3.1.

Three-Dimensional Localization of Suspended Microspheres

In combination with microscopy, DHM is able to produce single hologram containing all the information about the three-dimensional structure of micro-scale objects. In a first experiment, the applicability of DHM and autofocusing algorithm is investigated by performing three-dimensional profiling of polymer microspheres (Type 7510A, Duke Scientific Corporation, Palo Alto, California, mean diameter 9.6 μm). Figure 2(a) presents the hologram of microspheres in suspension captured by DHM. Angular spectrum method^1,2 is applied and the reconstructed amplitude image is shown in Fig. 2(b) where several in-focus and out-of-focus microspheres are seen at this reconstruction plane. The field of view is $90\times 90{\mu \mathrm{m}}^2$ with $464\times 464\text{pixels}$. The microsphere focuses the incoming light and forms a bright spot along the optical path. Therefore, the pattern of the microsphere appears to have a maximum intensity at the center near the in-focus plane and the surrounding is dark. Then, an autofocusing algorithm based on peak searching can be applied to identify the in-focus position of each particle.

Fig. 2

Three-dimensional profiling of suspended microspheres. Field of view of (a)–(d) is $90\times 90{\mu \mathrm{m}}^2$ with $464\times 464\text{pixels}$. (a) hologram; (b) amplitude image; (c) all-in-focus intensity profile; (d) depth position profile; (e) three-dimensional profile ($90\times 90\times 120{\mu \mathrm{m}}^3$) and grey-scale representation of intensity; (f) $XY$ view of three-dimensional profile ($90\times 90{\mu \mathrm{m}}^2$); and (g) $XZ$ view of three-dimensional.

The hologram is numerically reconstructed in longitudinal direction $Z$ from 100 μm to $-100\mu \mathrm{m}$, in $-2\mu \mathrm{m}$ steps. The autofocusing algorithm is then applied by searching on the reconstructed images for the planes that contain the objects with the peak and sharpest details to establish the all-in-focus profile.²⁰ At each pixel, the intensity variation in longitudinal direction is investigated to find out the peak intensity. The peak intensity value becomes a pixel of an in-focus profile and its location in longitudinal direction is recorded as a corresponding depth map value. The combination of all-in-focus intensity profiles Fig. 2(c) and the corresponding depth maps Fig. 2(d) enables to produce the three-dimensional visualization of microspheres and a threshold on the intensity allows for distinguishing objects from other elements, illustrated in Fig. 2(e). In fact, we take every center position of brightest points in Fig. 2(c) as $X\text{-}Y$ locations of particles and Fig. 2(d) is used to determine the $Z$-location of those particles identified in Fig. 2(c). Figure 2(d) is mostly noisy and $Z$ information of only the locations where the particles are present is actually used. Figures 2(f)–2(h) show the $XY$, $XZ$, and $YZ$ views of the three-dimensional profile, respectively. This straightforward autofocusing algorithm is able to determine focal planes for all the objects in the reconstructed volume automatically. The combination of depth map $Z$ and axial $X\text{-}Y$ position of objects allows for quantitative three-dimensional profiling.

The accuracy of this measurement system was estimated using a microsphere (2 μm) fixed on a glass slide.²¹ A series of holograms of the fixed particle was recorded at 10 fps, up to 8 s. Figure 3(a) is one hologram of the fixed particle and Fig. 3(b) shows the temporal changes for the measured $Z$ positions of the particle in a step of 0. 005 μm. The standard deviation of the measured $Z$ positions is estimated to be 0.039 μm, which is the depth accuracy of the system.

Fig. 3

Depth accuracy estimation of a fix particle (2 μm). (a) Hologram of the fix particle; (b) Temporal plot of in-focus $Z$ positions of the particle.

3.2.

Three-Dimensional Profiling of Stationary Microfibers

Nonwoven and randomly oriented microfibers were spun onto glass coverslips from co-poly ($\mathrm{L}\text{-}{\text{gluatmic acid}}_4$, $\mathrm{L}\text{-}{\mathrm{tyrosine}}_1$) (PLEY) dissolved in water and crosslinked as described previously.²² The diameter of fibers is 0.2–5.0 μm. Figure 4(a) presents a hologram of microfiber. The angular spectrum method is applied to obtain the reconstructed amplitude image in Fig. 4(b). The microfiber appears curved and elongated, consistent with our previous work.²² The field of view is $90\times 90{\mu \mathrm{m}}^2$ with $464\times 464\text{pixels}$. The hologram is then reconstructed from $Z=100$ to $-100\mu \mathrm{m}$, in $-1\mu \mathrm{m}$ steps. Adding these 201 reconstructed images of amplitude yields the axial projection shown in Fig. 4(c). The final image has the same number of pixels as each reconstructed image. The projected image is converted into binary by threshold and segmentation, shown in Fig. 4(d). The binary image is then axially back-projected along the path of amplitude over the set of reconstructed images.

