Hilbert–Huang single-shot spatially multiplexed interferometric microscopy (H2S2MIM) is presented as the implementation of a robust, fast, and accurate single-shot phase estimation algorithm with an extremely simple, low-cost, and highly stable way to convert a bright field microscope into a holographic one using partially coherent illumination. Altogether, H2S2MIM adds high-speed (video frame rate) quantitative phase imaging capability to a commercially available nonholographic microscope with improved phase reconstruction (coherence noise reduction). The technique has been validated using a objective in a regular Olympus BX-60 upright microscope for static, as well as dynamic, samples showing perfect agreement with the results retrieved from a temporal phase-shifting algorithm.
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Nowadays quantitative phase imaging (QPI) of biological specimens is a very useful tool for accurate analysis in medical diagnosis [1–3]. QPI is usually achieved by digital holographic microscopy (DHM), a technique that combines into a single platform high-quality imaging provided by microscopy, whole-object wavefront recovery ensured by holography, and numerical processing capabilities yielded by computers [4,5]. DHM allows visualization of phase samples using a noninvasive (no need for stained samples), full-field (nonscanning), real-time (on-line control), noncontact (no sample damage), and static (no moving components) operating principle.
In connection with this fact, today there is a growing interest in proposing new DHM configurations with improved capabilities in terms of simplicity, accuracy, robustness, price, and ease of use [6–22]. Among those methods, spatially multiplexed interferometric microscopy (SMIM) proposes a noncomplex and low-cost way to convert a commercially available regular microscope into a holographic one with only minimal modifications [19,20]. SMIM is based on a common-path interferometric layout and the modifications consist of (1) the replacement of the broadband light source by a coherent one, (2) the proper insertion of a one-dimensional (1D) diffraction grating in the microscope embodiment, and (3) saving a clear region at the input plane for reference beam transmission. The SMIM has been validated under an off-axis holographic recording with Fourier filtering  or slightly off-axis recording with a phase-shifting (P-S) algorithm . In Ref. , SMIM is merged with superresolution capabilities to overcome its main drawback: the field of view (FOV) restriction imposed by the need to transmit a clear transparent reference beam for holographic recording.
The recently proposed single-shot Hilbert–Huang phase microscopy (S2H2PM) method is composed of a fast, accurate, and robust QPI technique employing a single hologram . The S2H2PM is based on the Hilbert–Huang transform (HHT). Huang’s part—empirical mode decomposition (EMD)—originally employs adaptive dissection of analyzed 1D nonstationary and nonlinear signal into a set of so-called intrinsic mode functions representing signal features in different scales . Efficiently managing this set of sub-signals (discarding spurious and preserving informative ones), one can develop a capable signal processing tool. Moreover, each empirical mode oscillates around zero which makes them perfectly suited for a further Hilbert transform phase and amplitude demodulation based on the analytic signal concept . Recently, HHT was generalized into two-dimensional (2D), substantially enhanced in terms of both time consumption and accuracy extending its capabilities especially onto fringe pattern analysis, e.g., in phase microscopy [23,25–29]. As a result, S2H2PM uses an adaptive image-domain fringe filtering based on 2D enhanced fast EMD (first part of the 2D HHT) and 2D Hilbert spiral transform  with precise phase demodulation aided by the local fringe direction map estimation  (second party of the 2D HHT).
The S2H2PM retrieves sample phase distribution, regardless of the tilt angle between both interferometric beams and without the need to apply a temporal P-S algorithm. This capability is very appealing since it optimizes the space-bandwidth product of the system while allowing real-time imaging only limited by the sensor acquisition frame rate. In a few words, S2H2PM  improves and overcomes the limitations of previously introduced Hilbert transform for phase demodulation [25,26] regarding (1) a 1D operating principle needing perpendicular fringes to the analyzed direction; (2) reduced flexibility in the type of fringes to be analyzed (sinusoidal, with no nonlinearities, not closed or significantly bent, having a certain fringe period close to off-axis configuration); and (3) uniform and constant hologram background. The S2H2PM technique  surpasses all of those limitations by using an adaptive 2D fringe pattern filtering employing empirical decomposition [27,28] and modified automatic selective reconstruction . Only sharply extracted regions of each informative mode containing well-denoised fringes are preserved in the localized fringe filtering process.
