We propose a novel backscatter holographic imaging system, as a compact and effective tool for 3D near-wall flow diagnostics at high resolutions, utilizing light reflected at the solid-liquid interface as a reference beam. The technique is fully calibrated, and is demonstrated in a densely seeded channel to achieve a spatial resolution of near-wall flows equivalent to or exceeding prior digital inline holographic measurements using local tracer seeding technique. Additionally, we examined the effects of seeding concentration and laser coherence on the measurement resolution and sample volume resolved, demonstrating the potential to manipulate sample domain by tuning the laser coherence profile.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Three dimensional (3D) flow field measurements based on imaging tracer particle motions in a volume, referred to as 3D particle image velocimetry (PIV) or particle tracking velocimetry (PTV), has established itself as a powerful flow diagnostic technique, enabling access to an unprecedented level of information for fundamental research in fluid mechanics [1,2]. The commonly used 3D PIV/PTV techniques, i.e., Tomographic PIV (e.g., commercialized by LaVision, Inc.) and Defocusing PIV (e.g., commercial product V3V from TSI, Inc.), use multiple cameras to image a volume illuminated by a high power laser, resulting in an expensive system requiring a cumbersome calibration. Furthermore, these systems also impose limits on the spatial resolution of the measurement, particularly for near-wall flows.
In comparison, PTV based on Digital Inline Holography (DIH), referred to as DIH-PTV hereafter, has emerged as a compact and low-cost alternative to probe 3D near-wall flows, achieving unparalleled spatial resolutions . DIH-PTV employs a laser beam to illuminate a flow field with tracer particles, and a single digital camera to capture interference between the forward scattered light (from tracers) and the undiffracted portion of the beam (i.e., hologram). Particularly, the use of forward scattering lowers the requirements on laser power, while the recorded 2D hologram that encodes 3D information of tracers eliminates the need for multiple cameras. The hologram is numerically reconstructed by convolving with a diffraction kernel (e.g., Raleigh Sommerfeld or Kirchhoff - Fresnel kernel) which simulates the propagation of light. Such a process creates intensity cross sections of the illuminated particles in the sampling volume at different longitudinal distances, i.e., a reconstructed 3D intensity field. Finally, through appropriate thresholding & segmentation routines, we extract the particle positions and track them over time to extract the 3D flow field.
Despite the clear advantage of DIH-PTV over other 3D PIV/PTV techniques in terms of its compactness and cost, it still suffers from several major drawbacks limiting its broad application in flow measurements. The first and most important of these is the longitudinal elongation of the reconstructed particles caused by an extended depth-of-focus and signal truncation associated with discrete sampling in DIH, limiting longitudinal resolution . Secondly, the need for capturing interference places a limit on the maximum particle concentration, in order to maintain a sufficient level of signal-to-noise ratio (SNR). As we increase tracer concentration with the goal of achieving high spatial resolutions, the corresponding noise generated from cross interference of signal (i.e., from adjacent particles and particles outside the sampling volume, but within the laser path) goes up. Apart from an increase in cross interference, the dense suspension also attenuates the reference beam required to create the hologram, thereby decreasing the SNR. At its extreme limit of concentration, the recorded hologram contains only a statistical speckle pattern, with no clear way to discriminate individual particles for processing. In order to quantify the effect of particle concentration and sample depth on the SNR of holograms and extraction efficiency of DIH, shadow density (SD), i.e., (particle concentration, ‒ the thickness of the sampling volume and ‒particle diameter), is introduced [4,5]. Prior studies have reported a significant drop in hologram quality and particle extraction efficiency (~20%) as SD approaches 10% . Other PIV/PTV techniques do not suffer from similar speckle-based limitations as they directly record scattered intensity and not an interference pattern. Though some recent improvements in processing algorithms have substantially improved our ability to process holograms with relatively low SNR for DIH-PTV ([6,7]), they are still insufficient to handle the holograms with SD comparable to conventional PIV techniques. Such an increase in particle concentration, and through it the resolution, is required to resolve fine features of complex near-wall turbulent flows that range from µm to sub-mm scales.
