3D dynamic observation of human sperm by parallel phase-shifting digital holographic microscopy based on pixelated polarization

(a) Optical path for pixelated polarization digital holographic microscopy. (b) A pixelated micro-polarization array (PMA) is integrated into a photoelectric sensor in pixelated polarization imaging. Four pixels with micro-polarizers in a super-pixel unit record the different light intensities (I_0°, I_45°, I_90°, and I_135°) of the linearly polarized incident light in the four different directions. Then, the polarization state of the incident light can be determined based on the Stokes parameter reconstruction method.

B. Pixelated polarization imaging and pixelated polarization digital holography

1. Principle of pixelated polarization imaging

In pixelated polarization imaging, a pixelated micro-polarization array [PMA, Fig. 1(b)] consisting of period-parallel metal wires is integrated into the photoelectric sensor, which makes it possible to obtain light intensity and color information as well as polarization properties. When the metal grating period is smaller than the light wavelength, the cross-polarized light can pass through the metallic nanowires due to polarization and non-diffraction effects, while the parallel-polarized light is blocked.²⁸ Currently, the widely used pixelated micro-polarization array is composed of 0°, 45°, 90°, and 135° linear polarizer units arranged with a 2 × 2 periodic pattern (namely, a super-pixel), as shown in Fig. 1(b). Assuming that the polarizer and photodetector perform linearly, for each pixel, the image intensity I and the first three Stokes parameters (s₀, s₁, s₂) satisfy²⁹

I (x, y) = m_{0} (x, y) s_{0} (x, y) + m_{1} (x, y) s_{1} (x, y) + m_{2} (x, y) s_{2} (x, y),

(1)

where x and y are the horizontal and vertical pixel coordinates, respectively. m₀, m₁, and m₂ are the ideal modulation parameters of the Stokes parameter, denoted as

\{\begin{cases} m_{0} (x, y) = \frac{1}{2}, \\ m_{1} (x, y) = \frac{1}{2} \cos 2 θ (x, y) = \frac{1}{2} (\cos π x + \cos π y), \\ m_{2} (x, y) = \frac{1}{2} \sin 2 θ (x, y) = \frac{1}{2} (\cos π x - \cos π y), \end{cases}

(2)

where θ represents the orientation of the linear polarizer (0°, 45°, 90°, and 135°). We neglect the quantum efficiency of the camera as well as the gain in the expression for brevity.

The spatial resolution is reduced by a factor of 4 (a factor of 2 both in x and y) since it requires the combination of four adjacent pixels to obtain the complete polarization information (each pixel can only capture partial polarization properties). Many reconstruction methods [the Stokes parameter reconstruction is to deal with linear equations composed by Eq. (1)] have emerged to remedy this limitation. They can be classified primarily into spatial domain interpolation-based and frequency domain filtering-based methods. For the interpolation-based method, based on the four-channel structural design of the PPC, 0°, 45°, 90°, and 135° polarization images are extracted from the polarization images. Then, by interpolating the missing pixel values in the four separated images, the Stokes parameter estimations are determined from the interpolated images. The commonly used interpolation methods include nearest-neighbor, bilinear,³⁰ bicubic,^31,32 and bicubic natural spline³³ interpolations. Interpolation-based methods are broadly accessible for their simplicity and fast computing speed, but they introduce large interpolation errors at sharp boundaries in the polarization image. The filtering-based method extracts Stokes parameters by constructing proper filters (such as Hamming,³⁴ Gaussian,³⁵ and Planck³⁶ window functions) based on the distribution of the Stokes parameters in the frequency domain. The filtering-based method can theoretically reconstruct the Stokes parameters perfectly with adequate sampling. Nonetheless, selecting filtering parameters is empirical, and it is challenging to choose the optimal ones.

2. The principle of pixelated polarization digital holography

In digital holography, an object beam

U_{o} = E_{o} e^{i φ_{o}}

and a reference beam

U_{r} = E_{r} e^{i φ_{r}}

form a hologram I by interference (U_o and U_r denote the complex amplitudes of the object beam and the reference beam, respectively, E_o and E_r are the electric field strengths, and φ_o and φ_r are the electric field phases), and the hologram I is expressed as

\begin{align} I & = U_{o} U_{o}^{*} + U_{r} U_{r}^{*} + U_{o} U_{r}^{*} + U_{o}^{*} U_{r} \\ = E_{o}^{2} + E_{r}^{2} + 2 E_{o} E_{r} \cos (φ_{o} - φ_{r}), \end{align}

(3)

where

U_{o} U_{o}^{*} + U_{r} U_{r}^{*}

is the zero-order diffraction term,

U_{o} U_{r}^{*}

is the object beam diffraction term, and

U_{o}^{*} U_{r}

is the object beam conjugate term. * represents the conjugate. The zero-order term and the conjugate term are superimposed on the image of objects and interfere with the reconstruction of the object wavefront. Then, the phase-shifting method is usually adopted to eliminate the zero-order term and the conjugate term. In the parallel phase-shifting, the object beam U_o and the reference beam U_r are a set of linearly polarized lights with an orthogonal electric field vibration direction. Assume that the object beam is horizontally polarized and the reference beam is vertically polarized. After passing through a 1/4 waveplate at 45° orientation and a linear polarizer at an angle of θ, the object beam and the reference beam reach the photoelectric sensor. The Jones matrix J_q for the 1/4 waveplate is²⁷

J_{q} = \frac{1}{\sqrt{2}} [\begin{matrix} 1 & i \\ i & 1 \end{matrix}] .

