1. Introduction

Digital holographic particle image velocimetry (DHPIV) can measure 3D velocity fields of complex flow phenomena [1–5]. In DHPIV, holograms of tracer particles seeded in a flow are directly recorded by a digital camera. A pair of holograms or consecutive holograms is recorded, and then the holograms are numerically reconstructed to retrieve 3D particle fields to calculate 3D velocity field information. In-line digital holography combined with optical microscopy, which is called digital holographic microscopy (DHM), has been widely used to measure various micro-scale flow phenomena [4, 6–9]. The basic concept of DHM is to magnify a hologram using an optical lens system to identify microscopic fringes with high spatial resolution. The DHM technique is used to measure 3D velocity fields of micro-scale flows [5, 10].

Holographic microscopy can effectively measure 3D positional information of particles in a volume. However, as an inherent problem, the measurement accuracy of the depth-directional component is inferior compared with that of in-plane components. The in-plane resolution and measurement accuracy can approach the diffraction limit by using an objective lens with high numerical aperture (NA). However, the depth-directional accuracy reaches several times of that of the particle diameter. Inaccuracy in the measurement of the depth-wise displacement (Δz) of a tracer particle strongly influences the uncertainty of depth-directional velocity in DHPIV, especially when the time interval (Δt) between two consecutive holograms is very short.

Several methods have been proposed to resolve the inherent problem in the measurement of the depth-position of tracer particles. Some researchers suggested methods to determine particle position along the optical-axis by searching locations with maximum intensity [6, 8, 11] or by using complex amplitude [12, 13] in reconstructed particle images. In our previous studies, an operator that quantifies image sharpness of a reconstructed image, which is called focus function, is used to search the depth-position of a particle or biological cells [14, 15]. Recently, Guildenbecher, et al. [16] proposed a method to measure particle displacement between a hologram pair by searching the maximum cross-correlation of edge sharpness in local particle windows.

Digital holography is widely used in the measurement of the 3D position of particles and of 3D particle tracking to estimate particle distribution in a volume or 3D flow information [17–19]. Recently, digital holography has also been utilized to investigate 3D trajectories of biological samples such as red blood cells (RBCs), microorganisms, and other cells [14, 15, 20, 21]. In 3D flow analysis using DHPIV, polystyrene spheres or a silver-coated glass sphere is used as flow tracers. A polystyrene sphere is transparent, whereas a silver-coated glass sphere is opaque. The distinctive surface feature of these particles also influences their optical characteristics [22]. A transparent particle acts as a ball lens under laser illumination. For transparent particles, sharp and distinct fringe patterns are observed, which are attributed to the interference among the refracted beam, diffracted beam from a particle, and reference wave. However, for opaque particles, the fringe pattern is attributed to the diffraction caused by the edges of the particles. This difference can influence the determination of particle position in DHPIV.

For biological samples, they have diverse shapes and optical characteristics. In both particles and biological samples, the complex shapes or different optical characteristics contribute to the difficulty in measuring accurate 3D position and 3D displacement (or velocity) of particles and biological samples in a volume.

In our previous studies, we successfully demonstrated the practical adoption of focus function in determining the depth-position of transparent particles [17], RBCs [14] and microorganisms [15] in a volume. We obtained acceptable results in the measurement of depth-position. However, small Δt and measurement uncertainty in z₁ and z₂ resulted in unsatisfactory results in calculating depth-directional velocity using the formula ${\text{V}}_{\text{z}}\text{=}\left({\text{z}}_{\text{1}}\text{-}{\text{z}}_{\text{2}}\right)\text{/Δt}$. To minimize the measurement error in Δz, we suggest a method in which depth-displacement (Δz) is directly obtained from a cross-correlation of focus values between the consecutive holograms. In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of time. The focus value profiles of the consecutive holograms are highly correlated with each other. By calculating the cross-correlation of the focus values, the depth-displacement is directly obtained without determining the depth-position. In the present study, we demonstrate that the method provides precise depth-displacement of a particle or a cell. In addition, as a practical application, the method is adopted to measure the circulating flow inside a confined droplet.

2. Recording and reconstruction of a digital hologram

2.1 Digital hologram recording

A single beam in-line digital holographic microscopy (IDHM) technique was applied to measure the 3D particle position and 3D velocity information. Figure 1(a) shows a schematic diagram of IDHM, which comprises a continuous diode-pumped solid-state laser (100 mW, CrystaLaser), water immersion objective lens (40 × , NA = 0.80, Nikon), and a CCD camera (PCO2000, PCO-TECH) with 2048 × 2048 pixel array and 7.4 μm pixel size. Two kinds of spherical particles and RBCs were used as test samples. A polystyrene sphere (d = 15 μm, Polysciences) is transparent, whereas a silver-coated glass sphere (d = 15 μm) is opaque. Human RBCs were diluted in PBS solution and served as the biological sample.