In this contribution, we combine SMIM with a modified version of the S2H2PM algorithm into a single platform. Several major advancements are incorporated into the processing path of the S2H2PM technique. The modifications are aimed at increasing robustness and versatility of the reported scheme. The S2H2PM starts with adaptive filtering over the single fringe pattern (hologram, interferogram, etc.), and we propose the use of an automatic masking algorithm based on variational image decomposition (VID)  and Otsu thresholding. Other steps, i.e., local fringe direction map estimation, phase unwrapping, and further filtering, are modified accordingly and will be described in the experimental validation section. The result, named Hilbert–Huang single-shot SMIM (H2S2MIM), takes all the advantages of SMIM regarding an extremely simple, cost-effective, highly stable, and fast way to convert a standard microscope into a holographic one, in addition to the ones incoming from the modified version of S2H2PM and related to a robust and single-shot operational principle, even when partially coherent illumination is used in SMIM. This perfect combination equips a commercially available nonholographic microscope with accurate QPI capabilities and allows fast dynamic sample analysis only limited by the frame rate of the camera sensor.
The experimental validation of the proposed H2S2MIM method is implemented using the embodiment of a BX60 Olympus microscope, as in previous SMIM Refs. [19–21]. For the sake of simplicity, Fig. 1 depicts a scheme of the approach, where the main components of the microscope setup are taken into account. A superluminescent diode ([SLD] from Exalos, Model EXS6501-B001, 10 mW of optical power, a 650 nm central wavelength, a 6 nm spectral bandwidth) illuminates the input plane, where the useful FOV is spatially multiplexed into object/sample (O) and reference/clear (R) regions. The objective (Olympus UMPlanF infinity corrected ) and the tube lens system magnify the input plane spatially multiplexed distribution at the output port of the microscope, where a charge-coupled device (CCD) camera (Basler A312f, , 8.3 μm px size, ) is placed. In this way, conventional (nonholographic) imaging is recorded, but a 1D diffraction grating (Ronchi ruled grating, period) is introduced in the analyzer insertion slot just before the tube lens to allow output plane replicas and, thus, interferometric recording. Issues about proper selection of the 1D grating can be found in Ref. .
Using this optical layout, the microscope objective provides an infinity conjugated image which is brought into focus by the tube lens. By following the ray tracing at Fig. 1 of the on-axis (solid line) object (black) and reference (red) rays at the center of both O-R input plane regions, the 0th order of the 1D grating does not disturb the conventional imaging mode. However, the st diffraction order provides a shifted replica at the output plane allowing the overlapping at the lower CCD half-area of the conventional object beam, named , with the reference beam coming from the st grating’s diffraction order, named . Something similar happens at the upper CCD half-area when considering the conventional (nondiffracted) reference beam ray tracing, named R’(0th), with the object beam coming from the st grating’s diffraction order, named . In order to clearly show this FOV duplicity at the output plane, Fig. 1 contains the output distribution coming from a 90 μm diameter microbead as an input object. However, the selected region for final reconstruction comes from the lower CCD half-area, that is, , due to practical reasons (grating order efficiency, FOV constrictions at the input plane and vignetting issues).
Experimental validation of H2S2MIM has been divided into three parts. The first one involves a calibration stage in which a microbead of 90 μm in diameter is used to show the whole process step by step. This microbead is the one included at Fig. 1 so in the experimental results (see Fig. 2), we will only include the region of interest (ROI) marked with the white square at Fig. 1. To allow comparison, the sample must be static at least in the time where the P-S process is implemented. For the P-S procedure, the grating is placed on a motorized linear translation stage (Newport, model ESP300). A grating motion step of 2.5 μm between consecutive holograms is applied, meaning that 20 images integrate a full P-S cycle according to the grating’s period. Nevertheless, 80 images are recorded in order to perfectly select the frames integrating the full P-S cycle and to minimize uncertainty errors in phase determination.
Figure 2(a) presents a single hologram (first frame of the full P-S video movie, Visualization 1), while Fig. 2(b) includes its Fourier transform (FT) showing overlapping of the different hologram diffraction orders (marked with white circles), so P-S is needed for complex amplitude retrieval; Fig. 2(c) presents a complex amplitude pupil retrieved after P-S algorithm application; Fig. 2(d) shows the intensity image of the bead obtained as inverse FT operation from Fig. 2(c); and Figs. 2(e)–2(f) present the 2D phase distribution in wrapped and unwrapped visualization modes, respectively. The results provided by the H2S2MIM method are included at Figs. 2(g) and 2(h) concerning 2D wrapped and unwrapped phase distributions. Finally, Fig. 2(i) presents a cross section of the retrieved phase profiles coming from the P-S and H2S2MIM methods in blue and red colors, respectively. Aside from the background differences, the H2S2MIM method perfectly matches the result provided by P-S technique.
The second validation compares reconstructions incoming from H2S2MIM method with a conventional P-S algorithm for more complex, but still static, samples. Two different types of prostate cancer cells (PC-3 and LnCaP cell lines) are used. Both types of cells were cultured in an RPMI 1640 medium with 10% fetal bovine serum, Penicillin and streptomycine at standard cell culture conditions (37°C in 5% in a humidified incubator). Once the cells reach a confluent stage, they were released from the culture support and centrifuged. The supernatant fluid is discarded by centrifugation, and the cells are resuspended in a cytopreservative solution and mounted in a microscopy slide.