(4)

The Jones matrix J_p of the linear polarizer is denoted as

J_{p} = [\begin{matrix} \cos^{2} θ & \cos θ \sin θ \\ \cos θ \sin θ & \sin^{2} θ \end{matrix}],

(5)

where θ represents the orientation of the linear polarizer. After the object beam and the reference beam pass through the 1/4 waveplate placed at 45° and a linear polarizer, the Jones vector E_U is expressed as

\begin{align} E_{U} & = J_{p} J_{q} [\begin{matrix} U_{o} \\ U_{r} \end{matrix}] \\ = \frac{1}{\sqrt{2}} [\begin{matrix} \cos^{2} θ & \cos θ \sin θ \\ \cos θ \sin θ & \sin^{2} θ \end{matrix}] [\begin{matrix} 1 & i \\ i & 1 \end{matrix}] [\begin{matrix} E_{o} e^{i φ_{o}} \\ E_{r} e^{i φ_{r}} \end{matrix}] \\ = \frac{1}{\sqrt{2}} [\begin{matrix} \cos^{2} θ (E_{o} e^{i φ_{o}} + i E_{r} e^{i φ_{r}}) \\ + \cos θ \sin θ (i E_{o} e^{i φ_{o}} + E_{r} e^{i φ_{r}}) \\ \cos θ \sin θ (E_{o} e^{i φ_{o}} + i E_{r} e^{i φ_{r}}) \\ + \sin^{2} θ (i E_{o} e^{i φ_{o}} + E_{r} e^{i φ_{r}}) \end{matrix}] . \end{align}

(6)

The final recorded hologram I is represented as³⁷

\begin{align} I (x, y) & = E_{U} E_{U}^{*} = \frac{1}{2} [E_{o} {(x, y)}^{2} + E_{r} {(x, y)}^{2}] \\ + E_{o} (x, y) E_{r} (x, y) \sin [φ_{o} (x, y) - φ_{r} (x, y) + 2 θ (x, y)], \end{align}

(7)

where x and y denote the pixel coordinates of the camera and θ takes the values of 0°, 45°, 90°, and 135°.

3. Relationship between the Stokes parameters and the hologram components

In previous parallel phase-shifting works,^18–21 four polarization images with different polarization orientations (four holograms with different phase shifts) were separated from the polarization image. Then, the missing grayscale values in the four images were interpolated using the interpolation method. Finally, object wavefront reconstruction was performed based on these four interpolated full-resolution polarization images. This process is consistent with the interpolation-based Stokes parameter reconstruction method, which implies a close connection between the Stokes parameter and the hologram components. Therefore, we investigate the relationship between the Stokes parameter and the hologram components. The expansion of Eq. yields

\begin{align} I (x, y) & = \underset{m_{0}}{\underset{︸}{\frac{1}{2}}} \underset{s_{0}}{\underset{︸}{[E_{o} {(x, y)}^{2} + E_{r} {(x, y)}^{2}]}} \\ + \underset{m_{1}}{\underset{︸}{\frac{1}{2} \cos 2 θ (x, y)}} \underset{s_{1}}{\underset{︸}{2 E_{o} (x, y) E_{r} (x, y) \sin [φ_{o} (x, y) - φ_{r} (x, y)]}} \\ + \underset{m_{2}}{\underset{︸}{\frac{1}{2} \sin 2 θ (x, y)}} \underset{s_{2}}{\underset{︸}{2 E_{o} (x, y) E_{r} (x, y) \cos [φ_{o} (x, y) - φ_{r} (x, y)]}} . \end{align}

(8)

Comparing Eqs. – with Eq. , we find that in pixelated polarization digital holography, the zero-order diffraction term corresponds to s₀, the real parts of both the object beam diffraction term and the conjugate term correspond to s₂/2, and the imaginary parts of both the object beam diffraction term and the conjugate term correspond to s₁/2 and −s₁/2, respectively. Each hologram component corresponds to the Stokes parameter, so we can use the Stokes parameter reconstruction method to reconstruct the object wavefront. After the acquisition of s₁ and s₂ based on the Stokes parameter reconstruction method, the object beam diffraction term can be expressed as

U_{o} U_{r}^{*} = \frac{1}{2} s_{2} + \frac{i}{2} s_{1} .

(9)

Equation indicates that the object wavefront extraction is inherently equivalent to the Stokes parameter reconstruction. The Stokes parameter reconstruction method has been a research hotspot, and various reconstruction methods have emerged. We have analyzed the noise robustness of different Stokes parameter reconstruction methods in Ref. 38. For the interpolated-based methods, the nearest-neighbor interpolation method is the most sensitive to noise, and the noise robustness of the other methods decreases with increasing interpolation order. In previous work on parallel phase-shifting, object wavefront reconstruction was essentially achieved by the nearest neighbor interpolation method. It is not a good choice due to its lower computational accuracy and poor noise robustness. In holography, the imaging quality is more seriously affected by noise, so we choose the bilinear interpolation-based Stokes parameter reconstruction method with better noise robustness and higher accuracy to extract the object wavefront. We ignore the filtering-based methods because the filtering parameters need to be chosen manually based on the observed scenario.

