Comprehensive tool for a phase compensation reconstruction method in digital holographic microscopy operating in non-telecentric regime

Quantitative phase imaging (QPI) via Digital Holographic microscopy (DHM) has been widely applied in material and biological applications. The performance of DHM technologies relies heavily on computational reconstruction methods to provide accurate phase measurements. Among the optical configuration of the imaging system in DHM, imaging systems operating in a non-telecentric regime are the most common ones. Nonetheless, the spherical wavefront introduced by the non-telecentric DHM system must be compensated to provide undistorted phase measurements. The proposed reconstruction approach is based on previous work from Kemper’s group. Here, we have reformulated the problem, reducing the number of required parameters needed for reconstructing phase images to the sensor pixel size and source wavelength. The developed computational algorithm can be divided into six main steps. In the first step, the selection of the +1-diffraction order in the hologram spectrum. The interference angle is obtained from the selected +1 order. Secondly, the curvature of the spherical wavefront distorting the sample’s phase map is estimated by analyzing the size of the selected +1 order in the hologram’s spectrum. The third and fourth steps are the spatial filtering of the +1 order and the compensation of the interference angle. The next step involves the estimation of the center of the spherical wavefront. An optional final optimization step has been included to fine-tune the estimated parameters and provide fully compensated phase images. Because the proper implementation of a framework is critical to achieve successful results, we have explicitly described the steps, including functions and toolboxes, required for reconstructing phase images without distortions. As a result, we have provided open-access codes and a user interface tool with minimum user input to reconstruct holograms recorded in a non-telecentric DHM system.

L355-377: "Finally, we have compared the performance of the proposed method with the one provided by the subtraction method, which uses a blank hologram to compensate for the spherical aberrations caused by a non-telecentric alignment [21] via the direct subtraction between both reconstructed phase maps.Figure 8 shows the normalized reconstructed phase images of a USAF phase target for the proposed method (Fig. 8a) and the subtraction one (Fig. 8b), demonstrating the high agreement between both methods.The accuracy and resolution of both methods have been evaluated by plotting the normalized phase values along the vertical direction [indicated by the white arrows in Figs. 8(a) and (b)] of the 9 group, see Fig. 8(c).From these profiles, one can identify the minimum resolvable element of the USAF phase target is the 9-3 element, which corresponds to 0.78 µm.This value confirms that both methods provide reconstructed images operating at the system's coherent diffraction limit, d = λ/NA = 0.532 / 0.75 = 0.71 µm.Despite that the dual-shot subtraction method is simple to perform and computationally inexpensive, its effectiveness is highly dependent on the experimental conditions during data acquisition, requiring that the blank hologram must represent an exact replica of the spherical distortions to produce accurate phase distributions.If the experimental DHM system suffers from external factors such as vibrations and temperature fluctuations, the blank and sample holograms do not have the exact same distortion and the subtraction method will not produce accurate results, requiring the use of additional computational methods to reduce any residual distortion.This negative result is further exacerbated during the acquisition of large datasets, where the temporal and spatial changes are most likely to occur.However, the proposed method alleviates such constraints by taking advantage of the single shot nature of off-axis DHM." The following figures has been added to the revised manuscript: Comment #2: "The resolution measurement has not been performed.This could be done by analyzing the edge of the star-target image (Figure 6)." Response: We have used the reconstructed phase images of a USAF test target from the Benchmark Technologies Quantitative Phase Target (QPT™) by Benchmark Technologies to measure the resolution of the proposed method.Please see response to comment #1 for more details.
Comment #3: "More important, the phase measurement accuracy needs to be shown using samples of known refractive index values (e.g., polymer microspheres in index-matching liquid)." Response: We apologize for the miscommunication.The phase measurement accuracy is shown in Fig. 6.This figure shows the reconstructed 2D phase map of a star target from the Benchmark Technologies Quantitative Phase Target (QPT™).The refractive index of the star target is well known and given by the manufacturer (refractive index = 1.52).In particular, panel (b) in Fig 6 shows the radial phase profile at two different radii: r = 43.95μm (pink), and r = 73.25 μm (cyan).The gray-shaded area shown in Fig. 6(b) marks the nominal phase value based on the manufacturer's specifications (n = 1.52 and thickness = 350 nm).There is a high similarity between the experimental phase values and the nominal ones, demonstrating the accuracy of the proposed method to provide accurate quantitative phase imaging.
