically below 20 mλ. We use the method to compare different tube lens designs and to study the effect of objective lens aging. The results are in good agreement with direct measurement of aberrations based on Shack-Hartmann wavefront sensing with a typical error ranging from 10 mλ to 40 mλ.

© 2015 Optical Society of America

## 1. Introduction

The primary activity of a pathologist is to make diagnoses via microscopic examination of tissue and cells from a biopsy. In the emerging clinical practice of digital pathology [1–4] this is based on digital high-resolution images of tissue slides acquired with a high-resolution and high-throughput automated microscope, the Whole Slide Imaging (WSI) system. These systems are designed for imaging a postage stamp sized area on a glass slide with a sampling density down to 0.25 μm/pixel in about 1 min. Digital pathology comprises a number of applications such as connectivity for remote diagnosis in regional hospital cooperation or for expert consultations, connectivity for correlating pathology images with radiology data, workflow management and quality control [5], teaching and certification, and the use of Computer Aided Diagnostics (CAD) and Clinical Decision Support (CDS) [6–9].

A number of different optical architectures have been considered for WSI systems [10], from a step-and-stitch approach using standard widefield acquisitions, to array microscopy [11], and architectures based on continuous scanning with a line sensor (“pushbroom” scanning). The latter approach (see Fig. 1) is often implemented using Time Delay Integration (TDI) cameras [12], for increasing the Signal-to-Noise-Ratio (SNR). The SNR-increment originates from a more efficient use of the illumination etendue. The line scanner approach has appeared to be the most favoured architecture because of the mechanical simplicity and the reduced need for stitching (if any at all).

The optical quality of any WSI system is the key to guaranteeing the best possible image quality of the final digital slide images. For that reason there is a need for testing and monitoring optical quality in manufacturing and during the operational lifetime of the instrument. This systematic optical quality assessment must give access to many optical parameters, e.g. primary aberrations including astigmatism, field curvature, spherical aberration, coma, and chromatic aberrations. In addition, it must be cost effective and practical, i.e., run without end-user intervention or modification of the hardware setup of the scanner.

A non-invasive method that satisfies these requirements is based on the evaluation of the Modulation Transfer Function (MTF) derived from features such as edges or line patterns on a resolution target [13, 14]. A through-focus measurement is advisable to assess the effects of field curvature and tilt of the image plane around the scan direction. These may result in a focus mismatch between the center and the edge of a scanned lane and between two adjacent scan lanes, respectively. Although optical quality assessment can be done based on the MTF data alone, it is desirable to extract the basic optical information, i.e. to estimate the optical aberrations underlying a possible deterioration of the MTF. A general way to extract the aberrations is to fit a model MTF, that is a function of the aberrations, to the MTF data [15, 16]. The drawback of iterative aberration retrieval from MTF data is that convergence to the global optimum is not guaranteed, and that convergence to an optimum can be rather slow. Hence there is a need for a more simple and straightforward way to estimate aberrations from through-focus MTF data.

Here, we propose a method that addresses this need. First of all, we find the best focus as a function of both field position and spatial frequency from the through-focus MTF. In this way we can directly quantify astigmatism and field curvature (from the dependence on field position). We add to this the novel aspect of extracting the spherical aberration from the dependence of best focus on spatial frequency. Second, we introduce a new way to estimate coma, namely by measuring the full Optical Transfer Function (OTF). It turns out that coma can be estimated from the dependence of the argument of the OTF - the Phase Transfer Function (PTF) - evaluated at best focus on the spatial frequency. Few studies have been made into the use of the PTF for analyzing optical image quality [17, 18], none of them address the quantification of coma.

