We present a simple and effective method to eliminate system aberrations in quantitative phase imaging. Using spiral phase integration, complete information about system aberration is calculated from three laterally shifted phase images. The present method is especially useful when measuring confluent samples in which acquisition of background area is challenging. To demonstrate validity and applicability, we present measurements of various types of samples including microspheres, HeLa cells, and mouse brain tissue. Working conditions and limitations are systematically analyzed and discussed.
© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Quantitative phase imaging (QPI) is a rapidly growing technique for imaging biological samples [1,2]. Employing the principle of interferometric imaging, QPI measures the phase shift induced by a sample, from which 2D phase delay maps or 3D refractive index (RI) tomograms of the sample can be retrieved. Due to its non-destructive, label-free, and quantitative imaging capability , QPI has been widely used for study in various biological and medical fields, including hematology [4,5], immunology , neuroscience [7–9], cell biology [10,11], microbiology , histopathology [13,14], and nanotechnology [15–17].
The imaging quality in QPI is determined by speckle noise and aberration. Speckle noise, unwanted diffraction patterns due to the use of coherent illumination and multiple reflection or scattering from the surface of optical elements or dust particles, can be alleviated by various approaches. For example, the use of temporally or spatially incoherent illumination decreases the effects of speckle noise in QPI [18–21] or common-path interferometric geometry [22–25].
Compensation for system aberration has frequently been performed by the background subtraction method ; the aberration in a measured sample field is numerically subtracted using a field image obtained from a background or no-sample region under the same optical setup. Although the background subtraction method has been widely used, measurements of clear background regions are often not available in practical situations, especially for confluent biological cells and tissue slices. This constraint is unfortunate because QPI, with its unique label-free contrast modalities, its quantitative imaging capability, and its avoidance of damage to samples, has much to offer the fields of cell biology and histopathology. Alternatively, parametric approaches have been introduced in which system aberrations are simply modeled with a few parameters [27–30]. However, these approaches have limitations, especially when high orders of aberration are involved.
Here, we propose a simple but general method for the compensation of aberration. By exploiting the linear shift invariant property, aberration term is decoupled from captured holograms with lateral shifts. Three holograms are captured while laterally shifting the samples in orthogonal directions, and by the static part –aberration – is cancelled each other out. The spiral phase integration method [31,32] is utilized to reconstruct the aberration-free phase image from the shifted phase images. The applicability of the present method is demonstrated using various samples including polystyrene microspheres, eukaryotic cells, and a mouse brain tissue slice. Systematic analyses of working conditions and limitations are also discussed.
Figure 1 depicts the working principles of the present method. The quantitative phase image ϕ(x, y) retrieved from a hologram can be decomposed into an ideal aberration-free phase image ϕ0(x, y) and an aberration term ε(x, y) as ϕ(x, y) = ϕ0(x, y) + ε(x, y), as shown in Figs. 1(a)-1(b). In the background subtraction method , aberration term ε(x, y) is obtained by measuring the background hologram of an area without a sample [Fig. 1(c)]. Then, ε(x, y) is subtracted from the measured ϕ(x, y) in order to retrieve ϕ0(x, y), [Fig. 1(d)]. However, the background subtraction method has difficulty in finding clean background regions, especially for confluent cells or tissue slices.
Here, we propose a simple but require finding a background region to measure ε(x, y). Instead, the method retrieves ε(x, y) from three holograms with lateral shifts even in the presence of confluent samples. One hologram of a sample is recorded, and two other holograms are recorded after shifting the sample in orthogonal directions. From three measured holograms, original and shifted phase images are obtained. Subtracting the original phase image from shifted phase images, two differential phase images Δxϕ(x, y) and Δyϕ(x, y) are calculated as Δxϕ(x, y) = ϕ0(x + Δx, y) - ϕ0(x, y) and Δyϕ(x, y) = ϕ0(x, y + Δy) - ϕ0(x, y), respectively, [Figs. 1(e)-1(f)].
