# Visualizing the fine structure and dynamics of living cells with temporal polychromatic digital holographic microscopy

Mohamed Haouat, Céline Larivière-Loiselle, Marie-Ève Crochetière, Johan Chaniot, Maxime Moreaud, Erik Bélanger, Pierre Marquet

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## To cite this version:

Mohamed Haouat, Céline Larivière-Loiselle, Marie-Ève Crochetière, Johan Chaniot, Maxime Moreaud, et al.. Visualizing the fine structure and dynamics of living cells with temporal polychromatic digital holographic microscopy. Journal of the Optical Society of America. A Optics, Image Science, and Vision, 2024, 41 (11), pp.C109-C124. 10.1364/JOSAA.534150. hal-04887067

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# Visualizing the fine structure and dynamics of living cells with temporal polychromatic digital holographic microscopy

MOHAMED HAOUAT,<fnref n="1" />,<fnref n="2" /> CÉLINE LARIVIÈRE-LOISELLE,<fnref n="1" />,<fnref n="3" /> MARIE-ÈVE CROCHETIÈRE,<fnref n="1" />

JOHAN CHANIOT,<fnref n="1" />,<fnref n="2" /> MAXIME MOREAUD,<fnref n="1" />,<fnref n="4" /> ERIK BÉLANGER,<fnref n="1" />,<fnref n="3" />,<fnref n="7" />,† AND

PIERRE MARQUET<fnref n="1" />,<fnref n="2" />,<fnref n="5" />,<fnref n="6" />,<fnref n="8" />,†

Received 2 July 2024; revised 17 September 2024; accepted 3 October 2024; posted 3 October 2024; published 25 October 2024

Polychromatic digital holographic microscopy (P-DHM) has demonstrated its capacity to generate highly denoised optical path difference images, thereby enabling the label-free visualization of fine cellular structures, such as the dendritic arborization within neuronal cells in culture. So far, however, the sample must remain more or less stationary since P-DHM is performed manually, i.e., all actions are carried out sequentially over several minutes. In this paper, we propose fully automated, robust, and efficient management of the acquisition and reconstruction of the time series of polychromatic hologram sets, transforming P-DHM into temporal P-DHM. Experimental results have demonstrated the ability of the proposed temporal P-DHM implementation to non-invasively and quantitatively reveal the fine structure and dynamics of living cells. © 2024 Optica Publishing Group

under the terms of the Optica Open Access Publishing Agreement

https://doi.org/10.1364/JOSAA.534150

## 1. INTRODUCTION

Over the past few decades, quantitative phase imaging (QPI) has become a widely used tool in the field of label-free live-cell imaging [1,2] through various applications ranging from cell classification for diagnostic purposes to screening of new molecules in the context of drug discovery [3,4]. One of the most popular QPI approaches is digital holographic microscopy (DHM) due to its computational flexibility, which allows numerical reconstruction of the wave-front diffracted by the observed sample by fully simulating the diffraction propagation of a light-wave by a single digitalized hologram [5–8]. The light-wave diffraction propagation approach opens the possibility of autofocusing [9,10], thus compensating for thermal and experimental drifts [11], enabling total correction of aberrations and distortions [12–15], and allowing extended depth of focus [16]. Due to its interferometric nature, the quantitative phase signal

(QPS) of the DHM is extremely sensitive down to the nanometer scale [17,18]. Moreover, in order to preserve the size of the field of view in an off-axis configuration, and to record holograms with well-defined interference fringes [19–21], a light source with a certain degree of coherence is required. However, the coherence property of the light source itself generates coherent noise (CN), a recognized source of image degradation that mitigates some of the many advantages of DHM [22,23].

To date, several research teams have focused on developing strategies to reduce CN, primarily for the intensity signal. An overview of the available methods can be found in Ref. [24]. Up to now, the denoising of the intensity signal has been mainly focused on macroscopic objects with numerical approaches [25–27], although not strictly limited to said numerical approaches [28,29]. On the contrary, there are far fewer strategies for denoising the QPS compared to the intensity signal, mainly due to a non-negligible constraint, which is to strictly preserve its quantitative nature, which is not an easy

<footnote n="1">¹Centre de recherche CERVO, Québec, Québec G1E 1T2, Canada</footnote>

<footnote n="2">²Département de psychiatrie et neurosciences, Université Laval, Québec, Québec G1V 0A6, Canada</footnote>

<footnote n="3">³Département de physique, de génie physique et d'optique, Université Laval, Québec, Québec G1V 0A6, Canada</footnote>

<footnote n="4">⁴IFP Énergies nouvelles, Solaize 69360, France</footnote>

<footnote n="5">⁵Centre d'optique, photonique et laser, Québec, Québec G1V 0A6, Canada</footnote>

<footnote n="6">⁶Unité mixte internationale en neurodéveloppement et psychiatrie de l'enfant, Université Laval, Québec G1V 0A6, Canada, Centre des neurosciences psychiatriques, Département de psychiatrie, Centre hospitalier universitaire vaudois, Université de Lausanne, Lausanne 1008, Switzerland</footnote>

<footnote n="7">⁷erik.belanger.1@ulaval.ca</footnote>

<footnote n="8">⁸pierre.marquet@neuro.ulaval.ca</footnote>

<footnote n="1">¹These authors contributed equally to this work.</footnote>
matter. Among the most efficient strategies to reduce the CN of the QPS, there are multiple hologram techniques, also called multi-look approaches, mainly embodied by approaches based on optical instrumentation. As a general rule, these are based on the averaging of a given number of reconstructed quantitative phase images of the same field of view with uncorrelated CN fields, typically resulting in significantly reduced CN. In practice, the series of holograms required for the multi-look techniques can be obtained by a variety of active approaches, e.g., with different lateral positions of the camera [30] or a slight shift or rotation of the object [31,32], by using different longitudinal laser modes [33], or by modulating the wave-front curvature of the object beam with an electrically tunable lens [34]. To overcome many of the limitations of these methods, mainly the need for mechanical motion or deformation of at least one optical element, approaches are proposed to obtain a multi-wavelength series of holograms using a tunable light source. Kosmeier et al. used this approach with a tunable laser diode to record static quantitative phase images of cancer cells, i.e., a snapshot representation of these cells at a given point in time. However, with this light source, the range of available wavelengths is extremely limited, resulting in modest denoising, and the wavelength tuning is done with a stepper motor, which severely limits the possible frame rate and hinders the potential temporal capability of this particular implementation [35]. Following this approach, we have performed a proof-of-principle that the acquisition of holograms at different wavelengths distributed over a range of 350 nm makes it possible to produce optical path difference (OPD) images, that are almost free of any CN without compromising the quantitative aspect [36]. Such an approach called polychromatic DHM (P-DHM) is initially performed manually, i.e., all actions are carried out sequentially over several minutes with numerous clicks by an experienced operator using different software, thus precluding the possibility of accurately monitoring the dynamics of biological processes. However, since P-DHM is based on the use of an acousto-optic filter controlled by a radio frequency driver to select individual wavelengths from the wide bandwidth generated by a supercontinuum white-light laser, temporal P-DHM should be achievable with considerable speed. This technological advance of adding temporal resolution at the acquisition stage simply makes manual digital reconstruction of holograms impractical. Consequently, the digital hologram reconstruction stage, i.e., the management of spatial filtering (isolating the imaging term) and optical aberrations (causing distortion in the image), as well as defocus (probing different axial planes) inherent to temporal P-DHM and low coherence perturbations (causing background non-uniformities), must also be automated. Moreover, these are critical points because the slightest deviation in these areas can be highly detrimental to the final OPD image.

Specifically, in this work, we have developed an acquisition-reconstruction methodology tailored to the requirements of the temporal P-DHM implementation. This approach allows a fully automated, robust, and efficient management of the acquisition and reconstruction of the time series of polychromatic hologram sets. The hologram acquisition is performed using a user-free scripting approach. For hologram reconstruction, spatial filtering is facilitated by centering the imaging term with a digital reference wave, and phase aberrations are

corrected numerically with a 2D base surface in a very precise manner. In addition, defocus due to axial chromatic aberrations inherent to P-DHM and due to experimental perturbations (thermal expansion, vibrations, bulging of the coverslip and perfusion chamber, etc.), which are typical in temporal imaging, is also corrected numerically in a precise manner based on the determination of the optimal reconstruction distance for each wavelength at each time-point. Finally, small defects in the background uniformity of the quantitative phase images due to the low coherence of the light source are corrected by identifying an illumination field through a minimization algorithm based on B-spline fitting. After describing the performances of each step of the acquisition-reconstruction methodology, experimental results that reveal the dynamic reorganization of astrocytes processes (lamellipodia and a filipodia) are shown. These phenomena are not visible in classical single-wavelength DHM; thus, they demonstrate the ability of the proposed temporal P-DHM approach to non-invasively visualize fine cell structure and dynamics with an unprecedented level of detail.