In our paper we give two examples for the use of the proposed OTF-based optical image quality assessment. A first example concerns the evaluation of tube lens design, which is highly relevant to WSI-systems. Non-standard tube lenses are required in order to achieve a required sampling density in object space (the slide). For example, an Olympus 20 × /0.75 objective lens has a focal length of 180 mm/20 × = 9 mm. For a required sampling density of 0.25 µm/pixel and a camera pixel size of 7 µm a tube lens with a focal length of 252 mm is then needed. This is significantly different from the standard 180 mm tube lens focal length of Olympus. Similar arguments hold for other manufacturers of high-end objective lenses. Now the complication arises of balancing the low order aberrations (field curvature, astigmatism, axial and lateral color) between objective lens and tube lens, i.e. the objective lens may suffer from these aberrations as long as they are compensated by aberrations of the tube lens that are equal in magnitude but of opposite sign. This balancing may be done differently for different manufacturers. Now the question arises how well the custom tube lens with the required magnification matches the given off-the-shelve high-end objective lens. We apply our method to compare two simple tube lens designs, namely the use of a single catalogue achromat, and two back-to-back achromats of twice the required focal length. This example has implications for the use of home-built microscope systems around the world, as such simple tube lenses are often used. The second example relates to monitoring of aging in imaging systems and to detecting defective objective lenses. We show results for an old and possibly maltreated objective lens for quantifying the degree of sub-standard optical quality.

The proposed method of extracting aberrations is implemented for a push-broom scanning system based on a line sensor, but can in principle also be applied to an ordinary widefield non-scanning system based on an area sensor. The difference between the two optical architectures is that the former has translational invariance in the scan direction, whereas the latter does not. We have used this to our advantage by averaging over features (edges) of the resolution target in the scan direction, thereby increasing the SNR.

The outline of this paper is as follows. Section 2 describes the methods, in particular the design of the resolution target, the measurement of the OTF from a through-focus image stack, and the extraction of aberration coefficients from the measured OTF. Section 3 presents the results on the two aforementioned examples as well as on a validation experiment with a Shack Hartmann wavefront sensor. The paper is concluded in section 4 with a short discussion and outlook on possible follow-up research.

## 2. Methods

#### 2.1 Scanning and image acquisition

The WSI lab-system [Fig. 1] uses Dalsa Piranha HS-40-04k40 TDI line scan sensors (4096pixels, pixel size 7µm × 7µm, maximum frame rate 36kHz), Nikon 20X/NA0.75 Plan apochromat and Olympus 20X/NA0.75 UPlanSApo objective lenses, tube lenses assembled from the Thorlabs catalogue AC508-250-A-ML (250 mm focal length) and AC508-500-A (500 mm focal length) achromat lenses, a PI M-505 low profile translation stage (for positioning of the slide in the field direction), a Newport XM1000 ultra precision linear motor stage (for the scanning motion of the slide), a PI M-111 compact micro-translation stage (for coarse positioning of the objective lens in the axial direction), and a PI P-721.CL0 piezo nano-positioner (for fine positioning of the objective lens in the axial direction). The illumination was based on LEDs and a condenser matched in NA to the objective lens. It provides red, green, blue light with maximum power at 618, 565, and 435 nm respectively. The corresponding spectral full width at half maximum are 93, 67, and 27 nm.

#### 2.2 OTF measurement

The use of the edge response for measuring the MTF is a standard technique [13, 14, 19, 20]. Here, we use the through-focus edge response of a custom made resolution target [Fig. 2(a)] to measure the full OTF. The target pattern is etched in a thin chrome mask on a Soda Lime glass (B270) substrate with thickness of 1 ± 0.015 mm. A D236 glass cover slip of thickness 0.17 ± 0.015 mm is glued on top of it with UV-curing glue and coated with an antireflection coating. The etched lines have a width of 50 ± 0.8 µm giving a spatial frequency of 10 line pairs/mm. Typically, we use stacks of 170 images recorded at 0.1µm axial step size of the objective lens. Edges oriented along the scan and field directions are used for measuring the sagittal and tangential response. Different Regions Of Interests (ROIs) for measuring the OTF are shown in Fig. 2(a). The ROIs indicated in red give the response for the left, middle and right of the objective lens’ Field Of View (FOV). The ROIs indicated in blue serve to assess the field curvature and astigmatism. Prior to scanning, the glass slide is aligned to be perpendicular to a mechanical reference, namely the translation axis of the objective lens piezo stage. Keeping reference marks at the edges of the glass slide at the same focus level for a single position in the FOV of the objective lens when the slide is translated laterally does this.