The following text has been rewritten in the revised manuscript: L314-321: "The nominal phase values have been calculated based on the manufacturer's specifications, a refractive index and thickness equal to 1.52 and 350 nm, respectively.There is a high similarity between the experimental phase values and the nominal ones, demonstrating the accuracy of the proposed method to compensate spherical phase distortions.The results illustrated in Fig. 6 confirm that, within experimental errors, the spherical phase factor introduced by the non-telecentric configuration has been compensated, providing a linear shiftinvariant quantitative phase imaging tool." Comment 4: "The low-frequency phase distribution in the background region (i.e., non-sample region) would be useful to confirm the effectiveness of the proposed method."Response: We agree with the reviewer, and, for this reason, quantitative phase profiles along different background regions have been included in Fig. 7, see non-sample profiles colored in green, cyan, magenta.The comparison of these background profiles in Figs.7b&d confirms that the low frequency phase information is fairly uniform across the different directions, demonstrating the effectiveness of our proposed method to compensate spherical distortions in all directions.
The following sentence has been added in the revised manuscript: L330-335: "We have plotted some vertical and horizontal background phase profiles in Figs.Comment #0: "The authors report about a method for compensation of phase aberrations in quantitative phase imaging (QPI) with non-telecentric off-axis digital holographic microscopy (DHM).After an explanation of the underlying principles and characterization of the method by utilizing a phase test chart the application on a technical sample and red blood cells is illustrated.In general, the manuscript is motivated, organized, and includes adequate references.The experimental investigations appear to be accurately performed.The results are plausible.The authors address an important topic in QPI with DHM: The compensation of spherical phase aberrations which may be of interest for the field of DHM and the interdisciplinary areas of QPI and label-free biomedical imaging.In summary, the content of the manuscript appears to be suitable for the journal PLOS one.However, the authors should consider revisions." Response: We appreciate the kind words from the reviewer.We hope to have adequately addressed the reviewer's comments in the revised manuscript.
Comment #1: "Abstract: From the abstract the novelty aspects of the proposed phase compensation concept with respect to reference 35 becomes not fully clear.The authors should clarify the abstract concerning this topic." Response: We believe that the novelties of our contribution with respect to Ref.
[35] are the following.Firstly, the proposed method further simplifies the algorithm initially reported by Ref.
[35] by reducing the number of required parameters to only the camera pixel size and light source wavelength.Secondly, we provide a step-by-step guide for the practical implementation of the proposed method with specific design choices made explicit (i.e., toolboxes, functions, code, etc.) for each step.We believe that the proper implementation of a framework is critical to achieve successful results.Thirdly, the proposed algorithm offers an additional advantage by incorporating an optional minimization algorithm for quantitative reconstructed phase images.This thoughtful inclusion aims to alleviate the user's burden, streamlining the process and making it more user-friendly.Finally, but not least important, we have provided an open-access computational tool accessible to the public with code and application (GUI).The developed software tool will allow researchers in life and material sciences, even those without computational reconstruction knowledge, to analyze their results accurately, leading to new discoveries.
We have rewritten the abstract in the revised manuscript to clarify the novelties of our contribution: "Quantitative phase imaging (QPI) via Digital Holographic microscopy (DHM) has been widely applied in material and biological applications.The performance of DHM technologies relies heavily on computational reconstruction methods to provide accurate phase measurements.Among the optical configuration of the imaging system in DHM, imaging systems operating in a non-telecentric regime are the most common ones.Nonetheless, the spherical wavefront introduced by the non-telecentric DHM system must be compensated to provide undistorted phase measurements.The proposed reconstruction approach is based on previous work from Kemper's group.Here, we have reformulated the problem, reducing the number of required parameters needed for reconstructing phase images to the sensor pixel size and source wavelength.The developed computational algorithm can be divided into six main steps.In the first step, the selection of the +1-diffraction order in the hologram spectrum.The interference angle is obtained from the selected +1 order.Secondly, the curvature of the spherical wavefront distorting the sample's phase map is estimated by analyzing the size of the selected +1 order in the hologram's spectrum.The third and fourth steps are the spatial filtering of the +1 order and the compensation of the interference angle.The next step involves the estimation of the center of the spherical wavefront.An optional final optimization step has been included to fine-tune the estimated parameters and provide fully compensated phase images.Because the proper implementation of a framework is critical to achieve successful results, we have explicitly described the steps, including functions and toolboxes, required for reconstructing phase images without distortions.As a result, we have provided open-access codes and a user interface tool with minimum user input to reconstruct holograms recorded in a non-telecentric DHM system."