In order to retrieve the ideal phase image from the shifted phase images, we utilized the spiral phase integration [31,32]. The spiral phase integration enables a non-iterative reconstruction of a phase image from differential images, which has been utilized in differential interference contrast microscopy . According to Fourier shift theorem, the Fourier transforms of Δxϕ(x, y) and Δyϕ(x, y) are (e2πiΔxu − 1)Φ0(u,v) and (e2πiΔyv − 1)Φ0(u,v), respectively, where Φ0(u,v) is the Fourier pair of ϕ0(x, y), or FT[ϕ0(x, y)]. Because (e2πiΔxu − 1) and (e2πiΔyv − 1) are deterministic for the fixed spatial shift distances, Δx and Δy, Φ0(u,v) can easily be calculated by dividing (e2πiΔxu − 1) or (e2πiΔyv − 1) from the Fourier transforms of differential phase images. However, the division of (e2πiΔxu − 1) or (e2πiΔyv − 1) produces singularities at the zeroes of the functions. Thus, in order to minimize information loss near the singularities, we combine two spatial differential images in a complex form. Accordingly, we define G(u,v) and H(u,v) as follows,Appendix A for detailed derivations.
3.1. Optical setup
The experimental setup is presented in Fig. 2. A coherent plane-wave beam from an He-Ne laser (λ = 633 nm, HNL050R, Thorlabs Inc.) is split by a beam splitter into a reference and a sample beam. Samples were mounted on a motorized scanning stage (MLS203-1, Thorlabs Inc.) for the automatic shift. The light diffracted from the samples was collected by a high numerical aperture (NA) objective lens (NA = 1.2, water immersion, UPLSAPO 60XW, 60 × , Olympus, Inc., Japan). The sample beam was further magnified by a factor of two and interfered with a reference beam at the image plane. Interference pattern was recorded using a complementary metal–oxide semiconductor camera (DCC3240M, Thorlabs Inc.).
3.2. Sample preparation
Silica (SiO2) beads (n = 1.4570 at λ = 633 nm) with diameter of 3 μm were immersed in water and sandwiched by two cover slips. Polystyrene beads (n = 1.5875 at λ = 633 nm) with a diameter of 10 μm were immersed in index matching oil (n = 1.5279 at λ = 633 nm) and sandwiched by two cover slips. HeLa cells (human cervical cancer cell line) were cultured in DMEM (Dulbecco’s modified Eagle’s media) with 10% of FBS (fetal bovine serum) and 1% of penicillin-streptomycin, and fixed with 4% formaldehyde. Fixed cells were prepared in a Petri dish and immersed in PBS (Phosphate-buffered Saline). Mouse brain tissue was obtained from a 22-year-old male mouse. After fixation and dehydration, brain tissue was sliced and sandwiched between two cover slips with a mounting medium (n = 1.355).
4. Results and discussion
To demonstrate the validity of the present method, we captured phase images of the 3-μm-diameter silica beads. Phase images obtained by background subtraction method and by the present method are presented in Fig. 3. Figure 3(a) displays a raw phase image ϕ(x, y) retrieved from a single hologram. In the background subtraction method, an aberration phase image ε(x, y) is additionally measured. Then, ε(x, y) was subtracted from ϕ(x, y) to obtain an improved phase image ϕ0(x, y), as presented in Fig. 3(c).
In contrast, the present method utilizes two shift differential phase images Δxϕ(x, y) and Δyϕ(x, y) [Fig. 3(b)]. Spiral phase integration of the differential phase images yields the reconstructed phase image ϕ0(x, y), in which the aberration phase term ε(x, y) is automatically canceled out. Based on the reconstructed phase image, we can also retrieve ε(x, y) by subtracting the reconstructed phase image from the raw phase image [Fig. 3(c)]. involved.
The phase image reconstructed the conventional method contains high-frequency parasitic fringe patterns [Fig. 3(d)], mainly originated from multiple reflection. These patterns can be reduced by the conventional approach, but not always. For large or confluent samples, one should travel relatively long distance searching for the background (or non-sample) region. Accordingly, the multiple reflection patterns could be changed due to the slight changes in sample thickness, which cause remaining artifacts as shown in Fig. 3(d). In contrast, the proposed method does not need long distance travel regardless of the sample; therefore, the phase image improved by the present method showed relatively cleaner phase images. The standard deviation of the phase values in the flat no-sample area are 0.163 and 0.086 rad for the conventional method and the present method, respectively. The phase distributions of silica bead also agree well with the result from conventional method [Fig. 3(d)].
