2.1.

Polychromatic Digital Holographic Microscope

The polychromatic setup was assembled from a commercial transmission Mach–Zehnder interferometer-based DHM (DHM T-1003, Lyncée Tec), described previously.^26,27 The genuine internal coherent light source of the DHM was bypassed by a tunable wavelength laser system, consisting of a supercontinuum white-light laser (SWLL) (SuperK EXR-20, NKT Photonics) coupled with a multiwavelength acousto-optic tunable filter (AOTF) (SuperK SELECT VIS-nIR, NKT Photonics), injected in the microscope with a delivery optical fiber (OF) (FD7-PM, NKT Photonics), as shown in Fig. 1(a). The DHM setup works by splitting a coherent light source into two: the object beam, which passes through the sample, and the reference beam. The microscope objective (MO), located in the object arm, has a magnification of $20\times$ with a free working distance of $400\mu \mathrm{m}$ and an NA of 0.80 (HC PL APO, Leica). At the exit of the interferometer, the two beams interfere producing a hologram in the camera plane. The setup has an off-axis geometry, meaning that a small angle is introduced between the object and reference waves.²⁸ A digital camera equipped with a monochrome 2.3-megapixel Sony IMX174 CMOS sensor with a pixel size of $5.86\mu \mathrm{m}$ (acA1920, Basler) captured the holograms, which were integrated over a period varying between $200\mu \mathrm{s}$ up to 14 ms, depending on the operating wavelength.

Fig. 1

(a) Experimental setup injecting the SWLL and AOTF into the DHM using an OF, two mirrors (M1 and M2), and an iris (I). (b) Stack of holograms used to obtain (c) a stack of OPD images which are averaged to obtain (d) the P-DHM image.

2.2.

Methodology

The methodology begins with the manual acquisition of a stack of 36 holograms recorded sequentially at wavelengths ranging from 500 to 850 nm, in steps of 10 nm, as shown in Fig. 1(b). For each hologram at each wavelength: the exposure time was adjusted to optimize the use of the dynamic range of the camera and the holograms were numerically reconstructed with numerical refocusing to obtain QP images by simulating the illumination with a digital reference wave.²⁸ The hologram numerical reconstruction makes it possible to correct at each wavelength the aberrations of the object wavefront,^29–32 a fundamental step to obtain quality coherent-noise-reduced images. The exposure time adjustment, numerical refocusing, and aberrations compensation procedures are described in more detail in the Supplementary Material. The QP images thus reconstructed provide a QPS, corresponding to the phase delay induced by the observed quasitransparent cells. At a given wavelength $\lambda$, for each of the $(m,n)$-pixel coordinates, the QPS is given by

The multiwavelength stack of 36 OPD images, as shown in Fig. 1(c), are then averaged pixelwise to form a mean OPD image according to

3.1.

Polychromatic Digital Holographic Microscopy Denoising

Figure 2 highlights the denoising capability of the proposed approach by showing two averaged OPD images of the same field-of-view of a rat neuronal cell culture—one corresponds to the control image and the other to P-DHM. For each image group, the color bar is globally rescaled to properly appreciate the denoising outcome. A typical OPD image, obtained from a single-shot QP image at 540 nm using Eq. (2), is shown in Fig. 2(a), while Figs. 2(b) and 2(c) show the control and P-DHM images, respectively. The reduced granular noise pattern is clearly visible, especially in the background as shown in Figs. 2(b1) and 2(c2). This coherent noise reduction is also particularly noticeable in Figs. S1(a)–S1(c) in the Supplementary Material showing 3D rendered images with the OPD as vertical axis. In contrast, the images in Figs. 2(b3) and 2(c4) focus on an area rich in neuronal processes and the dashed white circles labeled (i)–(iii) draw attention to features unveiled solely by P-DHM. While the “X” shape formed by neuronal processes highlighted in (ii) can hardly be identified in the control image, it is sharply visible in the P-DHM image. In addition, a set of thin neuronal processes [emphasized in (i) and (iii) areas] are revealed by P-DHM but are not visible in the control image. These cell structures, taking the form of fine intersecting neuronal processes, are clearly visible in Figs. S1(b3) and S1(c4) in the Supplementary Material. Figures 2(d) and 2(e) show the individual OPD profiles extracted from the control [Fig. 2(b3)] and P-DHM images [Fig. 2(c4)], respectively, at the position indicated by the dotted white line. The OPD spatial fluctuations along the profile corresponding to the control image [Fig. 2(d)] preclude the possibility to detect the presence of the two thin neuronal processes, which have in contrast a clear signature in the form of two peaks [black arrows, Fig. 2(e) and white arrows, Fig. 2(c4)] along the P-DHM profile. This results from the fact that the background spatial fluctuations (exhibiting amplitudes comparable to those of the two peaks and mainly reflecting the presence of coherent noise) have been drastically reduced. For this proof-of-principle, the 36-hologram acquisition time was around 10 min reflecting, however, a manual approach, requiring performing several tens of clicks to operate both the software of the microscope and laser. An automated approach could easily achieve an acquisition time under a second.