## 2. WORKING PRINCIPLE

Since P-DHM is a relatively new microscopy modality, it is important to remind its operating principle before introducing the developments presented in this article. The coherent noise that affects quantitative phase images of DHM is highly dependent on the illumination wavelength $\lambda$ used. By recording holograms at multiple $\lambda$ and reconstructing them individually—while normalizing them to their respective $\lambda$, i.e., producing an OPD image—it becomes possible to generate a sequence of images of the sample, each with distinct coherent noise patterns. As long as the dispersion of the sample is quite close to that of the surrounding medium (usually a perfusion solution), averaging these images preserves the quantitative nature of the signal while effectively reducing the noise in the final OPD image [36].

The central issue of the present work is to propose an embodiment that allows the implementation of temporal P-DHM. This requires the automation of two sequential aspects of the technique, the live acquisition of the holograms (Section 2.A) and their offline reconstruction (Section 2.B). First, automation of the various actions required to acquire the holograms provides temporal resolution to the technique and is achieved through serial communication protocols and scripting. This automated acquisition increases the data quantity linearly with the number of temporal points acquired, i.e., $O(N)$, making manual reconstruction impractical. Therefore, it becomes imperative to automate the reconstruction of holograms. The different steps of this automated method are described in the following paragraphs.

### A. Automatic Acquisition of Holograms

The experimental setup used in this study is an extensively modified transmission digital holographic microscope (DHM; for the sake of simplicity, this acronym is used for both the imaging modality and the microscopy apparatus itself) constructed around a 4f off-axis Mach–Zehnder interferometer (T-1003; Lyncée Tec) [37,38]. An optical path retarder (OPR) consisting of a delay line made of two corner cubes (i.e., retroreflector
mirrors), one of which is motor-driven, can adjust the optical path length (OPL) of the reference arm of the interferometer. The microscope is coupled to a tunable-wavelength laser system, consisting of a supercontinuum white-light laser (SuperK EXR-20; NKT Photonics) combined with a double multi-wavelength acousto-optic tunable filter (AOTF) (SuperK SELECT, VIS(1x), and nIR1; NKT Photonics) operated by a radio frequency (RF) driver. The selected $λ$ is injected into the microscope using a combination of a delivery optical fiber (FD7-PM; NKT Photonics) and free-space propagation [36]. The microscope setup includes a 20×-magnification microscope objective (MO) with a working distance of 400 µm and a NA of 0.8 (HC PL APO; Leica), as well as a monochrome CMOS camera sensor with 1920 × 1200 pixels of size 5.86 × 5.86 µm² (IMX174; Sony).

In this study, contrary to Larivière-Loiselle et al. [36], hologram acquisition is achieved without any user intervention using an automated protocol. The supercontinuum laser system (SLS) and the DHM are fully time-sequenced and automated. As illustrated in Fig. 1(a), the AOTF of the SLS is controlled by the software development kit (SDK; NKT Photonics) through the RF driver, and the microscope is controlled by the remote access module based on a TCP/IP communication protocol (Remote TCP/IP; Lyncée Tec). Both are connected via USB interfaces. A script written in Python 3, illustrated as a flowchart in Fig. 1(b), sets the parameters of the SLS, defines the camera settings, and adjusts the OPL of the reference arm relative to the object arm. The script also grabs a frame (a hologram), saves the hologram as well as metadata (time stamps, hologram contrast, version of the different calibration tables) to disk, and performs all other related operations to achieve an automatic temporal acquisition (start, loop, sleep, stop, etc.). This script results in a data structure for holograms following a (t, λ)-organization, i.e., *q* time-points (1 to *Q*) and *p* colors (1 to *P*). Note that since the AOTF consists of two crystals with a single RF driver, the λ looping sequence shown in Fig. 1(b) pauses briefly to allow the said RF driver to change crystals when the maximum λ of the first crystal is reached before resuming the loop.

To maintain the optimal imaging conditions at each $λ$, two calibration tables are built prior to the imaging session: an OPR table and an exposure time table. Figures 1(c) and 1(d) show the OPR and exposure time tables in graphical form for a neural cell culture [39] with the 20×-MO with units in µm and ms, respectively. The OPR table compiles the reference arm lengths that are to be used at each $λ$ to optimize the fringe contrast of the hologram. The exposure time table, on the other hand, compensates for variations in light intensity caused by $λ$-dependent parameters, including the output laser power, microscope transmission, and camera sensitivity. It adjusts the integration time to maintain a constant hologram exposure over the entire chromatic range. It is theoretically possible to adjust the laser power instead of the integration time, but this option is not chosen because it offers less flexibility in this case as the power adjustment range is more limited. Thus, at each color iteration, the script selects the relevant entries in the tables and sets the current $λ$, exposure time, and OPR position.

With this automated protocol, the acquisition time for a P-DHM image composed of 36 $λ$ is now on the order of 30 s, a significant reduction compared to the manual version [36].

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Workflow of automatic acquisition of holograms for temporal P-DHM. (a) Schematic showing the connection of the DHM and the SLS with the computer for automatic sequential acquisition control. (b) Flowchart illustrating the automatic temporal P-DHM acquisition process comprising an inner loop for the *p* colors and an outer loop for the *q* time-points. (c) Typical curve of the OPR as a function of $λ$ for the 20×-MO measured for a neural live-cell culture, i.e., a neuron's cell culture. (d) Typical exposure (exp.) time curve as a function of $λ$ for the 20×-MO measured for a neuron's cell culture.</figcaption></figure>

A maximum acquisition rate of 2 images per min is achieved, which is sufficient to visualize dynamically unfolding biological phenomena, such as fine morphological reorganization during cell migration and translocation.

## B. Automatic Reconstruction of Holograms

In addition to the challenge of automation, the very nature of the experimental setup in P-DHM requires a more advanced spatial filtering procedure and generates optical aberrations as well as a low coherence effect that must be corrected in a robust and efficient way to obtain denoised aberration-free quantitative phase images. The first aberration encountered is the spherical phase aberration caused by the overall curvature of the lenses that make up the MO, resulting in a quantitative phase image background resting on a parabolic surface. Then, the tilt phase aberration caused by the off-axis recording geometry, which makes it possible to spatially separate in the Fourier plane (FP)
<figure><img src="image_5.png" /><figcaption>Fig. 2. Workflow of automatic reconstruction of holograms for temporal P-DHM. The process begins with the loading of object holograms $I_{H,\lambda_p}(k, l)$ and reference holograms $I_{R,\lambda_p}(k, l)$ and the multiplication of each by a digital reference wave $R_{D,\lambda_p}(k, l)$ for centering (ctrg) the +1 order in the FP, followed by FT and automatic spatial filtering. Both sets of filtered Fourier spectra undergo inverse FT to produce the filtered holograms $I_{H,\lambda_p}^F(k, l)$ and $I_{R,\lambda_p}^F(k, l)$. The latter is converted into a DPM $\Gamma_{\lambda_p}^H(k, l)$ and multiplied with $I_{H,\lambda_p}^F(k, l)$ to generate corrected wave-fronts $I_{H,\lambda_p}^C(k, l)$. Then, $d_{r,t_1}(\lambda)$, which allows the correction of axial chromatic aberrations, is determined based on the correlation coefficient (CC) criterion. Then, $d_r(t, \lambda)$, which allows autofocusing in the presence of various environmental perturbations, is obtained, still based on CC and offsetting, followed by numerical propagation to the said image plane and QPS extraction $\Phi_{\lambda_p}(m, n)$. Residual phase aberration correction is applied in the image plane to refine the retrieved QPS $\phi_{\lambda_p}^C(m, n)$. Finally, OPD averaging over the p colors and low coherence perturbation correction yield the final temporal P-DHM data.</figcaption></figure>

of the hologram the real image from the virtual image (also known as the twin images) and the zeroth order of diffraction, results in a quantitative phase image background resting on an inclined plane. As for the chromatic aberrations caused by the dispersion of the 4f MO-tube lens (TL) arrangement, i.e., the different $\lambda$ travels at different speeds while passing through a lens, they are expressed by colors seeing different focal lengths. These optical aberrations have two effects: they change both the magnification of the image and the axial position of the focal plane, and this, for each $\lambda$. Additionally, the defocusing caused by the experimental perturbations produces an axial drift over time. Finally, the low coherence of the light source impacts the visibility distribution of the fringes, which in turn can also affect the uniformity of the quantitative phase image background. All these phenomena taken together can significantly degrade the quality of the final P-DHM image; therefore, robust and efficient management of them is mandatory. A flowchart showing the main steps of the proposed automatic reconstruction method is shown in Fig. 2, while the different steps of the spatial filtering (Section 2.B.1), as well as of the management of both phase aberrations (Section 2.B.2) and defocus (Sections 2.B.3 and 2.B.4), propagation to the best focal plane (Section 2.B.5), and compensation of the low coherence effect (Section 2.B.6), are detailed in the following paragraphs.

