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A method to measure the orientations of transparent ellipsoidal particles using digital holographic microscopy (DHM) is proposed in this study. This approach includes volumetric recording and numerical reconstruction at different depths. Distinctive light scatterings from an ellipsoid with different angles of orientation are analyzed. A focus function is applied to obtain a reconstructed image that contains a bright line parallel to the major axis of the projected particle, which provides in-plane orientation information. An intensity profile is collected along the major axis of the projected particle in the direction of the optical axis, and this profile is then utilized to measure the out-of-plane orientation of the ellipsoid. After being verified for an ellipsoid with known orientations, the proposed method is applied to ellipsoids suspended in a pipe flow with random orientations. This DHM method can extract the essential information of ellipsoids and therefore has great potential applications in particle dynamics.
© 2016 Optical Society of America
1. Introduction
Most microscale particles or cells in nature and in industrial applications are in ellipsoidal in shape rather than being simple spheres. As such, the motion of ellipsoidal particles has been a great interest to researchers, who continue to investigate the effects of ellipsoids on fluid flow, as well as their motion. In addition, with the advances in microfluidics, such as lab-on-a-chip devices, the motion of an ellipsoid is gaining considerable attention for both scientific research and engineering applications. Since Jeffry [1] and Hinch and Leal [2] analytically investigated the motion of ellipsoids, numerous numerical and experimental studies have been conducted [3–8]. However, unlike numerical studies, experimental studies on ellipsoidal particles are seldom performed because the three dimensional (3D) location and orientation of each particle are difficult to measure. In general, at least two angles of observation are required to calculate the orientation of non-spherical particles. This stereoscopic observation is utilized because of its relatively simple concept [9–11]. Nonetheless, given the shallow depth of focus in optical imaging, the observable volume is restricted. Computed tomography (CT) method has also been utilized to measure the geometric parameters of ellipsoids [12–14]. However, this approach is unsuitable for real-time dynamic motion analysis because tomography requires multiple images and their reconstruction.
Various studies have utilized digital holographic microscopy (DHM) for micro-scale flow phenomena because this technique can reconstruct 3D volume information with a single shot of a hologram [15–19]. After volumetric recording, particles spread in various depth-wise locations can be precisely obtained by changing the reconstruction depth, thereby allowing for a deep observable depth. This advantage overcomes the weakness of conventional optical microscopy. Moreover, the numerical reconstruction procedure guarantees various quantitative analyses by adopting post-image processing techniques. In addition to these advantages, the experimental setup of DHM, particularly in-line-type DHM, is relatively simple compared with that of other 3D imaging techniques such as CT or stereoscopic method.
To utilize these advantages, the orientation measurement has been tried with DHM. Various previous studies can be classified into two groups: direct and indirect methods. Indirect method uses the hologram itself rather than a reconstructed image. Wang et al. [20] utilized the inverse-scattering analysis technique introduced by Lee et al. [21]. They obtained the best-fitted hologram by comparing a simulated hologram with an experimentally obtained one. Indirect method provides not only the size and location information of wavelength-scale particles but also their orientation. However, the generation of rotation libraries of micro-scale particles in the simulated hologram is a time-consuming process.
In the direct method, information is deduced through the use of reconstructed images and digital image processing techniques. Tthe processing time depends only on the reconstruction process, and the image analysis is intuitive. In the direct method, skeletonization method is employed to measure the size and orientation of opaque rod-shaped particles [22]. Nano-scale rod-shaped particles were analyzed in the similar way [23]. In previous studies, edge detection method has been frequently employed to find the best focus plane or boundary brightness jump. However, edge detection method is applicable only to limited cases where in the edge of the test samples clearly appears in the reconstructed image, such as in opaque particles or some bio-samples [15, 24]. Due to the lens effect of a transparent particle, it is hard to detect the edge of transparent particles such as polystyrene particles or biological cells whose interiors are filled with fluid. On the other hand, the lens effect provides distinctive optical features. We previously used these distinctive features to obtain the 3D information of ellipsoids [25, 26].