To demonstrate the applicability of the present method to biological samples, phase images of HeLa cells are measured [Fig. 4]. In the raw phase images, due to the system aberration, the quality of phase images is severely deteriorated [Fig. 4(a)]. In contrast, when the present method is applied, the background phase is removed and, then, the details of the sample can be clearly seen in the phase images [Fig. 4(b)]. The overall quality of the aberration-corrected images when using the present method is comparable to that of images measured with white-light QPI techniques [33–35].
To further demonstrate the potential of the present method, a phase image of mouse brain tissue slice is presented [Fig. 5]. Total 75 phase images were recorded to get the 5 × 5 stitched image. Then, the 25 phase images were manually stitched together to construct a large field-of-view phase image [Fig. 5(a)]. Representative raw and improved phase images are presented in Figs. 5(b)-(c). Raw phase images contain significant aberration artifacts, which can clearly be removed using the present method. Moreover, the corresponding aberration patterns are calculated from raw and improved phase images, as in Fig. 5(d). We found that the aberration images are generally similar, but still slight differences are present. This, it is possible to use the decoupled aberration pattern for the other patches for a limited range.
5. Summary and conclusion
In this Letter, we present a simple but powerful method of compensation for aberration in QPI. The present method will be applied to various fields, in particular the study of as confluent samples such as adherent biological cells and tissue slices.
Exploiting the spiral phase integration of differential phase images, we propose and experimentally demonstrate a method of compensation for aberration in QPI. This method does not require the measurement of any sample region, which limits the applicability of QPI for confluent samples such as high-density cell cultures and tissue slices. Instead, the present method measures three laterally translated phase images in the presence of samples, from which only set of sample phase information can be reconstructed; other static noises such as system aberration is systematically canceled out.
We demonstrate that the present method effectively removes aberrations from various types of samples including silica beads, eukaryotic cells, and tissue slices. We also compare the performance of the present method to that of the conventional background subtraction method.
For sparse or small samples, the background region is relatively easy to find, and the distance between sample and background region is usually close. Therefore, the aberration can be canceled out by conventional method . However, for confluent and large samples, the conventional method struggles since the background region is hard to find and/or far-off from the sample, which may induce the alteration of the aberration pattern [see the artifacts in Fig. 3(c)]. In contrast, the proposed method does not depend on the size, shape, or distribution of the sample, which could be especially beneficial for the large and confluent samples.
As shown in Figs. 4 and 5, the present method is also applicable to samples with complex contour, including cells and biological tissues. Moreover, phase images corrected with the present method can be successfully stitched together to reconstruct large field-of-view phase images. We also verify that our method is applicable to phase images that occupy the boundary of the field-of-view, including tissue samples.
One of the limitations of the proposed method is the requirement of the two additional hologram with lateral shifts in orthogonal directions. However, unlike the conventional method, the proposed method does not depend on the size, shape, or distribution of a sample. This advantage hugely reduces the total time, since there is no need for manually searching for a background region. Another limitation is that a sample should be stationary for the measurements of three holograms. Nonetheless, if the translation of a stage and the acquisition of three holograms is fast enough the present method can be applicable to common biological samples, as demonstrated in Figs. 4 and 5. Furthermore, the measurements of consecutive phase images only require one hologram for one corrected phase image. Because the present method is not limited by the type of instrumentation but is generally applicable to optical field measurement techniques, it can also be readily applied to various types of setup, ranging from digital holographic microscopy [37,38] to optical diffraction tomography [39,40].