Figure 3 presents in (a)–(d) the useful FOV (lower CCD half-area) provided by the proposed technique. In addition, these single images are the first frames, respectively, of two video movies (Visualization 2 and Visualization 3) containing the whole phase-shifted hologram sets. The 2D unwrapped phase distributions incoming from P-S and H2S2MIM algorithms are included through Figs. 3(b)–3(e) and Figs. 3(c)–3(f), respectively, for the ROIs marked with white solid-line squares at Figs. 3(a)–3(d). One can notice again that the values of the lateral scales coming from P-S and H2S2MIM are in perfect agreement.
Finally, maybe the most interesting capability of H2S2MIM relates to its single-shot operation principle. To validate this capability, we present a third experimental verification of the proposed method considering a sample integrated by two 45 μm diameter beads flowing in a 100 μm thickness counting chamber filled with water. Figure 4 includes the first frames of full video movies (Visualization 4 and Visualization 5) considering the ROIs (not the full available FOV) containing the 45 μm diameter beads and regarding (a) the recorded set of slightly off-axis holograms and (b) the retrieved 2D unwrapped phase distribution from H2S2MIM. Note that it is impossible to retrieve accurate bead phase information using a single hologram, since the diffraction orders are overlapping at spectral domain and no Fourier filtering can be applied. Hence, the single-shot working capability provided by H2S2MIM enables accurate QPI analysis of dynamic samples.
The modification of our previously reported H2S2PM algorithm  especially enables efficient analysis of the microbeads (extremely difficult case from the single-fringe-pattern analysis point of view). In both Figs. 2(a) and 4(a), one can observe that (1) the beads introduce a spurious circular diffraction pattern outside the bead region, making unidirectional carrier fringes difficult to analyze near the bead borders; (2) the fringes in the bead region have very significant spatially varying period and orientation; and (3) there are cumbersome fringe discontinuities alongside the border of the beads (discontinuous transition between background and bead shape coding fringes). Efficient phase demodulation of cumbersome bead holograms is possible due to the state-of-the-art modifications of the S2H2PM technique. The VID technique is used to extract a hologram background with a sharply edged bead region, and Otsu binarization is applied to complete the novel automatic masking procedure. In this way, we could accurately analyze both the bead region and carrier fringe region.
Significant spatial variation of the fringe period in the bead region is compensated for with a robust adaptive EMD technique . However, global parameters based on the mean amplitude and the mean spectral energy are introduced into mode reconstruction scheme, resulting in an increased signal-to-noise ratio of the reconstructed holograms. This is a crucial advancement enabling bead analysis.
The S2H2PM utilizes Hilbert spiral phase transform (HST)  for the phase demodulation of an adaptively bandpass filtered fringe pattern with a zero mean value (local oscillation around zero is ensured using the EMD). The neuralgic part of the HST is an estimation of the local fringe direction map to accommodate for cumbersome fringe orientation variations (in the bead case—full variability—the most troublesome case of closed fringes). We use a fringe direction estimator introduced in Ref.  and enhance it, employing EMD adaptive filtering of the very noisy first-estimate direction map. In this way, we obtain both accuracy and smoothness of the automatically estimated fringe direction map.
The modified HST algorithm generates phase map modulo to be unwrapped. A previously used scheme  failed in the studied cumbersome cases mainly due to fringe discontinuities. We have applied a recently proposed iterative phase unwrapping technique based on the transport of an intensity equation . It searches for a Poissonian smooth solution, which is its main advantage over the previous scheme  being very prone to local errors (especially in noisy areas with fringe discontinuities, e.g., beads borders, cell merger region, etc.) and their disastrous propagation. In this way a novel modified S2H2PM technique for one-stop-shop hologram quantitative phase determination is defined and implemented.
In summary, we have reported on a robust and single-shot principle working under partially coherent illumination for QPI in regular nonholographic microscopes. The method, named H2S2MIM, raises from a modification of our previously introduced S2H2PM method  applied to SMIM configuration [19–21]. H2S2MIM provides an extremely simple, cost-effective, highly stable, and fast way to convert a standard microscope into a holographic one working in a single illumination shot for QPI. Experimental validation has been presented and, although only a single microscope objective has been considered, the proposed method perfectly applies to other microscope lenses as SMIM does.
Secretaría de Estado de Investigación, Desarrollo e Innovación (SEIDI) (FIS2013-47548-P); Narodowe Centrum Nauki (NCN) (2017/25/B/ST7/02049); WUT statutory funds.
The authors want to thank Dr. José Antonio López-Guerrero from the Fundación Instituto Valenciano de Oncología-FIVO for PC-3 biosample preparation as well as Juan Martinez Carranza and Maria Cywinska for providing phase unwrapping and VID codes.
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