Fringe analysis: single-shot or two-frames? Quantitative phase imaging answers

Maciej Trusiak

doi:10.1364/OE.423336

1. Introduction

Modern full-field across-scales optical measurement techniques, very attractive in terms of non-invasive, fast and accurate evaluation, often rely on analyzing the fringe pattern [1–8]. Examples include classical solutions such as interferometry [1], fringe projection [2], moiré techniques [3], digital holography [4] and vey capable quantitative phase imaging QPI [5,6] (interferometric/holographic technology in label-free biomedical microscopy [7,8]). Fringe pattern – a quasi-periodic intensity distribution recorded by the camera – encodes in its phase term the information about the measured quantity, e.g., 3D shape of macroscale technical objects in fringe projection or cell refractive index micro-structure in quantitative phase imaging. It is, therefore highly, desirable to have at hand a numerical method enabling faithful and robust phase reconstruction. Thus, fringe analysis is a matured field [1] with constant advancements driven by new experimental challenges, i.e., high-speed high -resolution imaging of dynamic bio-events in natural environment and minimal light exposure for cancer label-free detection [9] or nanovesicles tracking [10]. Low signal-to-noise ratio (SNR) of recorded interferograms is often the most important obstacle and requires fast reliable processing.

Coherent full-field optical techniques (interferometry/holography) rely on the virtue of storing object information in an optical complex amplitude which is subsequently encoded in a digital image. The characteristic quasi-periodic intensity pattern – the hologram/interferogram (I) - is generated upon interference of object and reference beams which are clearly distinguishable in coherent (interferometric) coding. The hologram/interferograms, Eq. (1), comprises a sum of three fundamental intensity components: background (incoherent sum of intensities of interfering beams I₁ and I₂), noise (uncorrelated and/or structured, N) and coherent interference fringes constituted by a cosine function modulated in phase (θ) and amplitude (2(I₁I₂)^1/2):

(1)$$I = {I_1} + {I_2} + 2\sqrt {{I_1}{I_2}} \cos \theta + N. $$

Phase distribution (θ) is of interest and spatially manifests itself as a local shape of fringes – their period and orientation variations. Similar model can be used in incoherent techniques, i.e., fringe projection or moiré methods, assuming that a single harmonic is generated. Therefore, extremely important step of both coherent (and incoherent) coding comprises the fringe pattern phase demodulation understood as changing the domain from recorded intensity into calculated phase distribution. Carrier spatial frequency is introduced by establishing the inclination angle between object and reference beams in coherent coding. It modifies the fringe pattern Fourier spectrum and mutually translates three main spectral components: two object information carrying complex conjugate cross-correlation terms move away from the central auto-correlation peak (Fourier transform of the incoherent sum of beam intensities).

Phase reconstruction can be generally performed in three ways: (1) utilizing multi-frame approaches with 3+ interferograms in a phase-shifting sequence [1,11], (2) employing two-frame phase shifting [12–15] or (3) exploiting single-shot techniques [1,16]. Temporal phase shifting [1,11] enables the highest accuracy of phase reconstruction, especially taking into account long phase shifting sequences and capitalizing on the data redundancy and numerical approaches [17,18] to minimize all errors (e.g., phase shift miscalibration, fluctuations of the background illumination etc.). The temporal resolution of such optical measurement is impeded, however. The classical solution to this problem is a single-shot approach based on the Fourier transform (FT) [16]. Nonetheless it requires sufficiently high carrier spatial frequency to separate, otherwise overlapped, cross-correlation and auto-correlation spectral components. It thus inherently limits the space-bandwidth product (SBP) utilizing only a small fraction of the available detector bandwidth and imposes constraints on phase details of imaged bio-structure (object should be low-pass banded with respect to the carrier frequency). On the up-side, dynamic phenomena can be successfully analyzed. Advanced single-shot phase reconstruction techniques based on the Hilbert spiral transform (HST) [19,20] and the Kramers-Kronig (KK) analyticity [21] relax stringent requirements on the carrier frequency enabling spectral overlapping of cross-correlation terms with auto-correlation term (KK) and with each-other (HST), thus enlarging the available SBP. Recently emerged hybrid phase demodulation trade-off in terms of two-frame phase-shifting provides somewhat a merger of advantages of both classical approaches described above (single-shot and multi-frame demodulations) and supports new way of realizing QPI, i.e., slightly off-axis configurations not needing off-axis recording and thus offering higher SBP. Two frames with random phase shift can be recorded simultaneously (scarifying the field of view FOV) or in a very fast temporal sequence with minimal delay, while the carrier spatial frequency is not needed and thus it is well optimized in terms of the phase temporal SBP. Due to these advantages two-frame fringe-analysis techniques are under dynamic development [14,15], especially in slightly off-axis QPI (e.g., realized by digital holographic microscopy, DHM) [22–26]. In this contribution I will focus on the single-shot and two-frame techniques as they provide the highest temporal resolution in quantitative phase imaging for label-free bio-sensing [5–10].

The goal of this study is to systematically answer the question whether the effort in recording additional interferogram, thus sacrificing FOV or temporal resolution of the setup, is fully justified in QPI experimental reality. I will thus employ numerical models to simulate different experimental regimes and validate the numerical findings analyzing two sets of experimental interferograms recorded in slightly off-axis QPI configuration in (1) high SNR regime - liver sinusoidal endothelial cell, and (2) low SNR – RWPE human prostate cells.