Each ROI consists of several blocks of 100 × 100 pixels. To each block we apply a number of image processing steps [Fig. 2(b)]. The first step is averaging of the 100 line profiles to get a one dimensional Edge Spread Function (ESF) for each block. Possible misalignment of the orientation of the target around the optical axis (normal to the slide) may be neglected on the scale of the 100 lines. The second step is to reduce the noise in the upper and lower plateaus of the ESF without affecting the edge. The guided filter algorithm is used to this end [21]. This filter has four inputs: input line response, guidance image, local window size and regularization parameter. If the guidance image is equal to the filter input ESF then the filter acts as an edge preserving operator. Here the averaged edge response is the guidance image and the mean of the scaled standard deviations of the top and bottom plateaus is chosen for the regularization parameter. So the higher the noise, the more smoothing away from the edge. Possible biases in the ESF and the resulting OTF introduced by the denoising have been checked for by evaluating OTFs with and without application of the guided filter operation. No biases were found, only the expected difference in noise level of the resulting OTF data. The third step is to calculate the line spread function (LSF) from the denoised edge response by numerical differentiation. Here, the discrete difference must be divided by a correction factor sinc(*πaf/2*), with *f* the spatial frequency and $a$ the pixel size [22]. The fourth step computes the 1D Fourier Transform (FT) of the LSF to get a cross section through the origin of the OTF. The modulus and the argument of the OTF then yield the MTF and the PTF, respectively. A fifth step is needed to correct for a possible shift of the center of the LSF from the computational window of the FT. Such a shift *Δx* leads to a linear phase contribution *2πfΔx* to the PTF. This shift is estimated by:

where rem(a,b) indicates the remainder after division a/b. The weight function *W*(*f*) is taken to be equal to the MTF.

After computing the MTF and PTF for all individual blocks in a given ROI, the MTF of the ROI is calculated by averaging over all MTFs. Repeating the procedure with a different weight function increases the accuracy of the linear phase removal. Namely, now the weight is the inverse variance of the PTFs of the set of blocks in the ROI. In addition, the weights of the spatial frequencies beyond the onset of aliasing [18] are reduced with a factor of 10. After this refined linear phase removal the PTF of the ROI is found by averaging over all PTFs of the different blocks. The statistical error in the MTF and PTF is found by the standard deviation of the MTF and PTF over the blocks within the ROIs. Note that the current procedure does not provide a measurement of the OTF for all two-dimensional spatial frequencies (*f*_{x},*f*_{y}) but only the cross-sections OTF_{x} and OTF_{y} along the lines *f*_{y} = 0, and *f*_{x} = 0, respectively.

#### 2.3 Aberration extraction

The proposed method can be explained from the salient features of the through-focus MTF and PTF affected by the different primary aberrations, as shown in Fig. 3. Aberration values reported throughout this paper are standard Zernike coefficients [23], corresponding in magnitude with the Root Mean Square (RMS) value but with the sign maintained of the Zernike fringe coefficients. The route to aberration quantification runs via the extraction of the best focus (optimum MTF) as a function of spatial frequency and position in the FOV of the objective lens. The best focus for a given spatial frequency is found by a Weighted Least Squares (WLS) fit of a parabola to five data points (focus level for which the MTF is maximum and its four neighbors) with weights equal to the inverse of the measurement variances of the data points. The maximum of the fitted parabola yields the best focus position for that spatial frequency. The uncertainty is found from the measured MTF standard deviation by standard error propagation.

Spherical aberration introduces curvature in the best focus as a function of spatial frequency: the best focus is shifted toward the marginal rays’ focus for low and high spatial frequencies [Fig. 3(b)]. This effect can be quantified by fitting a parabola to the best focus line (using again a WLS fit):

*λ*is the wavelength and

*NA*the Numerical Aperture of the objective lens. Scaling of the spatial frequencies with

*NA/λ*and scaling of the best focus position with the depth-of-focus [24] ensures that the coefficients (

*p*

_{1},

*p*

_{2}

*,p*

_{3}) are largely independent of wavelength and

*NA*. We found that

*p*is linearly dependent on the Zernike spherical aberration coefficient (

_{1}*p*

_{1}

*/A*

_{40}= 0.0104

*/mλ*).