1. The formulation of the algorithm
Spiral phase integration was first introduced in  for differential interference contrast (DIC) microscopy. It has been widely applied for DIC microscopy  and quadriwave lateral shearing interferometry (QWLSI) [42,43]. The spiral phase integration assumes spatial periodicity in an image, because the shift theorem for finite Fourier transform is applied to circular shift to an image. When a differential phase image does not satisfy spatial periodicity, artifacts occur at the boundary of the image, as described in [31,43]. In this Letter, boundary parts of shift differential phase images Δxϕ(x, y) and Δyϕ(x, y) were modified to match spatial periodicity and prevent boundary artifacts. Although the method described in  also removes boundary artifact, that method requires 4 times larger data. In contrast, the present method modifies only a few rows or columns of the data, which does not significantly increase computation time. In the spiral phase integration, the division of G(u,v) by H(u,v) generates singularities at H(uc,vc) = 0, which occurs when
2. The effects of the shift size
The effects of the shift size on the quality of image improvement are investigated. Figure 6(a) presents a raw phase image and a phase image improved by the background subtraction. To test the effects of the shift size, various shift sizes were applied to the present method. As displayed in Fig. 6(b), the best improvement is achieved when the shift is comparable to the diffraction limit size of the system (527.5 nm). When the shift was smaller than the diffraction-limited size, low-frequency aberration and artifacts are present whereas the shift is greater than the diffraction limited size, high spatial frequency artifacts arise near samples.
To investigate the effects of the shift to the image quality, Fourier spectra were shown. The second row of Fig. 6(c) presents the intensity map of |H(u,v)|+|H’(u,v)|, and the third row presents the retrieved phase spectra in Fourier domain Φ0(u,v). When the shift size is smaller than the diffraction limit, |H(u,v)|+|H’(u,v)| attains small values near the center of the Fourier plane. In this case, division by H(u,v) and H’(u,v) leads to the exaggeration of the low-spatial-frequency errors, as shown in the reconstruction image. In contrast, when the shift size is larger than the diffraction limit, zeroes of |H(u,v)|+|H’(u,v)| other than (u,v) = (0,0) located inside the NA circle (the dotted circle in Fig. 6). Thus, some of the high-spatial-frequency components of the image are lost, as shown in the reconstruction image.
KAIST, BK21+ program; Tomocube; National Research Foundation of Korea (2015R1A3A2066550, 2017M3C1A3013923, 2014K1A3A1A09063027).
Mr. Lee, and Prof. Park have financial interests in Tomocube Inc., a company that commercializes optical diffraction tomography and quantitative phase imaging instruments and is one of the sponsors of the work.
References and links
1. G. Popescu, Quantitative phase imaging of cells and tissues (McGraw Hill Professional, 2011).
2. K. Lee, K. Kim, J. Jung, J. Heo, S. Cho, S. Lee, G. Chang, Y. Jo, H. Park, and Y. Park, “Quantitative phase imaging techniques for the study of cell pathophysiology: from principles to applications,” Sensors (Basel) 13(4), 4170–4191 (2013). [PubMed]
3. D. Kim, S. Lee, M. Lee, J. Oh, S.-A. Yang, and Y. Park, “Refractive index as an intrinsic imaging contrast for 3-D label-free live cell imaging,” https://www.biorxiv.org/content/early/2017/02/06/106328 (2017).