2. Description of selected fringe analysis methods

2.1 Single-shot techniques

Case of single interferogram needed for phase extraction allows for the highest throughput enabling dynamic events examination (i.e., cell division and interactions or dynamic response to treatment). Several groups of single-shot techniques can be listed:

(1) Intensity domain methods based on fringe skeletonizing, numbering and eventually polynomial fitting of the phase [27]; they are susceptible to noise and general low quality of fringes and provide low resolution in terms of intensity domain operation.
(2) Intensity domain methods based on functional regularization and optimization techniques, i.e., regularized phase tracking [28], implicit smoothing splines [29], autoregressive Kalman filtering [30]; they are highly capable methods, however the functional, the regularizator and the optimization solver need to be chosen appropriately. Thus, it can be a generally a formidable task to provide accurate, fast and versatile tool for highly differing phase functions often met in QPI, as cells have highly diversified and individualized structures.
(3) Integral transform domain methods based on the analytic signal construction (complex analytic fringe pattern) with easy Gabor-like access to phase/amplitude terms, e.g., classical and historically first Fourier transform [16], windowed Fourier transform [31], continuous wavelet transform [32,33], S-transform [34], monogenic transform [35], Riesz transform [36,37], Hilbert transform [38,39] and Hilbert-Huang transform [40,41].
(4) Phase shifting methods based on the pixel-to-pixel sequencing, i.e., spatial carrier phase shifting [42,43] requiring appropriate carrier spatial frequency or pixelated phase mask [44] based on a custom-made polarization-driven 4-pixel phase shifter mask resulting in an instant single phase pixel demodulation.
(5) Deep learning based methods [45], highly skilled and specialized in a given task requiring extensive training, large data sets, and correctly designed network architectures.

Phase maps reconstructed employing methods from groups 1–2 are continuous, whereas techniques from groups 3–5 tend to provide phase distribution in a wrapped form (deep learning from group 5 can be trained to return unwrapped maps or to perform the unwrapping itself [46]). Thus, modulo 2π phase fringes need subsequent unwrapping [47,48] to generate continuous distribution of phase values to be rescaled onto a measurand map.

Integral transform based techniques position themselves as the most versatile ones in terms of learning-free, optimization-free, accurate and deterministic single-shot phase reconstruction. Within this large group Fourier transform is the most known example, however at the same time it constitutes the most limited representative due to well-known downsides of the numerical Fourier transformation, i.e., global calculations – no ability to localize given harmonic in space, and error propagation, spectral leakage, Gibbs artifacts. As a result of global spectral filtering based phase demodulation only a small part of the detector bandwidth is used and stringent recording conditions are to be met – carrier frequency should be high enough to enable cross-correlation term separation and thus object should be generally smooth. Localized advancements of FT – windowed FT and continuous wavelet transform are highly capable techniques enabling instantaneous phase demodulation, however both are limited by space-frequency resolution tradeoff, use predefined harmonic basis, require large number of pre-set parameter values and generally prefer high spatial frequency open quasi-unidirectional fringes. Hilbert transform based phase demodulation can be seen as the most versatile one as it employs no rejection of spectral components, works well also for low carrier frequencies and provides access to the analytic (Gabor quadrature) signal with complex values easily associable to local phase and amplitude of the measured optical field. It requires a background-free fringe pattern with slowly varying amplitude term (Bedrosian theorem), however. Fringe patterns were analyzed using 1D Hilbert transform [38,39], scanning-induced incoherency between 1D signals constituting 2D image introduce jeopardizing errors especially in case of fringes with spatially varying orientation and period (as a result of complicated and challenging underlying phase function). Out of all 2D Hilbert transform generalizations the one proposed by Larkin, called the Hilbert spiral transform, can be seen as the most appealing due to the isotropic natural demodulation [19], which in turn needs additional information on estimated local fringe direction map. Required background removal can be ensured employing classical Gaussian high-pass filtering or more advanced image decomposition techniques, i.e., 2D empirical mode decomposition (EMD) [49–51] and variational image decomposition [52,53]. Previous studies suggest [54] that noise can and should be removed prior to phase demodulation employing classical Gaussian or median filtering or more advanced block matching 3D filtering (BM3D) [55] and non-local means [56], among other algorithms. The BM3D works especially well on fringe patterns due to their inherent spatial self-similarity, which facilitates 3D block matching [20,53]. Interestingly, this powerful technology proves efficient also for fringe-less cases, i.e., fluorescent super-resolved microscopy [57]. The fringe direction map (modulo 2π) can be calculated upon unwrapping of fringe orientation map (modulo π) obtained via, e.g., local plane fitting and gradient based operations [58], or accessed through generalized quadrature (Riesz) transform [59] and principal component analysis [26,41] etc.

In this study I will employ single-frame phase demodulation in terms of the Hilbert spiral transform enhanced with fringe pattern preprocessing by means of the BM3D noise reduction and the 2D EMD background rejection. Denoised fringe pattern is decomposed into three components: first two modes and residue, deploying the enhanced and fast empirical mode decomposition algorithm EFEMD [49], as it is currently the fastest empirical decomposition algorithm on the market. Adaptive background rejection is performed by summing up first two modes. Thus, preprocessed fringe pattern can be modeled as shown below in Eq. (2):

(2)$${I_P} = 2\sqrt {{I_1}{I_2}} \cos \theta.$$

The complex analytic fringe pattern (AFP, Eq. (3)) is calculated having its real part defined as preprocessed interferogram (I_P) and imaginary part as the Hilbert spiral transform of preprocessed interferogram designated as shown in Eq. (4),

(3)$$AFP = {I_P} + iHST({{I_P}} ),$$

(4)$$HST({{I_P}} )={-} i\textrm{exp} ({ - i\beta } ){F^{ - 1}}\left\{ {\textrm{SPF}\ast F\left\{ {2\sqrt {{I_1}{I_2}} \cos \theta } \right\}} \right\},$$

where β is the local fringe direction map [19,20,26,41,58,59], SPF denotes spiral phase function, F and F⁻¹ denote forward and inverse FT. Fringe direction map is estimated using plane fitting and gradient based algorithm [58] for orientation map generation (modulo π). It then undergoes unwrapping from modulo π to modulo 2π and sine/cosine filtering [20]. The complex analytic fringe pattern grants easy access to the phase map:

(5)$$\theta = angle({AFP} )= \textrm{atan}\left( {\frac{{HST({{I_P}} )}}{{{I_P}}}} \right),$$

where angle is a Matlab function computing the argument map of complex valued image, Eq. (5). Phase unwrapping [47] completes the phase demodulation path. Classical Fourier transform approach [1,16] will be deployed as a reference technique for benchmarking purposes.