Fig. 4

Three-dimensional profiling of stationary microfibers. The field of view of (a)–(f) is $90\times 90{\mu \mathrm{m}}^2$ with $464\times 464\text{pixels}$. (a) hologram; (b) amplitude image; (c) projection image; (d) binary image, microfiber 1 and background 0; (e) all-in-focus intensity profile; (f) depth position profile; (g) three-dimensional profile ($90\times 90\times 35{\mu \mathrm{m}}^3$) and gray-scale representation of intensity; (h) three-dimensional fit ($90\times 90\times 35{\mu \mathrm{m}}^3$); (i) $XY$ view of three-dimensional fit ($90\times 90{\mu \mathrm{m}}^2$); (j) $XZ$ view of three-dimensional fit ($90\times 35{\mu \mathrm{m}}^2$); and (k) $YZ$ view of three-dimensional fit ($90\times 35{\mu \mathrm{m}}^2$).

We apply the same autofocusing algorithm on the resultant images to record the maximum intensity value at each pixel along the reconstruction direction and the axial $XY$ and depth position $Z$ of where the maximum intensity occurred at the same time. The obtained all-in-focus intensity profile Fig. 4(e) and the corresponding depth map Fig. 4(f) are then combined to generate the three-dimensional visualization of microfiber and a threshold on intensity is chosen to highlight the area of regional interest. However, there still exist noisy point clouds resulted from noise in the three-dimensional profile Fig. 4(g). To address this, an average and polynomial curve-fitting algorithm accounting for errors is adopted. The fitted line of the point clouds (standard deviations in the $X$ and $Z$ directions are 0.21 and 0.06 μm) in the three-dimensional coordinate is present in Fig. 4(h). Figures 4(i)–4(k) show the $XY$, $XZ$, and $YZ$ views of the three-dimensional profile, respectively. $Z$ coordinate provides the depth information of real image of microfiber in the reconstruction volume. The length of the microfiber in three-dimensional volume is determined from $X$, $Y$, and $Z$ coordinates of real image of microfiber, that is 126 μm.

3.3.

Four-Dimensional Motility Tracking of a Moving Chilomonas Cell

Chilomonas are fast-moving cells, which sense and respond to their surroundings by swimming towards or away from stimuli. A time-lapse hologram movie of the movement of chilomonas was recorded at 30 fps (Video 1). The field of view is $90\times 90{\mu \mathrm{m}}^2$ with $464\times 464\text{pixels}$. An excerpt of 14 frames is taken and the difference between consecutive hologram pairs is calculated pixel by pixel $(h_1-h_2,h_3-h_4,\dots h_{n-1}-h_n)$ to eliminate background structure and retain only the object information, where $h_n$ is the $n$’th hologram of selected frames. The resulting seven different holograms (from a total of 14) are then summed $(h_2-h_1+h_4-h_3\dots +h_{13}-h_{14})$ into a single hologram Fig. 5(a), which contains all the information on moving cells.¹⁶ Reconstruction by the angular spectrum method is applied on the resultant hologram and Fig. 5(b) is the amplitude image reconstructed at the specific plane $Z=0$.

Fig. 5

Four-dimensional tracking of a moving chilomonas cell (Video 1, MOV, 188 KB) [URL: http://dx.doi.org/10.1117/1.JBO.19.4.045001.1]. The field of view of (a)–(d) is $90\times 90{\mu \mathrm{m}}^2$ with $464\times 464\text{pixels}$. (a) difference hologram; (b) amplitude image; (c) all-in-focus intensity profile; (d) depth position profile; (e) three-dimensional profile of the trajectory of the cell ($90\times 90\times 90{\mu \mathrm{m}}^2$) and gray-scale representation of intensity. (f) $XY$ view of three-dimensional trajectory ($90\times 90{\mu \mathrm{m}}^2$); (g) $XZ$ view of three-dimensional trajectory ($90\times 200{\mu \mathrm{m}}^2$); and (h) $YZ$ view of three-dimensional trajectory ($90\times 90{\mu \mathrm{m}}^2$).

The three-dimensional trajectory of the moving chilomonas was built up by firstly reconstructing the two-dimensional resultant hologram from $Z=80$ to $-120\mu \mathrm{m}$, in $-2\mu \mathrm{m}$ steps. The same algorithm of numerical autofocusing is then utilized, keeping the maximum intensity value at each pixel in the reconstructed image along $Z$ direction to obtain the all-in-focus intensity projection Fig. 5(c), and combining with the depth position where in-focus intensity occurred Fig. 5(d) to determine the focal planes for the entire trajectory in the reconstructed volume. A threshold of intensity is applied to eliminate the unnecessary background, and the center position and the mean intensity of every points cloud representing every trajectory of chilomonas are determined. The combination of depth positions $Z$ and $XY$ displacement of cells allows for quantitative three-dimensional motility tracking. The cell is estimated to move a total path length of 93 μm at the velocity of $198\mu \mathrm{m}/\mathrm{s}$. Figure 5(e) demonstrates the applicability of DHM for automatically four-dimensional tracking of living cells with temporal and spatial resolution at the subsecond and microlevel. Figures 5(f)–5(h) show the $XY$, $XZ$, and $YZ$ views of the three-dimensional trajectory, respectively. It is worth mentioning that cells can change their shape when they are moving, so one has to define the positions of cells to perform the cell tracking. “Center of mass” can be a reasonable definition of the position of cells, and morphology changes and center of mass methods have been discussed in the literatures, such as (Ref. 23) where the cells reported had a size of hundreds of micron. On the other hand, in our case, the chilomonas cell is relatively small and has a more rigid shape, changing of shape is not an issue here.