C. The effect of sampling interval on system resolution

In pixelated polarization imaging, since the pixel size cannot be infinitely small, the spatial bandwidth of the captured scene cannot be too wide to avoid confusion in the frequency domain.³⁴ In microscopic imaging, the spatial frequency is limited by the finite numerical aperture of the microscope objective.³⁹ These two factors together determine the resolution of the system in pixelated polarization digital holographic microscopy. We have investigated the effect of PPC pixel size on the system resolution based on spectral analysis. The Fourier transform

(F)

of Eq. yields³⁴

\begin{align} I (u, v) & = F \{U_{o} U_{o}^{*} + U_{r} U_{r}^{*}\} (u, v) \\ + F \{I m (U_{o} U_{r}^{*}) + R e (U_{o} U_{r}^{*})\} (u + \frac{1}{2}, v) \\ + F \{I m (U_{o} U_{r}^{*}) - R e (U_{o} U_{r}^{*})\} (u, v + \frac{1}{2}) . \end{align}

(10)

As shown in Eq. , the frequency components of the zero-order term

U_{o} U_{o}^{*} + U_{r} U_{r}^{*}

are modulated to the low-frequency region of

I

(⁠

I

is the Fourier transform of I) and the frequency components of Im

(U_{o} U_{r}^{*})

+ Re

(U_{o} U_{r}^{*})

and Im

(U_{o} U_{r}^{*})

-Re

(U_{o} U_{r}^{*})

are modulated to the horizontal and vertical high-frequency regions of

I

⁠, as shown in Fig. 2(a). Re represents the real part of the complex number, and Im represents the imaginary part. In microscopic imaging, the spatial frequency of the light field is limited by the numerical aperture of the microscope objective, which does not exceed NA/Mλ. M is the objective magnification, and NA is the numerical aperture of the microscope objective.³⁹ The magnification and the numerical aperture of a typical microscope objective are given in Table I.

FIG. 2.

$(a) Distribution of each diffraction term of the hologram image in the spectrum and its bandwidth. (b) Relationship between magnification and numerical aperture under the limiting resolution. The black dots represent typical microscope objectives, and the shaded area indicates that the diffraction limit resolution can be reached.$

(a) Distribution of each diffraction term of the hologram image in the spectrum and its bandwidth. (b) Relationship between magnification and numerical aperture under the limiting resolution. The black dots represent typical microscope objectives, and the shaded area indicates that the diffraction limit resolution can be reached.

TABLE I.

Magnification and numerical aperture of a typical microscope objective.

Magnification (M)

4

10

20

40

60

100

Numerical aperture (NA)

0.10

0.25

0.40

0.65

0.80

1.25

In in-line interference, the bandwidth (BW) of each diffraction term in the hologram spectrum after sampling by a pixelated polarization camera at intervals of Δ can be expressed as⁴⁰

\begin{gathered} B W (U_{o} U_{o}^{*} + U_{r} U_{r}^{*}) = \frac{4 N A Δ}{M λ}, \\ B W (I m (U_{o} U_{r}^{*}) + R e (U_{o} U_{r}^{*})) = \frac{2 N A Δ}{M λ}, \\ B W (I m (U_{o} U_{r}^{*}) - R e (U_{o} U_{r}^{*})) = \frac{2 N A Δ}{M λ} . \end{gathered}

(11)

Figure 2(a) shows the distribution of each diffraction term in the hologram spectrum and its bandwidth. To ensure that each diffraction term can be reconstructed without confusion, the diffraction terms in the spectrum should not overlap. Then, the microscope objective’s magnification M, numerical aperture NA and wavelength λ, and sampling interval Δ should satisfy

\frac{6 N A Δ}{M λ} \leq 1 .

(12)

When Eq. holds, the resolution of the digital holographic microscope system is limited only by the numerical aperture, and it can reach the limiting resolution of 0.82λ/NA under coherent illumination.⁴¹ The downsampling of 0, π/2, π, and 3π/2 phase-shifting images does not degrade the system resolution. Figure 2(b) shows that the microscope objectives enable the system to reach the limiting resolution at different sampling intervals (3.45, 6.45, and 7.4 μm, illumination at 532 nm). The black dots in Fig. 2(b) represent typical microscope objectives, and the three red, blue, and green lines represent the case when Eq. is taken as equal. The shaded area under the sloping line shows that microscope objectives allow the system to reach the limiting resolution at that sampling interval. For the sampling interval of 3.45 μm, the system can achieve limiting resolution with any microscope objective; for 6.45 μm, only 60× and 100× microscope objectives meet the requirements, while the limiting resolution is unattainable for 7.4 μm. In our pixelated polarization digital holographic microscope system, the numerical aperture (NA) of the microscope objective is 0.8 and the pixel size (sampling interval) of the pixelated polarization camera is 3.45 μm. The system can reach the limiting resolution under coherent illumination.

III. EXPERIMENTS AND RESULTS

A. Sample preparation

We built a parallel phase-shifting digital holographic microscope system based on a pixelated polarization camera and observed the 3D movement of sperm with it. The sperm sample was provided by the Reproductive Medicine Center, the First Affiliate Hospital of Anhui Medical University. The study received ethical preapproval from the Reproductive Medicine Ethics Committee of the First Hospital of Anhui Medical University, and written consent was obtained from the participants. Semen was obtained from the patient by masturbation and allowed to liquefy in an incubator (37 °C, 5% CO₂) for 30 min. One milliliter of semen was then taken with a pipette into a labeled tube, and 2 ml of SpermRinse buffer (Vitrolife, 10 101) was added before incubation in the incubator for 30 min. The upper layer of the liquid was transferred to a new tube and centrifuged at 1000 g for 10 min. Finally, the supernatant was discarded, and the sperm sample was obtained by resuspension after the injection of a properly equilibrated fertilization buffer (COOK, K-SIFM-50). If the sperm sample is dropped directly into the container, the droplet is not flat due to surface tension, which affects the measurement results. A special cuvette was made to solve this problem. It consists of a standard slide with a small hole (diameter of 6 mm) and a square coverslip (24 mm in length and 50 μm in thickness). The coverslip adheres to one side of the small hole with UV-curable adhesive. An appropriate amount of sperm sample was pipetted into a small well and then sealed with the same coverslip. In this way, the surface of the solution can be kept flat. Be careful to avoid the appearance of bubbles when sealing the coverslip. Finally, the cuvette was placed over the sample area [Fig. 1(a)] for observation.