## 1. Automatic Spatial Filtering in the Frequency Domain

The digital hologram intensity $I_H(x, y)$, resulting from the interference between a reference wave $R$ and an object wave $O$, is given by

$$I_H(x, y) = |R|^2 + |O|^2 + RO^* + R^*O, \quad (1)$$

where $x$ and $y$ represent the spatial coordinates in the hologram plane and $|R|^2$ and $|O|^2$ denote the square of the magnitude of the reference and object waves, respectively, i.e., the intensity, and * denotes the complex conjugate operator [5]. The first two terms contribute to the zeroth order of diffraction, and the third and fourth terms ($\pm 1$ orders of diffraction) encode the complex information of the wave transmitted by the sample and allow the reconstruction of the real and the virtual image, respectively. When a DHM system operates in an off-axis configuration, $R$ is slightly tilted and can be formally expressed as $R(x, y) = e^{j\frac{2\pi}{\lambda}(x \sin \theta_x + y \sin \theta_y)}$, where $(\theta_x, \theta_y)$ are the tilting angles. The Fourier transform (FT) of $R(x, y)$ is a shifted Dirac function represented by $\delta$. Therefore, the Fourier spectrum of $I_H(x, y)$ can be written as follows:

$$\begin{aligned} \text{FT}\{I_H(x, y)\} = & \tilde{O}(u, v) \otimes \tilde{O}^*(u, v) + \delta(u, v) \\ & + \tilde{O}^*(u, v) \otimes \delta\left(u + \frac{\sin \theta_x}{\lambda}, v + \frac{\sin \theta_y}{\lambda}\right) \\ & + \tilde{O}(u, v) \otimes \delta\left(u - \frac{\sin \theta_x}{\lambda}, v - \frac{\sin \theta_y}{\lambda}\right), \end{aligned} \quad (2)$$

where $\tilde{\sim}$ denotes the FT operator, $(u, v)$ are the spatial-frequency coordinates, and $\otimes$ is the convolution operator [40–42]. The zero-diffraction order corresponding to $\tilde{O}(u, v) \otimes \tilde{O}^*(u, v) + \delta(u, v)$ is placed in the center of the FP. Because of the tilt of $R(x, y)$, the spectral component (SC) of the real and the virtual images moves away from the center of the FP and is located symmetrically around the spatial frequencies ($\pm \frac{\sin \theta_x}{\lambda}, \pm \frac{\sin \theta_y}{\lambda}$), as shown in Fig. 3(a). From this spectral analysis, a first key feature can be understood. The spectral
separation depends on both ($\theta_x$, $\theta_y$) and $\lambda$ of the light source. Since ($\theta_x$, $\theta_y$) are fixed by the experimental configuration of the setup but here $p$ colors are used, in this case the position of the SC of the two imaging terms changes as $1/\lambda$, according to Eq. (2). As a result, the smaller the $\lambda$, the farther the two imaging terms are from the center of the FP. On the contrary, the larger the $\lambda$, the closer the two imaging terms are to the center of the FP, as shown in Figs. 3(a)–3(c) for three different $\lambda$ (500, 680, and 850 nm, respectively). The influence of the $\lambda$ change can also be observed in the spatial domain, as shown by the hologram excerpt (cyan solid contour line) appearing as an inset of Figs. 3(a), 3(b), and 3(c) for 500, 680, and 850 nm, respectively. The fringes of the three holograms are oriented in a specific direction $\gamma$ determined by ($\theta_x$, $\theta_y$) of $R(x, y)$ [see Fig. 3(b)], while the fringe spacing $i$, which is inversely proportional to the carrier frequency, increases with the increase of $\lambda$ [see Figs. 3(a)–3(c)]. Note that in order to minimize the detrimental overlap with the zero order, ($\theta_x$, $\theta_y$) must be carefully adjusted to have a trade-off between an efficient use of the sensor bandwidth and ensure sufficient separation between the three SCs for each $\lambda$. Thus, ($\theta_x$, $\theta_y$) are adjusted to be equal, ensuring that the centers of the SC of the two imaging terms are always aligned along the diagonal of the FP. They are also fine-tuned to ensure sufficient separation of the three SCs, especially for the larger $\lambda$, while avoiding aliasing [43].

From the spectral analysis of Eq. (2), a second key feature can be understood: the off-axis configuration allows a spectral separation of the three SCs of the hologram, which makes it possible to isolate one of the imaging terms by performing a filtering operation, either manually or automatically [7,44]. However, the variable shift of the SC of the two imaging terms in the FP caused by the use of multiple $\lambda$ complicates the filtering operation. This situation requires either the use of an independent filter window for each $I_H(x, y)$ or the use of a single filter window that shifts as a function of $\lambda$ [45]. To deal with this situation, the spatial filtering is performed as follows. First, it consists of shifting the SC of the term of interest (here, the +1 term) to the center of the FP for each of the $p$ monochromatic $I_H(x, y)$ of a time-point. For this, a digital reference wave conjugated to the experimental reference wave $R_D$ is generated for each $\lambda$. It is given by $R_{D,\lambda_p}(k, l) = e^{-j \frac{2\pi}{\lambda_p} (k \sin \theta_x + l \sin \theta_y)}$, with $\sin \theta_x$ and $\sin \theta_y$ estimated as

$$ \sin \theta_x = \frac{|u_0 - u_{+1,\lambda_p}| \lambda_p}{N \Delta x}, \qquad (3) $$

$$ \sin \theta_y = \frac{|v_0 - v_{+1,\lambda_p}| \lambda_p}{M \Delta y}, \qquad (4) $$

where $(k, l)$ are the discrete coordinates in the hologram plane, ($u_0, v_0$) is the position of the center of the zero order of diffraction in the FP, $N$ and $M$ are the vertical and horizontal number of pixels of the sensor, respectively, and $\Delta x$ and $\Delta y$ are the pixel pitch in each direction [42]. All these parameters are known and fixed. Therefore, the only parameter to be identified is ($u_{+1,\lambda_p}, v_{+1,\lambda_p}$), which is the position of the center of the +1 term in the FP. Since $\sin \theta_x$ and $\sin \theta_y$ are constant regardless of the change in $\lambda$, they are estimated from the spectrum of the first hologram of the first polychromatic series (first time-point) by

<figure><img src="image_6.png" /><figcaption>Fig. 3. Automatic spatial filtering in the frequency domain. (a)–(c) The spectra of holograms recorded at $\lambda$ of 500, 680, and 850 nm, respectively, along with their respective hologram excerpt (cyan solid contour line) appearing as the inset for a neural live-cell culture, i.e., a neuron’s cell culture. (d)–(f) The three previous spectra with the +1 order (term of interest, also known as the real image) centered in the FP. (g) The only filtering window used to perform the automatic spectral filtering operation. (h) An example of a centered and filtered spectrum at $\lambda$ of 500 nm.</figcaption></figure>

searching for the position of the peak of maximum amplitude in the quadrant containing the frequencies of the +1 term [46]. Once $\sin \theta_x$ and $\sin \theta_y$ are calculated, a $R_{D,\lambda_p}(k, l)$ is generated for each $\lambda$. Then, each monochromatic $I_{H,\lambda_p}(k, l)$ is multiplied by the corresponding $R_{D,\lambda_p}(k, l)$ to shift the +1 term at the center of the FP as shown in Figs. 3(d)–3(f) [47]. Now that each hologram spectrum is centered for all $\lambda$, a common filter window can be used. For simplicity, a circular shaped window
placed in the center of a matrix of zeros of size $N \times M$ is used [see Fig. 3(g)]. The radius of the window is taken to be 220 pixels as indicated by the dashed green circles in Figs. 3(d)–3(f). The optimal radius size of 220 pixels is visually determined through an empirical analysis. This radius effectively preserves the high-frequency content of the imaging term (+1 order diffraction, real image), producing a sharp image, while minimizing the superposition artifacts caused by overlap with the zero-order term (DC) at longer $\lambda$. Consequently, the fine details of the resulting image remain clearly visible with minimal degradation in image quality. The second step consists of multiplying the spectrum of the centered hologram by a filter window for each $\lambda$ as illustrated in Fig. 3(h). Therefore, for each $I_{H,\lambda_p}(k, l)$ recorded at one of the $p$ given $\lambda$, the spatial filtering procedure can be summarized by the following equation:

$$ \mathbf{I}_{H,\lambda_p}^{F}(k, l) = \mathrm{FT}^{-1}\{\mathrm{FT}\{I_{H,\lambda_p}(k, l)\mathbf{R}_{D,\lambda_p}(k, l)\}W(u, v)\}, \quad (5) $$

where $\mathbf{I}_{H,\lambda_p}^F(k, l)$ is the $p$th filtered hologram, which is equivalent to the conjugate of the object wave in the hologram plane, and $W(u, v)$ is the spatial filtering window in the FP. The typical computation time for the automatic spatial filtering in the frequency domain is approximately 0.5 s for each of the $p$ given $\lambda$. Finally, it is worth noting that in this way, a single filtering window can be used to filter all $p$ holograms at each of the $q$ time-points and the set of $\mathbf{R}_{D,\lambda_p}(k, l)$ generated for the first polychromatic series of holograms must be stored to be used for all other $q-1$ time-points.