The overall best focus is found by weighted averaging (weighted by the inverse square error) of the best focus line over the middle frequency range (0.90≥MTF≥0.10). Plotting the best focus as a function of the position in the FOV of the objective lens, for both the sagittal and tangential ROIs gives direct access to field curvature and astigmatism. The level of astigmatism can be quantified by the axial distance *Δz _{ob}* between the best focus for the sagittal and tangential regions. This distance can be related to a value for the Zernike standard aberration coefficient

*A*

_{22}using the method of minimum RMS value of the aberration function [25, 26]. The aberration function including astigmatism and defocus is:

*z*is the objective lens axial position, and

_{ob}*x*and

*y*are the pupil coordinates normalized with the pupil radius. The best sagittal and tangential focus positions are found by minimizing the RMS value of the aberration functions

*W*(

*x*,

*0*) and

*W*(

*0*,

*y*), i.e. along the lines

*y = 0*and

*x = 0*in the pupil plane. Details of the procedure are outlined in the appendix. The final outcome is a relation

*A*

_{22}

*= S*(

*NA*)

*Δz*. In the paraxial regime we have

_{ob}*S*(

*NA*)

*≈NA*, for the current objective lenses with

^{2}/4/Ö6*NA*= 0.75 we find

*S*(

*NA*) = 0.0669. The field curvature is measured from the overall best focus position for each sub block of the horizontal blue ROIs (see Fig. 2(a)). Ideally, this curve will be parabolic and symmetric with respect to the center of the FOV. Since we aligned the slide to be perpendicular to the mechanical symmetry axis of the objective lens, any asymmetry in the field curvature is indicative for tilt or decenter of individual optical components, such as the lenses within the objective lens assembly. The field curvature is quantified by the difference

*FC*between the average of the sagittal and tangential best focus values at the edge of the FOV from the best focus in the center of the FOV, after removal of any focus plane tilt.

_{edge}Coma does not alter the ideal straight best focus line, as opposed to the previously treated cases it is an odd aberration. Although the MTF at the best focus is reduced by coma, the PTF appears to be a better indicator for this aberration. It turns out that the PTF as a function of spatial frequency can be qualitatively described by a third order polynomial [Fig. 3(e)], with a distinct maximum and minimum in the range of spatial frequencies below the cutoff. Interestingly, it appears that the difference *ΔPTF = PTF _{max}* −

*PTF*depends linearly on the Zernike coefficient for coma (with coefficient

_{min}*ΔPTF*/${A}_{31}$ = 0.340 deg/mλ), but not on wavelength and NA, at least for aberration levels below approximately 120 mλ. The error in the measured coma directly follows from the standard deviation in the measured PTF.

#### 2.4 Shack-Hartmann measurements

The set of aberrations measured from the through-focus OTF are compared to Shack-Hartmann (SH) measurements for verification. We built a wavefront sensing setup using an Optocraft SHR-150-CL SH-sensor (12 bit, detection area 11.8 × 8.9 mm2, 78 × 59 microlens array) to measure the aberrations of the optical system under inspection [Fig. 4]. A pinhole (5 µm diameter) mounted in between a glass slide and a coverslip is used as a point source object. In order to provide a (near) parallel wavefront to the SH sensor an additional collimating lens (Melles Griot 01LAO536 - 120mm focal length) is placed at its focal distance from the system’s image plane. We measured the aberrations of this collimating lens separately to offset the final results. The aberrations for different positions in the FOV of the objective lens can be probed by laterally translating the pinhole slide.

## 3. Results

#### 3.1 Effect of the tube lens design

WSI-systems require custom tube lenses in order to achieve a required sampling density, as discussed in the introduction. We considered two simple tube lens designs: a single achromat (Thorlabs AC508-250-A-ML, 250 mm focal length) and double back-to-back achromat design (two Thorlabs AC508-500-A, 500 mm focal length). Simulation of these designs using Zemax ray tracing software (see Fig. 5) shows that the astigmatism and field curvature for the double back-to-back achromat design are −0.57 µm and −0.21 µm at 0.5 mm field position, respectively, compared to + 1.6 µm and + 1.99 µm for the single achromat design. This is a reduction with a factor 2.8 in astigmatism and 9.5 in field curvature.

