One advantage of digital in-line holography is the ability for a user to know the 3-D location of a moving particle recorded at a given time. When the time exposure is much larger than the time required for grabbing the particle image at a given location, the diffraction pattern is spread along the trajectory of this particle. This can be seen as a convolution between the diffraction pattern and a blurring function resulting from the motion of the particle during the camera exposure. This article shows that the reconstruction of holograms recorded under such conditions exhibit traces that could be processed for extracting 3D trajectories.
© 2013 OSA
Digital holography has now largely replaced conventional holography for diagnostics in flows, and there are a significant number of articles concerning the numerical reconstruction of digital holograms [1–7]. With the recent development of CCD/CMOS cameras, it is now possible to record holograms in real time. For a few years, it has been known that the 3-D location of particles or fibers can be determined accurately by numerical reconstruction of holograms that are recorded directly by a camera. The in-line configuration is convenient for metrology in fluids where optical access is not easy. When the medium is illuminated by short pulses (typically a few microseconds for velocities up to several meters per second), the characteristics of a seeded flow (particle size, velocity vector fields) can be determined at a given time [8–14]. Multiple exposures have also been carried out to characterize 3D trajectories . For this purpose, original algorithms have been developed to extract motion or behavior of particles or microorganisms [16,17]. In the case of long pulse duration, for example a time exposure of 1 ms, a sharp trace of particle images is reconstructed [18,19]. These traces can also be used to extract the 3-D trajectories of particles. The aim of the present paper is not to propose a new method for tracking particles recorded by several short pulses holograms. This work has already been well developed by several authors. Here, we show that reliable information is given by long time exposure holograms. In particular, by expressing such images by a convolution operation between a short-pulse particle hologram and a displacement function, the signal-to-noise ratio (SNR) of the reconstructed images can be predicted. Some experimental results are shown.
2. Hologram recording of moving particle
2.1 Digital recording of the hologram
Here, wavelet formalism, used in our laboratory for a decade, is used for the description of the recording and reconstruction steps. Let be the binary amplitude distribution of an opaque object illuminated by a monochromatic plane wave, located at a distance ze from a quadratic sensor, as shown in Fig. 1. Note that the case of a plane wave is used for the sake of clarity, but as shown in , the reasoning can be extended to the case of a diverging beam.
From the Huygens-Fresnel integral, the complex amplitude in the detector plane can be expressed as the following 2-D convolution operation 11].
2.2 Influence of the displacement during the exposure
In industrial environments, particles may move across the object field with high velocities. This is generally considered as a drawback because the displacement during the exposure time induces modification of the fringe pattern recorded by the hologram. Such holograms can be described easily by a simple convolution operation between the diffraction pattern of a fixed image and a point spread function supposed to represent the displacement of the image . Consider an object moving with the velocity V. For this reasoning, V is supposed to be a constant and parallel with respect to the camera sensor during the exposure time τ (i.e., V = (Vx,Vy,0)). The particular case of zero velocity in z direction is first considered for the sake of clarity. Displacement in z direction will be introduced later in subsection 2.3. The angle between V and the x-axis is noted as . The CCD exposure during time τ is given byEq. (5) can also be expressed as a convolution operation of the exposure of a fixed pattern with a displacement function :Eq. (3) into Eq. (9), Eq. (8) gives:Eq. (11). Δ(x,y) represents the displacement of this particle during the exposure time.
2.3 Hologram reconstruction
With regard to the recording step, the intensity distribution in the reconstructed plane, located at a distance zr from the camera, can be calculated by a convolution operation:Eq. (9), Eq. (12) can be rewritten as:20]. As a result, the hologram reconstruction of fast moving particles is not very different from a time resolved hologram. The only difference observed in the reconstruction plane is the elongation of particle images with respect to the moving direction. This elongation is simply given by a convolution operation with a displacement function Δ(x,y). The recorded distance ze may also vary during the recording, especially if a long exposure time is considered. In that case, the 3-D trajectory of particles should be considered by interrogating several planes. Afterwards, a tilted plane adapted to the main direction of the particle should be defined in order to visualize the entire trajectory of the object. This situation is not different from the case where fiber holograms are recorded under arbitrary angles . However, it must be kept in mind that the lateral resolution is generally much better than the depth resolution . Consequently, the trajectory would not be estimated with the same accuracy along the x,y and z direction. Another important consideration concerns the Signal to noise ratio (SNR) of holographic images. Even if the noise term is reduced by a time average during the exposure, Eq. (13) shows also that the image contrast is reduced by the convolution operation with Δ(x,y). These contradictory effects are studied in the following section.