## 2. Correction of Phase Aberrations in the Hologram Plane

A quantitative phase image obtained with DHM is usually affected by phase aberrations, as described in Section 2.B. Typically, the amplitude of these aberrations is more pronounced than the QPS itself, which degrades its accuracy. Correcting for these aberrations is therefore a fundamental step in obtaining a denoised aberration-free quantitative phase image. Thanks to the numerical flexibility offered by digital holography, these phase aberrations can be corrected numerically either in the hologram plane and/or in the image plane by fully simulating the light-wave propagation and wave-front conditioning [15]. In the following strategy, a correction is applied in both planes.

Here, the reference-conjugated hologram (RCH) method is used, which allows the first correction in the hologram plane to be made [12]. A polychromatic series of reference holograms (blank holograms, i.e., without sample or imaging chamber) $I_{R,\lambda_p}(k, l)$ is recorded before the experiment. These are spatially filtered in the same way as described in Section 2.B.1 to obtain a series of $p$ filtered reference holograms $\mathbf{I}_{R,\lambda_p}^F(k, l)$, and then for each $\lambda$ a digital phase mask (DPM) $\mathbf{\Gamma}_{\lambda_p}^H(k, l)$ commonly known as a numerical parametric lens (NPL) is computed:

$$ \mathbf{\Gamma}_{\lambda_p}^H(k, l) = e^{j\arg(I_{R,\lambda_p}^F * (k,l))}. \quad (6) $$

Note that the set of $\mathbf{\Gamma}_{\lambda_p}^H(k, l)$ must be stored in memory for subsequent use in correcting each of the $p$ holograms at each of the $q$ time-points. Then, each $\mathbf{\Gamma}_{\lambda_p}^H(k, l)$ is multiplied by the corresponding $\mathbf{I}_{H,\lambda_p}^F(k, l)$ to obtain the corrected filtered

hologram $\mathbf{I}_{H,\lambda_p}^C(k, l)$ defined as

$$ \mathbf{I}_{H,\lambda_p}^C(k, l) = \mathbf{\Gamma}_{\lambda_p}^H(k, l)\mathbf{I}_{H,\lambda_p}^F(k, l). \quad (7) $$

The typical computation time for correcting phase aberrations in the hologram plane is considered negligible compared to the other steps of the reconstruction process, as it primarily involves an element-wise multiplication. In an ideal case, where the quadratic phase curvature generated by the MO and the tilt introduced by the off-axis geometry are perfectly captured by $I_{R,\lambda_p}(k, l)$, the positioning of this NPL in the hologram plane (see Eq. 7) allows full numerical compensation of the various phase aberrations [12]. However, the experimental conditions for live-cell imaging [39] differ from the ideal case; therefore, it is unrealistic to systematically capture the full extent of phase aberrations within $I_{R,\lambda_p}(k, l)$. Indeed, $I_{R,\lambda_p}(k, l)$ is recorded without the imaging chamber that $I_{H,\lambda_p}(k, l)$ has. It is important to note that the imaging chamber is closed by two coverslips [48,49], which can bend very slightly under active perfusion, and even this small deformation is sufficient to produce different phase curvatures. Moreover, considering the typical duration of a temporal acquisition (here, 90 or 128 min) and the exacerbation by vibrations and mechanical stresses caused by various environmental perturbations (building vibrations not fully damped by the optical table, valve opening and closing, heat, gas, and humidity control, etc.), it is impractical to fully correct phase aberrations for an entire recording. As a result, $I_{R,\lambda_p}(k, l)$ alone is not enough to completely correct phase aberrations, and there are always some residual phase aberrations that must be corrected numerically using additional methods. Considering that in the proposed strategy the tilt caused by the off-axis geometry is already compensated by $\mathbf{R}_{D,\lambda_p}(k, l)$ introduced in the spectral filtering process (see Eq. 5), and since the RCH method alone is not enough to completely correct phase aberrations, the main motivation for using the RCH method is its effectiveness in correcting diffraction patterns caused by motionless dust particles in the optical path of either of the two beams, and any other irregularity or distortion in the object wave-front caused by stationary dirt or imperfections in the optical elements [12]. As for the correction of the residual phase aberrations, it is proposed to correct them in the image plane with a second DPM [15], as described in Section 2.B.5.

## 3. Chromatic Aberration Correction

There are two types of chromatic aberrations that come into play. First, lateral chromatic aberrations can cause a difference in the magnification of the images produced by the imaging system at different $\lambda$ [50–52]. However, in practice, lateral chromatic aberrations can be corrected by adjusting the imaging system in the object beam to operate in a telecentric system (also known as an afocal system) [53–55], i.e., the distance between the MO and the tube lens (TL) is adjusted so that the back focal point of the MO coincides with the front focal point of the TL. In this way, the wave-front emerging from the TL is practically parallel, and the magnification remains nearly constant as $\lambda$ changes. This is the approach chosen here to minimize lateral chromatic aberrations.
Second, the axial chromatic aberrations can lead to the formation of images of the sample originating from different axial planes for each $\lambda$ among the p colors, as schematically shown

reconstruction strategy. This criterion measures the degree of similarity between two images taken at two different $d_z$ and is calculated as

$$CC = \frac{\sum_m \sum_n (I(m, n) - \overline{I(m, n)}) (I'(m, n) - \overline{I'(m, n)})}{\sqrt{\sum_m \sum_n (I(m, n) - \overline{I(m, n)})^2 \sum_m \sum_n (I'(m, n) - \overline{I'(m, n)})^2}} \quad (8)$$

where $I(m, n)$ and $I'(m, n)$ are the reconstructed intensity images (calculated as in Yang et al. [58]) at distance $d_z$ and distance $d'_z$, respectively. The axial chromatic aberrations for a polychromatic series of holograms can be corrected numerically by calculating the $d_r$ in the following manner.

For the central color ($\lambda_{p/2}$) of the first time-point ($t_1$), a set of $I(m, n)$ is reconstructed along the z-axis at different $d_z$, as shown at the bottom of Fig. 5(a). Then, a CC-based pairwise comparison is performed between two intensity images distanced by (striding across or stepping over) a number of intermediate intensity images $\Delta C_z$ (e.g., 10 images) while scanning $d_z$, as schematically shown at the bottom of Fig. 5(a). The operating principle of this step of the technique is based on two facts supported experimentally by Mohammed et al. [57]. First, $I(m, n)$ reconstructed symmetrically on either side of the focus plane are more similar in the case of a pure amplitude object

in Ferraro et al. [51]. In this situation, if only a single reconstruction distance $d_z$ is used that corresponds adequately to the focus plane $d_r$ of a specific $\lambda$, all the quantitative phase images of the remaining p-1 colors will be out of focus. Therefore, these axial chromatic aberrations produce a resulting blurred P-DHM image [i.e., the final OPD image obtained by averaging over the p colors; see Eq. (20)], as shown in the split-view configuration of Fig. 4. This blurring is particularly evident in the width of the neural network branches and in the diffuse appearance of the QPS, as seen in the upper left of the white dashed diagonal in Fig. 4 (without correction), as well as emphasized in the inset corresponding to the area delimited by the white solid square. Therefore, the complete compensation of axial chromatic aberrations plays a crucial role in obtaining denoised aberration-free quantitative phase images, and the correction of these aberrations is one of the most important steps to automate the P-DHM reconstruction. Indeed, when axial chromatic aberrations are properly corrected, the resulting P-DHM image is much sharper, as seen in the lower right of the white dashed diagonal of Fig. 4 (with correction), as well as emphasized in the inset corresponding to the area delimited by the white solid square. Since each of the p wave-fronts scattered by the sample and recorded out of focus must already be propagated to the correct $d_r$ to obtain a quantitative phase image in focus, it is convenient to introduce compensation for axial chromatic aberrations into this propagation process.