4. J. Hur, K. Kim, S. Lee, H. Park, and Y. Park, “Melittin-induced alterations in morphology and deformability of human red blood cells using quantitative phase imaging techniques,” Sci. Rep. 7(1), 9306 (2017). [PubMed]
5. N. T. Shaked, L. L. Satterwhite, M. J. Telen, G. A. Truskey, and A. Wax, “Quantitative microscopy and nanoscopy of sickle red blood cells performed by wide field digital interferometry,” J. Biomed. Opt. 16(3), 030506 (2011). [PubMed]
6. J. Yoon, Y. Jo, M. H. Kim, K. Kim, S. Lee, S.-J. Kang, and Y. Park, “Identification of non-activated lymphocytes using three-dimensional refractive index tomography and machine learning,” Sci. Rep. 7(1), 6654 (2017). [PubMed]
7. S. A. Yang, J. Yoon, K. Kim, and Y. Park, “Measurements of morphological and biophysical alterations in individual neuron cells associated with early neurotoxic effects in Parkinson’s disease,” Cytometry A 91(5), 510–518 (2017). [PubMed]
8. P. Jourdain, N. Pavillon, C. Moratal, D. Boss, B. Rappaz, C. Depeursinge, P. Marquet, and P. J. Magistretti, “Determination of transmembrane water fluxes in neurons elicited by glutamate ionotropic receptors and by the cotransporters KCC2 and NKCC1: a digital holographic microscopy study,” J. Neurosci. 31(33), 11846–11854 (2011). [PubMed]
9. M. E. Kandel, D. Fernandes, A. M. Taylor, H. Shakir, C. Best-Popescu, and G. Popescu, “Three-dimensional intracellular transport in neuron bodies and neurites investigated by label-free dispersion-relation phase spectroscopy,” Cytometry A 91(5), 519–526 (2017). [PubMed]
10. K. Kim, S. Lee, J. Yoon, J. Heo, C. Choi, and Y. Park, “Three-dimensional label-free imaging and quantification of lipid droplets in live hepatocytes,” Sci. Rep. 6, 36815 (2016). [PubMed]
11. B. Kemper, J. Wibbeling, L. Kastl, J. Schnekenburger, and S. Ketelhut, “Continuous morphology and growth monitoring of different cell types in a single culture using quantitative phase microscopy,” in SPIE Optical Metrology, (International Society for Optics and Photonics, 2015), 952902.
12. Y. Jo, S. Park, J. Jung, J. Yoon, H. Joo, M. H. Kim, S. J. Kang, M. C. Choi, S. Y. Lee, and Y. Park, “Holographic deep learning for rapid optical screening of anthrax spores,” Sci. Adv. 3(8), e1700606 (2017). [PubMed]
13. M. Lee, E. Lee, J. Jung, H. Yu, K. Kim, J. Yoon, S. Lee, Y. Jeong, and Y. Park, “Label-free optical quantification of structural alterations in Alzheimer’s disease,” Sci. Rep. 6, 31034 (2016). [PubMed]
14. T. H. Nguyen, S. Sridharan, V. Macias, A. Kajdacsy-Balla, J. Melamed, M. N. Do, and G. Popescu, “Automatic Gleason grading of prostate cancer using quantitative phase imaging and machine learning,” J. Biomed. Opt. 22(3), 036015 (2017). [PubMed]
15. D. Kim, N. Oh, K. Kim, S. Lee, C.-G. Pack, J.-H. Park, and Y. Park, “Label-free high-resolution 3-D imaging of gold nanoparticles inside live cells using optical diffraction tomography,” http://www.sciencedirect.com/science/article/pii/S1046202317301792 (2017).
16. G. Baffou, P. Bon, J. Savatier, J. Polleux, M. Zhu, M. Merlin, H. Rigneault, and S. Monneret, “Thermal imaging of nanostructures by quantitative optical phase analysis,” ACS Nano 6(3), 2452–2458 (2012). [PubMed]
17. N. A. Turko, D. Roitshtain, O. Blum, B. Kemper, and N. T. Shaked, “Dynamic measurements of flowing cells labeled by gold nanoparticles using full-field photothermal interferometric imaging,” J. Biomed. Opt. 22(6), 66012 (2017). [PubMed]
18. Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. U. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express 19(2), 1016–1026 (2011). [PubMed]
19. Y. Park, W. Choi, Z. Yaqoob, R. Dasari, K. Badizadegan, and M. S. Feld, “Speckle-field digital holographic microscopy,” Opt. Express 17(15), 12285–12292 (2009). [PubMed]
20. F. Dubois, M.-L. N. Requena, C. Minetti, O. Monnom, and E. Istasse, “Partial spatial coherence effects in digital holographic microscopy with a laser source,” Appl. Opt. 43(5), 1131–1139 (2004). [PubMed]
21. S. Shin, K. Kim, K. Lee, S. Lee, and Y. Park, “Effects of spatiotemporal coherence on interferometric microscopy,” Opt. Express 25(7), 8085–8097 (2017). [PubMed]
22. Y. Park, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Diffraction phase and fluorescence microscopy,” Opt. Express 14(18), 8263–8268 (2006). [PubMed]
23. Y. Kim, H. Shim, K. Kim, H. Park, J. H. Heo, J. Yoon, C. Choi, S. Jang, and Y. Park, “Common-path diffraction optical tomography for investigation of three-dimensional structures and dynamics of biological cells: erratum,” Opt. Express 23, 18996 (2015). [PubMed]
24. A. S. Singh, A. Anand, R. A. Leitgeb, and B. Javidi, “Lateral shearing digital holographic imaging of small biological specimens,” Opt. Express 20(21), 23617–23622 (2012). [PubMed]
25. P. Girshovitz and N. T. Shaked, “Compact and portable low-coherence interferometer with off-axis geometry for quantitative phase microscopy and nanoscopy,” Opt. Express 21(5), 5701–5714 (2013). [PubMed]