B. Sperm 3D location

Before sperm observation, we first needed to determine the actual magnification of the system by observing a standard USAF calibration plate. Figure 3(a) shows the zero-order term (corresponding to the Stokes parameter s₀) of the observed USAF calibration plate with 2048 × 2048 pixels². The signal-to-noise ratios of the object beam diffraction term and the conjugate term are relatively low, so the zero-order term image is used for magnification calibration. There are three straight lines (the line width is 6.96 μm) in the horizontal and vertical directions in Fig. 3(a). The intensity distribution of lines 1 and 2 in Fig. 3(a) is shown in Fig. 3(b). The number of pixels occupied by the lines (∼133 pixels) is determined based on image grayscale variation. Combined with the physical size of the pixels (3.45 μm), the actual magnification of the system is 65.9.

FIG. 3.

$(a) The zero-order term image of the USAF calibration plate. (b) The intensity distribution along lines 1 and 2 in (a). (c) Amplitude image of sperm. The dashed boxes of different colors represent different area sizes. (d) Relationship between the TC coefficient and the diffraction distance for different area sizes in (c).$

(a) The zero-order term image of the USAF calibration plate. (b) The intensity distribution along lines 1 and 2 in (a). (c) Amplitude image of sperm. The dashed boxes of different colors represent different area sizes. (d) Relationship between the TC coefficient and the diffraction distance for different area sizes in (c).

The 2D coordinate (X and Y coordinates) of the sperm is determined based on the sperm head centroid, while the depth perpendicular to the camera plane (Z coordinate) is obtained based on the numerical focusing algorithm. In the system, the distance between the camera plane and the microscope objective is constant; thus, the focusing plane is also at a fixed depth. With the focusing plane as the reference plane, the distance from the sperm to the focusing plane is considered as the depth of sperm perpendicular to the camera plane (taken as positive away from the objective and negative near the objective). First, the complex amplitudes of the object wavefront at different diffraction distances away from the camera plane are reconstructed by the angular spectrum diffraction algorithm.⁴² Then, the TC coefficient⁴³ is used to estimate the in-focus diffraction distance. Finally, the distance of the sperm deviation from the focal plane is calculated with the system magnification. The TC coefficient is defined as

T C = \sqrt{\frac{σ (I)}{⟨ I ⟩}},

(13)

where σ(I) and ⟨I⟩ represent the standard deviation and the mean of image I, respectively. The sperm head may be brighter or darker than the background in a frame during motion, and it is difficult to directly distinguish the sperm from the background in all frames by simple binarization. Since sperm swims continuously and the displacement between two sequential frames is in a range (less than 100 pixels in our experiments), we can determine the 2D coordinates of the sperm in the current frame based on the position of the sperm in the previous image. In the last frame, select a square area of appropriate size centered on the sperm head centroid. If the square area is too small, the sperm may move out of the region; if the size is too large, too high a percentage of background may lead to an incorrect reconstruction distance. We explored the effect of the selected region size on the reconstruction distance. Figure 3(c) shows the original sperm amplitude images with different sizes, and Fig. 3(d) shows the TC coefficient curves obtained from the selected regions in Fig. 3(c). Figure 3(d) indicates that the reconstructed distances obtained with different size regions are all approximately −41 mm, and all of them can obtain accurate results. However, the size affects the robustness of the TC criterion. When the selected area is small, the sperm occupy a high percentage of the whole field with a distinct minimum point (which points to the in-focus diffraction distance) on the TC coefficient curve. As the area chosen increases and the background increases, the TC coefficient curve changes flatly, and the minima become less pronounced. To ensure that the sperm do not swim out of the selection range and to obtain accurate reconstructed diffraction distances, we selected a square area of 300 × 300 pixels². Then, the complex amplitude of the in-focus image is acquired by a numerical focusing algorithm. The sperm head region is extracted in the phase map by image binarization (the threshold chosen as 1.2 rad) and morphological opening and closing operations. Finally, the position of the sperm head centroid in the 2D plane is determined. In this way, we can obtain the 2D coordinates of the sperm head in all frames once it is located in the first frame. For the first frame, a point on the sperm head can be selected as the sperm head centroid in the 0th frame image (assumed), and then, the same method above can be used to determine the sperm head centroid in the first frame. It is worth noting that the sperm concentration should not be too high to avoid crossing of the sperm trajectories in the 2D plane. For the sperm depth information, although the numerical focusing algorithm has been performed to determine the diffraction distance when determining the sperm 2D coordinate, the selected area (300 × 300 pixels²) is so large at that time that too much background will interfere with the accuracy of the diffraction distance. To obtain a more reliable diffraction distance, we select the sperm head region determined from the phase map and perform the numerical focusing algorithm again to determine the diffraction distance of the sperm head. Then, we calculate the distance of the sperm head from the focal plane with the system longitudinal magnification (the square of the lateral magnification) and thus obtain the 3D coordinates of the sperm. Movie S1 of the supplementary material shows the motions of the sperm before and after the numerical focusing.