In recent years, a considerable number of criteria have been proposed to numerically determine the correct $d_r$ in digital holography. It is not the purpose of this article to summarize or compare these strategies, as this work has already been done in several articles [56,57]. Based on the study reported by Mohammed et al. [57], the correlation coefficient (CC) criterion is one of the most robust metrics to determine $d_r$ in the case of off-axis digital holography. Therefore, CC is adopted as the focus determination metric in the proposed automatic

<figure><img src="image_7.png" /><figcaption>Fig. 4. Split-view P-DHM image (white dashed diagonal) of a neural live-cell culture, i.e., a neuron's cell culture, without (upper left-hand side) and with (lower right-hand side) chromatic aberration correction emphasized in the inset (to the right) corresponding to the area delimited by the white solid square (at the left). The scale bar indicates 55 µm.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5. Determination of $d_{r,t_1}(\lambda)$ to subsequently correct for axial chromatic aberrations. (a) The schematic description of the calculation of $d_r$ using CC as the focus determination metric for the central color ($\lambda_{p/2}$) of the first time-point ($t_1$), i.e., $d_{r,t_1,p/2}$ for a neural live-cell culture, i.e., a neuron's cell culture. (b) The schematic description of the calculation of $d_r$ using CC as the focus determination metric for all the other p-1 holograms ($\lambda_{v_p\{1/p/2\}}$) of the first time-point ($t_1$), leading to $d_{r,t_1}(\lambda)$.</figcaption></figure>
and more dissimilar in the case of a pure phase object. Second, when this pair of images is gradually off-centered with respect to the focus plane, i.e., while scanning $d_z$, it becomes increasingly dissimilar in the case of a pure amplitude object and increasingly similar in the case of a pure phase object. Therefore, since living cells are essentially pure phase objects, the resulting curve of CC versus $d_z$ is concave, at least in the vicinity of a local minimum corresponding to the focus distance $d_r$, as shown at the top of Fig. 5(a). Since the computation time is directly proportional to the search distance interval and inversely proportional to the distance step size [i.e., $O(N)$], a two-step search strategy is used to minimize the computation time. This strategy consists of a coarse scan followed by a fine scan. The coarse scan is performed using a large search interval (e.g., from $d_z = -3.0$ cm to $d_z = -0.5$ cm) with a large step size (e.g., $\delta d_z = 0.5$ mm) and aims to make a rough estimate of $d_r$, i.e., $\hat{d}_r$. The coarse scan uses a relatively large search interval and step size. These are first estimated qualitatively and then adjusted, if necessary, to obtain a few data-points on either side of the optimal focus plane. This step is critical because the reduced search interval is limited to a subset of the larger search interval. The typical computation time for the coarse scan is approximately 12 s. Once this distance is estimated, it is used to identify a reduced search interval to lessen the overall search time. Typically, the reduced search interval is in the range of $\hat{d}_r - 2$ mm to $\hat{d}_r + 2$ mm. The second

maximum CC values, the experimental curve $d_{r,t_1}(\lambda) = f_{t_1}(\lambda)$ to be applied to correct axial chromatic aberrations is then known (purple solid line), as shown in Fig. 5(b$_{iii}$). The typical computation time for $d_{r,t_1}(\lambda)$ is approximately 4–5 min. This curve is fitted to a theoretical model based on the assumption that the imaging system in the object arm is represented by an equivalent thin lens. In this context, $d_r^{th}(\lambda)$ is defined as $d_r^{th}(\lambda) = d_{img}(\lambda) - d_{l-s}$, where $d_{img}(\lambda)$ represents the image distance, and $d_{l-s}$ denotes the distance between the lens and the sensor. According to Horn et al. [60], $d_{img}(\lambda)$ is expressed as follows:

$$d_{\text{img}}(\lambda) = \frac{R_1 R_2 d_{\text{obj}}}{((R_2 - R_1)(n(\lambda) - 1)) d_{\text{obj}} - R_1 R_2} \quad (9)$$

The constants $R_1$ and $R_2$ represent the radii of curvature of the lens, while $d_{obj}$ denotes the object distance. Additionally, the spectral refractive index, $n(\lambda)$, is expressed using the Sellmeier equation:

$$n(\lambda) = \sqrt{\frac{B_1 \lambda^2}{\lambda^2 - B_2} + \frac{C_1 \lambda^2}{\lambda^2 - C_2} + \frac{D_1 \lambda^2}{\lambda^2 - D_2} + 1}, \quad (10)$$

where $B_1$, $B_2$, $C_1$, $C_2$, $D_1$, and $D_2$ are empirical constants. The final expression for $d_r^{th}(\lambda)$ is derived from Eqs. (9) and (10) and is expressed as follows:

$$d_r^{th}(\lambda) = \frac{R_1 R_2 d_{\text{obj}}}{((R_2 - R_1) \left( \sqrt{\frac{B_1 \lambda^2}{\lambda^2 - B_2} + \frac{C_1 \lambda^2}{\lambda^2 - C_2} + \frac{D_1 \lambda^2}{\lambda^2 - D_2}} + 1 - 1) \right) d_{\text{obj}} - R_1 R_2} - d_{l-s}. \quad (11)$$

scan is performed similarly to the coarse scan but with a small search step size (e.g., with a search step size $\delta d_z = 10$ µm) over the reduced search interval and allows the precise identification of $d_z$ that corresponds to $d_r$, i.e., $d_{r,t_1, p/2}$. For the fine scan, a notably smaller step size is selected, ensuring that it is less than the smallest variation in focus distance between each pair of neighboring $\lambda$. In this case, further reduction of the step size does not improve the quality of the final image and only leads to an increase in computation time. The typical computation time for the fine scan is approximately 16 s.

Then, for all the other $p-1$ holograms ($\lambda_{\forall p \setminus \{p/2\}}$) of the first time-point ($t_1$), a sequence of $I(m, n)$ is generated at many different $d_z$ using the same interval and step size as in the fine scan of the previous step. Then, a CC-based pairwise comparison is performed between the in-focus $I(m, n)$ of the central color ($\lambda_{p/2}$) of the first time-point ($t_1$) (shown in green) and the $I(m, n)$ for each $d_z$ values within the reduced search interval of all the other $p-1$ holograms ($\lambda_{\forall p \setminus \{p/2\}}$) of the first time-point ($t_1$) (illustrated in cyan, blue, yellow, and red), as shown schematically in Fig. 5(b$_i$). For the sake of simplicity, only four of the remaining 35 $\lambda$ are illustrated here. Typically, the intensity images reconstructed from different $\lambda$ will be more dissimilar when they are farther away from their own focal plane and will be more similar when they are closer to their own focal plane, as supported by Zhong et al. [59]. Consequently, the resulting curves of CC versus $d_z$ are convex, at least in the vicinity of a local maximum corresponding to the focus distance $d_r$, as shown in Fig. 5(b$_ii$). By collating all these $d_r$ values corresponding to the

The least squares method is used to minimize the difference between $d_r^{th}(\lambda)$ and $d_{r,t_1}(\lambda)$ by optimizing the said empirical constants. The resulting fitted curve is depicted by the solid purple bold line in Fig. 5(b$_iii$). This fitting approach minimizes the influence of potential outliers. It is also important to note that it is possible to obtain such excellent quality fitted curves from limited data sets (only half or even a third of the data-points). In this way, the fitted curve is used to predict the deliberately omitted data-points of $d_{r,t_1}(\lambda)$, thus reducing the calculation time by a factor of two to three, i.e., down to 1–3 min. At this stage, all the $d_{r,t_1}(\lambda)$ corresponding to the $p$ colors of the first time-point ($t_1$) are adequately determined, ultimately leading to the formation of the first denoised aberration-free P-DHM image (Section 2.B.5). Note that this process of chromatic aberration correction could in principle be performed starting from any $\lambda$. Finally, the next step is to process all $q-1$ remaining time-points.

### 4. Autofocusing in the Presence of Environmental Perturbations

In practice, however, the presence of various environmental perturbations (thermal expansion, optomechanical instabilities, bulging of the coverslips and perfusion chamber, etc.) can cause focus drift [11,39]. Therefore, $d_r$ must be continuously adjusted frame-by-frame to keep an observed cell in focus over time. From a phenomenological point of view, the variation of $d_r$ as a function of $\lambda$ and $t$, i.e., $d_r(t, \lambda)$, is shown in Fig. 6. First, the variation of $d_{r,t_q}(\lambda)$ as a function of $\lambda$ is monotonic
<figure><img src="image_9.png" /><figcaption>Fig. 6. Determination of $d_r(t, \lambda)$ to subsequently autofocus in the presence of various environmental perturbations. Typical variation of $d_r$ as a function $t$ and $\lambda$ is shown for a neural live-cell culture, i.e., a neuron's cell culture. As a function of $\lambda$, the variation represents the effect of axial chromatic aberrations (purple solid line, dashed dark orange and magenta lines), as in Fig. 5(b). The observed changes in $t$ both increase and decrease $d_r$ in response to environmental perturbations (green solid line, cyan, blue, yellow, and red dashed lines).</figcaption></figure>

and represents the effect of axial chromatic aberrations (purple solid line, dashed dark orange, and magenta), as in Fig. 5(b(iii)). However, the temporal variations of $d_{r,\lambda_p}(t)$ are not monotonic. The variations both increase and decrease in response to environmental perturbations (green solid line, cyan, blue, yellow, and red dashed lines). However, the $\lambda$ dependence of $d_r$ does not show significant changes over time. Consequently, a simple shift of one of these $d_{r,tq}(\lambda)$ for each time-point is sufficient to obtain $d_r(t, \lambda)$.