 

Fig. 1 (a) Schematic diagram of in-line holographic microscopy to measure micro-scale fluid flows. (b) Typical holograms of a transparent particle, an opaque particle, and a red blood cell.

Download Full Size | PDF

The focal plane of the objective lens was positioned slightly above the test sample to allow forward scattering from the sample. The superposition of the unaffected reference wave and the wave diffracted by a particle or cell (object wave) created fringes (i.e., hologram) on the image plane. The hologram images were recorded by a digital camera.

Figure 1(b) shows the holograms of a transparent particle, opaque particles, and RBC. The transparent and the opaque particles exhibit different fringe patterns despite having the same size and shape. The distinct fringe pattern is observed in both inside and around the transparent particle, whereas the dim fringe pattern is observed around the opaque particle. The transparent particle acts as a ball lens under the laser illumination. The fringe pattern in the transparent particle is attributed to the interference among the refracted beam, diffracted beam, and reference wave. However, the fringe pattern observed in the opaque particle is attributed to the interference between the reference wave and the diffracted beam caused by the edge of the particle. This difference can influence the accuracy in the determination of the 3D position of a particle and the extraction of 3D velocity in DHPTV.

2.2 Digital hologram reconstruction

The hologram image is numerically reconstructed by employing an angular spectrum algorithm [23]. If $E_0(x_0,y_0;0)$ is the wavefield at the hologram plane (z = 0), the corresponding angular spectrum $A(k_x,k_y;0)$ can be obtained by deriving its Fourier transform as follows:

The complex wavefield at any plane perpendicular to the propagating z-axis can be reconstructed as follows:

The hologram is reconstructed with the interval of z = 1 μm in depth-direction. If the reconstruction volume ranges between z = 1 μm and z = 300 μm with 1 μm interval, 300 reconstructed images are generated from one hologram.

2.3 Extraction of in-plane position

After the numerical reconstruction of each hologram, the sequent in-plane (x, y) position and the depth (z) position of particles are determined to acquire 3D velocity information in DHPTV. The reconstructed images, with 1 μm reconstruction interval for each hologram, are projected into one image (projection image) to find the in-plane positions of the particles. In the reconstruction image or projection image, a bright spot is observed at the center of a sphere particle, as shown in Fig. 2. From this perspective, a peak-intensity-searching algorithm can be used to determine the in-plane position of a particle. The thresholding and image-processing method can also be applied to the projection image to search for the in-plane position of a particle or biological cell. Detailed information on the determination of the in-plane position of a particle or a biological sample is presented in our previous study [14, 15, 17, 20].

 

Fig. 2 Reconstructed images for each hologram are projected into one image (projection image) to find the in-plane positions of particles. Red circles represent the centers of particles, which are determined by using a peak-searching algorithm. Red squares indicate the segmented regions whose focus value requires evaluation by scanning the reconstructed images of each particle.

Download Full Size | PDF

3. Identification of depth-position using a focus function

3.1 Transparent particle

After obtaining the in-plane position of each particle, a list of rectangular sections enclosing each particle is generated by segmenting the particles in the projection image, as shown in Fig. 2. Each reconstructed image is cropped with the rectangular sections, and the focus value of each section is calculated by adopting a focus function [14]. The focus function quantifies the image sharpness by scanning the reconstructed image of each particle. This process generates a list of focus values along the z-direction for each particle. The focus value typically increases as the reconstruction plane approaches the actual position of a particle. Hence, the z-position of each particle can be regarded as the peak position of each focus value profile.

In our previous study, we tested several focus functions to find the optimal one having a higher accuracy [14, 15]. The operators, Laplacian (LAP) and Variance (VAR), are found to be effective in determining the z-position of a particle and biological samples. LAP and VAR are defined as follows:

The efficiency of a focus function is evaluated to determine the z-position of a transparent particle. Transparent particles are dispersed between slide glasses into a single layer. The first frame hologram captures the particles positioned approximately 100 μm apart from the focal plane using a translation stage. The second frame hologram captures the particles precisely positioned at 100 μm apart from the first hologram along the z-direction. In the reconstructed images shown in Fig. 3(a), the most focused image of a particle is observed at z₁ = 80 μm and z₂ = 180 μm in the first and second holograms, respectively. This finding matches with that obtained from the focus value profiles. Figures 3(b) and 3(c) represent focus value profiles obtained by scanning the reconstructed images of the particle from z = 1 μm to z = 300 μm, with the application of LAP and VAR, respectively.