3.4.

Three-Dimensional Displacement of Microfiber by Swimming Paramecium

A hologram movie of paramecium moving through a microfibers mesh was recorded at 17 fps (Video 2). A cell is seen to swim through fibers and cause obvious displacement of fibers. Two hologram frames showing cell approaching to fibers Fig. 6(a) and swimming away after pulling fibers Fig. 6(b) are extracted. The field of view is $200\times 200{\mu \mathrm{m}}^2$ with $768\times 768\text{pixels}$. The different hologram Fig. 6(c) clearly shows the displacement of the fiber due to the movement of cell and the trajectory of cell. The reconstructed amplitude images from angular spectrum method at $Z=0$ and $Z=-8\mu \mathrm{m}$ are shown in Figs. 6(d) and 6(e), respectively. Different parts of the fiber appear to be in focus at different reconstructed planes. Autofocusing algorithm at the reconstruction volume from $Z=60\mu \mathrm{m}$ to $-70\mu \mathrm{m}$ is applied and the all-in-focus intensity and depth position information are obtained in Figs. 6(f) and 6(g).

Fig. 6

Three-dimensional displacement of microfiber by swimming paramecium (Video 2, MOV, 384 KB) [URL: http://dx.doi.org/10.1117/1.JBO.19.4.045001.2]. The field of view of (a)–(h) and (j) is $200\times 200{\mu \mathrm{m}}^2$ with $768\times 768\text{pixels}$. (a) hologram 1; (b) hologram 2; (c) difference of holograms 1 and 2; (d) amplitude reconstructed at $Z=0$; (e) amplitude reconstructed at $Z=-8\mu \mathrm{m}$; (f) all-in-focus intensity profile; (g) depth position profile; (h) all-in-focus intensity of the first track of the displacement of fiber; (i) three-dimensional fitted line of (h) ($200\times 200\times 20{\mu \mathrm{m}}^3$); (j) All-in-focus intensity of the second track of the displacement of fiber; (k) three-dimensional fitted line of (j) ($200\times 200\times 20{\mu \mathrm{m}}^3$); (l) three-dimensional fitted line of the displacement of fiber ($200\times 200\times 20{\mu \mathrm{m}}^3$) (Video 3, MOV, 64 KB) [URL: http://dx.doi.org/10.1117/1.JBO.19.4.045001.3]); (m) $XY$ view of (l) ($200\times 200{\mu \mathrm{m}}^2$); (n) $XZ$ view of (l) ($200\times 20{\mu \mathrm{m}}^2$); and (o) $YZ$ view of (l) ($200\times 20{\mu \mathrm{m}}^2$). For better visualization, the scales of $X\text{-}Y$ and $Z$ coordinates are not in ratio.

To best visualize the displacement of the fiber in three-dimensional volume, we analyze the two tracks of the displacement of the fiber independently. For brevity, most of the following descriptions refer to the first track. Numerical segmentation and thresholds are applied to Fig. 6(f) and the all-in-focus intensity image containing only one track of fiber is generated in Fig. 6(h). Combined with the obtained depth position profile Fig. 6(g) and fitting algorithm accounting for errors, the fitted line of the point clouds (standard deviations in $X$ and $Z$ directions are 1.27 and 0.27 μm) in three-dimensional coordinate is present in Fig. 6(i). Similar results for the second track of fiber are shown in Figs. 6(j) and 6(k) and the standard deviation of its corresponding point clouds in the $X$ and $Z$ directions are estimated to be 0.80 and 0.26 μm. The two tracks of the fiber are then plotted in one three-dimensional coordinate, Fig. 6(l) and Figs. 6(m)–6(o) show the $XY$, $XZ$, and $YZ$ views of the three-dimensional plot, respectively. The length and the displacement of the fiber are estimated to be 210.1 and 11.1 μm. A movie of three-dimensional displacement of the fiber within 0.5 s period is also presented (Video 3). The three-dimensional displacement of an individual microfiber due to interaction with paramecium cell has been measured as a function of time at subsecond and micrometer level.

To the best of our knowledge, this is the first quantitative profiling and tracking by DHM of the curvature and displacement of individual microfiber by swimming cells in three dimensions. Displacement could vary with cell type, physiological state of the cells, microfiber preparation, etc. The prospects seem excellent for DHM for particle flow analysis and three-dimensional imaging of randomly oriented microfibers, in cases where quantitative measurement of characteristics (size, length, orientation, speed, and displacement, etc.) is of great interest.