The 3D coordinates of the sperm head centroid in each frame are obtained; that is, the 3D spatial trajectory of the sperm is obtained. Figures 4(a) and 4(b) show the trajectory maps of the sperm in the 2D plane and 3D space, respectively, and Figs. 4(d)–4(f) are the projections of the 3D trajectory [Fig. 4(b)] in each plane. Based on the obtained sperm trajectory map, we can calculate its motion parameters, such as straight-line velocity (VSL), curvilinear velocity (VCL), and linearity. The VCL represents the total distance that the sperm head covers in the observation period. The VSL is determined from the straight-line distance between the first and last points of the trajectory and gives the net space gain in a period. The linearity represents the comparison of the straight-line and curvilinear paths, VSL/VCL. VSL is 51.66 μm/s, VCL is 135.30 μm/s, and linearity is 0.38 based on the 2D trajectory in Fig. 4(a). VSL is 52.76 μm/s, VCL is 165.31 μm/s, and linearity is 0.32 based on the 3D trajectory in Fig. 4(b). Since the sperm swims near a plane parallel to the camera, the two VSLs are roughly equal. However, the VCLs show a significant difference. Sperm motility features are positively correlated with fertility, which is an important criterion for assessing whether the sperm can successfully fertilize the egg. The motility parameters obtained from the 2D planar information of conventional imaging are somewhat different from the actual situation (3D trajectory) and cannot accurately measure the motility properties of the sperm. Pixelated polarization-based parallel phase-shifting digital holographic microscopy, which is rapid, mechanical scan-free, and label-free, can accurately obtain the 3D trajectory and the actual motility parameters of sperm for better sperm viability assessment. In addition, based on the 3D trajectory obtained, it is possible to determine the sperm motion pattern, which conventional imaging cannot determine. The sperm head rotates around the central axis at a very stable speed, forming a good right-hand helix. The lateral displacement (perpendicular to the direction of sperm advancement) distribution of its head is shown in Fig. 4(c), with an average value of 2.33 μm. Accordingly, the swimming pattern can be determined to be helical.

FIG. 4.

(a) and (b) Trajectory maps of sperm in 2D plane and 3D space. (c) Lateral displacement distribution of the sperm head, perpendicular to the direction of sperm advancement. (d)–(f) Projection of spatial 3D trajectory (b) in each plane.

Furthermore, we analyzed the movement of the sperm in different directions (X, Y, and Z), as shown in Fig. 5. The trajectory of the sperm during motion is spiral-shaped. The projections in different directions show sinusoidal curves along the motion axis, as shown in Figs. 5(a)–5(c). The hollow circle represents the actual sperm motion trajectory, and the chain line is its linear fit. The sperm in Fig. 5 shows straight-line advancement. The slope of the fitting straight line reflects the average rate of sperm advancement in the direction. Figures 5(d)–5(f) represent the projection of sperm lateral displacement in each direction (the actual sperm trajectory minus the linear fitting). The Fourier transform of Figs. 5(d)–5(f) can be performed to obtain the beating frequency of sperm, as shown in Figs. 5(g)–5(i). Figures 5(g) and 5(h) show three main peaks (8.37, 21.34, and 29.74 Hz), which quantifies sperm beating rates. In Fig. 5(i), although the fluctuations in the spectrum are relatively large due to some errors in the determination of Z-direction coordinates, it is still possible to identify three main peaks of the spectrum, corresponding to frequency values in Figs. 5(g) and 5(h). The motion in the X and Y directions can be directly obtained from the 2D image, and the results are accurate and reliable. The movement in the three directions (X, Y, and Z) is the decomposition of the forward movement of the sperm, and theoretically, they should have the same characteristic frequency. The characteristic frequency in the Z direction is calculated based on the coordinates we acquired in the Z direction, and the results match better with the X and Y directions. This is consistent with our expectations and to some extent indicates the accuracy of the coordinates we obtained in the Z direction. The 3D trajectory of sperm is acquired by determining the 3D position of the sperm head centroid. Then, the motility parameters and swimming patterns of the sperm are determined from the 3D trajectory. The sperm beating rates are extracted through the Fourier transform of the projections of sperm lateral displacement in each direction. Sperm movement can be characterized by motility, swimming patterns, and oscillation rates. 2D sperm tracking using conventional imaging cannot obtain information on the depth of sperm (Z coordinate). Moreover, to ensure that the spermatozoa are within the effective depth of field during motion, the sperm sample is restricted to a controlled depth range (10 or 20 µm). This restriction in the depth direction can alter the form of sperm movement, and the observed sperm movement is not the normal movement of sperm.⁴⁴ Compared with conventional imaging, our method is more comprehensive and accurate in characterizing sperm motion properties and has a greater potential for utilization.

FIG. 5.

(a)–(c) Variation of sperm displacement in different directions (X, Y, and Z). (d)–(f) Variation of sperm lateral displacement projected in different directions. (g)–(i) are the Fourier transforms of (d)–(f).

C. Sperm head 3D dynamic reconstruction

During motion, the posture of the sperm head is constantly changing with flagellar beating. In previous work on 3D sperm tracking,^13–17 the 3D trajectory of sperm was often obtained by tracking the centroid of the sperm head. In this process, the sperm head is often treated as a mass point, and the posture information of the sperm head is neglected. The sperm will deviate from the focal plane during motion, which makes the sperm image defocused and blurred, and thus, the sperm contour cannot be accurately extracted. To reconstruct the sperm head, first, numerical focusing algorithms (including angular spectral diffraction and TC image focusing criterion) are used to obtain in-focused sperm images, which solves the issue of sperm defocus in conventional imaging. Next, we model the sperm head as an ellipsoid, determine the size of the sperm head based on the 2D projection ellipse of the sperm head after numerical focusing, and then determine its azimuthal angle based on the inclination and the variation of the major and minor axes of the projection ellipse, thus obtaining complete information on the sperm head.

