Single shot quantitative phase gradient estimation using Wigner-Ville distribution in digital holographic microscopy

Ankur Vishnoi; Rajshekhar Gannavarpu; Rajshekhar Gannavarpu

doi:10.1364/OSAC.431940

1. Introduction

Biological cells mostly behave as phase objects, and are characterized by weak scattering and low absorption under visible light [1]. Hence, bright-field imaging of these cells under a microscope provides poor contrast, and usually requires an invasive procedure of addition of labels such as fluorophores, stains or dyes to improve the image contrast [2]. For non-invasive and label-free microscopy of biological cells, quantitative phase imaging has emerged as an important methodology in the domain because the phase of the scattered wave encodes information about the test specimen’s properties such as refractive index and structure [3]. Accordingly, several quantitative phase imaging techniques such as digital holographic microscopy (DHM) [4,5], spatial light interference microscopy [6], diffraction phase microscopy [7], Hilbert phase microscopy [8], transport of intensity imaging [9], tomographic deconvolution phase microscopy [10], phase-sensitive optical coherence microscopy [11], fast Fourier phase microscopy [12] and Fourier ptychography [13] have been proposed. In addition to phase, another quantity of great interest in the domain is the phase gradient, as it directly provides relevant information about morphological features such as edges and sharp structures in the test specimen [14–16]. Recently, several techniques have been reported for phase derivative and gradient estimation such as gradient field microscopy [17], quantitative phase gradient microscopy [18], gradient light interference microscopy [19,20], asymmetric illumination-based differential phase contrast microscopy [21], oblique back illumination microscopy [22] and color-coded LED microscopy [23]. However, most of these techniques rely on specialized setups, sophisticated equipment such as spatial light modulators, specific illumination configurations and qualitative assessment using intensity approximations. This could lead to additional experimental complexity and potential requirement of trained operator. Another popular approach which does not require specialized experimental modalities to extract the quantitative phase derivative information is to perform numerical differentiation operation [24,25]. The main idea in such approach is to initially estimate the phase map, and apply a finite difference operation to estimate the phase derivative. However, such operation is susceptible to noise and usually requires filtering procedures [26]. Further, phase gradient computation requires the estimation of partial derivatives with respect to horizontal and vertical dimensions. Hence, simultaneous measurement of both phase derivatives is strongly desired in single shot quantitative phase gradient measurement.

In this paper, we propose a method for direct and single shot measurement of the phase gradient in quantitative phase imaging without the need of additional experimental complexity, numerical differentiation and filtering operations. The method is based on space-frequency analysis of the complex wavefield using Wigner-Ville distribution to simultaneously compute the phase derivatives along horizontal and vertical dimensions. In the domain of non-destructive deformation testing, Wigner-Ville distribution has been applied for measurement of displacement derivatives and strain of a deformed object [27–30]. However, its applicability for studying biological cells and micro-structure samples via quantitative phase gradient microscopy has not been hitherto explored to the best of our knowledge. The proposed method is presented in the context of digital holographic microscopy, and both transmission configuration for imaging biological samples and reflection or epi-illumination configuration for imaging material samples are investigated. The paper is organized as follows. The details about the experimental configuration and hologram processing are described in section 2. The experimental results are demonstrated in section 3. Finally, the discussions and conclusions are presented.

2. Experimental setup & hologram processing

For our study, we built custom off-axis digital holographic microscopy setups for both reflective and transmissive configurations as depicted in Fig. 1. A coherent laser (LS) having wavelength of 532 nanometers is used as the illumination source. The light beam originating from source is expanded using beam expander (BE) and split in two beams by a beamsplitter (BS1). One beam acts as the reference beam which traverses directly to the camera; whereas the other beam is used to illuminate the sample. A microscope objective (MO) is used to produce the magnified image of the object under inspection. Depending on the test sample and the area to be imaged, we used microscope objectives with different magnification ($5\times$ with $NA=0.1$, $20\times$ with $NA=0.4$ and $40\times$ with $NA=0.65$) in our experiments. The beam scattered from the object, also known as object beam, and the reference beam interfere at the camera using combining beamsplitter (BS2) in Mach-Zehnder configuration. A small tilt is introduced in the reference beam to create an offset angle between reference and object beam, which results in an off-axis configuration [31]. A monochrome CMOS camera (Allied Vision: Mako U-503B) is used to record the interference pattern or hologram. Next, the complex amplitude of the object wavefield, denoted by $U_1(x,y)$, is reconstructed by numerical processing of the hologram using angular spectrum approach [32]. The parameters used in numerical reconstruction are as follows: pixel resolution or image size of $2592 \times 1944$ pixels, pixel size of $2.2 \,\mu m \times 2.2 \,\mu m$, and the propagation distance as 0.1 mm. For our analysis, we also record a reference or background hologram without the test sample and perform numerically reconstruction to obtain another complex amplitude signal $U_2(x,y)$. Next, we perform a complex superposition of the two wavefields to minimize the effect of aberrations [33]. The resulting complex signal of interest can be mathematically represented as ,