In our experiment we used Nikon 20 × /0.75 objective lenses (focal length 200mm/20 × = 10 mm) leading to a sampling density of 0.28 µm/pixel for a 250 mm focal length tube lens. The reason for selecting Nikon objective lenses is the reputed absence of balancing of low order aberrations between objective lens and tube lens, implying that the aberrations of the tube lens should match with the aberrations of the composite objective plus tube lens system. The measurements were done with green light with peak wavelength λ = 565 nm. Our experimental result for the single achromat tube lens (case A) and the double back-to-back achromat tube lens design (case B) are shown in Fig. 6 and Fig. 7. The measured sampling densities were 0.28 and 0.27 µm/pixel for case A and B, respectively, close to the design value.

The best focus lines in Fig. 6(a) and Fig. 7(a) show little curvature, which is indicative for low values of spherical aberration (measured values 16 ± 10 and 35 ± 8 mλ averaged over the two middle ROIs). The erratic behavior of the best focus line at spatial frequencies close to the cutoff in the tangential right ROI of Fig. 6(a) is due to the low and noisy MTF values at those frequencies (signal to noise ratio below ~10dB). The measured PTFs (Fig. 6(b) and Fig. 7(b)) indicate that the level of coma is also low. The measured values for the left, middle and right ROIs are for case A: 7 ± 10 mλ, 7 ± 12 mλ, and 5 ± 10 mλ in the sagittal area, 48 ± 8 mλ, −7 ± 12 mλ, and −57 ± 8 mλ in the tangential area, and for case B: 9 ± 10 mλ, 13 ± 12 mλ, and 9 ± 10 mλ in the sagittal area, 42 ± 3 mλ, −8 ± 9 mλ, and −37 ± 3 mλ in the tangential area.

There is, however, a significant amount of astigmatism and field curvature. First, it is noted that the tangential and sagittal best focus curves are significantly asymmetric [Figs. 6(c) and 7(c)], while the slide tilt is near zero [Figs. 6(d) and 7(d)], pointing to a tilt of the objective lens’ optical axis w.r.t. its mechanical symmetry axis on the order of 1-2 mrad. We measured a field curvature *FC _{edge}* = 1.38 µm for case A, compared to −0.13 µm for case B, also in reasonable agreement with the Zemax design. The astigmatism quantified by the average difference in sagittal and tangential best focus at ± 0.5 mm field position is 0.99 µm for case A, compared to −0.41 µm for case B, in reasonable agreement with the Zemax design. The astigmatic aberration coefficients for the single achromat were found to be 154 mλ, 3 mλ, and 79 mλ with an uncertainty of 3 mλ for the ROIs at the left, middle and right of the FOV, respectively, whereas we measured corresponding values of 36 mλ, 7 mλ, and 61 mλ with an uncertainty of 2 mλ for the double back-to-back achromat design, which amounts to an average reduction in astigmatism for the double back-to-back achromat design with a factor 2.0.

#### 3.2 Statistical errors

The statistical error of the measured aberrations is found by error propagation from the standard deviation of the MTF and PTF within the ROIs, as described in section 2. Figure 8 shows typical examples of the error in MTF measurement (around 1%) and PTF (ranging from below 1 deg for small spatial frequencies to above 10 deg for high spatial frequencies). The best focus as a function of spatial frequency can subsequently be determined with an error of around 100 nm, the overall best focus with an error of around 10 nm. This translates into statistical errors in the measured aberration coefficients ranging typically from 5 to 20 mλ.

#### 3.3 Effect of an aging objective lens

Objective lenses, as any piece of hardware, suffer from aging. This manifests itself in misalignment of the individual lenses within the mounted objective lens, giving rise to increased levels of aberrations. In order to test the aging effect, we measured the aberrations of an 8-year old Olympus 20 × /NA0.75 objective lens with undocumented use in these 8 years, in combination with a single achromat tube lens (case C, Fig. 9).