2.2 Two-frame phase-shifting techniques

Two-frame phase-shifting techniques emerged quite recently as the ones merging advantages of fast single-shot processing and accurate temporal phase-shifting demodulation [14,15]. Two randomly phase-shifted fringe patterns can be recorded with a small temporal delay [22,24] or simultaneously employing two cameras or multiplexing the field of view of a single camera [23,25]. Two recorded fringe patterns are mathematically modeled (Eqs. (6) and (7)) as:

(6)$${I_0} = {I_1} + {I_2} + 2\sqrt {{I_1}{I_2}} \cos \theta + {N_0},$$

(7)$${I_\delta } = {I_1} + {I_2} + 2\sqrt {{I_1}{I_2}} \cos ({\theta + \delta } )+ {N_\delta },$$

where mutual phase shift between them is termed as δ. Historically first two-frame approach proposed by Kreis in 1992 utilized additional frame to estimate the phase sign map and used it to correct Fourier transform phase jumps due to high local fringe direction variations [60]. It took almost 20 years for the topic to flourish on a worldwide scale with a vast number of new techniques reported in the last decade [12–15]. Examples include but are not limited to: optical flow algorithm [12], Gram-Schmidt orthonormalization [13,61], extreme values of interference method [62], tilt-shift error estimation technique [63], independent component analysis [64], Gabor filter bank [65], Lissajous figure and ellipse fitting technology [66], orthogonal polynomials and global optimization [67], and state space analysis [68]. Applications concern, e.g., dual wavelength interferometry [69] and interferometric dual-channel study of droplet evaporation [70]. Very recently hybrid techniques emerged [71–73] capitalizing on merging various concepts, i.e., Lissajous, Gram-Schmidt, and least square fitting. Since only 2 interferograms are available, fringe pattern preprocessing (employed separately to both frames/interferograms) plays crucial role in two-frame techniques. Fringe pattern background rejection cannot be employed through subtraction of out-of-phase frames, like in a regular temporal phase shifting [1,11]. Another common feature of all two-frame algorithms is the fact that they work for random phase shift, in such a way that they do estimate the step prior to the phase demodulation with extended formula [62,63,74] or force the random phase step to assume ideal π/2 value [13,61,66]. The closer the initial random phase step is to π/2 the better, as ‘starting’ orthogonality of two fringe signals naturally increases. Phase step equal to π is not valid because then two fringe signals are linearly dependent, and thus act like a single one.

Taking into account previous works [12–15,61–74] the Gram-Schmidt orthonormalization (GS) [13,61] aided by the 2D EMD pre-filtering has been selected as the first very capable representative of the two-frame phase-shifting methods. It treats fringe patterns as vectors and performs orthogonal projection of one onto another thus creating π/2 shifted frames out of initial randomly shifted fringe patterns. Background rejection is crucial, hence 2D EMD algorithm plays important role here [61]. The optical flow algorithm (OF) [12] is chosen as the second representative of two-frame phase shifting phase reconstruction methods. Here, Gaussian filtering is employed for background term removal and additional frame serves the purpose of two-shot fringe direction map estimation basing of the regularized optical flow of intensity between two phase shifted fringe patterns (in each point it is normal to the fringe, thus points onto local fringe direction). Phase demodulation is performed upon the Hilbert spiral transform, applied to Gaussian-filtered fringe pattern and aided by the fringe direction map. Both techniques are well-established and utilize completely different paradigms, thus they very-well cover the broad range of two-frame phase demodulation concepts.

Hilbert spiral transform in two-frame regime (OF) will be compared with single-frame Hilbert-Huang based approach described in section 2.1 (HST employed to EMD-filtered fringe pattern). Comparative studies will constitute a closer look into HST and are aimed at helping to grasp the differences between both frameworks. For the sake of the completeness of the envisioned Hilbert transform benchmarking analysis additional phase demodulation technique is incorporated. Two fringe patterns are subtracted, Eq. (8), to create a difference pattern (I_D) with boosted signal-to-noise ratio due to experimentally eliminated background and improved amplitude modulation of interference fringes. It can be modelled as, for δ=π:

(8)$${I_D} = {I_0} - {I_\delta } = a + 4\sqrt {{I_1}{I_2}} (\cos \theta ) + {N_D},$$

where a denotes difference pattern residual background term (one expects a=0 as a consequence of identical background present in two fringe patterns) and N_D denotes the resulting image noise. Similar approach of data enhancement through two interferograms subtraction was reported to improve the amplitude demodulation process in interferometric full-field vibration studies [75], for volumetric bio-imaging using structured illumination microscopy and optical sectioning [76], for phase demodulation in multi-beam Fizeau interferometry [77] (all three used temporal multiplexing to register two frames) and for phase map calculation improvement in quantitative phase microscopy [23,25] (with spatial multiplexing of two frames recorded simultaneously within the two halves of the camera FOV). Studies presented in [77] fully corroborated δ=π as the best case scenario with a +/- π/3 range of significant enhancement around it. Importantly, this approach shifts the classical paradigm from two-shot phase extraction aided by separate single-frame pre-filtering into single-frame phase reconstruction aided by the two-shot merged pre-filtering.