Fig. 2

(a) Single-shot image acquired with 540-nm illumination, (b) control image, and (c) P-DHM image. The white dotted lines in (b) and (c) represent line profiles further analyzed in Fig. 3. (1) and (2) Background section of the control and P-DHM images, respectively. (3) and (4) Neuronal process area of the control and P-DHM images, respectively. (i)–(iii) The white dashed circles highlight cell features and the asterisk identifies a small nerve extension further analyzed in Fig. 4. (d) and (e) Height profiles extracted from the white dotted lines in the bottom left corner of (3) and (4). Scale bar indicates $10\mu \mathrm{m}$.

Fig. 3

Analysis of the height profile from Figs. 2(b) and 2(c). The mean profiles of (a) the control and (b) P-DHM are shown in black, and the single-shot images used for the frame averaging are displayed in color, corresponding to the wavelength used for acquisition. Black dotted lines correspond to the ${\sigma }_b$ of each specific profile, in this case 5.4 and 1.2 nm for the control case and P-DHM, respectively. The inset in (b) is an enlarged portion of the line profile. (c) ${\sigma }_b$ of the line profile for the (a) control case in green and (b) P-DHM in orange as a function of $N$. The black line is the $1/\sqrt{N}$-curve with $A$ chosen as the first data point. (d) ${\sigma }_b$ of another line profile extracted from a different location in the same neuron image, displaying a different behavior as a function of $N$.

Fig. 4

P-DHM image as $N$ increases from (a) $N=1$ to (h) $N=36$. Scale bar indicates $10\mu \mathrm{m}$.

3.2.

Polychromatic Digital Holographic Microscopy Denoising Quantification

To better understand coherent noise reduction, Figs. 3(a) and 3(b) show both the monowavelength average and P-DHM OPD profiles corresponding to a linear height profile within the background [dotted white lines, Figs. 2(b) and 2(c)] accompanied in each case by the 36 individual OPD background profiles that were used to calculate these two average profiles. The values of standard deviations of both background average profiles ${\sigma }_b$, indicated by dashed black lines in Figs. 3(a) and 3(b), provide a quantification of the drastic reduction in coherent noise achieved by P-DHM. Conversely, the 36 individual background profiles of the control case [Fig. 3(a)] are remarkably similar to each other while those of P-DHM [Fig. 3(b)] (each of them corresponding to a specific wavelength) are very different from each other. This illustrates the fact that the change of wavelength is likely to significantly decorrelate the coherent noise between each of the OPD images, therefore explaining why the OPD frame averaging procedure in P-DHM is highly effective in reducing coherent noise.

A set of more than 50 OPD line profiles randomly selected in the background have been analyzed and Figs. 3(c) and 3(d) show the two most representative ${\sigma }_b(N)$ of this OPD line profile set for both the control case and P-DHM. Notably, regarding P-DHM, ${\sigma }_b(N)$ [despite a behavior that can be erratic for values of $N\le 15$; Fig. 3(d)] always accurately follow a $1/\sqrt{N}$-decrease [Fig. 3(c), resulting from the line profile of Fig. 3(b)]. In contrast, the monowavelength case has a noise reduction curve exhibiting a much smaller decrease, stabilizing already after the averaging of five images. Based on these 50 OPD profiles, an average background standard deviation of ${\overline{\sigma }}_{b,540}=5.7\pm 1.4\mathrm{nm}$, ${\overline{\sigma }}_{b,\mathrm{ctrl}}=4.1\pm 0.9\mathrm{nm}$, and ${\overline{\sigma }}_{b,\mathrm{P}\text{-}\mathrm{DHM}}=1\pm 0.3\mathrm{nm}$, respectively, for the single-shot OPD image at 540 nm, the control case, and P-DHM is obtained. These results demonstrate a noise reduction by a factor of $\sim 1.4$ for the control case and by a factor close to 6 for the P-DHM when using 36 OPD images. Considering $1\text{-}\mu \mathrm{m}$-wide thin neuronal processes (corresponding to $\sim 3$ to 4 pixels with the 0.8-NA MO of $20\times$ magnification) and having a characteristic OPD of 1 to 9 nm, this results in a signal-to-noise ratio (SNR) of $\sim 5$ at the single-pixel level. As for the mid-size neuronal extensions and larger axons, single-pixel SNRs of 20 to 75 are attained. This approach was also tested on other cell types, including fibroblasts, and a similar advantage in terms of denoising was systematically observed.

Interestingly, an examination of the increasingly denoised P-DHM image as a function of the number of averaged images (ascending order of wavelengths) as shown in Fig. 4 reveals that the section of the thin neuronal process identified by an asterisk at the bottom of circle (i) in Fig. 2(c4) requires at least a three-image average to become apparent (as a dotted line) to the naked eye, and five images for its full length to emerge clearly.

3.3.

Dispersion of the Observed Cells

According to Eqs. (1)–(3), averaging OPD images acquired at different wavelengths will provide an accurate averaged OPD image if the dispersion of the observed cells does not significantly differ from that of the surrounding medium, i.e., water. In this situation, the values of the OPD generated by the observed cell must remain constant whatever the considered wavelength. The results of Fig. S2 in the Supplementary Material (showing the OPD measurements as a function of the wavelength for the left-hand side neuronal cell body in Fig. 2) support this fact. This is an expected result, previously demonstrated by Rappaz et al.,³³ and remains valid for mammalian nonpigmented cells usually composed of $\sim 80\%$ water and for which the refractive index dispersion specifically due to their constituents can be neglected.