(1)$$\begin{aligned}O(x,y)&=U_1 U_2^*\\ &=a(x,y)e^{j \phi_{obj}(x,y)} + \eta(x,y) \end{aligned}$$

where $\phi _{obj}$ represents the phase change introduced by the test specimen, $a(x,y)$ is the amplitude term, $\eta$ represents the noise and "$*$" denotes complex conjugate operator. Given $O(x,y)$, our goal is to estimate the phase gradient $\nabla \phi _{obj}$, which is mathematically expressed as

(2)$$\nabla \phi_{obj} = \sqrt{\left(\frac{\partial \phi_{obj}}{\partial x}\right)^2 + \left(\frac{\partial \phi_{obj}}{\partial y}\right)^2}$$

where $\partial \phi _{obj}/\partial x$ and $\partial \phi _{obj}/\partial y$ are the partial phase derivatives with respect to $x$ and $y$. It is evident that we need to estimate both partial derivatives to compute the phase gradient in digital holographic microscopy. In our case, this is achieved using Wigner-Ville distribution, which is essentially a space-frequency transform that provides joint space and frequency representation of the signal [34]. For the given complex signal, the Wigner-Ville distribution is mathematically expressed as,

(3)$$W(x,y,f_x,f_y) =\iint g(x',y') O(x+x',y+y') O^*(x-x',y-y') e^{{-}j4\pi (f_x x' + f_y y')} \,dx'\,dy'$$

where $g(x',y')$ represents a real symmetric Gaussian window. In the above equation, the Wigner-Ville distribution $W$ is a function of both space ($x$ and $y$) and frequency ($f_x$ and $f_y$) parameters. Thus, the above equation signifies a joint space-frequency transform. For our approach, we assume that the amplitude term is relatively constant within the window. Accordingly, we obtain,

(4)$$\begin{aligned} &W(x,y,f_x,f_y)\\ &=a^2(x,y)\iint g(x',y') e^{j[\phi_{obj}(x+x',y+y')-\phi_{obj}(x-x',y-y')]} e^{{-}j4\pi (f_x x' + f_y y')} \,dx'\,dy' \end{aligned}$$

Next, we model the phase as a second order polynomial within the window using Taylor series approximation. Consequently, we have,

(5)$$\phi_{obj}(x+x',y+y')- \phi_{obj}(x-x',y-y') \approx 2x' \frac{\partial\phi_{obj}(x,y)}{\partial x} + 2y'\frac{\partial\phi_{obj}(x,y)}{\partial y}$$

Using the above approximation, Eq. (4) reduces to the form,

(6)$$\begin{aligned} &W(x,y,f_x,f_y)\\ &=a^2(x,y)\iint g(x',y') e^{{-}j4\pi\left [ f_x -\frac{1}{2\pi}\frac{\partial\phi_{obj}}{\partial x} \right ]x'} e^{{-}j4\pi\left [ f_y -\frac{1}{2\pi}\frac{\partial\phi_{obj}}{\partial y} \right ]y'} dx' dy' \end{aligned}$$

The above equation can be further simplified as,

(7)$$W(x,y,f_x,f_y) = a^2(x,y) \mathcal{G} \left[2\left(f_x - \frac{1}{2\pi}\frac{\partial \phi_{obj}}{\partial x}\right), 2\left(f_y- \frac{1}{2\pi}\frac{\partial \phi_{obj}}{\partial y}\right) \right]$$

where $\mathcal {G}$ represents the two dimensional Fourier transform of window function. Considering the low pass nature of Gaussian signal, the peak of its Fourier transform occurs near the zero frequency. From the above equation, this low pass behavior signifies that whenever $2 \pi f_x= \partial \phi _{obj}/\partial x$ and $2 \pi f_y = \partial \phi _{obj}/\partial y$, the Wigner-Ville spectrum attains its peak. Hence, the phase derivative can be evaluated by computing the spatial frequencies $(f_x,f_y)$ at which the Wigner-Ville spectrum $|W(x,y,f_x,f_y)|$ has maximum value. In other words, we obtain,

(8)$$\left[ \frac{\partial \phi_{obj}}{\partial x} , \frac{\partial \phi_{obj}}{\partial y} \right] = \mathop{\textrm{arg}\,\textrm{max}}\limits_{(2\pi f_x,2\pi f_y)}| W(x, y, f_x , f_y )|$$

It can be noted that the Wigner-Ville method offers simultaneous estimation of the two partial derivatives of phase map with respect to the horizontal and vertical dimensions. Once these derivatives are obtained, the phase gradient is computed using Eq. (2).