In the present study, in-line DHM technique is used to measure the two orientations of a transparent ellipsoid. Figure 1 shows the two orientations measured in this study. The first is the in-plane orientation, which is the angle between the x-axis and the major axis of the projected ellipsoid. The other is the out-of-plane orientation,which is the angle between the xy plane and the major axis of the ellipsoid. Out-of-plane orientation is usually disregarded in observations with a conventional 2D camera. This paper suggests a new method of measuring the two orientations of an ellipsoid by analyzing the distinctive light scattering features in the image reconstructed by DHM technique. The proposed method is verified for an ellipsoid with known orientations. Moreover, the potential application of the proposed method is demonstrated by applying it to randomly oriented ellipsoids in a pipe flow. This technique would be useful for analyzing the dynamic behaviors of ellipsoidal particles and ellipsoid-shaped bio-cells.
2. Methods and materials
2.1 Sample preparation
Ellipsoidal particles were fabricated in accordance with a previously known protocol [27]. 10% fully hydrolyzed poly vinyl alcohol (PVA, Sigma-Aldrich) was dissolved in water at 75°C. Polystyrene spherical particles (diameter: 15μm, Polysciences, Inc.) were suspended in the solution at a concentration of 0.04%wt./vol. 1% glycerol was added to plasticize the films. The solution became a film by drying on a flat surface at room temperature (25°C) for 12h. The film was stretched by a custom-made device in oil at 120°C. The stretched film was dissolved in 30% isopropanol water solution at 75°C. The particles generated from the dissolved film were washed using the same solution at least five times to remove PVA residue on the particle surface. The sizes of the longest and the shortest axes of the particle were 24 and 12μm, respectively.
The fabricated particles were suspended in polydimethylsiloxane (PDMS, Sylgard 184, Dow corning) and cured at 80°C for 3h. After the location of the particles is checked, a small PDMS block (with a thickness of 1 mm) containing the suspended particles was cut out. Given that the ellipsoidal particle was fixed inside the PDMS block, its orientations can be easily controlled with the use of the outer rotating devices. Because the ellipsoid embedded in a PDMS block has random orientation, the orientation of the ellipsoid should be calibrated. For this, the sizes of the major and minor of the ellipsoid are measured using an optical microscopy. At zero degree, the ellipsoid lies horizontally and the measured sizes should be the same as the known dimensions. Therefore, the angle at which the measured sizes become the same as the known dimensions is defined as zero degree.
A refractive index (RI) matching solution [28, 29] was used to avoid any unwanted optical distortion introduced by different refractive indices of the PDMS and the surrounding fluid. By carefully adding sodium iodide (Samchun, Korea) in water by 70%wt./wt., the RI of the solution became 1.414, which was close to that of PDMS (1.413). The RI matching solution was filled to the acrylic box where the particle-embedded PDMS block was located. The optical distortion at the edge of the PDMS block was significantly reduced in this experimental setup.
2.2 Hologram recording and numerical reconstruction
Figure 2 illustrates the experimental setup used in this study. The laser beam (λ = 532 nm, 100mW; Crystal Laser, USA) was spatially filtered and collimated to form an on-axis reference wave. The particle-embedded PDMS was connected to the rotating device with an angular precision of 1°. The area around the PDMS block was filled with a RI matching solution. A water-immersion microscopic objective lens (40x, Nikon, Japan) was attached to magnify the hologram of the particle. The focal plane of the objective lens was set to 80μm above the exact location of the test particle. The hologram was recorded with a high-speed charge-coupled device camera (Ultima-APX, Photron, Japan). The corresponding pixel resolution (Δ) of the camera was 0.425μm in the hologram plane. Holograms were acquired by rotating the PDMS block from 0° to 90° at intervals of 10°.