Therefore, considering the experimental data just presented, the following procedure is proposed to determine the set of the reconstruction distances corresponding to a time series of polychromatic hologram stacks monitoring a dynamic biological phenomenon. First, $d_r$ is determined as a function of $\lambda$ using the fitting approach for the first polychromatic image $d_{r,t_1}(\lambda) = f_{t_1}(\lambda)$, i.e., the first time-point ($t_1$) among the $q$, as explained in Section 2.B.3 and schematically shown in Fig. 5(b(iii)). Then, for each other polychromatic image $q$ of the time series, only a single reconstruction distance $d_{r,t_{q,\lambda_p}}$ corresponding to a given $\lambda$ is calculated among the $p$ ones, according to the first step of the axial chromatic aberration correction of Section 2.B.3 and shown schematically in Fig. 5(a). The different $d_{r,t_{q}} = f_{t_{q}}(\lambda)$ curves, although differing by a vertical offset as a function of $t$ due to environmental perturbations, as experimentally demonstrated in Fig. 6, retain the same $\lambda$ dependence since it originates from the axial chromatic aberrations, which are for all intents and purposes constant over $t$. In practice, the specific $\lambda$, for which the single reconstruction distance $d_{r,t_{q,\lambda_p}}$ is calculated, is chosen near the center of the spectral range, i.e., 680 nm or $\lambda_{p/2}$. However, other choices can be made for this specific $\lambda$. Thus, the curve $d_{r,t_{q}}(\lambda)$ can be estimated from the curve $d_{r,t_1} = f_{t_1}(\lambda)$ as follows:

$$d_{r,t_{q}}(\lambda) = f_{t_1}(\lambda) - d_{r,t_1,\lambda_{p/2}} + d_{r,t_{q},\lambda_{p/2}}, \quad (12)$$

where $-d_{r,t_1,\lambda_{p/2}} + d_{r,t_{q},\lambda_{p/2}}$ is the offset. This approach reduces the computation time from 1 to 3 min for the first set of polychromatic holograms to approximately 28 s for any subsequent set without compromising the sharpness of the resulting P-DHM image.

## 5. Numerical Propagation to the Best Focal Plane and Residual Phase Aberration Correction in the Image Plane

The propagation of the corrected object wave from the hologram plane to the best focal plane, i.e., the image plane, by using the reconstruction distances $d_{r,t_{q}}(\lambda)$ obtained in Sections 2.B.3 and 2.B.4 is performed by using the convolution formulation [12]:

$$\Psi_{I,\lambda_p}(m, n) = \mathbf{A} \cdot \text{FT}^{-1} \left\{ \text{FT} \left[ \mathbf{I}_{H,\lambda_p}^C(k, l) \right] e^{-j\pi\lambda_p d_r \lambda_p} \left[ \left( \frac{k}{N\Delta x} \right)^2 + \left( \frac{l}{N\Delta y} \right)^2 \right] \right\}. \quad (13)$$

$\Psi_{I,\lambda_p}$ is the reconstructed wave-front in the image plane for $\lambda_p$, and $\mathbf{A} = \frac{e^{(j2\pi d_r \lambda_p / \lambda_p)}}{j\lambda_p d_r \lambda_p}$. The typical computation time for numerical propagation is considered negligible compared to the other steps of the reconstruction process, as it primarily involves two FT operations. The quantitative phase image of the sample $\Phi_{I,\lambda_p}(m, n)$ can be extracted from $\Psi_{I,\lambda_p}(m, n)$ such that [6]

$$\Phi_{\lambda_p}(m, n) = \tan^{-1} \left\{ \frac{\text{Im} \left[ \Psi_{I, \lambda_p}(m, n) \right]}{\text{Re} \left[ \Psi_{I, \lambda_p}(m, n) \right]} \right\}. \quad (14)$$

At this stage of the reconstruction process, it is important to note that part of $\Phi_{\lambda_p}$ comes from phase aberrations that are not fully corrected in the hologram plane (see Section 2.B.2). Consequently, these residual phase aberrations $\Phi_{ab,\lambda_p}$ are corrected in the image plane. It is also important to remember that $\Phi_{\lambda_p}$ is affected by coherent noise $\Phi_{\text{noise},\lambda_p}$, which is not corrected in any way by the RCH method. Consequently, $\Phi_{\lambda_p}$ is a mixture of three components, the true QPS, i.e., $\Phi_{\lambda_p}$, $\Phi_{ab,\lambda_p}$ and $\Phi_{\text{noise},\lambda_p}$, and can be written as follows [61,62]:

$$\Phi_{\lambda_p}(m, n) = \Phi_{\lambda_p}(m, n) + \Phi_{ab,\lambda_p}(m, n) + \Phi_{\text{noise},\lambda_p}(m, n). \quad (15)$$

Here, the use of $\Phi_{\lambda_p}(m, n)$ to estimate $\Phi_{ab,\lambda_p}(m, n)$ for further correction is proposed, as explained later in this section. But, before going into the details, note that depending on the magnitude of $\Phi_{ab,\lambda_p}(m, n)$, $\Phi_{\lambda_p}(m, n)$ is often phase-wrapped (i.e., exceeds $[-\pi, \pi]$), so a phase unwrapping algorithm must be used to unfold it [63].

In recent years, several methods have been developed to correct phase aberrations in DHM quantitative phase images. In a recent paper, Sirico et al. [64] summarized and compared the available methods. Their classification results in four categories: optical methods, digital methods, hybrid methods, and artificial intelligence-based methods. According to this study, digital and artificial intelligence-based approaches are the most effective for phase aberration correction. Most of the approaches belonging
to these two categories are based on the estimation of a DPM by a polynomial fitting procedure, either with 2D canonical polynomials or with Zernike polynomials. Polynomial fitting is actually a well-suited approach in the present case, because, as explained in several references [15,64], the phase aberrations in a DHM quantitative phase image have a polynomial form that is explained by the physics involved. In the case of a canonical polynomials, the DPM has the following form [14]:

$$ \Gamma_{\lambda_p}^I(m, n) = e^{-j \sum_{\alpha=0}^{\nu} \sum_{\beta=0}^{\nu} S_{\alpha, \beta} m^{\alpha} n^{\beta}}, \quad (16) $$

where $\nu$ represents the polynomial order and $S_{\alpha, \beta}$ defines a set of reconstruction parameters. Several strategies are proposed for their estimation. Colomb et al. [14] proposed to estimate them with a 2D polynomial curve fitting procedure along manually selected regions where the sample contribution to the QPS can be considered as constant, i.e., a QPS iso-surface. However, this method requires several manual operations to select these regions. Other researchers have proposed to use segmentation procedures, either by classical segmentation methods [65] or by artificial intelligence [66,67], to automatically select these zones. The success of phase aberration correction with these methods depends strongly on the determination of the background regions. In certain instances, however, the application of these methods is challenging or consumes user's time and efforts. For example, when the cell population in the quantitative phase image is very dense, it is very difficult to properly segment the background areas from the foreground, or at least the background coverage determined by segmentation is insufficient to obtain a good fit.

In this work, a new, simple, and very efficient approach to estimating the DPM is presented. This approach does not require prior knowledge of the sample, nor does it require segmentation or manual intervention to select the appropriate flat areas in a quantitative phase image. Rather, the DPM can be estimated from the entire quantitative phase image. This approach is inspired by the techniques used to correct the uneven baseline commonly encountered in various signals from experimental techniques such as Raman spectroscopy, X-ray diffraction, and X-ray fluorescence.

In analogy to the work of Lieber and Mahadevan-Jansen [68], the residual phase aberrations $\Phi_{ab,\lambda_p}(m, n)$ can be considered as the base surface, i.e., a surface of arbitrary shape, underlying the raw QPS $\Phi_{\lambda_p}(m, n)$. A 1D baseline correction by an iterative polynomial fitting algorithm, originally developed by Lieber and Mahadevan-Jansen [68] and then slightly modified by Gan et al. [69], is here fully generalized in 2D and adapted to correct the residual phase aberrations with a base surface in a quantitative phase image. The principle of this approach is based on three main steps that are performed iteratively: signal input and fitting, signal comparison and reassignment, and signal output and stop criterion evaluation. First, the input signal, i.e., the original quantitative phase image in the case of the first iteration [see Fig. 7(a)], is fitted with a low order 2D polynomial. Since the QPS usually has a complicated shape [blue solid line, Fig. 7(b)], this polynomial surface will not provide an accurate fit of the entire QPS, but rather a rough estimate of the base surface [red dashed line, Fig. 7(b)] [68]. Note that for simplicity,

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Residual phase aberration correction in the image plane. (a) The original quantitative phase image of a neural live-cell culture, i.e., a neuron's cell culture, to be corrected. (b) Description of the first iteration showing the raw signal as the input signal (blue solid line) corresponding to the white dashed line along one of the diagonals of (a), the low order polynomial fit of the input signal (red dashed line), and the reassigned signal (green solid line). (c)–(d) Idem as (b) but for the second and the last iteration of the iterative process. (e) The final 2D polynomial fitting surface considered as the base surface, i.e., the estimated residual phase aberrations $\Phi_{ab,\lambda_p}(m, n)$ of the quantitative phase image. (f)–(g) Quantitative phase image reconstructed after residual phase aberration correction (f), and the corresponding pseudo 3D view (g). The scale bar indicates 50 µm.</figcaption></figure>

only a 1D profile extracted from one of the diagonals of the input signal [white dashed line, Fig. 7(a)] is used to illustrate the principle of this approach, while in practice a 2D surface fit is used to estimate the base surface. The resulting 2D curve [red dashed line, Fig. 7(b)] is then compared on a pixel by pixel basis to the input signal [blue solid line, Fig. 7(b)] to generate the reassigned signal [green solid line, Fig. 7(b)] from which the next 2D fit is estimated in the following iteration. Each pixel of the input signal [blue solid line, Fig. 7(b)] that has a higher value than the corresponding pixel in the estimated 2D fit [red solid line, Fig. 7(b)] is reassigned to the value of the estimated 2D fit [red solid line, Fig. 7(b)]. In contrast to the previous comparison, when the value of the input signal [blue solid line, Fig. 7(b)] is smaller than that of the 2D fit (red solid line, Fig. 7(b)), the value of the input signal [blue solid line, Fig. 7(b)] is retained. The result of this two-step comparison process gives the reassigned signal [green solid line, Fig. 7(b)]. The newly generated signal [green solid line, Fig. 7(b)] is the signal output of the current iteration. The current signal output is taken as the signal input of the next iteration. This process is repeated iteratively
[Figs. 7(b)–7(d), first, second, and last iterations, respectively] until the true QPS, i.e., $\phi_{\lambda_p}(m, n)$, is eliminated from the raw QPS, i.e., $\Phi_{\lambda_p}(m, n)$, leaving only the base surface [red dashed line, Fig. 7(d)], i.e., $\Phi_{ab,\lambda_p}(m, n)$.