Fig. 1. Schematic of DHM experimental setup for (a) transmissive sample and (b) reflective sample

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3. Results

For digital holographic microscopy experiment in transmission configuration, we used human epithelial cheek cells and onion cells as the test specimens. For imaging, the specimen was placed on a glass slide and protected with a cover slip. These cells were imaged without adding any labels or contrast agents. Next, we performed numerical reconstruction of the recorded holograms using angular spectrum approach and obtained the complex signal of interest as described in section 2. Subsequently, we applied the proposed method for phase gradient estimation. For comparison, we also computed the phase gradient using the standard numerical differentiation approach. This process involves two steps: (1) phase estimation by computing the argument of the complex signal followed by phase unwrapping [35], and (2) numerical computation of the phase derivatives using finite difference approach.

In Figs. 2(a,e,i), we show the recorded holograms corresponding to several cheek cells. These cells are of varied shapes and sizes. In Figs. 2(b,f,j), we show the phase gradient (radians/pixel) estimated using the proposed method. As part of the first step in the numerical differentiation approach, the phase maps obtained using phase unwrapping are shown in Figs. 2(c,g,k). Similarly, we show the phase gradient (radians/pixel) estimated using the numerical differentiation method in Figs. 2(d,h,l). Further, we show the recorded holograms corresponding to onion cells in Figs. 3(a,e). In Figs. 3(b,f), we show the phase gradient estimated using the proposed method. We also show the unwrapped phase maps in Figs. 3(c,g) and corresponding phase gradient estimated using the numerical differentiation method in Figs. 3(d,h).

Fig. 2. Imaging of epithelial cheek cells. (a,e,i) Recorded Holograms. (b,f,j) Phase gradient estimated using the proposed method. (c,g,k) Unwrapped phase in radians. (d,h,l) Phase gradient estimated using the numerical differentiation method.

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Fig. 3. Imaging of onion cells. (a,e) Recorded Holograms. (b,f) Phase gradient estimated using the proposed method. (c,g) Unwrapped phase in radians. (d,h) Phase gradient estimated using the numerical differentiation method.

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Next, we conducted second set of experiments using digital holographic microscope in reflection configuration. We used standard calibration test targets such as the USAF1951 (Thorlabs, R1DS1P) sample and micro-pillar structures (Micromasch,TGXYZ02) for imaging. We show the recorded holograms corresponding to USAF target in Figs. 4(a,e). In Figs. 4(b,f), we show the phase gradient estimated using the proposed method. Similarly, we show the unwrapped phase maps in Figs. 4(c,g) and corresponding phase gradient estimated using the numerical differentiation method in Figs. 4(d,h). Similarly, the recorded holograms corresponding to micro-pillar structures are shown in Figs. 5(a,e). The corresponding phase gradient estimated using the proposed method is shown in Figs. 5(b,f). In addition, we show the unwrapped phase maps in Figs. 5(c,g) and phase gradient estimated using the numerical differentiation method in Figs. 5(d,h).

Fig. 4. Imaging of USAF target. (a,e) Recorded Holograms. (b,f) Phase gradient estimated using the proposed method. (c,g) Unwrapped phase in radians. (d,h) Phase gradient estimated using the numerical differentiation method.

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Fig. 5. Imaging of micro-pillar sample. (a,e) Recorded Holograms. (b,f) Phase gradient estimated using the proposed method. (c,g) Unwrapped phase in radians. (d,h) Phase gradient estimated using the numerical differentiation method.

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It is evident from these results that the proposed method exhibits comparatively better structural details than the standard numerical differentiation approach. This is an important advantage of the proposed method. In addition, as described before, both the phase derivatives are obtained in a single shot fashion without the need of additional filtering and unwrapping operations.