3. Numerical evaluation and discussion

In this section I test algorithms of interest under carefully designed and simulated conditions. The goal is to evaluate the performance of the selected single-shot and two-frame methods against strong variability of the carrier spatial frequency and the fringe signal-to-noise ratio. Underlying phase function is chosen from a real life quantitative phase imaging of HeLa cells [78–80] to stay as close to reality as possible. Quantitative phase imaging is of special interest nowadays due to its versatility in bio-imaging very diversified cell cultures. Polynomial based underlying phase maps, regularly tested in two-frame and single-shot fringe analysis oriented studies [12–15,17,18], are on purpose omitted in the numerical testing as they oversimplify the generally challenging QPI phase demodulation problem and make the numerical part not coherent with the QPI experimental reality.

Algorithms taking part in the numerical evaluation process are:

(1) Hilbert spiral transform single-shot phase demodulation: termed HST1 (EFEMD + BM3D interferogram pre-filtering);
(2) Two-frame Hilbert spiral transform, understood as HST1 employed onto a difference of two phase-shifted interferograms: HST2;
(3) Two-frame Gram-Schmidt orthonormalization: GS (EFEMD + BM3D interferograms pre-filtering);
(4) Optical flow two-frame interferometry: OF;
(5) The OF algorithm with additional fringe pattern filtering (BM3D + EFEMD described in next paragraph): OFf;
(6) The Fourier transform method: FT.

I have simulated all interferograms employing the Eq. (9):

(9)$$I = A[1 + X(\cos (\textrm{fx} + \theta + \delta )] + {\textrm{N}_I} \cdot \textrm{rand}(2048), $$

where [x y] = meshgrid(linspace(-pi,pi,2048)) generates an interferogram domain, N_I controls the noise factor, f is responsible for carrier spatial frequency modification, A constitutes the background intensity term, X steers the interferograms signal-to-noise ratio (the bigger the X in comparison with N_I the higher the SNR), phase denotes underlying phase function (dynamic range of 14π radians) and shift describes the phase shift. Background term, Eq. (10), is set as:

(10)$$A = zeroone({fspecial({^{\prime}gaussian^{\prime},2048,1350} )} )+ 0.5,\;$$

where zeroone is a function rescaling intensity range to [0 1]. Noise parameter N_I is set to 0.4, while f attains three specific values: 80 for low frequency regime (corresponding to the on-axis QPI architecture with spectral overlapping of auto-correlation and cross-correlation terms), 160 for mid-frequency regime (corresponding to the slightly off-axis QPI architecture with spectral overlapping of auto-correlation with separated cross-correlation terms) and 320 for high frequency regime (corresponding to the off-axis QPI architecture with full spectral separation appropriate for Fourier phase demodulation). No background nor amplitude modulation fluctuation between two simulated phase shifted frames has been assumed; phase shift is considered to be free of tilt-shift error. In two-frame HST the best value of phase shift is π, however the set value equals to 0.8π in order to numerically account for the possible experimental errors. For GS and OF π/2 phase shift value is theoretically the most suitable, however introduced 0.8π/2 value accounts for experimental error.

From the SNR point of view the analysis is divided into 4 representative cases.

(1) First one artificially assumes perfect cosine terms as input images – this way a basic error of phase reconstruction algorithm is studied as no pre-filtering is used.
(2) Second case corresponds to the high SNR in terms of X=1.
(3) Third case describes medium SNR assuming X=0.5.
(4) Fourth case explores low SNR regime by setting X=0.1.

Figure 1 shows simulated underlying phase map with HeLa cells in different stages of mitosis (proliferation) and three exemplifying interferograms visualized to represent designed testing conditions. Figure 2 presents their Fourier transform magnitude maps, corroborating the spectral components shape and mutual localizations in each testing condition associated with the different QPI architectures (on-axis, slightly off-axis and off-axis). Reduced SNR results in smaller cross-correlation spectral spots as noise covers object high spatial frequencies.

Fig. 1. (a) numerically tested phase map (in radians); exemplifying interferograms with (b) high carrier frequency and high SNR, (c) mid-frequency and medium SNR and (d) low carrier frequency and low SNR.

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Fig. 2. Fourier spectra of interferograms presented in Fig. 1: (a) high carrier frequency and high SNR and (b) low carrier frequency and low SNR.

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The preprocessing employed prior to the phase reconstruction consists of the BM3D algorithm based noise minimization (sigma parameter set to 20) and EFEMD-driven background term removal (adaptive high-pass filtering understood as dissecting and summing up first two empirical modes). Results of the quantitative evaluation are presented in four tables: pure cosine terms are tested in Table 1, high SNR interferograms are studied in Table 2, medium SNR data is examined in Table 3, while low SNR patterns are investigated in Table 4. It is to be noted that all values presented in Tables 1–4 are in radians and constitute root mean squared errors calculated comparing reconstructed phase (after unwrapping and plane fitting to remove linear carrier term) with simulated ideal one. Values in brackets refer to the case of 50 pixels border cut (∼2,5% each side) to minimize the border errors coming from limited interferograms spatial domain. Eliminating border errors in such a way helps to judge the degree to which these errors contribute to overall phase accuracy budget for each method, which can be seen as additional gain of the presented analyses. Exemplifying phase maps reconstructed under pure cosine terms and high, medium and low SNRs are depicted in Figs. 3, 4, 5 and 6, respectively.

Fig. 3. Reconstructed phase maps and phase error distributions obtained for pure cosine and low carrier spatial frequency regime via: (a-b) HST1, (c-d) GS, (e-f) OF and (g-h) FT methods. All maps are depicted in radians.