However, specific workarounds have been developed to enable high-resolution near-wall measurements using DIH-PTV. For example, Sheng et al.  implemented a localized seeding approach, which introduces particles in a jet just upstream of the region of interest to capture near-wall 3D coherent flow structures in a smooth-wall channel over a sampling volume of 1.5 mm × 2.5 mm × 1.5 mm. Similarly, through the use of local seeding, Talapatra & Katz  identified U-shaped coherent structures generated by individual pyramidal elements in a rough-wall channel flow in a 3.1 mm × 2.1 mm × 1.8 mm sampling volume. Apart from introducing disturbances to the flow field, the local seeding method results in significant particle concentration fluctuations in the field of view (355-1770 particles/mm3 in  and 426-853 particles/mm3 in , respectively), both of which are detrimental to achieving accurate and consistent vector field measurements. Alternatively, Allano et al.  employed a fiber optic probe to position the laser illumination close to the wall of a wind tunnel, recording signals from the tracers confined within a small depth to limit cross interference noise. The technique is not only intrusive, influenced by probe vibrations, and requiring careful calibration (due to spherical illumination beam), but is ultimately limited by a low tracer concentration, i.e., ~7 particles/mm3 within a 5 mm × 3 mm × 5 mm volume.
Though not directly implemented for near-wall flow measurements, several other approaches have also been reported to improve SNR of recorded holograms, e.g., side-scatter holography and LED holography. Cao et al.  recorded holograms from a 8 mm × 5 mm × 6 mm sampling volume (4 particles/mm3), by interfering side scattered light from particles with a separate reference beam in an off-axis geometry. Thus, a sampling volume can be placed at any position within the flow domain to capture a weaker side-scatter signal, but at the cost of increased laser power. Furthermore, a larger path length difference between the particle signal and the reference beam requires a laser with larger coherence length. In contrast, Petruck et al.  and El Mallahi et al.  used LED light to limit speckle noise recorded on the hologram. The low coherence of LED sources eliminates any interference from objects outside the sampling volume, but effectively limits the depth of field to around tens of microns.
In this paper, we propose a novel approach for near-wall flow measurements based on capturing holograms of backscattered signal from particles, which can overcome some of the key challenges discussed. The traditional recording geometry (both in-line & off-axis) for generating holograms from backscattered signals uses a separate reference beam to produce the interference pattern. The predominant uses of backscatter holography have been for characterizing surface topographies and as a microscope [14–17]. Herrmann & Hinsch [18,19] reported the implementation of backscatter holographic PTV for flow measurement using film recording medium. By making use of low coherence laser light and an off-axis recording geometry, they were able to restrict the information from particles at different depths from cross-interference, separating them laterally on the film, and extracting ~0.5 particles/mm3 in a 24 mm × 18 mm × 29 mm volume. However, the complicated setup and limited resolution make the technique unviable for near-wall measurements. Our proposed approach eliminates the need for off-axis recording by utilizing the light reflected at the solid-fluid interface (i.e., wall-fluid boundary) as the reference beam.
The paper is organized in separate sections as follows. In Section 2, we present a description of the approach along with the basic principles of the technique. Subsequently, Section 3 provides the results from proof-of-concept experiment, a measurement calibration, as well as the effects of particle concentration and laser coherence. Finally, we will conclude this paper through a summary and a brief discussion of the results in Section 4.
The general setup and working principle of the proposed approach, referred to as Digital Fresnel Reflection Holography (DFRH), are illustrated in Fig. 1. The setup [Fig. 1(a)] consists of a laser, beam splitter, sample container with an imaging window, objective lens, and a camera. Light from the laser passes through the beam splitter reaching the sample, where a part of the beam reflected at the imaging window (i.e., Fresnel Reflection) serves as the reference wave, interfering with light scattered by particles in the sampling volume to form the hologram. Whenever light reaches an interface with a discontinuity in refractive index, a fraction of the beam is reflected back while the rest passes through . As shown in Fig. 1(b), such a reflection occurs at both the outer and inner surfaces of the window, where the intensity of reflected light depends on the difference in the index of refraction (e.g., ~3.8% for air-acrylic interface and ~0.32% for the acrylic-water interface for normal incidence of light). Furthermore, since the illumination laser is incident normal to the window and the backscatter signal of particles and the reference light from the reflection recorded by the camera are close to the normal direction, polarization of the light source does not yield appreciable impact on our holograms . In order to maintain a high fringe contrast on the hologram, we require the path length difference between the particle signal and the reflected reference wave to be within the coherence length of the source (the smaller the better) and the intensity difference between the interfering waves be as close as possible. The weaker reflection from the inner wall produces particle fringes with the highest contrast, as it is closer in intensity and path length difference to the particle signal. In comparison, the stronger reflection from the outer wall does not contribute to interference with the particle signal, as their path length difference lies in the low visibility range (refer to Sec. 3.5) and the difference in intensities is larger. Hence, the outer wall reflection acts as a source of noise, increasing the background intensity, and causes a reduction in overall fringe contrast. Due to a significant reduction in scattering efficiency in the backward direction, the technique does require us to increase the laser power. Typically, inline holographic systems can work with laser powers in the μW to mW range owing to high forward scattering efficiencies. In comparison, DFRH capturing backscattered intensities that are almost 2-3 orders weaker than forward scattering (for 10 μm particles in water) might require up to ~100 mW in laser power. Therefore, the SNR in DFRH can be improved by either increasing scattered signal strength (e.g., using silver-coated particles or increasing index difference between particle and fluid) or reducing the reflection at the front wall (e.g., applying anti-reflection coatings).
Compared to the hologram formation process for a single particle in conventional DIH [represented by Eq. (1)] (see ), the recorded intensity for DFRH [Eq. (2)] yields an additional phase difference of , where are the distances travelled by the reference (reflected at interface) and object waves (backscattered at particle) respectively. However, presence of this added phase will not alter intensity reconstructions from the hologram, and hence would not affect the positions extracted through processing. The equations presented are for a single particle which can be easily extended to multiple particles through linear superposition of independent delta function representing each (neglecting the interference between weak scattered signals of different particles).
Similar to conventional DIH, the processing algorithm of DFRH consists of image preprocessing, numerical reconstruction, and segmentation. Image pre-processing includes Fourier domain filtering and time-averaged background subtraction to enhance fringe contrast. As the separation between the two walls of the imaging window are within the coherence length of the laser, we capture interference patterns between their corresponding reflections. The captured holograms contain periodic variation in intensity (sinusoidal) over the entire image, which corresponds to discrete peaks in the Fourier domain. We suppress the interference pattern by selectively masking out these peaks. Next, we subtract a time-averaged background from every image to remove stationary artifacts such as dirt and defects in the optical path and to boost the SNR of the fringes. The enhanced holograms are then processed using a GPU-based compressive holography algorithm to identify a sparse representation of the 3D intensity field that generates the recorded hologram [23,24]. Specifically, we employ FISTA (Fast Iterative Shrinkage Thresholding Algorithm) to solve the ill-posed linear optimization problem of the hologram formation with an l1 norm regularization function (Refer to [23,25] for a complete overview of the algorithm). A manual threshold segments the reconstructed 3D intensity field and the particles extracted as intensity weighted centroids. Finally, to perform 3D-PTV, the extracted positions are tracked using a nearest neighbor routine to generate trajectories of particles from which velocities can be calculated . Apart from generating trajectories, we also use the tracking operation to validate the extracted particles and eliminate any noise introduced by the thresholding operation. With the particles validated and tracked, we measure the size of our sampling volume along the depth direction in DFRH by plotting a histogram of all particle positions (ensemble) and using a 5% cut-off on either side as the limits.
3.1. Proof-of-concept experiment
A proof-of-concept experiment is first conducted to determine the feasibility of DFRH. The experimental setup as shown in Fig. 2(a), consists of a laser (OptoEngine 80 mW 532 nm solid state laser), a beam splitter, an acrylic sample chamber (50 mm × 50 mm × 10 mm with a wall thickness of 3 ± 0.3 mm), an objective lens (Mitutoyo 10X Long Working Distance objective 0.28 NA), and a camera (Flare 2M360, 2048 × 1088 pixels, 30 fps with 200 µs exposure time). A particle suspension (13 µm silver-coated hollow glass spheres) of a fixed weight of particles (0.12 g) in a 20 ml distilled water, corresponding to a concentration of ~3000 particles/mm3 (SD ~400%), is prepared. The use of silver-coated particles enhances the SNR of the fringe patterns by increasing the strength of the backscattered signal. Additionally, a commercially available anti-reflection (AR) coated acrylic (Acryllite Inc.) is used in making the sample container to reduce reflections from the outer walls. A sample raw hologram, 512 µm × 512 µm field of view (1 µm/pixel), and its Fourier spectrum are shown in Figs. 2(b) and 2(c) respectively, where the strong peaks corresponding to the interference pattern can be clearly seen (marked by arrows) in the spectrum. After identifying the spatial frequency of the noise, we implement a band-stop filter centered at this frequency to suppress the fringe information from the hologram. The filtered hologram, enhanced through a time-averaged subtraction [Fig. 2(d)], is processed following the algorithm specified in the methods section. We have included a video of a sample reconstruction for Fig. 2(d) as Visualization 1. Please note that for all visualizations, we have stretched the contrast for better illustration.
Due to the fluctuation of numbers of particles in the sampling domain across a sequence of holograms, the key metrics evaluating the performance of DFRH, including sampling depth, particle concentration, effective resolution (averaged particle separation in the sampling domain), are calculated as ensemble-averaged quantities. In addition, the uncertainties of these metrics are represented using one standard deviation from the mean. Accordingly, as shown in Fig. 2(e), the reconstructed 3D tracks (see Visualization 2) span a depth of ~300 µm. Please note the shorter length of tracks at larger depths, caused by particles moving away from the wall (due to random flow in the cuvette), and the corresponding drop in signal strength leading to loss of particles. However, in spite of such limitations, we successfully extracted and tracked up to 93 ± 10 particles over a sequence of 300 holograms, resulting in an effective resolution of ~95 ± 7 µm. In comparison, through local seeding, Sheng et al.  and Talapatra & Katz  achieve resolutions of 112 ± 59 µm and 119 ± 27 µm respectively. Even though sampling depth is limited in DFRH, by eliminating local seeding, we are able to achieve a much higher and more consistent particle concentration (1240 ± 133 particles/mm3 in current experiment vs 1062 ± 707 particles/mm3 in Sheng et al.  and 639 ± 213 particles/mm3 in Talapatra & Katz , respectively).
3.2. DFRH measurement calibration
To provide further validation of DFRH, we perform a calibration experiment comparing reconstruction results to a ground truth, obtained through 3D scanning of the sample. Specifically, a gelatin (water based) seeded with particles (13 µm silver-coated hollow glass spheres) is prepared and allowed to cool in an AR-coated acrylic container, similar to the one used in the proof-of-concept experiment. Once the gel hardens, we place the sample in the DFRH system and perform a scan, utilizing the same optical path and recording parameters (1 µm/pixel). The scanning operation moves the image focal plane through the sample, capturing laser light scattered from the fixed particles without requiring a numerical reconstruction. Furthermore, the translation is performed with a linear stage, over a range of 600 µm at a resolution of 10 µm/step, limited by the resolution of the micrometer. After completion of the scan, we return the sample to its initial position and record a sequence of backscatter holograms while moving the sample in the lateral direction (x) in order to calculate an average background for image enhancement.
A comparison of the scanned and DFRH reconstruction results is provided in Fig. 3. Specifically, Fig. 3(a) presents x-y and y-z slices from the scanned volume with three specific particles marked by boxes. The insets illustrate the corresponding slices around the three selected particles (based on highest intensity in the reconstructed volume) indicating a clear overlap between the two in all three directions. Note that the particles seem to appear as binary objects in the grayscale image due to the reconstruction algorithm (FISTA) enhancing the sparsity of the intensity field. Additionally, we also plot the longitudinal intensity profiles through the center of the three selected particles [Fig. 3(b)], which highlight the close match between their peaks. The peaks of the reconstruction and scanned results are within ± 10 µm of each other, which corresponds to a value below the uncertainty for linear stage used. The comparison shows that our method (DFRH) does provides accurate reconstruction (both lateral and longitudinal) of particle positions in 3D.
3.3. Flow test case
To assess the feasibility of DFRH for near-wall flow measurements, we perform an experiment in a small-scale flow channel, with a test section 250 mm in length and a square cross section (10 mm × 10 mm) [Fig. 4(a)]. The channel is made of the same AR coated acrylic used in the proof-of-concept experiment. The imaging window is located near the end of the channel to ensure a fully developed velocity field that does not change in the streamwise direction. We conduct the experiment with the same silver-coated hollow glass tracers (13 µm), at a concentration of ~1000 particles/mm3 (SD ~150%). The camera records holograms at 700 fps and 250 µs exposure time with a pixel resolution of 0.8 µm/pixel. The flow speed is set to limit the maximum particle displacement to be ~15 pixels (under the current recording condition) within two consecutive frames.
The reconstructed 3D trajectories, illustrated in Fig. 4(b), span a volume of ~400 µm × 400 µm × 270 µm capturing ~44 ± 6 particles on average (970 particles/mm3) over a sequence of 4000 holograms (~5.7 s), giving us a mean particle separation of ~99 ± 5 µm. In comparison, our proof-of-concept experiments performed under a higher seeding concentration (~2.5 times) yielded a particle separation of ~95 ± 7 µm. Next, the calculated unstructured velocities are used generate a 3D velocity field on a regular grid [Fig. 4(c)] of 50 µm spacing. The velocity vector at each grid point is estimated with data points located within a fixed search window (typically a multiple of grid size), by numerically solving a set of linear equations that models Taylor expansion in all three directions [8,9].
3.4. Effect of particle seeding concentration
To evaluate the effect of seeding concentration on the effective resolution of DFRH measurements, we investigate samples of different particle concentrations using the setup presented in the proof-of-concept experiment (section 3.1). Here we characterize the concentration in terms of shadow density (SD), specifically to offer a comparison between DFRH and DIH-PTV, even though SD does not necessarily serve as a proper metric for holograms formed from backscattered signals. The recorded holograms are processed and the corresponding measurement metrics (i.e., extracted particle concentration and effective resolution) are calculated. Additionally, we also compute the extraction efficiency, defined as the ratio of extracted particle concentration to the seeding concentration, to offer comparisons with prior DIH experiments.
Figure 5 summarizes the variation of extracted particle concentration, the extraction efficiency, and effective resolution with changes in shadow density. Specifically, as the seeding concentration (characterized by SD) increases, the extracted particle concentration from DFRH [shown in Fig. 5(a)] first rises to a peak of ~1200 particles/mm3 at a SD of 450%, then drops sharply with further increase of SD, due to a decline in SNR at extremely high SD. The corresponding extraction efficiency exhibits a similar trend, but reaches its peak value of 0.7 at slightly lower SD (~150%). In comparison, Malek et al. (2004), using inline holography, report a maximum extraction of ~0.2 under a shadow density that is less than 10%. Likewise, the effective resolution [Fig. 5(b)] first improves with increasing SD and reaches its minimum value of ~95 µm (SD ~450%) under the current settings. With further increase in SD the resolution plateaus, due to the combination of a smaller particle count and reduced sample volume (caused by the weak SNR of particles located deeper in the sample).
3.5. Effect of coherence
Using the same setup presented in Section 3.1, the effect of coherence on the sampling domain of DFRH is investigated using two lasers of different coherence profiles, i.e., Qphotonics 660 nm red diode laser (as a low coherence case) and 532 nm green diode laser employed in the previous sections (as a relatively high coherence case). The coherence profile for each laser is represented as the variation of fringe visibility on a Michelson Interferometer, i.e., where and denote the maximum and minimum intensity associated with the central fringe, respectively . Such changes in fringe visibility directly influences the SNR of recorded holograms, with higher visibility leading to higher SNR.
As shown in Fig. 6(a), the coherence profile for the green laser consists of a wide central peak with a full-width-half-maximum (FWHM) of ~1.6 mm at the zero optical path-length-difference (OPLD = physical distance × refractive index of medium), and several weak secondary peaks located further away (4.8 and 6.4 mm respectively). In contrast, the red diode is characterized by a narrow primary peak (FWHM ~0.37 mm) and a significantly stronger secondary peak (of similar width) at an OPLD of ~6.2 mm. The second peak in the profile (red diode) corresponds to another longitudinal mode setup in the cavity, given the separation is of similar order as the cavity size, i.e., order of mm. A theoretical estimate of the exact spacing is difficult due to lack of specific information (e.g., laser gain medium properties under specific operation temperature and current) from the manufacturer.
The sampling domain of DFRH, illustrated by the maximum intensity projection in the y-z plane [in Figs. 6(b) and (c)], highlight the stark differences between the two sources. Specifically, the domain under the low coherence (red) laser, is clustered around two specific depth positions (right next to the inner wall & 0.8 mm away from the inner wall), spanning a thickness of ~0.18 mm each, with no particles reconstructed in the region between them. On the other hand, the sample volume resolved by the green diode extends over ~0.8 mm with no apparent gaps in the reconstructed particle positions. Furthermore, the depth span of each sample domain closely matches the corresponding half-widths of the coherence peaks, i.e., 0.8 mm for the green diode and 0.18 mm for the red. Interestingly, for the red laser case, the sampling domain farther from the inner wall is not a result of the interference between the particle signal and the inner wall reflection, as it falls in the low visibility range of the laser coherence profile, with an OPLD of ~0.8 × 1.33 ~1 mm. However, the OPLD between the particle signal and the outer wall reflection does end up falling within the secondary peak of high visibility. We estimate the distance for this scenario as ~3.3 mm × 1.5 + 0.8 mm × 1.33 ~6.04 mm at the largest value of wall thickness, with an uncertainty limit specified by the manufacturer ( ± 10%). We attribute the weaker reconstructed intensity at this depth to a reduced particle signal and a decrease in the reference beam intensity due to the presence of the AR coating on the outer wall. Overall, the above investigation suggests a potential of manipulating the sample domain in the flow by tuning the coherence profile of the illumination laser.
4. Conclusion and discussion
In this study, we have introduced DFRH as a novel backscatter holographic imaging system for near-wall 3D flow measurements. By utilizing the reflection at the solid-liquid interface as a reference beam, the approach significantly simplifies hologram recording in a backscatter configuration by eliminating the need for a separate reference beam. Furthermore, the similarities of the current approach to conventional DIH enable the use of processing algorithms developed for DIH-PTV without loss of generality. A proof-of-concept experiment has demonstrated that DFRH can achieve increased particle concentration (1240 ± 133 particles/mm3) and superior resolution (95 ± 7 µm) in comparison with prior studies using local seeding. The accuracy of DFRH is also assessed through a comparison with the results from a manual scan of particles fixed in a gel. Moreover, DFRH is implemented for 3D velocity field measurements in a flow channel, resolving the near-wall flow field with a spatial resolution of ~99 ± 5 µm. Finally, the effect of seeding concentration and laser coherence on several selected metrics of DFRH (e.g., sampling depth, extracted particle concentration, extraction ratio and effective resolution) are examined.
Although DFRH has shown great promise as a compact and effective tool for near-wall flow diagnostics, it is still limited in the sampling depth and resolution for more demanding situations (e.g., high-Reynolds number turbulent flows). The goal of improving the measurement resolution beyond the current limits (~100 μm) would involve the use of smaller particles, which increase backscatter efficiencies, or modification of the coherence of the laser, increasing the depth of imaging. However, the reduction in particle size leads to an associated drop in the particle cross section that requires a higher imaging magnification to resolve, leading to a sacrifice of the captured FOV. Furthermore, while an increase in coherence can improve our imaging depth, the drop in the spherical particle wave strength with distance (inverse square law dependence) places the ultimate limit on the maximum value. Thus, any further improvements of these metrics hinges primarily on the SNR of recorded holograms. The simplest approach to boost signal strength involves increasing laser power, which currently is still very low compared to other laser based flow diagnostics techniques that capture an even weaker side-scattered signal. In contrast, we can improve the backscattered signal strength without a high-power laser (and silver-coated particles) just by ensuring a larger difference between the refractive index of particle and the surrounding medium e.g., glass particles in air. Apart from the above, optimal design of the AR-coating for a specific laser wavelength, as opposed to a broadband commercial coating, can help suppress undesired reflection from the outer wall, enhance the reflection at the inner wall, and boost the amount of light transmitted into the sample, thereby increasing particle SNR. Such improvements would be vital towards designing an optical probe capable of utilizing the reflected light from particles to perform flow diagnostics in a field environment where the use of conventional DIH-PTV is still challenging.