Before the hologram was reconstructed, bicubic interpolation method was applied to eachhologram image to obtain fine detail information. Interpolation method causes an image to consist of more points, thereby allowing for subpixel analysis to be performed. In this way, more precise information about light scattering is obtained from the particles.
A hologram image was numerically reconstructed by employing the Fresnel-Kirchhoff diffraction formula. This diffraction formula consists of the convolution of the hologram function and the diffraction kernel. The following convolution theorem was employed in the reconstruction process:
where F and F−1 represent the fast Fourier transform (FFT) and the inverse FFT, respectively; x and y denote the spatial coordinates in the hologram plane, and and correspond to that of the reconstructed plane. Angular spectrum method was used to obtain the FFT of the diffraction kernel as follows [30]:where and denote the spectral coordinates, d is the distance between the hologram and the reconstructed planes, and is the wavelength of laser light. Detailed information on the DHM and the reconstruction method adopted in the present study can be found in previous studies [16, 17, 31].3. Results and discussion
3.1 Interpolation of the hologram
Bicubic interpolation method was applied to the captured raw holograms. This approach allowed for sub-pixel analysis to be performed, thereby overcoming the limitations of the conventional experimental setup. Before interpolation method was applied, the hologramsexhibited rough patterns, and the light intensity profiles along the major axis of the projected ellipsoid in the direction of the optical axis contained sparse information, as shown in Figs. 3(a) and 3(c). Given that the light intensity profile is obtained from the reconstructed hologram images, the density of the information in the hologram image is directly related to the intensity profile. Moreover, the sparse information in the intensity profile leads to a rough measurement of the out-of-plane orientation of the ellipsoid because the intensity profile is utilized in its measurement. Therefore, a rough raw hologram image causes a rough measurement of out-of-plane orientation. After interpolation method is applied, a hologram image has more points and therefore contains detailed patterns. Consequently, the intensity profile has dense information, and the out-of-plane orientation can be more precisely measured. Figures 3(b) and 3(d) show that the hologram image and the intensity profile have become densely formed after interpolation, and that they contain more detailed information compared with those before interpolation method was applied.
Fig. 3 Hologram images of an ellipsoid before the interpolation (a), after the interpolation (b). Intensity profile along the major axis of the projected ellipsoid in the direction of the optic axis before the interpolation (c), after the interpolation (d). Scale bar is 5μm.
3.2 Reconstructed image and measurement of in-plane orientation
The images reconstructed from the captured holograms of ellipsoids exhibit distinctive light scattering information. Figures 4(b)–4(d) depict the raw hologram and the reconstructed images at different depths. In Fig. 4(d), a bright line appears along the major axis of the ellipsoid. This bright line is induced by the lens effect of a transparent particle [26, 32]. A laser beam is focused onto the focal point when a transparent particle works as an optical lens. As a result, a bright point appears on the focal length away from the particle location. Given the elongated shape of an ellipsoid, the bright points become connected and appear as a brightline. Owing to its distinctive brightness, the location of the bright line can be detected quantitatively by applying a focus function to the reconstructed images. The autofocus function VAR(z) is evaluated from the intensity as follows:
where I represents the intensity distribution in the segmented reconstructed image, and Nx and Ny denote the image dimensions. Figure 4(e) shows a typical variation of the focus function value. Two peaks are clearly observed. Depending on the order of appearance, the two distinct peaks are named the first and second peaks. The reconstructed images corresponding to the two peaks have bright points that are located along the minor and major axes of the ellipsoid, respectively. The bright points at the second peak are located closely, and the line connecting them clearly appears, unlike those at the first peak.Fig. 4 (a) Optical image of an ellipsoid tested in the study. (b) Hologram image of the same ellipsoid. (c-d) Reconstructed images at different reconstruction depths of z = 21.5, 42μm, at which the focus value profile has clear peaks (e). Scale bar is 10μm.