Comment #2: "Introduction:
To complete the description of the-state-of-the-art the authors may consider adding that non-telecentric arrangements can simplify the combination of DHM with commercial optical microscopes as, for example, reported in Drug Deliv.and Transl.Res. 12, 2207Res. 12, -2224Res. 12, (2022))." Response: We appreciate the suggestion and have included the mention citation as part of the introduction when mentioning potential reasons for non-telecentric arrangements.
The following sentence has been added to the revised manuscript: L72-74: "The third reason is related to the integration of DHM imaging modality with commercial microscopic systems to extend the reach of the DHM technique to a broader community and enable dual-mode fluorescent imaging with QPI [20]." The following reference has been added to the revised manuscript "20. A. Marzi, K. Eder, A. Barroso Response: Following the reviewer's comment and the novelties of our work described in Comment #1, we have rewritten the introduction in the revised manuscript.
L118-139: "In 2017, Min et al. proposed a single-shot computational approach to estimate the spherical wavefront of non-telecentric DHM systems based on a spectral analysis of the recorded holograms [36].This work extends Min's approach by further reducing the number of required parameters needed for reconstructing phase images.The input parameters of our computational method are the sensor's pixel size and the wavelength of the light source.
Because the proper implementation of a framework is critical to achieve successful results, the accuracy of the algorithm proposed by Min et al. is highly dependent on the methods used for the thresholding and segmentation of the ±1 diffraction terms and their compact support size.A preliminary computational function to reconstruct non-telecentric holograms was recently implemented in the pyDHM library under the CNT function [37].This function provides reconstructed phase images without or with minimum phase distortions after finding the best curvature of the spherical wavefront along the two lateral spatial coordinates using two nested for loops, leading to a high processing time.However, in this work, we have explicitly described the steps, including functions and toolboxes, required for reconstructing phase images without distortions.In contrast to Min's method, we have included iterative minimization algorithms [18] as an optional final optimization step to fine-tune the estimated parameters of the spherical wavefront and provide fully compensated phase images.The implementation for each step of the proposed method has been thoughtfully investigated, aiming to develop a generalized computational tool in DHM.Our approach is independent of the sample's size (i.e., not requiring sample-free field of view within the hologram).Additionally, we validate the performance of this approach for several microscopic samples, including biological and non-biological samples, distorted with a spherical wavefront with different curvature.The proposed method has been implemented in MATLAB 2021a and Python 3.7.1 and is publicly available via GitHub [38], offering an open-source reconstruction tool (i.e., codes and GUI) for the DHM community." Comment #4: "Section 2: The authors may consider removing "operating" from the section title." Response: The title from Section 2 has been changed to "Off-axis Digital Holographic Microscopy in non-telecentric mode." Comment #5: "Section 2: In general, the explanations in this section include many details and are partly difficult to understand.The author may consider shortening and clarifying the text and to further emphasize the most important statements/topics."Response: We appreciate the reviewers' suggestions regarding Section 2. However, we believe that this section should be preserved as it is.Whereas this section may be quite dense, the details explained on it are important for new researchers (i.e., graduate students).This is especially important since there are not reported works providing this information for nontelecentric systems.One of the goals for our research groups is increasing the knowledge in Optical Engineering, enhancing career opportunities for undergraduate and graduate students, including underrepresented students.Nonetheless, if the reviewer still believes that this section is to complex, we are happy to reduce it and provide this information as an appendix.