Ideally, the 2D polynomial fitting procedure converges quickly after only a few iterations to fit the base surface of the quantitative phase image, i.e., $\Phi_{ab,\lambda_p}(m, n)$. In addition, the procedure stops when all pixel values of the input signal are equal to or less than the 2D fit curve. However, due to the coherent noise in the QPS and, to a lesser extent, other intrinsic artifacts, this ideal case is rarely reached numerically in practice. Therefore, a stop criterion is established based on a phenomenological observation. The stop criterion used is based on the observation that in practice, in a quantitative phase image of a neural cell culture, backgrounds (i.e., the in-plane space between cells) make up at least 10% of the field of view. Thus, if at the kth iteration a cumulative total of 90% of the pixel values of the first input signal are reassigned, the process should be stopped. At this point, additional iterations will not significantly improve the fit. The stopping criteria can be adjusted by the user depending on the confluence of the cell culture. It is acknowledged that as cell confluence approaches 100%, the effectiveness of the technique may potentially decline due to the lack of a background. Conversely, when confluence is lower than the current level (90%), no loss of efficiency of the technique is anticipated. The final 2D polynomial fit surface [Fig. 7(e)] is considered as the base surface, i.e., the estimated residual phase aberrations $\hat{\Phi}_{ab,\lambda_p}(m, n)$, of the quantitative phase image, and it is used to construct the DPM such that

$$ \Gamma_{\lambda_p}^I(m, n) = e^{-j \hat{\Phi}_{ab,\lambda_p}(m,n)}. \qquad (17) $$

Once $\Gamma_{\lambda_p}^I(m, n)$ is computed, it is multiplied by the reconstructed wave-front $\Psi_{I,\lambda_p}(m, n)$ as [15]

$$ \Psi_{I,\lambda_p}^C = \Gamma_{\lambda_p}^I(m, n) \Psi_{I,\lambda_p}(m, n). \qquad (18) $$

In this way, all residual phase aberrations can be efficiently numerically compensated. The typical computation time for residual phase aberration correction is approximately 0.3 s. Figure 7(f) shows a quantitative phase image reconstructed after numerical correction of residual phase aberrations using this approach. Also, a corresponding pseudo 3D view, in which the residual phase aberrations are fully corrected and therefore the image background is very flat, is shown [Fig. 7(g)] with improved overall image quality compared to the initial image [Fig. 7(a)]. This procedure must be applied to all $\lambda$ among the $p$ colors of each $q$ time-points. It should be noted that the proposed iterative approach using polynomials for phase aberration correction can be applied either in the plane of the hologram or in the plane of the reconstructed image.

In the last step, the corrected quantitative phase images $\phi_{\lambda_p}^C(m, n)$ are calculated from $\Psi_{I,\lambda_p}^C(m, n)$ [see Eq. (14)] and converted to OPD images as follows [35]:

$$ \mathrm{OPD}_{\lambda_p}(m, n) = \frac{\phi_{\lambda_p}^C(m, n)\lambda_p}{2\pi}. \qquad (19) $$

Then, the average OPD image corresponding to the denoised image is calculated as [36]

$$ \langle \mathrm{OPD}_{\lambda_p}(m, n) \rangle = \frac{1}{P} \sum_{p=1}^{P} \mathrm{OPD}_{\lambda_p}(m, n). \qquad (20) $$

## 6. Low Coherence Perturbation Correction

As mentioned in Section 2.A, an AOTF is employed to sequentially select a plurality of bands of a specified width (i.e., $VIS(1 \times 0.5–1.85$ nm and $nIR1 1.8–5$ nm) from a broad spectrum of $\lambda$ emitted from a SLS to generate the polychromatic hologram series. The primary advantage of the AOTF is that it enables the unmanned and rapid selection of $\lambda$, thereby facilitating the achievement of fully automated and rapid acquisition. Conversely, the use of this technology can potentially affect the visibility distribution of the fringes, which is primarily related to the coherence length of the filtered light, a phenomenon known as low coherence perturbations. In fact, the coherence length of a light source is inversely proportional to its spectral bandwidth. A short coherence length causes a complex coherence envelope that modulates the distribution of the interference fringes of the hologram. Depending on the width of this complex envelope, its effect can be variable. In the most extreme cases where its width is very small, which is not the case here, this can potentially lead to a limited field-of-view (FOV) for off-axis DHM. In milder cases, where the complex envelope width is sufficiently large, this can lead to non-uniformities of varying magnitude in the background of the quantitative phase image [70–73].

In the present case, the effect is weak, even very weak, at most $\lambda$. In other words, it is only visible at a few specific $\lambda$. At this stage, it is possible to mitigate the impact of the low coherence perturbations by precisely regulating the OPR of the interferometer, which centers the modulation envelope on the hologram diagonal and minimizes its falloff. The visibility of the fringe distribution at $\lambda = 850$ nm is shown in Fig. 8(a). In Fig. 8(b), enlarged areas of the region bounded by the dark orange, green, and pink solid squares along the diagonal of the hologram (white dashed line) in Fig. 8(a) are shown. It can be observed that the fringes are clearly visible in the center of the hologram (green solid square) and less visible at the edge of the hologram (dark orange and pink solid squares). The general trend of this effect can be seen in the profile extracted from the hologram diagonal (white dashed line) as shown in Fig. 8(c). The magnitude of the low coherence perturbations at $\lambda = 850$ nm is shown in Fig. 8(d). As mentioned above, the effect is very small and even difficult to observe without further analysis. On the other hand, Fig. 8(e) shows a quantitative phase image reconstructed from the most affected hologram, which corresponds to $\lambda = 710$ nm, where the low coherence perturbations result in a non-uniform background. In most cases, such an effect is observed at a single or at most two $\lambda$. When all the OPD images are averaged together, the result is a denoised P-DHM image with a regular background, showing only some distortions of very small magnitude [Fig. 8(f)]. To correct this small irregularity in the background, it is considered as an inhomogeneity in the image. Therefore, a minimization algorithm based on B-spline fitting, as described by Lindblad and Bengtsson [74], is used to estimate the non-uniformity field, as shown in Fig. 8(g). The resulting illumination function is then subtracted from the P-DHM image to obtain a final
<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Low coherence perturbation correction. (a) Hologram acquired at $\lambda = 850$ nm and three regions bounded by dark orange, green, and pink solid squares along the diagonal of the said hologram (white dashed line) for a neural live-cell culture, i.e., a neuron's cell culture. (b) Enlarged parts from the three regions bounded to the hologram in (a) along with their corresponding line profiles. (c) Entire line profile along the diagonal of the hologram (white dashed line) shown in (a). (d) OPD image reconstructed from the hologram in (a) before low coherence perturbation correction. (e) OPD image reconstructed from a hologram acquired at $\lambda = 710$ nm before low coherence perturbation correction. (f) Average OPD image from all the *p* colors. (g) Estimated non-uniformity field obtained from (f) with a minimization algorithm based on B-spline fitting. (h) Final P-DHM image after low coherence perturbation correction showing that the background is practically free of any irregularity. The scale bar indicates 50 µm.</figcaption></figure>

P-DHM image with a very flat, i.e., truly homogeneous background, as shown in Fig. 8(h). The typical computation time for low coherence perturbation correction is approximately 15 s. This step is repeated for each of the *q* time-points of a temporal acquisition.