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Table 1. Pure cosine terms analysis (RMS errors in radians; in brackets the case of 50 pixels border-cut; underlined values mark the cases depicted in Fig. 3).

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3.1 Interferograms of ideal signal-to-noise ratio

In the case of pure cosine terms under study, Table 1, one can observe that all techniques provide more accurate results for increasing carrier spatial frequency. HST1 and HST2 methods yield very similar accuracy which is a logical outcome since interferograms were already background free and normalized (to [−1 1] cosine range), thus two-frame subtraction comes with no viable advantage. The same conclusion can be derived comparing the OF and OFf methods, although this time additional filtration, not subtraction, has no influence on phase reconstruction ability. The GS technique produces constantly the best accuracy of phase demodulation, whereas the FT method is consequently the less accurate one (with the highest border errors minimized upon border-cuts, see results in brackets). This analysis helps to grasp the underlying error connected with the sole method of phase reconstruction, assuming ideal input data. Error values presented in Table 1 should be seen as baseline phase accuracy achievable by tested algorithms. In Fig. 3 exemplifying phase maps for low frequency regime are depicted (as mid and high frequency range yield minimal errors).

3.2 Interferograms of high signal-to-noise ratio

Analyzing the case of high SNR (X=1), Table 2, one can apprehend that accuracy scales proportionally to the carrier spatial frequency. HST1 and HST2 methods ensure similar outcomes, as high SNR eases the pre-filtering. The OFf method aided by BM3D and EFEMD preprocessing outperforms classical OF technique, which originally uses Gaussian filtering. FT method returns disqualifying errors in low carrier frequency regime due to the spectral overlapping of information carrying lobes. The GS technique provides, consequently, the best phase reconstruction accuracy regardless the carrier spatial frequency regime. This fact additionally promotes BM3D/EFEMD pre-filtering, which is also employed prior to the GS demodulation. Especially in low frequency regime the advantage of GS is clearly observable (2x better in terms of RMS error than HST1/HST2/OFf). In high frequency range all techniques, besides the classical OF, provide very similar and highly precise phase reconstruction, which is indeed symptomatic for single/two shot demodulation. It is worth showcasing that single-shot HST1 provides similar accuracy to two-frame GS and OFf in high frequency regime (taking into account border errors free maps the similarity occurs also in the mid-frequency regime). Figure 4 contains exemplifying phase error maps (basing on presentations in Fig. 3 it is hard to judge visually the phase reconstruction quality basing on the map itself, therefore more informative 8 error maps with high contrast will be being depicted).

Fig. 4. Reconstructed phase error distributions obtained for high SNR regime via: (a) HST2, (b) GS, (c) OF, (d) OFf, (e) FT techniques – all for low frequency range; and (f) HST1, (g) GS and (h) FT methods for mid-frequency range. All maps are depicted in radians.

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Table 2. High SNR of interferograms, X=1 (RMS errors in radians; in brackets the case of 50 pixels border-cut; underlined values mark the cases depicted in Fig. 4).

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3.3 Interferograms of medium signal-to-noise ratio

In the case of medium SNR (X=0.5), Table 3, the HST2 technique is significantly more accurate in comparison with the HST1, as subtraction of two phase shifted interferograms constitutes very capable ‘experimental’ preprocessing technique (efficient background term removal and amplitude improvement – resulting in SNR boost). In the low frequency regime the GS technique compares favorably with other methods, especially inherently limited FT. This follows from the fact that GS method works well regardless the local fringe shape as it treats interferograms like vectors and performs orthogonal projection of one onto the other. The HST1/HST2 and OF/OFf techniques deploy strategy with local direction map estimation, which works better for denser fringes. Image based interferogram filtering in terms of EFEMD background removal also exhibits higher performance for denser fringes. In mid-frequency regime the HST2 and GS methods are the most recommendable ones, whereas the OFf and HST1 are not far behind, especially taking into account the border-free case. In high frequency regime the HST2 takes the lead (being consequently better than HST1 throughout all frequency regimes), while HST1 ensures the same phase reconstruction quality as GS, which is worth emphasizing as HST1 employs a single-shot operation (!). Quantitative evaluation under these specific conditions allows for counterintuitive conclusion that second frame recording is not beneficial and pre-filtered Hilbert spiral transform thus outperforms pre-filtered Gram-Schmidt demodulation in terms of higher temporal resolution and similar accuracy level. It is also worth mentioning that for all frequency regimes HST1 assures the same level of phase reconstruction accuracy as two frame OFf method. The FT method is competitive only in high frequency regime, as it needs full spectral separation of cross-correlation term. The OF algorithm reports disqualifying errors due to insufficient Gaussian filtering. Large error maps would be similar to Figs. 4(c) and 4(e) and thus visualizing them do not add viable information into discussion. Exemplifying error maps are presented in Fig. 5.

Fig. 5. Reconstructed phase error distributions obtained for medium SNR regime via: (a) HST1, (b) HST2, (c) GS – all for low frequency range; and (d) HST1, (e) HST2, (f) GS, (g) OFf and (h) FT methods for high frequency range. All maps are depicted in radians.

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Table 3. Medium SNR of interferograms, X=0.5 (RMS errors in radians; in brackets the case of 50 pixels border-cut; underlined values mark the cases depicted in Fig. 5).