Comment #6: "Section 2: Fig. 1: The authors show the sketch of a specific experimental arrangement in which sample illumination and reference wave are plane waves.However, in practice, e.g., a sample illumination via a condenser lens (for illustration see, e.g., ref. 35 of the manuscript or Drug Deliv. and Transl. Res. 12, 2207-2224(2022)) or utilization of a spherical reference wave can also result in a spatial frequency spectrum as illustrated in Fig. 2 of the manuscript.The authors thus may consider adding a discussion concerning the possible transfer of their approach to a more general regime."Response: We appreciate the reviewer's comment.The performance of the proposed approach is not restricted to non-telecentric imaging systems and reference plane waves.Our computational methods can be used for any off-axis DHM system in which a spherical wavefront distorts the complex object distribution in the recorded hologram.This distorting spherical wavefront can come from the object illumination, the reference illumination, the imaging system or all the above.We have validated this statement by introducing a non-telecentric relay system in the reference arm of our off-axis Mach Zehnder DHM system (Fig. 1 in the manuscript) and reconstructing the phase image with our computational tool.Even though, the object distribution is now affected by an additional spherical phase factor coming from the object, we were able to remove it properly.
The following sentences have been changed in the revised manuscript: L82-83: "Spherical wavefront distortions must be compensated to provide accurate phase measurements across the imaged field of view, converting the non-telecentric DHM system into a linear shift-invariant phase tool.These distortions can arise from several different sources and can be compensated via physical or numerical methods." L468-452: "Although the proposed computational approach has been validated with nontelecentric DHM imaging systems and reference plane waves, it can be used for any off-axis DHM system in which a spherical wavefront distorts the complex object distribution in the recorded hologram.This distorting spherical wavefront can come from the object illumination, the reference illumination, the imaging system or all the above."4)] with a digital reference wave (rD).A spherical wavefront related to using a non-telecentric imaging system still distorts the reconstructed raw phase image." Comment #8: "Section 3: Figs.7 and 8: The investigated sample "wedding cakes" should be explained with more details.The authors may consider adding cross-section plots through the phase images in Figs.8c and 9a." Response: We agree that a cross-sectional profile is a useful way of visualizing the "wedding cakes" sample.We have included a cross-sectional plot along the pink direction of Fig. 9c in Fig. 9e.This direction has been selected to highlight that the phase values in the wedding cakes are the same, independently of the field of view.
The panel (e) has been added to the following figure:

Fig 8 .
Fig 8. Reconstructed phase images of a USAF phase target using the (a) proposed method and (b) subtraction method.(c) Phase profile along the vertical direction (marked by the color arrows in panels a and b) through the horizontal lines of group 9.
7 (b) and (d) measured at the colored lines over the phase images in Fig 7(a) and (c).The comparison of these background profiles in Figs.7(b) and (d) confirms that the low frequency phase information is fairly uniform across the different directions Aside from minimal discrepancies from a complete flat background, these results demonstrate the effectiveness of our proposed method to compensate spherical distortions in all directions."Reviewer 2 , A. Wågbrø, Y. Mørch, A. Hatletveit, T. Visnes, et al., Drug Deliv Transl Res 12(9) (2022), 2207-2224.https://doi.org/10.1007/s13346-022-01207-5." Comment #3: "Introduction: In the last paragraph of the introduction the novelty aspects/extensions of the proposed phase compensation concept with respect to reference 35 become not fully clear.The authors should consider adding further clarifying details."

Comment # 7 :
"Section 3: Figs. 3 and 5 (major point): From the explanations it becomes not fully clear how the spherical phase aberrations are compensated and what are the differences in the procedure reported in reference 35 of the manuscript.For, example: Is equation 10 multiplied or subtracted from phase images like shown in Fig. 5a?The authors should add substantial clarifying information concerning this topic (and perhaps may extend Fig. 4 for an additional illustrating/clarifying sub figure?)." Response: We apologize for the confusion.Please see reply comment #1 to understand the main difference between Ref. [35] and this work.Figure 5a, which is the reconstructed phase image distorted by a spherical wavefront, is obtained by multiplying the complex amplitude distribution of the inverse Fourier Transform of the filtered hologram spectrum [Eq.(5)] with the complex amplitude distribution of the digital reference wave [Eq.(10)].We have remade Fig 4 to clarify this confusion.

Figure 4
Figure 4 has been modified in the revised manuscript: