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# Development of a defocused PTV technique for 3D3C measurements using a single camera. Application to the dynamics of a pulsed laminar jet

Kayvantash Tony, Cordier Mathieu, Duval Hervé, Laurent Zimmer

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Kayvantash Tony, Cordier Mathieu, Duval Hervé, Laurent Zimmer. Development of a defocused PTV technique for 3D3C measurements using a single camera. Application to the dynamics of a pulsed laminar jet. 16th International Symposium on Particle Image Velocimetry, ISPIV, Jun 2025, Tokyo, Japan. hal-05403389

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https://hal.science/hal-05403389v1

Submitted on 8 Dec 2025

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# Development of a defocused PTV technique for 3D3C measurements using a single camera. Application to the dynamics of a pulsed laminar jet

T. Kayvantash<sup>1,2,3*</sup>, M.Cordier<sup>1</sup>, H. Duval<sup>2</sup>, L. Zimmer<sup>3</sup>

## Abstract

This work aims to demonstrate the accuracy of an optical technique using defocused images of particles and a single view. To avoid overlap of the images, a special mask is inserted between the lens and the camera to limit the signal of each particle to the external ring only. To determine the effects of the mask on the uncertainties and accuracies of the technique, a special optical setup is developed. It is composed of a continuous laser (532nm, 500mW). The laser beam is transformed into a thin laser sheet using a plano-cylindrical lens. A transparent resin bloc containing millions of particles is then illuminated. A high-speed camera placed at 90 degrees captures the images through a lens of 105 mm and an aperture of 2.8. f. To simulate a flow, movements are imposed with a mechanical translation.

The effects of moving towards the bloc have been quantified using a Particle Tracking approach. Results show a constant increase of the apparent size with respect to the distance to the focal plane. We tried to investigate displacement towards the bloc.

The defocalisation technique is then applied to a real case of oil droplets within a pulsed laminar air jet. The forcing frequency is well-defined to retrieve the vortices by the pulsation of the flow. The camera used for the Particle Tracking Velocimetry (PTV) is placed at a forward angle to compensate for the decrease in intensity. A second camera is placed at 90 degrees to measure the velocity field with Particle Image Velocimetry (PIV). The 3D path of three example particles is plotted using results from the calibration.

## 1 Introduction

Following particles within a flow in 3D typically requires 2 high-speed cameras. This represents a large quantity of data to post-process as well as a significant financial cost. Different techniques have been investigated to perform a 3D particle tracking velocimetry (PTV) with a single camera. For example, Hoyer et al. (Hoyer et al., 2005) used a 3D scanning PTV to track particles within a homogeneous turbulent flow. More recently, Steinmetz et al. (Steinmetz et al., 2025) used periodic coded aperture to estimate the 3D position with one camera and performed particle image velocimetry on airflow near a ducted ship propeller and Xu et al. (Xu et al., 2025) used a convolutional neural network to handle a higher particle concentration.

Another technique, which originates from Damaschke et al. (Damaschke et al., 2002), is to deduce the depth of the particle within the flow using defocalisation.

A consequence of the defocalisation technique is that the particles occupy more pixels on the image, and thus can overlap and hide each other. To avoid this, Lin et al. (Lin et al., 2008) had the idea to introduce a circular mask between the lens and the camera's sensor to hide the center of the defocalised particle-image, only leaving an annular shape. With this method, they were able to obtain a precision of 23 µm in depth over a range of 1 mm. This method has also been used by Baudoin & Zimmer (Baudoin

<footnote n="1"><sup>1</sup> IFP Energies nouvelles, Institut Carnot IFPEN Transports Energie, 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison, France</footnote>

<footnote n="2"><sup>2</sup> LGPM, CentraleSupelec, Universite Paris Saclay, Gif-sur-Yvette, France</footnote>

<footnote n="3"><sup>3</sup> Universite Paris-Saclay, CNRS, CentraleSupelec, Laboratoire EM2C, Gif-sur-Yvette, France</footnote>

<footnote>* tony.kayvantash@ifpen.fr</footnote>
and Zimmer, 2015), who observed that the mask reduced the luminosity of the signal by 23%.

The first instance of the PTV algorithm was developed by Maas et al. (Maas et al., 1993). PTV algorithms are typically made of two phases: a particle detection phase and a following phase. This overall structure is still found in more recent works such as Jeon et al. (Jeon et al., 2021).