4. Discussion

For the transmissive cell samples, we observe that the phase gradient highlights the morphological details. For the reflective samples, the phase gradient readily provides information about the sharp features such as the edges in the test specimens. Hence, reliable measurement of phase gradient has important application potential in digital holographic microscopy. The conventional techniques in quantitative phase gradient imaging have mostly relied on specialized imaging modalities to obtain the information about the phase gradient. In contrast, our approach relies on numerical processing of the hologram signals using Wigner-Ville distribution to obtain the phase gradient. Consequently, the proposed method is suitable to be applied in a conventional digital holographic microscopy setup without the need of specialized equipment or specific experimental modifications. Further, the standard numerical differentiation approach relies on extracting the unwrapped phase map as its first step, and thus the accuracy of this method is limited by the accuracy of the unwrapping algorithm used. In other words, the numerical differentiation approach is highly susceptible to unwrapping errors and the proposed method offsets this limitation by direct estimation of the phase derivative without unwrapping. The main limitation of our approach is the large computational cost associated with computing the Wigner-Ville distribution. Though, graphics processing unit based computing [36] has emerged as a potential solution to this problem in the recent years and can provide exciting computational gains due to high parallel processing capabilities.

5. Conclusion

In conclusion, we demonstrated a quantitative phase gradient estimation approach in digital holographic microscopy. The method is based on the application of Wigner-Ville distribution and offers the feasibility of single shot phase gradient measurement. The results validate the utility of the proposed method for imaging transmissive cell samples as well as reflective metrological specimens. The authors believe that proposed method can enhance the applicability of digital holographic microscopy for cell imaging and non-invasive optical metrology.

Funding

Department of Science and Technology, Ministry of Science and Technology, India (DST/NM/NT/2018/2).

Disclosures

The authors declare that there are no conflicts of interest related to this paper.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. M. Mir, B. Bhaduri, R. Wang, R. Zhu, and G. Popescu, “Quantitative phase imaging,” Prog. Opt. 57, 133–217 (2012). [CrossRef]

2. R. Kasprowicz, R. Suman, and P. O’Toole, “Characterising live cell behaviour: Traditional label-free and quantitative phase imaging approaches,” The Int. J. Biochemistry & Cell Biol. 84, 89–95 (2017). [CrossRef]

3. Y. Park, C. Depeursinge, and G. Popescu, “Quantitative phase imaging in biomedicine,” Nat. Photonics 12(10), 578–589 (2018). [CrossRef]

4. M. K. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev. 1(1), 018005 (2010). [CrossRef]

5. P. Marquet, C. Depeursinge, and P. J. Magistretti, “Review of quantitative phase-digital holographic microscopy: promising novel imaging technique to resolve neuronal network activity and identify cellular biomarkers of psychiatric disorders,” Neurophotonics 1(2), 020901 (2014). [CrossRef]

6. 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). [CrossRef]

7. B. Bhaduri, C. Edwards, H. Pham, R. Zhou, T. H. Nguyen, L. L. Goddard, and G. Popescu, “Diffraction phase microscopy: principles and applications in materials and life sciences,” Adv. Opt. Photonics 6(1), 57–119 (2014). [CrossRef]

8. T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. 30(10), 1165–1167 (2005). [CrossRef]

9. T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997). [CrossRef]

10. M. H. Jenkins and T. K. Gaylord, “Three-dimensional quantitative phase imaging via tomographic deconvolution phase microscopy,” Appl. Opt. 54(31), 9213–9227 (2015). [CrossRef]

11. C. G. Rylander, D. P. Davé, T. Akkin, T. E. Milner, K. R. Diller, and A. J. Welch, “Quantitative phase-contrast imaging of cells with phase-sensitive optical coherence microscopy,” Opt. Lett. 29(13), 1509–1511 (2004). [CrossRef]

12. N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, “Quantitative phase imaging of live cells using fast fourier phase microscopy,” Appl. Opt. 46(10), 1836–1842 (2007). [CrossRef]

13. X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via fourier ptychographic microscopy,” Opt. Lett. 38(22), 4845–4848 (2013). [CrossRef]

14. M. Pluta, “Nomarski’s DIC microscopy: a review,” Proc. SPIE 1846, 10–25 (1994). [CrossRef]

15. S. A. Prahl, A. Dayton, K. Juedes, E. J. Sánchez, R. P. López, and D. D. Duncan, “Experimental validation of phase using nomarski microscopy with an extended fried algorithm,” J. Opt. Soc. Am. A 29(10), 2104–2109 (2012). [CrossRef]