 

Fig. 3 (a) Holograms and reconstructed images of a transparent particle. In the first frame of hologram, the particle is positioned approximately 100 μm apart from the focal plane. The second frame of hologram captures the particle precisely positioned 100 μm apart from the first hologram along the z-direction. The red squared images represent optimally focused reconstructed images of a particle. The degree of focusing state is evaluated by applying the focus functions (b) LAP and (c) VAR. The peak represents the z-position of the particle, at which the reconstructed image is optimally focused. (d) Holograms and reconstructed images of an opaque particle. The actual displacement of particle between the two holograms is 100 μm. (e) Typical focus value profiles obtained by adopting LAP. The focus value profiles of an opaque particle do not exhibit a sharp peak such as that of a transparent particle. However, the two focus value profiles of the first and second holograms have a strong similarity.

Download Full Size | PDF

The sharp peaks are observed at z₁ = 84 μm and z₂ = 178 μm in LAP and at z₁ = 83 μm and z₂ = 179 μm in VAR. These results well match those shown in Fig. 3(a).

The results show that the focus function can be effectively used to determine the z-position of a transparent particle.

3.2 Opaque particle

In the hologram of an opaque particle, a dim fringe pattern is observed around the particle, which is unlike that in the hologram of a transparent particle. The fringe pattern is caused by the interference between the diffracted beam induced from the particle edge and the reference beam. Intensity distributions of the reconstructed images along the optical axis are quite different between the transparent and opaque particles [22]. These differences also manifest in the focus value profiles of transparent and opaque particles and can influence the determination of z-position with the use of a focus function.

The efficiency of the focus function is evaluated to determine the z-position of an opaque particle. Similar to the experiment using a transparent particle, the first frame hologram captures an opaque particle positioned approximately 100 μm apart from the focal plane. The second frame hologram captures the particle precisely positioned 100 μm apart from the first hologram along the z-direction. In the reconstructed images shown in Fig. 3(d), the most focused images of the opaque particle are observed at z₁ = 120 μm and z₂ = 220 μm in the first and second holograms, respectively.

The focus value profiles are evaluated by scanning the reconstructed images of an opaque particle from z = 1 μm to z = 300 μm with the use of LAP. In Fig. 3(e), the focus value is demonstrated to rapidly increase as the reconstructed plane approached the focal plane. However, a sharp peak in the focus value profile for the opaque particle, similar to that of the transparent particle, is difficult to obtain. The focus value profiles shows a peak of around z₁ = 120 μm and z₂ = 218 μm in the first and second holograms, respectively. The near-peak regions, where the focus values are larger than 90% of the peak value (z_90%), are z_1,90% = 120 ± 3 μm and z_2,90% = 218 ± 5 μm in the first and second holograms. We estimate that the particles are positioned with an error of ± 5 μm from its actual position. Although the method of using a focus function for an opaque particle is not as effective as that for a transparent particle, the method provides acceptable results for the determination of z-position of an opaque particle.

3.3 Biological sample

Biological samples usually have a variety of shapes and inner structures, which create a complex hologram unlike that of a sphere particle. Therefore, determining z-position of a biological sample is more difficult than that of a sphere particle. By adopting a focus function in DHPTV, we successfully demonstrate 3D tracking of RBCs in a microtube flow [14] and 3D trajectories of free-swimming microorganisms in a volume [15]. In our studies, we have achieved acceptable results in the determination of z-position of RBCs and biological samples. However, in the measurements of z-directional velocity, cumulative errors in the measurement of z-position and short time interval (Δt) can induce an unacceptable errors.

In the present study, we demonstrate the process of determining the z-positions of RBCs and obtaining their accurate z-displacement. RBCs diluted in PBS solution are dispersed between two slide glasses into a thin layer. RBCs are positioned at 50, 100, and 150 μm apart from the focal plane along the optical axis. In the reconstructed images of the first, second, and third frames of the holograms [Fig. 4(a)], the optimally focused images of RBCs are observed at approximately z = 50, 100, and 150 μm,.

 

Fig. 4 (a) Reconstructed images of RBC at z = 50, 100, and 150 μm. The z-displacement of the RBC in consecutive holograms are precisely controled to 50 μm. The red squared images indicate optimally focused reconstructed images of the RBC. (b) The degree of focusing state is evaluated by adopting the focus function VAR in the reconstructed images. The peak represents the z-position of the RBC, at which the reconstructed image is optimally focused. The focus value profiles of the RBC show more complex curves than those of a sphere particle. However, the focus value profiles well-reflect the degree of focusing state and the curves are highly correlated.

Download Full Size | PDF

The focus value profiles are evaluated by scanning the reconstructed images of the RBCs from z = 1 μm to z = 250 μm by applying VAR. The results are shown in Fig. 4(b). The focus value profiles of RBCs have more complex curves than those of a sphere particle. However, the focus value profile reflects the focusing state of the RBC. The focus value profiles have a peak at z = 52 ± 3, 98 ± 4, and 151 ± 3 μm in the first, second, and third holograms, respectively. These results match well with those shown in Fig. 4(a).

4. Depth-displacement obtained from cross-correlation

After determining the z-position of a particle, the z-directional velocity is obtained by dividing depth-displacement $(z_2\text{-}{\text{z}}_{\text{1}})$with the time interval $\Delta t$ in conventional HPTV technique. Despite the acceptable results in the determination of a particle position, the cumulative errors in z₂ - z₁ and the short time interval Δt can cause significant errors in the measurement of z-directional velocity. To minimize the measurement errors in Δz, we propose a method in which depth-displacement is directly extracted from the cross-correlation of focus values between the two frames of holograms.

The focus value profiles of particles or biological samples located at different z-positions in consecutive holograms have a strong similarity, as shown in Fig. 3 and Fig. 4. By using this similarity of the focus function profiles, the depth-displacement between particles or biological samples in a consecutive hologram can be directly extracted by calculating the cross-correlation of the focus values without the procedure of determining the z-position. The cross-correlation of the focus values of consecutive holograms is calculated by the following formula:

Figure 5(a) shows typical cross-correlation curves drawn from the focus values of transparent and opaque particle, as depicted in Figs. 3(b), 3(c), and 3(e). The second holograms of the transparent and opaque particles are precisely positioned at 100 μm apart from the first hologram. The results of the cross-correlation show that the peak positions are shifted to 100 μm away from the center. The depth-displacement identified from the cross-correlation of focus values well matches the actual displacement of particles. The depth-displacement of a transparent particle, as obtained by the cross-correlation of the focus value with the application of LAP and VAR, respectively, are 98 and 97 μm. For an opaque particle, the depth-displacement obtained by the cross-correlation is 101 μm.

 

Fig. 5 Cross-correlation curves of the focus values depicted in Figs. 3 and 4. The depth-displacements of (a) a transparent particle, (b) an opaque particle, and (c) RBC are obtained by the cross-correlation of focus values. The distance from the center to the peak represents the depth-displacement of a particle or a biological sample. The distance of a particle between two consecutive holograms is precisely controlled to 100 μm apart. RBCs are positioned with intervals of 50 μm along the z-direction in consecutive holograms. The depth-displacements obtained by the cross-correlation of focus value profiles well-matched with their actual displacements.

Download Full Size | PDF

We evaluate the effectiveness of the method using the cross-correlation of focus values in determining the depth-displacement of RBCs. RBCs are located with 50 μm interval along the z-direction, as shown in Fig. 4. The peak positions are shifted to 50 μm away from the center in the cross-correlation of $F_1\circ F_2$ and$F_2\circ F_3$, as depicted in Fig. 5(b). F₁, F₂ and F₃ represent the focus values of the first, second, and third frames of the holograms of the RBCs. The depth-displacement of the RBC obtained by cross-correlation of the first and second holograms is 49 μm, and that between the second and third holograms is 50 μm. For the first and third holograms, the depth-displacement obtained by the cross-correlation is 98 μm, which well-matched the actual displacement.

The depth-displacement of more than 100 particles or biological samples are identified to compare the performance of the cross-correlation method ($\Delta z=F_2\circ F_1$) with that of a conventional method ($\Delta z=z_2-z_1$). In the conventional method, the z-directional position of each particle is determined, and then depth-displacement is obtained from the difference among the z-directional position of particles in consecutive holograms. In the cross-correlation method, however, the cross-correlation of focus values obtained from consecutive holograms is calculated to directly determine depth-displacement directly, as mentioned above.

Figure 6 illustrates the results of the statistical analysis for the depth-displacement of transparent particles, opaque particles, and RBCs. The actual depth-displacements of transparent and opaque particles are precisely controlled at 100 μm. For transparent particles, the conventional method provided acceptable results, because a focus function is effectively working in determining the z-position with small errors, as shown in Figs. 4(b) and 4(c). However, the cross-correlation method provides better results than the conventional method. For opaque particles, the measurement errors in z-position are larger than those of transparent particles, as shown in Fig. 4. Therefore, the errors in depth-displacement are considerable, when the depth-displacement is determined from z₂ - z₁, as shown in Fig. 6. However, the cross-correlation method provide noticeably reduced errors in the depth-displacement measurement.