The in-plane orientation of the ellipsoidal particle is directly measured by using the reconstructed image at the second peak. Figure 5(a) shows a magnified reconstructed image atthe second peak. By using the global image threshold method developed by Otsu [33], the image is converted into a binary image that contains only the information of bright points, as shown in Fig. 5(b). The in-plane orientation () of the ellipsoid is determined by measuring the angle between the horizontal axis and the major axis of the ellipse that contains these bright points.
Fig. 5 Experimental procedure to measure in-plane orientation. (a) Reconstructed image with a bright line. (b) Binary image obtained by applying threshold method. (c) Measurement of in-plane orientation () using ellipse fitting. Scale bar is 10μm.
3.3 Light scattering from an ellipsoid particle and measurement of out-of-plane orientation
The distinctive light scattering from an ellipsoid particle is well depicted in the intensity profile along the major axis of the projected ellipsoid in the z-direction. Figures 6(a)–6(f) show the intensity profiles from the ellipsoid positioned at various out-of-plane orientations. When the ellipsoidal particle lies horizontally (), the intensity distribution is symmetric about the vertical center line. As the angle increases, the intensity distribution of bright spots becomes gradually skewed. In addition, the distance between the left-end and right-end bright points is decreased. The skewness of intensity profiles can be explained by the following optical principle. Because the sliced planes of an ellipsoid are circles with differentdiameters, the focal lengths caused by these circle are different as depicted in Fig. 7. However, there are two circles whose focal lengths are the same because ellipsoid is symmetric with respect to the minor axis. When a particle rotates about the minor axis of the ellipsoid, the locations of a specific circle and corresponding bright point are elevated. At the same time, the locations of the other circle lens located at the opposite side (with the same focal length)and corresponding bright point are dropped down with equivalent scale. Therefore, as a particle rotates, the locations of bright points which lie symmetrically about the centerline move toward the opposite z-directions in the intensity profile. For the case of a vertically oriented particle (), the intensity distribution is also symmetric; however, the bright points are located in a narrower region compared to those of a horizontally orientated particle ().
Fig. 6 Variations of intensity profiles along the major axis of the projected ellipsoid at various tilted orientations.
Fig. 7 Variations of measured out-of-plane orientation according to tilted angles for different threshold values.
The out-of-plane orientation is determined from the intensity profile. The skewness of the intensity profile is quantitatively evaluated by applying the global threshold method to the intensity profile image. When the intensity value at a point is higher than a certain threshold value determined by the maximum value in a profile image, it is selected as the bright point. Among those bright points, the bottom points of each column are collected. A polynomial curve of the bottom bright points is obtained by searching the curve with the lowest coefficient of determination (R2). Then, the angle between the line connecting the left-end and right-end points of the polynomial curve with respect to the horizontal axis is measured as the out-of-plane orientation () of the ellipsoid. Figure 8 demonstrates the measurement of the out-of-plane orientation of an ellipsoid with a tilted angle of 30°. The measured out-of-plane orientations of the ellipsoid are summarized in Table 1. The proposed DHM method can measure particle orientation with error of less than 4°. For a vertically orientated particle, the proposed method is difficult to be applied because the two side-end points have a flawed angle of zero as a result of the symmetry of the intensity profile. Therefore, an exceptional treatment is required. If the distance between the bottom end points is shorter than the effective pixel size of the experimental setup, the out-of-plane orientation is considered as 90°.
Fig. 8 Experimental procedure to measure out-of-plane orientation. (a) Intensity profile along the major axis of the projected ellipsoid with a tilted angle of 30°. (b) Selected points whose intensity is higher than the threshold value are depicted as black dots. A fitted curve to measured the out-of-plane orientation () is colored with red.