1. Determination of sperm head size

In Ref. 17, Dardikman-Yoffe et al. adopted a high-speed off-axis digital holography system to achieve the extraction of 3D refraction index profiles of sperm heads through optical diffraction tomography theory. They confirmed the validity of the ellipsoid model for sperm head orientation recovery. To determine the posture of the sperm head in each frame, we also model it as an ellipsoid [as shown in Fig. 6(a)], denoted as

{(x / A)}^{2} + {(y / B)}^{2} + {(z / C)}^{2} = 1,

(14)

where A, B, and C are the axis lengths of the ellipsoid in different directions, and it can be assumed that B ≫ A > C. To determine the parameter constants of the ellipsoid, we can obtain them from the planar projection. First, a numerical focusing process (including angular spectral diffraction and TC image focusing criterion) was performed on each hologram to obtain the complex amplitude of sperm. This process results in a focused image of the sperm, which helps in the extraction of the sperm contour. The phase map with 500 ×500 pixels² obtained from the complex amplitude is shown in Fig. 6(b). The sperm head region was segmented by binarization with a suitable threshold (we chose 1.2 rad). Then, the ellipse fitting algorithm in OpenCV was used to determine the major and minor axes of the ellipse as well as the orientation angle (corresponding to the yaw angle below) of the ellipse [Fig. 6(c)]. The planar projection of the sperm head during its movement is an ellipse. The ellipsoid parameters are related to the major and minor axes of the 2D projection ellipse. The major axis maximum of the projected ellipse corresponds to constant B, and the maximum and minimum of the minor axis correspond to constants A and C. The variation in the major and minor axes of the projected ellipse with the number of image frames is obtained, as shown in Fig. 6(d). The small circle with a cross represents the raw measurement data. We first perform a spline fit to the raw data and determine each axis parameter by averaging the extreme values of the fitted curve, where A = 1.23 μm, B = 2.14 μm, and C = 0.83 μm. The planar ellipse represents the projection of the sperm head, where the major axis of the projection ellipse corresponds to constant B of the sperm head. In cases where the sperm head is parallel to the camera plane, the major axis length is equal to B, while in other cases, it is smaller than B. To better reconstruct the sperm pose, we choose a sperm closer to the focal plane for the reconstruction. Movie S2 of the supplementary material shows the motion of the chosen sperm before and after numerical focusing. In movie S2 of the supplementary material, the displacement of the sperm in the Z-direction is much smaller than that in the X- and Y-directions. The forward direction of the sperm is almost parallel to the camera plane, which ensures that the sperm head parallel to the camera plane will be present during the movement. Therefore, the maximum value obtained by the 2D projection ellipse can indicate constant B. When the angle between the forward direction of the sperm and the camera plane (the pitch angle) is too large, the pitch angle will change during sperm motion, but it is still impossible for the sperm head to be parallel to the camera plane (i.e., pitch angle equals to 0). Therefore, the major axis length of the projected ellipse is always smaller than constant B and the major axis maximum value of the projected ellipse cannot be used to determine B in such cases.

FIG. 6.

(a) Ellipsoidal model. (b) Phase map after numerical focusing. (c) The sperm head region in the phase map and the ellipse fit. (d) Variation of the fitted ellipse major axis, minor axis, and orientation angle (yaw angle) with the number of image frames.

2. Sperm head azimuth determination

Apart from information on the 3D position of the sperm, we must also identify its orientation to characterize the sperm head posture. We define the yaw angle as the rotation angle in the (X–Y) plane of the camera [rotated about the z-axis in Fig. 6(a)], and its magnitude is defined as shown in Fig. 6(c). It can be identified directly from the orientation angle of the ellipse fit [Fig. 6(d)]. The roll angle is defined as the angle of rotation around the longitudinal axis of the sperm head [y-axis in Fig. 6(a)]. During free movement of the sperm, the sperm head rolls periodically, as shown in Fig. 6(d). The angle at the extreme point can be determined based on the periodicity (the roll angle is an integer multiple of 180° at the local maximum with an integer multiple of 90° at the local minimum). Under the ellipsoidal assumption, the structure of the sperm is symmetric, and there is essentially no difference between taking 0° or 180° for the first maximal point. Thus, we take the first local maximal point to be 0° and the first local minimal point to be 90°. The roll angle of other frames in a period can be inferred from the local minor-axis maximum and the minor-axis length of the frame. Although the minor axis shows good periodicity in Fig. 6(d), the extreme point magnitude still fluctuates slightly for the image noise-introduced reconstruction distance deviation and the inappropriate selection of the threshold in sperm head segmentation. In the local region, it is more appropriate to use the local extreme than the constants A and C. After determining the size of the roll angle for each frame, we also need to know the direction of its rolling (clockwise rotation or counterclockwise rotation). In the sperm coordinate (o-xyz), the sperm head rotates around the y-axis [Fig. 6(a)], and the direction of rotation can be determined from the location coordinate vectors of the sperm head in the x and z directions. We first calculate the gradient g_x of the coordinate vector in the x-direction and then use it to determine two parameters: V_H, which is the sum of all z-coordinate values when g_x is greater than 0, and V_L, which is the sum of all z-coordinate values when g_x is less than 0. If V_H is higher than V_L, then we determine the direction of rotation to be clockwise and vice versa. The detailed logical relationship can be seen in Fig. 7(a). If the sum of z values obtained for a positive change in x (blue arrows) is higher than for a negative change in x (red arrows), the rotation direction is clockwise, as shown in Fig. 7(a); otherwise, it is the case of Fig. 7(b). The sperm coordinate o-xyz satisfies the rotational translation transformation with the image coordinate O-XYZ [Fig. 6(c)]. The position coordinate vectors x and z tend to change in the same way as X and Z, and we can determine the roll direction from coordinate vectors X and Z during practical processing. The pitch angle is defined as the elevation of the sperm head relative to the camera (X–Y) plane [rotation around the x-axis in Fig. 6(a)]. The pitch angle is related to the major axis of the projection ellipse. When the sperm head is parallel to the camera (X–Y) plane, the major axis is a maximum, and the pitch angle size is 0. The pitch angle obtains a maximum when the major axis is a minimum. For the other frames, the pitch angle can be determined in the same way as the case of the roll angle. Note that the primary forward motion is along the spindle during swimming [y-axis in Fig. 6(a)], and the pitch angle does not exceed 60° in general. If this is not the case, this method may fail as the major axis of the projection ellipse may turn into the minor axis for larger pitch angles. The approximate ellipsoidal structure of sperm allows for a certain symmetry in its motion in the x-y and y–z planes. Figure 6(d) shows that the variation of the sperm direction in the x–y plane (yaw angle) is within 40°. This indicates that the change in pitch angle (y–z plane) is also no more than 40°. The forward direction of the sperm is almost parallel to the camera plane, which ensures that the pitch angle does not exceed 60°. Furthermore, we compare the depth of the sperm head and the neck (where the head and flagellum meet, its depth is determined by the anterior point of the sperm flagellum) to determine the positive and negative pitch angles. If the head is higher than the neck, the pitch angle takes a positive value; otherwise, it takes a negative value. Then, we obtain the 3D coordinates and spatial azimuth of the sperm head and determine the head posture of the sperm, as shown in Fig. 7(b). The combination of 3D trajectory and posture can fully characterize sperm motility and facilitate a more comprehensive assessment of sperm behavior.

FIG. 7.

(a) Principle of the rolling direction determination of the sperm head. (b) Spatial postures of the sperm head in different frames. The gray lines represent the trajectory of sperm movement.

D. Sperm flagellum 3D dynamic reconstruction

Sperm motion is the result of the beat of the sperm flagellum. The front part of the sperm flagellum (the midpiece of sperm) contains spirally arranged mitochondria that produce the energy required for sperm motion. As the propulsive waves initiated at the midpiece are propagated along the sperm flagellum, the sperm is propelled forward, and the sperm head is forced to rotate along the axis of the direction of travel. The sperm flagellum determines sperm motility, and it is more appropriate to focus on the sperm flagellum than on the sperm head to study sperm motility. However, the complex shape and motion of the sperm flagellum are disregarded in sperm observation due to its high beat frequency, low visibility, and 3D nature. To more completely characterize the motility of the sperm, we further reconstructed the sperm flagella during motion.

1. Determination of the sperm flagellum region in the 2D plane

To mitigate the effect of image noise, subtract the background hologram from each frame of the acquired polarization hologram. Then, select the in-focus amplitude image [shown in Fig. 8(a), normalized within 0–255] to determine the sperm flagellum region in the camera plane as the amplitude image has higher contrast than the phase image without phase wrapping. Due to the multilevel reflection at different interfaces, there are apparent stripes in the amplitude image [as shown in Fig. 8(a)]. Figure 8(b) is the log-scale spectrum of the amplitude image. There are two symmetrical ring regions in the low-frequency region [corresponding to the streaks, Fig. 8(c)]. Figure 8(d) shows the filtered amplitude image after filtering out the streaks by constructing a suitable filtering transfer function based on their distribution characteristics in the frequency domain. Following this, the sperm and the background can be roughly distinguished, but binarization cannot distinguish the sperm directly due to low contrast. To segment the sperm contour, we keep only the area near the sperm based on a manual mask, while the other is set to zero. The sperm binary map is then obtained by local adaptive binarization, image opening and closing operations, and manual modification, as shown in Fig. 8(e). Note that the binary map can only cover part of the sperm region for low contrast. Rotate the binary image, so the sperm is horizontal and then crop the background, keeping only the sperm region [as shown in Fig. 8(f) above]. The sperm centerline is obtained based on the principle of intermediate axis transformation [as shown in Fig. 8(f)]. The obtained centerline contains the sperm head, and we need to identify the sperm head and flagellum based on the endpoint (near the flagellum) of the sperm head fitting ellipse. The centerline is taken as the sperm flagellum, and thus, the 2D coordinates of the flagellum are obtained.

FIG. 8.

(a) Sperm amplitude image after 0–255 normalization. (b) The log-scaled spectrum of the amplitude image. (c) Enlarged view of the square region in (b). (d) Amplitude image after the removal of interference streaks. (e) Sperm binary map. (f) The cropped sperm binary map after rotation and the sperm centerline.

2. The depth of each voxel of the flagellum

The depth (Z coordinates) of each voxel of the flagellum is different during beat. Both the amplitude (image grayscale) and width of each voxel of the sperm flagellum image vary with different diffraction distances due to the diffraction effect, as shown in Fig. 9(a). We determined the Z coordinates of each voxel by finding the maximum amplitude within a square region centered at the voxel (a square area of 11 × 11 pixels² perpendicular to the flagellum direction) in the image stack. This image stack is obtained based on the angular spectrum diffraction algorithm with different diffraction distances. When the image is in focus, the image amplitude is maximum, and the depth corresponds to the ideal in-focus plane. The depth of the sperm head is determined directly by the TC criterion. Assume that the depth difference between two adjacent voxels on the flagellum does not exceed 2 μm based on continuity and smoothness considerations. The depth of the former voxel can be obtained within a range of 2 μm from the depth of the latter voxel based on the above focusing method. For the first point, the depth of the sperm head is set as the reference. If the focusing approach is not applicable due to a low signal-to-noise ratio (presence of multiple peaks or no peaks), we take the depth of the previous voxel as the in-focus depth of the current voxel. The in-focus depth of the part flagellum [the centerline determined in Fig. 8(f)] can be located in this way. The image contrast of the latter half of the flagellum is poor due to speckle noise and image artifacts [Fig. 9(a)], which makes the reconstruction results of the latter half discrepant from the actual values. Later, we can use partial coherent light illumination and coated lenses to improve the image. For the spatial smoothness of the flagellum, the depth with different voxels is fitted by a sinusoidal function. The 3D coordinates of a part of the flagellum are shown in Fig. 9(b). Together with the sperm head information, the 3D sperm posture can be more fully reconstructed [Fig. 9(c)]. The sperm flagellum not only produces the energy required for sperm motility but also transmits the propulsive waves generated by the midpiece of sperm. The obtained sperm flagellum posture can represent the sperm motility pattern more realistically and accurately. Based on the sperm motion posture, it is also possible to study the sperm dynamic properties, promising guidance for research on infertility in sperm. The complete sperm motion is shown in movie S3 of the supplementary material.