 

Fig. 6 Statistical analysis of depth-displacement of transparent particles, opaque particles, and RBCs. Depth-displacement of more than 100 particles or of RBCs are obtained by employing a conventional method ($\Delta z=z_2-z_1$) and a cross-correlation method ($\Delta z=F_2\circ F_1$).

Download Full Size | PDF

The actual depth-displacements of RBCs are precisely controlled to 50 and 100 μm, respectively, as shown in Fig. 5. The standard deviations of depth-displacement obtained by the conventional method are 4.8 and 4.9 μm for z₂ - z₁ and z₃ - z₁, respectively, as shown in Fig. 6. However, the standard deviations largely reduce to 1.6 μm ($F_2\circ F_1$) and 1.8 μm ($F_3\circ F_1$) with the application of the cross-correlation method.

The method using a cross-correlation of focus values can be used to directly obtain depth-displacement without determining z-position. The cross-correlation method provides more precise results for extracting depth-displacement of particles or cells than a conventional method. The cross-correlation method can be practically used for a 3D flow analysis using a HPTV.

5. Feasibility test: axial flow in a confined droplet

The method proposed in this study is applied to 3D flow visualization using HPTV to demonstrate its feasibility and usefulness. An evaporation-induced flow in a confined droplet is selected as a practical test case because of its strong axial-flow component.

Figure 7 represents a schematic diagram of the evaporation-induced flow in a confined droplet. A NaCl solution diluted with deionized water served as the working fluid. The molar concentration of the diluted solution was 1 mol. Polystyrene particles with a mean diameter of 4 μm were seeded in the solution as flow tracers. A small amount of the solution was placed between two polydimethylsiloxane (PDMS)-coated glasses to form a confined droplet. The surface treatment allowed the confined droplet to have an approximately 90° contact angle at the edge, which resulted in an almost rectangular cylindrical shape. The kinematic viscosity of the test fluid was approximately ν = 0.94 mm²/s. The room temperature was maintained at 25 °C ± 1 °C. Images were recorded at a frame rate of 14.7 fps, which corresponded to Δt = 0.068 sec.

 

Fig. 7 Schematic diagram of the evaporation-induced flow inside a confined droplet.

Download Full Size | PDF

As the fluid evaporates from the free surface of the confined droplet, a large-scale circulating flow is induced inside the droplet as a result of the concentration gradients between the edge and center regions of the droplet, as depicted in Fig. 7.

Figures 8(a) and 8(b) show a typical hologram of tracer particles seeded inside the confined droplet and a side view of the droplet. After the hologram reconstruction, 3D particle position and displacement are obtained by applying the focus function and cross-correlation. Two-frame particle tracking algorithm is adopted to obtain the 3D trajectories of particles [24]. Figure 8(c) represents particle trajectories induced by the circulation at the center plane of the confined droplet. By adapting the proposed method in the study, velocity field inside the confined droplet is obtained, as shown in Fig. 8(d). A large-scale circulating flow is clearly observed inside the confined droplet in the side view. The maximum upward axial velocity occurs at the core of the confined droplet, while downward flow is observed at the edge of the droplet.

 

Fig. 8 (a) A typical hologram of tracer particles seeded inside the confined droplet. (b) Side view of the confined droplet. (c) Particle trajectories and (d) velocity field inside the confined droplet as seen at the side view.

Download Full Size | PDF

In this experiment, we demonstrate the feasibility of the proposed method in measuring depth-position and depth-displacement of particles in a fluid flow that has a strong 3D velocity component.

6. Conclusions

We propose a method using a focus function and its cross-correlation to determine depth-position and depth-displacement of particles or biological samples. The focus function quantifies the degree of focusing state by scanning the sharpness of reconstructed images. The focus function provides acceptable results in determining the depth-position of a transparent particle, an opaque particle, and RBC.

Despite of the acceptable results in the determination of depth-position of a particle, cumulative errors in depth-displacement, z₂ - z_1, and a short time interval Δt can contribute to unacceptable results in identifying the depth-directional velocity in HPTV. To minimize the errors in Δz, we suggest a method that directly extracts depth-displacement from the cross-correlation of focus values in consecutive holograms. The focus value profiles of consecutive holograms are highly correlated with each other. By calculating the cross-correlation of focus values, the depth-displacement is directly obtained without the procedure of depth-position determination. The method provides precise depth-displacement of a particle or a biological sample. As a practical application, a circulating flow inside a confined droplet that has a strong depth-directional velocity component is successfully measured by using the HPTV technique.