We observed higher levels of all aberrations. The spherical aberration increased to 102 ± 10 mλ. We measured coma values of 62 ± 14 mλ, 46 ± 15 mλ, 30 ± 14 mλ in the sagittal area from left to right, and corresponding values of −130 ± 14 mλ, −80 ± 11 mλ, −30 ± 7 mλ in the tangential area. The measured of levels astigmatism were 295 ± 8 mλ, −17 ± 8 mλ, and 296 ± 8 mλ for the ROIs at the left, middle and right of the FOV. This is significantly higher than the result for case A, but this could also be due to design choices of the manufacturer for balancing the lowest order aberrations between the objective and the custom tube lens in their microscope systems. The field curvature and astigmatism are 0.48 µm and 2.5 µm, respectively. Also in this case we have observed a significant asymmetry in the tangential and sagittal best focus lines, arising from tilt of the optical axes w.r.t. the mechanical reference.

#### 3.4 Validation with Shack-Hartmann wavefront sensor

A summary of the aberrations measured from the through-focus OTF is shown in Fig. 10(a). The results of the validation measurement with the SH-setup are shown in Fig. 10(b). The two independent sets of measurements agree with typical errors ranging from 10 mλ to 40 mλ, which is reasonable in view of the alignment uncertainty in the different mechanical and optical components and the statistical error in measuring the aberrations from the through-focus OTF. The agreement is less good for spherical aberration, which we attribute to the sensitivity to the definition of the pupil radius in the SH measurement, which gives rise to a bit overestimated aberration coefficient compared to the through-focus OTF based measurement. The pupil radius in the SH measurement can be set with an accuracy of about 0.5 mm. We extracted the spherical aberration for different pupil radius settings and found variations of about 30 mλ for 0.5 mm difference of the pupil radius. Our proposal to quantify the spherical aberration from the curvature of the best focus line is therefore not validated with high accuracy.

In all three cases we observed an apparent tilt in the best focus curves, although there was no slide tilt w.r.t. the mechanical reference axis of the objective lens. This points to possible misalignments in the system, which may be linked to intricate patterns in the magnitude and orientation of astigmatism and coma across the FOV of the objective lens [27–29]. In order to test this link we have measured the aberrations across the FOV (see Fig. 11), which indeed shows complicated dependencies of magnitude and orientation of both astigmatism and coma on the field position. One typical example is shown in Fig. 11(c), where the astigmatic aberration field complies with so-called “binodal astigmatism”, i.e. the astigmatism is zero for two, diametrically opposite, points in the FOV. As a consequence, the tangential and sagittal best focus curves will not touch at the center of the FOV, but rather cross at two nonzero field positions, in agreement with what is seen in Fig. 9(c). It also implies there is a small (34mλ) on-axis astigmatism, which we attribute to relative misalignment of objective lens and tube lens.

Another interesting observation is related to the direction of coma in the FOV. According to Fig. 11(a) and Fig. 11(b), the direction of the coma component in the scan direction (sagittal coma) does not change. In contrast, the coma component in the field direction (tangential coma) is flipped from left to right. This behavior can be clearly observed in Fig. 6(b) and Fig. 7(b) in which all the sagittal PTF curves have the same shape, but the left tangential PTF curve has the opposite shape of the right tangential PTF curve.

#### 3.5 Chromatic aberration

Through-focus OTFs and aberrations have also been measured for red and blue light. Figure 12 shows the measured sagittal and tangential best focus lines for the three color channels for the single achromat tube lens (case A) and the double back-to-back tube lens design (case B). They are in good agreement with the Zemax simulation result (Fig. 5). For case B, however, there is a change in sign in tangential curvature for the blue channel compared to the Zemax results. We also observed a significant increase in coma in the blue channel compared to the green and red channel.

## 4. Discussion

In summary, we have presented a systematic method to assess the optical quality of a WSI system using measurement and analysis of the through-focus OTF for different positions in the FOV. The analysis of MTF and PTF enables the determination of spherical aberration, coma, astigmatism, and field curvature. The method has been benchmarked to SH-wavefront sensing results, and has been shown to be in good agreement with those. Our analysis tool will facilitate the reliable testing and monitoring of the optical quality of WSI systems, using only the custom resolution target and image analysis software, without the need for additional optical or mechanical hardware.

The non-invasiveness of the technique provides the key advantage compared to more conventional aberration measurement methods such as SH-wavefront sensing. The advantage compared to iterative aberration retrieval from through-focus MTF data is the simplicity and directness of the approach. In order to make a fairer comparison we have constructed an algorithm for doing iterative aberration retrieval based on a linear least squares fit of the measured OTF’s by the incoherent scalar diffraction OTF, where the lowest order aberrations defocus, astigmatism, coma and spherical aberration are the fit parameters. The algorithm was implemented with MATLAB’s fminsearch routine. We found that the iterative aberration retrieval algorithm has indeed problems with convergence, especially for aberration levels that exceed the diffraction limit. A drawback of the proposed method is that no distinction is made between lower and higher order aberrations of a certain type. For example, when the system is affected by sizeable amounts of higher order coma (*A*_{51}) in addition to lowest order coma (*A*_{31}) the method will give an effective value for the coma coefficient, not necessarily equal to the underlying Zernike values.

We would like to stress that the proposed method assesses the overall image quality of our home-built WSI system, including our tube lens configurations. The results obtained for systems with different objective lenses may therefore not be interpreted as characterizations or comparisons of these objective lenses alone.

The extension of the proposed optical quality assessment to widefield non-scanning imaging systems requires changes with respect to our implementation. First of all, averaging over the edge response in a certain direction is prohibited, as the translational invariance of the line scanning system is lost. Second, a different design of resolution target may be better suited to the optical architecture of the widefield system. We envision that an array of square blocks would provide the same set of mutually orthogonal edges (the top and bottom edges vs. the left and right edges of the square) for the different regions in the FOV of the objective lens.

We foresee two lines of research as follow-up of the currently reported research. In this work we have implicitly assumed that we deal with an incoherent imaging system, whereas in reality bright field microscopy systems are partially coherent. Approximating partially coherent imaging systems as being fully incoherent is reasonable for a condenser NA about equal to or higher than the objective lens NA, so that in fact the current treatment is sound. However, investigating the effects of partial coherence by tuning the condenser NA seems a fruitful direction for follow-up experiments. A second line of investigation is related to the analysis of full-field aberration maps as shown in Fig. 11 using the methods of [27–31]. This may help to quantify possible misalignments, and thereby may lead to a new way of finding the root causes of different aberrations that may be present or to novel methods of calibrating the alignment of different optical and mechanical components in the system.

## Appendix

The aberration function including astigmatism and defocus along the pupil line$y=0$is:

*x*. This results in:

## References and links

**1. **S. Al-Janabi, A. Huisman, and P. J. Van Diest, “Digital pathology: current status and future perspectives,” Histopathology **61**(1), 1–9 (2012). [CrossRef] [PubMed]

**2. **J. R. Gilbertson, J. Ho, L. Anthony, D. M. Jukic, Y. Yagi, and A. V. Parwani, “Primary histologic diagnosis using automated whole slide imaging: a validation study,” BMC Clin. Pathol. **6**(1), 4 (2006). [CrossRef] [PubMed]

**3. **J. Gu and R. W. Ogilvie, *Virtual microscopy and virtual slides in teaching, diagnosis, and research*, (Taylor & Francis, 2005).

**4. **R. S. Weinstein, M. R. Descour, C. Liang, A. K. Bhattacharyya, A. R. Graham, J. R. Davis, K. M. Scott, L. Richter, E. A. Krupinski, J. Szymus, K. Kayser, and B. E. Dunn, “Telepathology overview: from concept to implementation,” Hum. Pathol. **32**(12), 1283–1299 (2001). [CrossRef] [PubMed]

**5. **J. Ho, A. V. Parwani, D. M. Jukic, Y. Yagi, L. Anthony, and J. R. Gilbertson, “Use of whole slide imaging in surgical pathology quality assurance: design and pilot validation studies,” Hum. Pathol. **37**(3), 322–331 (2006). [CrossRef] [PubMed]

**6. **A. H. Beck, A. R. Sangoi, S. Leung, R. J. Marinelli, T. O. Nielsen, M. J. van de Vijver, R. B. West, M. van de Rijn, and D. Koller, “Systematic analysis of breast cancer morphology uncovers stromal features associated with survival,” Sci. Transl. Med. **3**(108), 108ra113 (2011). [CrossRef] [PubMed]

