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### Optical setup for interferometric phase microscopy

Figure S6 presents the optical setup used in this paper. Light from a helium-neon laser source illuminates the sample in an inverted microscope, composed of a 100× oil-immersion microscope objective (Olympus UPLSAPO 100 × O; numerical aperture, 1.4), and an achromatic tube lens of focal length 150 mm. The resulting sample beam then enters the off-axis external interferometric module (*27*, *28*), where it is split into two beams of equal intensity by a 50:50 beam splitter. The first beam is focused onto a laterally shifted retroreflector mirror, RR1, by the first module achromatic lens, L1, of focal length 100 mm, causing a small shift in the illumination angle on the camera, producing off-axis interference. The second beam exiting the beam splitter is focused by lens L1 onto a 15-μm pinhole placed in the Fourier plane of the lens, thereby removing all the high spatial frequencies containing the sample information and thus creating a clean reference beam. The reference beam is then reflected back to the beam splitter by retroreflector mirror RR2. The two beams then merge in the beam splitter and, after passing through another achromatic lens L2 (focal length, 150 mm) and an additional magnifying 4f system composed of achromatic lenses L3 (focal length, 30 mm) and L4 (focal length, 75 mm), an off-axis image hologram is created on an ultrafast digital camera (FASTCAM Mini AX200, Photron; square pixels of 20 μm each, 1024 × 1024 pixels, 2000 fps). The entire optical system has a total magnification of ×328 and a resolution limit of 452 nm.

### Biological preparation

A semen sample was collected in accordance with Tel Aviv University’s institutional ethical committee, from healthy 18- to 45-year-old donors, after undergoing 24 hours of abstention. After ejaculation, the sample was allowed to liquefy for 30 min. Following this, sperm cells were isolated from the semen fluid using a PureCeption bilayer kit (ORIGIO, Målov, Denmark) in accordance with the manufacturer’s instructions. In short, 0.5 ml of semen was placed on top of a 40 and 80% silicon bead gradient and centrifuged for 25 min at 1750 rpm. After centrifugation at 1250 rpm for 5 min, the supernatant was discarded, and the pellet containing the living sperm cells was washed with 10 ml of HTF (Irvine Scientific, CA, USA). After additional centrifugation, the supernatant was removed, and cells were resuspended in 2 ml of HTF supplemented with 7% PVP, 360,000 molecular weight (mw) (PVP360, Sigma-Aldrich). One milliliter of the cell solution was then placed in a chamber (CoverWell, PC1L-0.5; 32-mm width by 19-mm length by 0.6-mm depth, 1.5-mm diameter ports) for imaging. For the comparison in movie S5, PVP was not added, and after imaging the cells in their natural state, 5 mM caffeine was added.

### Algorithm

*Wavefront propagation*. An interferometric system is able to record the entire wavefront that propagated through a sample, rather than just its intensity, by interfering it with a reference wavefront, yielding a digital hologram or interferogram (*29*). The resultant hologram is given by the following expression∣Es(x,y)+Er∣2=∣Es(x,y)∣2+∣Er∣2+As(x,y)Arexp[jφs(x,y)−jφr]+As(x,y)Arexp[jφr−jφs(x,y)](1)where *j* denotes the imaginary unit, and *E*_{s}(*x*,*y*) = *A*_{s}(*x*,*y*)exp[*j*φ_{s}(*x*,*y*)] and *E*_{r} = *A*_{r}exp[*j*φ_{r}] are the sample and reference complex wavefronts, respectively (the latter assumed to be of constant phase and amplitude). Thus, the wavefront that propagated through the sample is fully conserved, although its extraction remains difficult. In off-axis holography, one of the interfering beams is titled at a small angle relative to the other, creating a linear phase shift that allows separation of the field intensity from the two complex-conjugate wavefront cross-correlation terms in the spatial frequency domain, thus allowing reconstruction of the complex sample wavefront from a single off-axis digital hologram. In this work, the complex wavefront was extracted from the raw hologram in the spatial frequency domain using a Fourier space–filtering algorithm (*30*), which cropped one of the cross-correlation terms, and then applied an inverse Fourier transform to obtain the complex wavefront. For efficiency reasons, the images were left at their cropped dimensions, four times smaller than the original hologram dimensions. Following this, the Rayleigh-Sommerfeld (RS) propagator was used to reconstruct the complex field at various distances from the recorded plane of focus (*31*), a propagation method suitable for weakly scattering objects (such as the sperm cell flagellum) (*32*). We took advantage of the transition to the Fourier space needed for isolating the wavefront and used the Fourier formulation of the Rayleigh-Sommerfeld propagator, simply requiring a pixel-wise multiplication of the cropped cross-correlation term with the following transfer function (*31*)HRS(u,v,z=d)={exp[j2πnmdλ1−(λu)2−(λv)2],foru2+v2<1λ0, otherwise(2)where *d* is the distance to be propagated, *n*_{m} is the RI of the surrounding medium, *j* is the imaginary unit, and λ is the illumination wavelength. Thus, the wavefront propagation may be regarded as a linear, dispersive, spatial filter with a finite bandwidth. The refocused wavefront at *z* = *d* can then be obtained by applying an inverse 2-D Fourier transform on the result. The spacing between the different reconstruction planes was chosen to be exactly the effective pixel size of the image (which was identical for horizontal and vertical coordinates), resulting in a pseudo-volume with isotropic sampling frequency. The boundaries of the propagation distance can be determined by the maximal depth allowed by the physical restriction used in the experiment. To prevent excessive memory use, a propagation distance of 16 μm is used in this paper, taking into consideration that the human flagellum is approximately 60 μm long, assuming that the sperm is not swimming toward the camera.

*Handling high noise levels*. One of the greatest challenges of tracing the 3-D location of the flagellum per each frame is finding the pixels associated with the tail in the 2-D image, i.e., performing 2-D segmentation. This is a complex task due to the low-phase values of the flagellum, which are similar to the noise level when using a coherent light source, especially for out-of-focus segments. We thus constructed a targeted, adaptable, locally conserving cleaning (TALCC) function, which identifies the important parts of the image that need to be locally conserved. What makes this function highly targeted and adaptable is its multiple inputs, including not only the original phase image but also the current binary segmentation map, row and column of interest, slope in region of interest, and noise level mode. The noise level mode, adapted automatically in our analysis, starts at a value of 0, indicating a neutral input, and can take higher positive values for dealing with increasingly higher noise levels that require robust cleaning or lower negative values for dealing with thin, low-contrast segments that need delicate cleaning. The row and column of interest, together with the slope in region of interest, are used to define a 3 × 6 pixel environment around the row and column of interest that needs to be preserved, defined as 1 pixel to each side, 4 pixels forward and 1 pixel backward, rotated in the direction of the slope. The input phase map is first thresholded to obtain a binary image, with a threshold value respective to the noise level mode; then, both the pixel group defined by the 3 × 6 pixel environment and the pixel group marked as the object in the current (and previous frame, if available) segmentation map are marked as the object in the binary image. At this point, morphological opening and closing (*33*) are subsequently applied to clean noise and close holes, respectively, with parameters according to the automatically detected noise level mode. The structuring elements used are lines with a slope either parallel or orthogonal to that of the region of interest, thus preserving segments with similar direction and erasing others. In the case of low-contrast segments, indicated by a negative noise level mode, the closing operation is applied before the cleaning operation, and the structuring elements used are disk shaped.

*Seed points*. To aid the segmentation process in the low signal-to-noise-ratio conditions, we find a good (*x*,*y*,*z*)…

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