Table 1. Measured out-of-plane orientation and its error according to tilted angle
The intensity value used to determine the out-of-plane orientation depends on the given experiment conditions. To apply the proposed method to another case with different experimental conditions, a normalized value is used in the analysis of the intensity profile. However, threshold value is still subjectively determined. If a threshold value is too low or high, many or few number of points are involved in the orientation measurement because the threshold value determines the number of points used to measure the out-of-plane orientation. Therefore, the selection of adequate threshold value is important and the threshold value of 0.73 works successfully. A robust angle measurement should have a negligible dependency on the threshold value. Thus, the effect of the threshold value on the measured orientation was investigated for varying threshold values. Figure 9 shows the variation of the measured out-of-plane orientations according to the tilted angles for different threshold values. The standard deviations in the measured orientations are less than 2°.
Fig. 9 Variations of measured out-of-plane orientation according to tilted angles for different threshold values.
3.4 Application to suspended ellipsoids in a pipe flow
The proposed method was applied to suspended ellipsoids in a pipe flow. For this experimental demonstration, the same laser source and objective lens were used. Ellipsoid particles with an aspect ratio of 2 were suspended in water. The suspended ellipsoids were infused in a transparent fluorinated ethylene propylene (FEP, n = 1.338) microtube of 200 μm in diameter. This FEP tube was used to minimize the optical distortion caused by RI mismatch. Flow rate was adjusted to maintain a Reynolds number of 3. Holograms were consecutively acquired at a frame rate of 1,000 fps with an exposure time of 15μs. The focal plane of the objective lens was adjusted to 150μm above the center plane of the tube. Figure 10 shows the images captured with the proposed method for a suspended ellipsoid in the pipe flow. Table 2 summarizes the measured orientations of the suspended particles. The measured orientations clearly show that the ellipsoid rotates in the pipe flow. This demonstrational application supports that the proposed method is useful for measuring the 3D orientations of randomly suspended particles in streaming conduit flows.
Fig. 10 (a) Consecutive holograms of a rotating ellipsoid in a pipe flow. (b) Reconstructed images used to measure in-plane orientation. (c) Intensity profiles along the major axis of the projected ellipsoid. (d) 3D perspectives of the randomly oriented ellipsoid, reconstructed by measurement of in-plane and out-of-plane orientations. Scale bar is 10μm.
Table 2. Measured in-plane and out-of-plane orientations of suspended ellipsoids in a pipe flow
The proposed method utilizes the advantages of the DHM method that measures 3D volumetric information. Without multiple observations or several view angles, the proposed method precisely measures the in-plane and out-of-plane orientations of ellipsoidal particles. Combined with the previously developed method for measuring 3D positional information of ellipsoids [26], the proposed method can be used to analyze the dynamic behaviors of ellipsoidal particles under various conditions. In addition, as long as particles are in ellipsoidal in shape, the proposed method does not require prior size information to measure their orientations. Therefore, the present results would be helpful for experimental analysis of 3D motions of ellipsoid-shaped particles or biological cells. Due to intrinsic limitation of DHM technique, the maximum number of particles in a unit volume of the captured hologram is limited [34, 35]. Therefore, the proposed method is ideal for dynamic analysis of a single particle although it is applicable to more particles with low number density.
4. Conclusion
A new DHM method for measuring the in-plane and out-of-plane orientations of transparent ellipsoids is proposed. Distinctive light scatterings and intensity profiles are obtained along the major axis of the projected ellipsoid at different orientations. The intensity profile of a tilted ellipsoid is skewed. The skewness is evaluated to obtain the orientation of the ellipsoid. The potential of the proposed method is confirmed by measuring the orientations of an ellipsoid with prior known orientation and subsequently suspended ellipsoids with random orientations in a pipe flow. Utilizing the advantages of DHM technique, including volumetric recording and numerical reconstruction, the proposed DHM method overcomes the technical limitations of short measurement depth and multiple view observations encountered in previous methods. Therefore, the proposed method has a strong potential application in the dynamic analysis of ellipsoid-shaped particles or biological cells.
Acknowledgment
This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government (MSIP) (No. 2008-0061991).
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