### 3. APPLICATION

This section presents the effectiveness of the fully automated temporal P-DHM methodology in the context of live-cell imaging. In particular, the ability to visualize the fine structure and dynamics of various cellular phenomena with an unprecedented level of detail is highlighted. More precisely, the subtle dynamics of living astrocytes will be investigated, focusing on phenomena involving lamellipodia motility and filopodia extension, both

key steps in cellular translocation. Two astrocyte cell cultures are monitored every minute for 90 and 128 min, respectively. Astrocytes are selected for this example because of their thin morphology, which makes them particularly difficult to visualize effectively in DHM, due to the relatively low signal-to-noise ratio. At each of the *q* time-points, a set of polychromatic holograms (*p* colors) is acquired and reconstructed using the automated acquisition and reconstruction protocol described in Section 2 to provide a set of denoised aberration-free P-DHM images. The same data set is also reconstructed using the same procedure as in Section 2, but this time at a single $\lambda$, i.e., a classical DHM reconstruction at 660 nm, for comparison purposes.

A comparison of DHM and P-DHM videos reveals a striking contrast. The first example shows the formation of a lamellipodium, a phenomenon essential for cell migration. This is a thin projection at the leading edge of the cell that is characterized by forward movement and rapid retraction. In the DHM video, this phenomenon is virtually invisible (Visualization 1, left-hand side). In contrast, membrane ruffling is clearly visible in the temporal P-DHM video (Visualization 1, right-hand side). Figure 9(a) shows a P-DHM image of an astrocyte at time zero, with an inset over a region where the lamellipodium is particularly active. To the right of this image, a series of four time-points ($t=0$ min, $t=26$ min, $t=66$ min, and $t=90$ min) of the same inset is shown in both P-DHM [Figs. 9(b)–9(e)] and DHM [Figs. 9(f)–9(i)] to emphasize that in DHM the lamellipodium is indistinguishable and, therefore, not quantifiable in its dynamics.

The second example illustrates the formation of filopodium, fine finger-like cell projections that typically follow the lamellipodium-seeking phase of the cell adhesion mechanism. Again, this phenomenon is not visible in the DHM video (Visualization 2, left-hand side). The situation is quite different in the temporal P-DHM video, where the anchoring phenomenon is clearly visible (Visualization 2, right-hand side). More specifically, Fig. 10(a) depicts a P-DHM image of an astrocyte at time zero, with an inset over a region where the filopodium is particularly active. To the right of this image, a series of three time-points ($t=0$ min, $t=54$ min, and $t=114$ min) of the same inset is shown in both P-DHM [Figs. 10(b)–10(d)] and DHM [Figs. 10(e)–10(g)] to emphasize that in DHM the filopodium is indistinguishable and, therefore, not quantifiable in its dynamics.

In support of these observations, it should be recalled that a proof-of-principle demonstrates that the noise level in P-DHM, more precisely the standard deviation of the OPD in the background areas $\sigma_b$, decreases proportionally to $1/\sqrt{N}$, i.e., by a factor of six when $36 \lambda$ is used [36]. As expected, this observation still holds true for the videos presented here. In the case of the lamellipodium (Fig. 9 and Visualization 1), a reduction in noise by a factor of approximately 5.7 is observed, while in the case of the filopodium (Fig. 10 and Visualization 2), a reduction in noise by a factor of approximately 5.4 is observed. In both cases, the typical $\sigma_b$ for P-DHM is 1.1 nm. To further substantiate the denoising observed here, the mean OPD in the lamellipodium and filopodium regions $\mu_s$ as well as the mean OPD in the background areas $\mu_b$ are obtained to calculate the contrast-to-noise ratio (CNR = $(\mu_s - \mu_b)/\sigma_b$) for both DHM
<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Temporal P-DHM video acquired and reconstructed with the proposed methodology illustrating the formation of lamellipodium in a neural live-cell culture, i.e., an astrocyte's cell culture (see Visualization 1). (a) P-DHM image of an astrocyte at *t* = 0 min. (b)–(i) Zoomed-in region corresponding to the white solid rectangle in (a) showing the lamellipodium formation at four different time-points (*t* = 0 min, *t* = 26 min, *t* = 66 min, and *t* = 90 min) in both P-DHM (b)–(e) and DHM (f)–(i). The scale bar indicates 10 µm.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Temporal P-DHM video acquired and reconstructed with the proposed methodology illustrating the formation of filopodium in a neural live-cell culture, i.e., an astrocyte's cell culture (see Visualization 2). (a) P-DHM image of an astrocyte at *t* = 0 min. (b)–(g) Zoomed-in region corresponding to the white solid square in (a) showing the filopodium formation at three different time-points (*t* = 0 min, *t* = 54 min, and *t* = 114 min) in both P-DHM (b)–(d) and DHM (e)–(g). The scale bar indicates 10 µm.</figcaption></figure>

and P-DHM. Typically, the size of the regions of interest for these two analysis ranges from approximately 1 to a maximum of 100 µm². This quantification revealed that the increase in CNR is three to nine times larger for P-DHM than for DHM.

On the other hand, the theoretical framework necessary to determine the exact diffraction limit in the field of QPI has not yet been fully established. Consequently, a comprehensive characterization of the lateral optical resolution in P-DHM, or even in DHM in general, lies beyond the scope of this article. Nonetheless, in a pragmatic attempt, the lateral optical resolution in P-DHM is estimated by analyzing OPD transitions between the background and lamellipodium or filopodium structures. These transitions are selected due to their resemblance to steep objects, which are commonly used to assess lateral optical resolution in microscopy. Multiple line profiles are analyzed, revealing that the transitions typically span 2 to 3 pixels. Considering the sensor pixel size of 5.86 µm and the 20× magnification of the MO, the lateral optical resolution is experimentally estimated to range between 600 and 900 nm. In brief, the denoising observed in P-DHM sharpens the final OPD image without modifying the inherent lateral optical resolution.

Also, it is important to note that the reconstruction of the holograms is done offline, rather than online. Consequently, it is not currently possible to view live video from temporal P-DHM. It should be noted that live reconstruction of holograms, as in Girshovitz and Shaked [75], could prove highly beneficial for a number of live-cell imaging applications, especially in the context of monitoring biological processes that take place over long periods of time, or providing real-time feedback to users during experiments that study fast biological processes.

In addition, although the proposed method is demonstrated for monitoring lamellipodium motility and filopodium extension, it can also be used to capture a range of other dynamic cellular processes that occur over minutes, hours, or even days. These processes include neurite growth and branching, as well as dendritic spine formation and pruning. However, faster biological events, such as axonal transport, evoked neuronal responses, and spontaneous neuronal activity [76], take place on much shorter time scales (seconds to milliseconds). Capturing these rapid processes requires further advances in the automation and optimization of P-DHM recording hardware, in particular the implementation of an all-electronic control.

In summary, Figs. 9 and 10 illustrate two key points: first, that these cell structures are clearly visible in P-DHM (in contrast to
DHM), and second, that they are highly dynamic, demonstrat-
ing the effectiveness of the proposed temporal P-DHM method
for non-invasively visualizing fine cell structures and dynamics
with an unprecedented level of detail. This temporal P-DHM
methodology has the potential to dramatically expand the range
of applications in cell biology, especially in mechanobiology,
which can be addressed in a relevant manner.

4. CONCLUSION

In conclusion, we have presented fully automated, robust, and
efficient management of the acquisition and reconstruction
of time series of polychromatic hologram sets, which allows
the formation of denoised and focused quantitative phase
images. This technological advance makes it possible to achieve
temporal resolution in P-DHM, namely temporal P-DHM.
Specifically, automatic spatial filtering in the frequency domain
is performed with a single filter window after centering the real
image SC using a digital reference wave, which greatly simpli-
fies the operation. Then, phase aberrations in the hologram
plane are corrected using the RCH method. Lateral chromatic
aberrations are then corrected by a telecentric system and axial
chromatic aberrations by calculating the reconstruction distance
corresponding to the focal plane. With regard to autofocusing in
the presence of various environmental perturbations, these are
also managed at the moment of calculating the reconstruction
distance corresponding to the focal plane. At this point, the
numerical propagation to the image plane is performed, and
residual phase aberrations are corrected by a 2D base surface
correction through an iterative polynomial fitting algorithm.
As a final step in the reconstruction process, low coherence per-
turbations are corrected in the image plane by a minimization
algorithm based on B-spline fitting. Finally, temporal P-DHM
is applied to live-cell imaging, which allows for the observation
of the fine structural reorganization over time of a lamellipodia
and filipodia of astrocytes. This demonstrates the ability of
the proposed temporal P-DHM approach to non-invasively
visualize fine cell structure and dynamics with an unprecedented
level of detail.

**Funding.** Canada Excellence Research Chairs (CERC) Program (Neurophotonics); Canada Foundation for Innovation (34265, 36689); Natural Sciences and Engineering Research Council of Canada (NSERC) (BESC D-466483059, RGPIN-2018-06198, RGPIN-2024-06756); Consortium Québécois sur la Découverte du Médicament (CQDM) (FACS-ANCDM); International Joint Unit on Neurodevelopment and Child Psychiatry between Université Laval, Canada, and Université de Lausanne, Switzerland.

**Acknowledgment.** The authors sincerely acknowledge Björn Kemper for fruitful discussions on chromatic aberrations.

**Disclosures.** PM declares a potential conflict as co-founder of Lyncée Tec. However, this independent study was performed in academic laboratories.

**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.

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