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3.4 Interferograms of low signal-to-noise ratio

In the case of low SNR (X=0.1), Table 4, one can readily observe that HST2 method provides the best phase reconstruction accuracy throughout all carrier frequency regimes, being especially instrumental for mid/high carrier frequency. The GS and OFf techniques ensure similar level of phase demodulation efficiency, which is lower than for single frame HST1 method in mid/high frequency range. Again, counterintuitively, this single-shot technique is corroborated to be more accurate that two-shot methods, making it especially interesting for high speed high accuracy QPI in slightly off-axis and off-axis configurations. The GS/OFf methods, in comparison with HST1, do accumulate errors in pre-filtering of two frames in low SNR regime. The FT method is competitive in mid and high frequency regime, as it needs full spectral separation of cross-correlation term and Fourier filtering does transfer noise. On additional note, capabilities of the FT method scale proportionally with the size of the image, as the spectral resolution increases, while HST method is not limited to cutting out a part of spectral bandwidth and thus has important advantage of spectral-resolution-invariance. Spectral filtering shortcomings are also visible in low frequency regime, where FT exhibited disqualifying errors (see Tables 1–4) and for low SNR (high noise transferred via Fourier domain). The OF algorithm reports disqualifying errors due to insufficient Gaussian filtering. Exemplifying phase error maps are presented in Fig. 6.

Fig. 6. Reconstructed phase error distributions obtained for low SNR regime via: (a) HST1, (b) HST2, (c) GS, (d) OFf – all for low frequency range; and (e) HST1, (f) HST2, (g) GS methods for mid-frequency range; and (h) FT method for high frequency range. All maps are depicted in radians.

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Table 4. Low SNR of interferograms, X=0.1 (RMS errors in radians; in brackets the case of 50 pixels border-cut; underlined values mark the cases depicted in Fig. 6).

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Analyzing the results one can observe that overall phase demodulation accuracy for all studied algorithms decreases with the decrease of the interferogram SNR, regardless the carrier spatial frequency, which is a very logical conclusion. Increased carrier spatial frequency does not always mean higher phase demodulation accuracy, as dense fringes mix with high frequency noise and tend to deform the cosinusoidal profile due to undersampling. All techniques have been also examined from the computational load point of view. Table 5 shows processing time required to complete phase demodulation utilizing regular PC unit (CPU 2.6 GHz RAM 16GB). All processing times look rather considerably large, nevertheless this can be mainly associated to low cost personal computing unit and high image size (2048 × 2048 pixels). The size was on purpose not limited to a regularly seen 512 × 512 pixels, as it artificially distances the numerical analysis from experimental QPI reality in which cameras have matrices bigger than 3–4 MPx.

Table 5. Processing time juxtaposition (time in seconds). Unwrapping and plane fitting takes around 2.71s. HST itself executes in around 0.91s (direction map estimation not included).

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4. Experimental verification and discussion

To corroborate the numerical findings achieved in Section 3 I will proceed by examining raw experimental interferograms. Phase shifting sequences, each including 5 randomly phase shifted interferograms, come from QPI studies of liver sinusoidal endothelial cells (LSEC) [80] and prostate cells of RWPE line [20]. Figure 7 contains exemplifying phase shifted interferograms from LSEC cell analysis (Fig. 7(a)-(b)) and RWPE cell studies (Fig. 7(d)-(e)), whereases Figs. 8–11 present calculated phase maps and error distributions.

Fig. 7. Experimental LSEC phase imaging: (a) first interferogram and (b) second randomly phase shifted interferogram after BM3D denoising. Experimental interferograms acquired during RWPE phase imaging: (d) first interferogram and (e) second randomly phase shifted interferogram after BM3D denoising. Ground truth phase maps obtained employing principle component phase shifting demodulation [17]: (c) LSEC and (f) RWPE cases. Phase maps (c) and (f) are depicted in radians.

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Fig. 8. Experimental LSEC cells phase imaging results employing (a) HST1, (b) HST2, (c) GS, (d) OF, (e) OFf and (f) FT reconstruction techniques: retrieved phase maps. All maps are depicted in radians.

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Fig. 9. Experimental LSEC cells phase imaging results employing (a) HST1, (b) HST2, (c) GS, (d) OF, (e) OFf and (f) FT reconstruction techniques: phase error distribution maps (calculated in comparison with the ground truth multi-frame PCA demodulation result). All maps are depicted in radians.

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Fig. 10. Experimental RWPE cells phase imaging results employing (a) HST1, (b) HST2, (c) GS, (d) OF, (e) OFf and (f) FT reconstruction techniques: retrieved phase maps. All maps are depicted in radians.

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Fig. 11. Experimental RWPE cells phase imaging results employing (a) HST1, (b) HST2, (c) GS, (d) OF, (e) OFf and (f) FT reconstruction techiques: phase error distribution maps (calculated in comparison with the ground truth multi-frame PCA demodulation result). All maps are depicted in radians.

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LSEC cells were studied in Linnik interferometer with pseudo-thermal light source and silicon reference mirror with sample secured on top of another silicon mirror (please see [79] for details of the setup and optical study and [80] for sample preparation). Period of the interference pattern is around 2 microns for Fig. 7(a). RWPE cells were studied in spatially multiplexed interference microscopy setup [23,25]. Period of the interference pattern is around 1.5 micron in Fig. 7(d). RWPE prostate cells were cultured in RPMI 1640 medium with 10% fetal bovine serum, 100 U/ml penicillin and 0.1 µg/ml streptomycin at standard cell culture conditions (37 °C in 5% CO2 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 on a microscope slide. It is worth mentioning that presented study of phase demodulation techniques is versatile and all of the demodulation methods can be implemented regardless the bio-object under study and the experimental setup employed, as long as the information is encoded in a fringe pattern with some level of carrier frequency (not in an uniform field mode).

Principle component analysis (PCA) based random phase shifting algorithm [17] is employed to obtain ground truth phase maps. Two cases are studied differing in additional BM3D filtering (sigma 10) of phase shifted interferograms sequence prior to PCA based multi-frame ground-truth phase reconstruction. Ground truth phase map obtained without BM3D noise removal (LSEC) is presented in Fig. 7(c), whereas ground truth phase map obtained with BM3D noise removal (RWPE) is depicted in Fig. 7(f). Results of the experimental evaluation are juxtaposed in Table 6, in terms of root mean square phase error calculated comparing reconstructed phase maps under various algorithmic solutions against ground truth phase distributions.

Table 6. LSEC and RWPE cells phase imaging results. Annotations 1–2 refer to the usage (2) or avoidance (1) of additional BM3D filtering of the interferograms for ground truth phase calculation and for each phase demodulation technique. Underlined values mark cases shown in Figs. 8,9 (LSEC1) and Figs. 10,11 (RWPE2).

View Table | View all tables in this article

RMS errors for LSEC bio-phase demodulation are of generally lower level due to higher signal-to-noise ratio of acquired holograms (experimental noise minimization through pseudo-thermal light source [78–80]), while the RWPE bio-phase analysis corresponds to the more challenging case of low signal-to-noise ratio interferograms. All RMS error values are decreased for BM3D preprocessed ground-truth PCA-based phase demodulation (annotation 2, i.e., LSEC2 and RWPE2), in comparison with BM3D-free ground-truth phase demodulation (annotation 1, i.e., LSEC1 and RWPE1). Level of error minimization due to BM3D filtering is higher for RWPE case due to generally lower SNR of these experimental interferograms.

Presented experimental verification fully corroborates novel numerical discoveries. The HST2 method is recommended as the most accurate for both LSEC and RWPE examination. The GS and OFf techniques represent similar level of accuracy, however, it is around 2–3 times lower than the HST2 with/without BM3D, and, somehow counterintuitively, 1.5 times lower than single-shot HST1 algorithm. Due to rather high carrier spatial frequency the FT method provides nice outcomes, slightly higher RMS than the outcomes of HST1, however still significantly lower than HST2 (2 times lower for LSEC and 1.5–2 lower for RWPE). The classical OF algorithm closes the stakes, as it was the case in numerical evaluation. Calculated phase maps and phase error fields can be appreciated in Figs. 8,9 for the high SNR LSEC cell analysis, and in Figs. 10,11 for low SNR RWPE cell examination. Visual observations and qualitative evaluation validated quantitative phase error analysis outcomes. It is worth mentioning that visually FT phase reconstructions exhibit significant detail ullage in comparison with the HST1, which is not fully manifested by RMS values in Table 6. This fact verifies the claim that quantitative and qualitative evaluations are somewhat complementary and only merged together present a full picture and allow for reliable reasoning.

5. Closing remarks

In this contribution I discussed the shortcomings and favorable features of classical and adaptively pre-filtered single-shot and two-frame phase reconstruction methods applied in quantitative phase imaging. They are crucial in full-field optical bio-measurement as they allow for highest temporal resolution of the phase imaging. Hilbert spiral transform and classical Fourier transform have been selected as a single-shot methods. Gram-Schmidt orthonormalization and the optical flow methods have been chosen as capable representatives of two-shot phase shifting techniques, and employed for the first time to QPI. Presented study consists in comprehensive analyses and allows for systematically answering the question on a potential gain in recording two phase shifted fringe patterns and applying two-frame phase demodulation. The price one has to pay is certain – spatial multiplexing introduces field of view limitation and temporal multiplexing lowers by 50% available throughput of the phase imaging.

In simulations, I tested cases of low, medium and high spatial frequency of fringes in a function of the fringe pattern signal-to-noise ratio. Pre-filtering in terms of EFEMD image-based background removal and BM3D denoising has been incorporated to facilitate phase reconstruction. The BM3D has been implemented as it already proved to be valuable asset in fringe analysis [53], however other algorithms could be employed here with similar gain [81,82]. Novel results allowed for increased understanding of single-shot and two-frame phase demodulation for QPI. Counterintuitively, single-shot Hilbert spiral transform demodulation exhibits higher accuracy than two-frame phase shifting methods in medium and low interferogram SNR regimes, especially for mid and high carrier spatial frequencies. The main reason is the following: pre-filtering errors are doubling in two-frame phase demodulation techniques, with respect to single-shot methods. Locally closed fringes are especially cumbersome as, on top of the pre-filtering error concentration, they are very similar for two phase shifter patterns, thus linearly dependent (which jeopardizes the demodulation). GS algorithm outperforms classical OF and pre-filtered OFf technique. Fourier transform, working correctly in high carrier spatial frequency regime, has significantly limited ability for phase detail preservation and is very susceptible to low-medium carrier spatial frequency.

It is important to note that two-frame phase demodulation schemes, until now, all employed single-frame pre-filtering of both fringe pattern separately and then utilized two-frame phase demodulation (most frequently by estimating unknown phase step). In this contribution I evaluated a shifted paradigm. It consists in employing single-shot phase demodulation after two-frame pre-filtering, understood as subtraction of two phase shifted fringe patterns aimed at ‘experimental’ background rejection and augmentation of fringe amplitude (contrast). Subtraction of two phase shifted fringe patterns thus results in increase of SNR, nevertheless difference image can be further filtered for noise reduction (e.g., by the BM3D algorithm) or residual background term correction (e.g., by the EFEMD algorithm). In a vast majority of cases studied within this contribution the best option was to subtract two frames and feed the resulting background-free and SNR-augmented image to a single-shot Hilbert spiral transform demodulation and to refrain from exploiting two-frame phase shifting calculation at all. This is also by far the best solution for real-life applications spanning wide range of SNR, carrier spatial frequency and underlying phase function variability as experimental fringe pattern analysis for quantitative phase imaging of LSEC and RWPE cell lines successfully corroborated all important and novel numerical findings.