In order to show the interest of such a method, holograms with long exposure time have been simulated by applying Eq. (5). For this purpose, holograms have been simulated in the same way of multiple exposure holograms as explained in Ref. 15 where the time interval between pulses (here 2 µs) has been reduced to 2 µs. Figure 2 shows a hologram produced by a 60-μm particle located at a distance ze = 258 mm from the CCD sensor and moving with a constant velocity Vx = 0.025 m/s, Vy = 0.5 m/s during an exposure time of . Here, the noise level, simulated with a Box-Muller method , has been adjusted in order to recover the experimental SNR observed in a real configuration. As we can see on Fig. 2, the hologram pattern is smoothed along the velocity direction and outlines the convolution operation given by Eq. (9).
The images of Fig. 3 show the reconstructed holograms for several velocities in the range [0.1–3 m.s−1].
These images show the effect of the convolution operation between the object function and the displacement function Δ(x,y) as introduced in section 2. Figure 4 show the variations of the SNR of the reconstructed images with respect to the object velocity and for several exposure durations, i.e., τ = 200, 500 and 1000 µs. The SNR is defined here by the ratio of the image intensity to the standard deviation of the background intensity. As shown on this figure, the SNR remains unchanged for object velocities under 0.05 m.s−1. This result is not surprising because under this limit, the particle displacement remains roughly below the particle size (d = 60 µm). Therefore, the smoothing effect due to the convolution operation with the function is not noticeable while the SNR is improved by the time averaging. Conversely, when the velocity is higher than 0.1 m.s−1 a decrease of -10 dB/decade is observed. This is in good accordance with Eq. (13), where the convolution operation leads to a reduction by a factor of the image contrast. Considering that the SNR has to be higher than 7 dB  for an automatic processing, this simulation suggests that the displacement must be smaller than 2 mm for a particle diameter d = 60 µm.
The experimental points plotted on Fig. 4 will be discussed in the next section.
4. Experimental results
The experimental setup used for the validation is shown in Fig. 5. A laser diode illuminates the probe volume at wavelength λ = 635 nm. This light source is controlled by a synchronization device that enables the generation of a TTL signal. Although a diverging beam is used, the equivalent plane wave scheme of Fig. 1 can be used because that the far field conditions are assumed .
The resulting pulse duration of the laser diode τ can be adjusted continuously from 5 to 1000 μs. The moving objects are bubbles produced in a cavitation tunnel and observed through a transparent pipe. More details can be found in Ref . Note that the distances bubbles-camera are large . Hence, as shown by Slimani et al. , the diffraction patterns produced by the bubbles can be safely approximated holograms recorded with opaque objects. Holograms are digitized by a standard computer vision system with a lensless 1280 × 1024 CMOS camera. Figure 6 shows a set of experimental diffraction patterns recorded by the probe for different pulse durations τ in the interval [10-1000 µs]. In order to compare the experimental results with the model, we have chosen images of bubbles with diameters around 60 µm. The mean velocities are roughly evaluated by calculating the ratio between the estimated length of the traces and the time exposure.
The experimental results obtained from this set of holograms (SNR versus estimated velocity) are plotted in Fig. 4 and can be compared to the theoretical SNR obtained from the simulations. It must be mentioned here that both object size and recording distance ze play an important role in the image contrast , but that it was not possible to fix these parameters for this experiment. This can explain the discrepancy of experimental values from theory. However, in order to evaluate the effect of the time exposure, we have reported these values on a representation versus time exposure τ. The graph of Fig. 7 shows that the −5dB behavior of the SNR(τ) graph is rather well confirmed by the experimental points.
It must be mentioned that long time exposure holography can be considered as the limiting case of multiple exposure holograms where the time interval between pulses tends to zero. Consequently, it is comforting that similar results are obtained to as those presented in  (i.e., −5dB/decade slope of the SNR).
Digital in-line holography has been tested with a long time exposure. For high particle velocities, the smoothing effect applied to the diffraction pattern can be expressed as a convolution operation between the intensity function of the hologram and a displacement function Δ(x,y). We have shown that this function can also be applied on the reconstructed images. The simulations show that this simple model can be applied and the comparison with experimental results is satisfactory. Then, the velocities can be measured easily by the analysis of particle image trajectories. Note that in the present case, the sense of the velocity vectors is known a priori because the objects studied here are confined in a pipe (laminar flow). However, if the velocity sense is unknown, it can be extracted by using a double pulse illumination with a suitable sequence of pulse duration (short-long for example). The evolution of the SNR is essential to define the limits of such a method. The present study, realized under far-field approximation, shows that particles with velocities of up to 3 m/s can be processed easily with a time exposure in the range [10–1000 µs]. These results are probably strongly related to the numerical aperture used here. It could be interesting to extend this model to lager apertures.
The authors are grateful for the financial support from the European Network of Excellence Hydro Testing Alliance under the sixth Research Framework Program (TNE5-CT-2006-031316). This work was also supported by the program “3D” of the LABEX EMC3 (Energy Materials and Clean Combustion Center).
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