26. C. M. Vest, “Holographic interferometry,” New York, John Wiley and Sons, Inc., 1979. 476 p. (1979).
27. A. Stadelmaier and J. H. Massig, “Compensation of lens aberrations in digital holography,” Opt. Lett. 25(22), 1630–1632 (2000). [PubMed]
28. P. Ferraro, S. De Nicola, A. Finizio, G. Coppola, S. Grilli, C. Magro, and G. Pierattini, “Compensation of the inherent wave front curvature in digital holographic coherent microscopy for quantitative phase-contrast imaging,” Appl. Opt. 42(11), 1938–1946 (2003). [PubMed]
29. T. Colomb, F. Montfort, J. Kühn, N. Aspert, E. Cuche, A. Marian, F. Charrière, S. Bourquin, P. Marquet, and C. Depeursinge, “Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy,” J. Opt. Soc. Am. A 23(12), 3177–3190 (2006). [PubMed]
30. L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. Nicola, “Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram,” Appl. Phys. Lett. 90, 041104 (2007).
31. M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214(Pt 1), 7–12 (2004). [PubMed]
32. D. Fu, S. Oh, W. Choi, T. Yamauchi, A. Dorn, Z. Yaqoob, R. R. Dasari, and M. S. Feld, “Quantitative DIC microscopy using an off-axis self-interference approach,” Opt. Lett. 35(14), 2370–2372 (2010). [PubMed]
33. T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat. Photonics 8, 256–263 (2014).
34. Y. Baek, K. Lee, J. Yoon, K. Kim, and Y. Park, “White-light quantitative phase imaging unit,” Opt. Express 24(9), 9308–9315 (2016). [PubMed]
35. B. Bhaduri, H. Pham, M. Mir, and G. Popescu, “Diffraction phase microscopy with white light,” Opt. Lett. 37(6), 1094–1096 (2012). [PubMed]
36. S. Chowdhury, W. J. Eldridge, A. Wax, and J. Izatt, “Refractive index tomography with structured illumination,” Optica 4, 537–545 (2017).
37. M. K. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev. 1, 018005 (2010).
38. P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. 30(5), 468–470 (2005). [PubMed]
39. K. Kim, J. Yoon, S. Shin, S. Lee, S.-A. Yang, and Y. Park, “Optical diffraction tomography techniques for the study of cell pathophysiology,” J. Biomed. Photon. Eng. 2, 020201 (2016).
40. K. Kim, H. Yoon, M. Diez-Silva, M. Dao, R. R. Dasari, and Y. Park, “High-resolution three-dimensional imaging of red blood cells parasitized by Plasmodium falciparum and in situ hemozoin crystals using optical diffraction tomography,” J. Biomed. Opt. 19(1), 011005 (2014). [PubMed]
41. S. V. King, A. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, “Quantitative phase microscopy through differential interference imaging,” J. Biomed. Opt. 13, 024020 (2008).
42. S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14(9), 3792–3805 (2006). [PubMed]
43. P. Bon, S. Monneret, and B. Wattellier, “Noniterative boundary-artifact-free wavefront reconstruction from its derivatives,” Appl. Opt. 51(23), 5698–5704 (2012). [PubMed]