**7. **S. Doyle, M. Feldman, J. Tomaszewski, and A. Madabhushi, “A boosted Bayesian multiresolution classifier for prostate cancer detection from digitized needle biopsies,” IEEE Trans. Biomed. Eng. **59**(5), 1205–1218 (2012). [CrossRef] [PubMed]

**8. **M. N. Gurcan, L. E. Boucheron, A. Can, A. Madabhushi, N. M. Rajpoot, and B. Yener, “Histopathological image analysis: a review,” IEEE Rev Biomed Eng. **2**, 147–171 (2009). [CrossRef] [PubMed]

**9. **J. P. Vink, M. B. Van Leeuwen, C. H. Van Deurzen, and G. De Haan, “Efficient nucleus detector in histopathology images,” J. Microsc. **249**(2), 124–135 (2013). [CrossRef] [PubMed]

**10. **M. G. Rojo, G. B. García, C. P. Mateos, J. G. García, and M. C. Vicente, “Critical comparison of 31 commercially available digital slide systems in pathology,” Int. J. Surg. Pathol. **14**(4), 285–305 (2006). [CrossRef] [PubMed]

**11. **R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. **35**(11), 1303–1314 (2004). [CrossRef] [PubMed]

**12. **H. Netten, L. J. van Vliet, F. R. Boddeke, P. de Jong, and I. T. Young, “A fast scanner for fluorescence microscopy using a 2‐D CCD and time delayed integration,” Bioimaging **2**(4), 184–192 (1994). [CrossRef]

**13. **G. D. Boerman, *Modulation transfer function in optical and electro-optical systems*, (SPIE, 2001).

**14. **T. Williams, *The optical transfer function of imaging systems*, (CRC, 1998).

**15. **J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**(15), 2758–2769 (1982). [CrossRef] [PubMed]

**16. **J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. **32**(10), 1737–1746 (1993). [CrossRef] [PubMed]

**17. **V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Image-based measurement of phase transfer function,” in *Digital Image Processing and Analysis**,* (2010).

**18. **V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Effects of sampling on the phase transfer function of incoherent imaging systems,” Opt. Express **19**(24), 24609–24626 (2011). [CrossRef] [PubMed]

**19. **J. C. Mullikin, L. J. Vanvliet, H. Netten, F. R. Boddeke, G. Vanderfeltz, and I. T. Young, “Methods for CCD camera characterization,” Proc. SPIE **2173**, 73–84 (1994). [CrossRef]

**20. ***Photography–Electronic Still Picture Cameras–Resolution Measurements,* ISO Standard 12233:2000.

**21. **K. He, J. Sun, and X. Tang, “Guided image filtering,” IEEE Trans. Pattern Anal. Mach. Intell. **35**(6), 1397–1409 (2013). [CrossRef] [PubMed]

**22. **I. A. Cunningham and A. Fenster, “A method for modulation transfer function determination from edge profiles with correction for finite-element differentiation,” Med. Phys. **14**(4), 533–537 (1987). [CrossRef] [PubMed]

**23. **M. Born and E. Wolf, *Principles of Optics*, (Cambridge University, 1999).

**24. **C. J. R. Sheppard, “Depth of field in optical microscopy,” J. Microsc. **149**(1), 73–75 (1988). [CrossRef]

**25. **S. Stallinga, “Compact description of substrate-related aberrations in high numerical-aperture optical disk readout,” Appl. Opt. **44**(6), 849–858 (2005). [CrossRef] [PubMed]

**26. **S. Stallinga, “Finite conjugate spherical aberration compensation in high numerical-aperture optical disc readout,” Appl. Opt. **44**(34), 7307–7312 (2005). [CrossRef] [PubMed]

**27. **R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE **0251**, 146–153 (1980). [CrossRef]

**28. **R. Tessieres, “Analysis for alignment of optical systems”, MSc Dissertation (University of Arizona, Tucson, 2003).

**29. **K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A **22**(7), 1389–1401 (2005). [CrossRef] [PubMed]

**30. **P. L. Schechter and R. S. Levinson, “Generic misalignment aberration patterns in wide-field telescopes,” Publ. Astron. Soc. Pac. **123**(905), 812–832 (2011). [CrossRef]

**31. **K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A **26**(6), 1503–1517 (2009). [CrossRef] [PubMed]