It is also worth mentioning that not only HST technique can benefit from the two-frame preprocessing (upon two phase shifted fringe patterns subtraction). Virtually every phase demodulation method would experience a gain in terms of interferogram background term reduction and fringe amplitude enhancement related to the two-frame subtractive preprocessing. I focused on the effect of two-frame preprocessing on the HST technique, as it allows for the widest range of fringe spatial frequencies to be analyzed successfully [20], whereas classical FT method is limited to open fringes and does not cope with locally concentric/closed fringes. These locally closing fringes are resulting from too significant local phase deviation in comparison with local carrier spatial frequency, and can be observed in simulation part of the paper (Fig. 1(d) and following numerical experiments). Moreover, the HST method works locally and employs no spectral filtering thus is independent from the image size, whereas FT performs better for bigger images (spectral resolution is enhanced with the increase of sampling points and data pixels).

Presented research advances the important field of fringe analysis solutions for optical full-field measurement methods with widespread bio-engineering applications (with seamless transition to technical and non-biological object examination). It allows for increased understanding of the dynamic (single-shot and two-frame) phase demodulation processes and enhance them by pinpointing the most efficient computational framework. Numerical and experimental complementary evaluation forges viable reference for further development of fringe pattern based full-field optical measurement, especially in quantitative phase imaging technology which is revolutionizing in recent decade the fundamental field of optical microscopy. I formulated important challenge for future works, i.e., approaching the theoretical accuracy of two-frame techniques (which supersede single-shot methods) in real life applications. Studies presented in the submitted manuscript showed that to achieve this goal new tools for fringe pattern pre-filtering will be needed.

Funding

National Science Center Poland (2020/37/B/ST7/03629); FOTECH-1 project granted by Warsaw University of Technology under the program Excellence Initiative: Research University (ID-UB).

Acknowledgments

Author would like to cordially thank Dr Azeem Ahmad and Professor Balpreet Ahluwalia from the Arctic University of Norway for providing access to experimental data from Figs. 7(a) and 7(b). Special thanks go also to Dr Jose-Angel Picazo Bueno and Professor Vicente Mico for providing access to experimental data from Figs. 7(d) and 7(e).

Disclosures

The author declares no conflicts of interest.

Data availability

All data and codes to reproduce this study will be made available upon reasonable request.

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	HST1	HST2	GS	OF	OFf	FT
L	0.108 (0.097)	0.117 (0.103)	0.003 (0.003)	0.102 (0.092)	0.102 (0.092)	0.385 (0.296)
M	0.027 (0.016)	0.034 (0.017)	0.003 (0.003)	0.028 (0.016)	0.028 (0.016)	0.052 (0.032)
H	0.017 (0.013)	0.016 (0.012)	0.003 (0.003)	0.017 (0.004)	0.017 (0.004)	0.023 (0.010)

	HST1	HST2	GS	OF	OFf	FT
L	0.133 (0.120)	0.124 (0.097)	0.061 (0.056)	0.277 (0.235)	0.126 (0.112)	2.103 (2.090)
M	0.045 (0.029)	0.039 (0.025)	0.030 (0.026)	0.254 (0.214)	0.046 (0.029)	0.134 (0.113)
H	0.034 (0.029)	0.032 (0.028)	0.028 (0.025)	0.281 (0.213)	0.032 (0.025)	0.046 (0.028)

	HST1	HST2	GS	OF	Off	FT
L	0.137 (0.126)	0.091 (0.085)	0.070 (0.065)	2.317 (1.232)	0.132 (0.120)	4.104 (2.901)
M	0.056 (0.043)	0.046 (0.034)	0.046 (0.042)	7.693 (4.003)	0.057 (0.043)	0.346 (0.237)
H	0.052 (0.046)	0.038 (0.032)	0.048 (0.044)	10.418 (8.731)	0.051 (0.044)	0.054 (0.042)

	HST1	HST2	GS	OF	Off	FT
L	0.223 (0.197)	0.181 (0.162)	0.196 (0.172)	3.846 (3.771)	0.206 (0.181)	2.149 (2.154)
M	0.164 (0.145)	0.127 (0.109)	0.190 (0.161)	5.594 (5.876)	0.179 (0.160)	0.448 (0.459)
H	0.177 (0.162)	0.135 (0.121)	0.185 (0.167)	12.003 (12.344)	0.184 (0.167)	0.207 (0.189)

	HST1	HST2	GS	OF	Off	FT
LSEC1	0.044 (0.035)	0.030 (0.020)	0.060 (0.052)	0.121 (0.099)	0.057 (0.047)	0.050 (0.041)
LSEC2	0.041 (0.031)	0.027 (0.016)	0.059 (0.050)	0.116 (0.095)	0.057 (0.044)	0.046 (0.036)
RWPE1	0.184 (0.181)	0.132 (0.130)	0.262 (0.260)	0.397 (0.389)	0.225 (0.221)	0.187 (0.182)
RWPE2	0.168 (0.168)	0.099 (0.094)	0.262 (0.261)	0.348 (0.340)	0.216 (0.213)	0.174 (0.168)

Fringe analysis: single-shot or two-frames? Quantitative phase imaging answers

Abstract

1. Introduction

2. Description of selected fringe analysis methods

2.1 Single-shot techniques

2.2 Two-frame phase-shifting techniques

3. Numerical evaluation and discussion

3.1 Interferograms of ideal signal-to-noise ratio

3.2 Interferograms of high signal-to-noise ratio

3.3 Interferograms of medium signal-to-noise ratio

3.4 Interferograms of low signal-to-noise ratio

4. Experimental verification and discussion

5. Closing remarks

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (11)

Tables (6)

Equations (10)

Optics Express