16. D. D. Duncan, D. G. Fischer, A. Dayton, and S. A. Prahl, “Quantitative carré differential interference contrast microscopy to assess phase and amplitude,” J. Opt. Soc. Am. A 28(6), 1297–1306 (2011). [CrossRef]

17. T. Kim, S. Sridharan, and G. Popescu, “Gradient field microscopy of unstained specimens,” Opt. Express 20(6), 6737–6745 (2012). [CrossRef]

18. H. Kwon, E. Arbabi, S. M. Kamali, M. Faraji-Dana, and A. Faraon, “Single-shot quantitative phase gradient microscopy using a system of multifunctional metasurfaces,” Nat. Photonics 14(2), 109–114 (2020). [CrossRef]

19. M. E. Kandel, C. Hu, G. N. Kouzehgarani, E. Min, K. M. Sullivan, H. Kong, J. M. Li, D. N. Robson, M. U. Gillette, C. Best-Popescu, and G. Popescu, “Epi-illumination gradient light interference microscopy for imaging opaque structures,” Nat. Commun. 10(1), 4691–4699 (2019). [CrossRef]

20. T. H. Nguyen, M. E. Kandel, M. Rubessa, M. B. Wheeler, and G. Popescu, “Gradient light interference microscopy for 3d imaging of unlabeled specimens,” Nat. Commun. 8(1), 210–219 (2017). [CrossRef]

21. S. B. Mehta and C. J. Sheppard, “Quantitative phase-gradient imaging at high resolution with asymmetric illumination-based differential phase contrast,” Opt. Lett. 34(13), 1924–1926 (2009). [CrossRef]

22. T. N. Ford, K. K. Chu, and J. Mertz, “Phase-gradient microscopy in thick tissue with oblique back-illumination,” Nat. Methods 9(12), 1195–1197 (2012). [CrossRef]

23. D. Lee, S. Ryu, U. Kim, D. Jung, and C. Joo, “Color-coded led microscopy for multi-contrast and quantitative phase-gradient imaging,” Biomed. Opt. Express 6(12), 4912–4922 (2015). [CrossRef]

24. C. Quan, D. Balakrishnan, W. Chen, and C. Tay, “Phase retrieval and phase derivative determination in digital holography,” in Advancement of Optical Methods in Experimental Mechanics, Volume 3, (Springer, 2014), pp. 241–250.

25. D. Khodadad, “Phase-derivative-based estimation of a digital reference wave from a single off-axis digital hologram,” Appl. Opt. 55(7), 1663–1669 (2016). [CrossRef]

26. Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal fourier analyses of a digital hologram sequence,” Appl. Opt. 46(23), 5719–5727 (2007). [CrossRef]

27. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Strain, curvature, and twist measurements in digital holographic interferometry using pseudo-wigner–ville distribution based method,” Rev. Sci. Instrum. 80(9), 093107 (2009). [CrossRef]

28. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Adaptive window wigner-ville-distribution-based method to estimate phase derivative from optical fringes,” Opt. Lett. 34(20), 3151–3153 (2009). [CrossRef]

29. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Polynomial wigner–ville distribution-based method for direct phase derivative estimation from optical fringes,” J. Opt. A: Pure Appl. Opt. 11(12), 125402 (2009). [CrossRef]

30. A. Vishnoi and G. Rajshekhar, “Rapid deformation analysis in digital holographic interferometry using graphics processing unit accelerated wigner–ville distribution,” Appl. Opt. 58(16), 4420–4424 (2019). [CrossRef]

31. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24(5), 291–293 (1999). [CrossRef]

32. A. Anand, V. K. Chhaniwal, and B. Javidi, “Real-time digital holographic microscopy for phase contrast 3d imaging of dynamic phenomena,” J. Disp. Technol. 6(10), 500–505 (2010). [CrossRef]

33. 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). [CrossRef]

34. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Estimation of displacement derivatives in digital holographic interferometry using a two-dimensional space-frequency distribution,” Opt. Express 18(17), 18041–18046 (2010). [CrossRef]

35. M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41(35), 7437–7444 (2002). [CrossRef]

36. Wenjing Gao and Qian Kemao, “Parallel computing in experimental mechanics and optical measurement: A review,” Opt. Lasers Eng. 50(4), 608–617 (2012). [CrossRef]

Single shot quantitative phase gradient estimation using Wigner-Ville distribution in digital holographic microscopy

Abstract

1. Introduction

2. Experimental setup & hologram processing

3. Results

4. Discussion

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (8)

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