FIG. 9.

$(a) Images of sperm flagellum at different diffraction distances. The orange box moves along the flagellum from the head to the distal end, finding the ultimate focus at each region. (b) Depth distribution of part voxels of the sperm flagellum, where the color map indicates the recovered depth (z coordinate) (c) Postures of sperm (both the sperm head and flagellum) in space.$

(a) Images of sperm flagellum at different diffraction distances. The orange box moves along the flagellum from the head to the distal end, finding the ultimate focus at each region. (b) Depth distribution of part voxels of the sperm flagellum, where the color map indicates the recovered depth (z coordinate) (c) Postures of sperm (both the sperm head and flagellum) in space.

A pixelated polarization digital holographic microscope system based on a PPC was used to reconstruct the sperm 3D trajectories and the posture of the sperm head and flagellum, which helps to comprehensively and accurately assess the behavioral characteristics of sperm. Due to the limited experimental conditions, only partial sperm flagellum can be reconstructed, and the reconstruction accuracy is degraded. However, the energy for sperm motility is provided by the midpiece (the front part of the sperm flagellum), and the flagellum reconstructed based on our method contains this part. It is adequate for analyzing the beating of the sperm flagellum, which helps to investigate the motility and dynamics characteristics of sperm as they pass through the female reproductive tract and cross the zona pellucida to fertilize the egg cell. If the image quality can be further optimized, the whole sperm can be accurately reconstructed, providing a powerful tool for sperm dynamics analysis and behavioral characterization. The system also has significant potential for 3D dynamic observation of micro-organisms.

IV. CONCLUSION

In this paper, a methodology for sperm 3D dynamic observation based on parallel phase-shifting digital holographic microscopy with a pixelated polarization camera (PPC) is presented. The methodology has the advantages of dynamic reconstruction and high modulation efficiency compared with conventional digital holography. Based on this, we retrieved the 3D trajectory and motion parameters of sperm and reconstructed the sperm head orientation and the thin, highly dynamic flagellum (∼30–40 μm).

From the theoretical derivation, we found that in pixelated-polarized digital holography, the diffraction wave in the hologram corresponds to the Stokes parameter, which indicates that it is possible to reconstruct the object wavefront by Stokes parameter extraction. Among various Stokes parameter reconstruction methods, we adopted the bilinear interpolation reconstruction method to recover the object wavefront, considering the computational accuracy and image noise. Meanwhile, by spectral analysis, we gave the relationship between the PPC sampling interval and objective parameters that should be satisfied to make the system reach the diffraction limit resolution in parallel phase-shifting digital holography.

Furthermore, we built a parallel phase-shifting digital holographic microscope system based on a PPC for sperm observation. The 3D sperm trajectories were extracted by the numerical focusing algorithm and image recognition. The sperm head and parts of the sperm flagellum were reconstructed based on image segmentation and the numerical focusing criterion under the ellipsoidal assumption. The sperm motion parameters can be obtained from the spatial 3D trajectory, which can be used to evaluate sperm activity. The reconstruction of the sperm head and flagellum can be used to analyze the behavioral characteristics of sperm during swim, helping to study the motility and dynamics of sperm as they pass through the female reproductive tract and cross the zona pellucida to fertilize the egg cell. Parallel phase-shifting digital holographic microscopy based on pixelated polarization enables 3D sperm trajectory tracking and pose reconstruction, providing a new and effective methodology for studying sperm motility, which is of great importance for male reproduction research.

SUPPLEMENTARY MATERIAL

In the supplementary material, movements of sperm before and after numerical focusing are given in detail. One of the sperm is close to the focal plane and has a smaller displacement in the z-direction (perpendicular to the camera plane). The other one has a large displacement in the z-direction. The posture of sperm during movement in different views is given as well.

ACKNOWLEDGMENTS

The authors acknowledge the First Affiliate Hospital of Anhui Medical University for providing the samples. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12232017, 12072339, and 11627803) and the University Graduate Scientific Research Program of Anhui Province (Grant No. YJS20210266).

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Chuanbiao Bai: Conceptualization (equal); Investigation (equal); Methodology (equal); Software (lead); Validation (lead); Visualization (lead); Writing-original draft (lead). Zhaoxiang Jiang: Conceptualization (equal); Investigation (equal); Software (supporting). Jiangcheng Zhao: Investigation (supporting); Software (supporting). Shangquan Wu: Conceptualization (equal); Methodology (equal); Supervision (supporting); Writing - review & editing (lead). Qingchuan Zhang: Conceptualization (equal); Funding acquisition (lead); Methodology (equal); Project administration (lead); Supervision (lead); Writing-review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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