Office of Naval Research (Program Manager, Dr. Thomas Fu) under Grant No. N000141612755 and McKnight Award from University of Minnesota.
The authors would like to thank Kevin Mallery for his assistance in the data processing and Brandon Shadakofsky for his help with the channel flow experiment.
References and links
1. M. P. Arroyo and K. D. Hinsch, “Recent developments of PIV towards 3D measurements,” Top. Appl. Phys. 112, 127–154 (2008). [CrossRef]
2. S. Discetti and F. Coletti, “Volumetric velocimetry for fluid flows,” Meas. Sci. Technol.in press.
3. J. Katz and J. Sheng, “Applications of Holography in Fluid Mechanics and Particle Dynamics,” Annu. Rev. Fluid Mech. 42(1), 531–555 (2010). [CrossRef]
4. M. Malek, D. Allano, S. Coëtmellec, and D. Lebrun, “Digital in-line holography: influence of the shadow density on particle field extraction,” Opt. Express 12(10), 2270–2279 (2004). [CrossRef] [PubMed]
5. S. Kim and S. J. Lee, “Effect of particle number density in in-line digital holographic particle velocimetry,” Exp. Fluids 44(4), 623–631 (2008). [CrossRef]
7. M. Toloui, K. Mallery, and J. Hong, “Improvements on digital inline holographic PTV for 3D wall-bounded turbulent flow measurements,” Meas. Sci. Technol. 28(4), 44009 (2017). [CrossRef]
8. J. Sheng, E. Malkiel, and J. Katz, “Using digital holographic microscopy for simultaneous measurements of 3D near wall velocity and wall shear stress in a turbulent boundary layer,” Exp. Fluids 45(6), 1023–1035 (2008). [CrossRef]
9. S. Talapatra and J. Katz, “Three-dimensional velocity measurements in a roughness sublayer using microscopic digital in-line holography and optical index matching,” Meas. Sci. Technol. 24(2), 24004 (2013). [CrossRef]
10. D. Allano, M. Malek, F. Walle, F. Corbin, G. Godard, S. Coëtmellec, B. Lecordier, J. M. Foucaut, and D. Lebrun, “Three-dimensional velocity near-wall measurements by digital in-line holography: calibration and results,” Appl. Opt. 52(1), A9–A17 (2013). [CrossRef] [PubMed]
11. L. Cao, G. Pan, J. de Jong, S. Woodward, and H. Meng, “Hybrid digital holographic imaging system for three-dimensional dense particle field measurement,” Appl. Opt. 47(25), 4501–4508 (2008). [CrossRef] [PubMed]
12. P. Petruck, R. Riesenberg, and R. Kowarschik, “Partially coherent light-emitting diode illumination for video-rate in-line holographic microscopy,” Appl. Opt. 51(13), 2333–2340 (2012). [CrossRef] [PubMed]
13. A. El Mallahi, C. Minetti, and F. Dubois, “Automated three-dimensional detection and classification of living organisms using digital holographic microscopy with partial spatial coherent source: application to the monitoring of drinking water resources,” Appl. Opt. 52(1), A68–A80 (2013). [CrossRef] [PubMed]
18. S. F. Herrmann and K. D. Hinsch, “Light-in-flight holographic PIV (LiFH-PIV) for wind- tunnel applications : Off-site reconstruction of deep- volume real particle images,” Appl. Opt. 15, 1–9 (2004).
19. S. F. Herrmann and K. D. Hinsch, “Light-in-flight holographic particle image velocimetry for wind-tunnel applications,” Meas. Sci. Technol. 15(4), 613–621 (2004). [CrossRef]
20. E. Hecht, “Optics 4th edition,” Opt. 4th Ed. by Eugene Hecht Read. MA AddisonWesley Publ. Co. 2001 1, 122 (2001).
21. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 2008).
22. T.-C. Poon and J.-P. Liu, Introduction to Modern Digital Holography: With MATLAB (Cambridge University Press, 2014).
25. A. Beck and M. Teboulle, “A Fast Iterative Shrinkage-Thresholding Algorithm,” Soc. Ind. Appl. Math. J. Imaging Sci. 2, 183–202 (2009).
26. J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179(1), 298–310 (1996). [CrossRef]