The detection and PTV algorithm is adapted from the works of Baudoin & Zimmer (Baudoin and Zimmer, 2015) and Parant & Zimmer (Parant et al., 2022). The algorithm detects then follows the detected particles until they are lost. The detection is made by generating a series of patterns of varying diameters and creating a correlation map for each pattern size. The algorithm searches for the highest correlation peak to identify the particles. Thus, the diameter of the pattern associated with the correlation peak is attributed.

This algorithm will be applied to the present work, in which the feasibility of the defocalisation method is demonstrated in an environment where the displacement of the particle is controlled. Then the calibration for following oil droplets within a pulsed air jet will be detailed. The frequency of the pulsed laminar jet is well known as the set-up is mostly adapted from (Lee et al., 2024). Finally, the path of a few particles within the flow will be represented in 3D to showcase the performances of the technique.

In the future, the defocalisation technique is planned to be applied to biphasic flow, which typically requires lower seeding, hence why an approach such as Shake The Box was not used here.

## 2 Proof of concept in a controlled environment

The defocalised technique is first applied to a controlled environment. Particles are still and contained within a resin block. They are illuminated by a laser sheet created using a continuous laser (532nm, 500mW) and a plano-cylindrical lens. A high-speed camera (Fastcam SA5 Photron) with a 105 mm lens (AF-DC Nikkon) is placed perpendicularly to the laser axis and captures the particles using a lens of 105 mm and an aperture of 2.8 f. Multiple masks were tested as displayed in figure 2, although mask n°2 ended up obfuscating too much light to be usable. The position of the particles relative to the objective is controlled using a micro-controller MM2500 Newport with a ±0.5 µm precision by moving the camera perpendicularly to the laser sheet.

<figure><img src="image_3.png" /><figcaption>Figure 1: Experimental setup</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 2: Masks tested alongside their dimensions (mm).</figcaption></figure>

An example of a particle-image as it appears using mask n°1 for different distance to the focal plane can be observed in figure 3.

<figure><img src="image_5.png" /><figcaption>Figure 3: Typical images for different distances with respect to the focal plane (from left to right and from top to bottom 0-10-20-30 mm)</figcaption></figure>

The perpendicular movement controls the defocalisation of the particle-images as they gradually grow further away from the focal plane. It should be noted that a minimum defocusing is necessary to enable good accuracy of the apparent size determination. Therefore, apparent sizes below 5 px in diameter are not presented as not useful for later experiments.

The camera-to-block distance is first adjusted to have the illuminated particles in the focal plane. Then, the camera is moved away from the focal plane by increments of 1 mm, where a picture of the illuminated particles is taken.

This process is repeated for 81 planes ranging from a distance to the focal plane of 0 to 80 mm, and for each mask. The diameter of the particle-images is then measured by correlation with generated patterns using Matlab.

The images are then correlated with generated patterns of varying diameter (from 5 to 50 px). The patterns follow a Gaussian profile and are made to be as close to the particle-images as possible, as displayed in figure 4.
<figure><img src="image_6.png" /><figcaption>Figure 5: Apparent diameter of the largest particle identified for each position and for masks n°1, 3 and 4.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 4: Examples of generated patterns compared to particle-images of the same size obtained with mask n°1. On the left: 15 px diameter, on the right: 30 px diameter</figcaption></figure>

Let's note $C_i$ the pattern of diameter i px. A Fast Fourier Transform is applied to all $C_i$ and correlated to the image to create the correlation maps $M_i$. An additional map $M_{max}$ is created from the maximum correlation value within all maps $M_i$ for each pixel. Correlation peaks are researched within $M_{max}$ to identify the position and diameter particle-images. Furthermore, identified particles are followed on the next 2 subsequent planes.

It should be noted that the mean pixel displacement is slightly changing when the camera is moved towards the bloc, a consequence of the change in pixel to mm conversion factor. The signal to noise ratio stays also relatively constant for the different focal planes. For each plane and for every measured displacement, the calculated SNR were within a 2 to 3 range, which demonstrates that the correlation quality remains good despite the defocalisation.

For masks n°1, 3 and 4, the apparent diameter of the particle-images relative to their distance from the focal plane is plotted in figure 5. The higher the slope, the more precisely we would be able to deduce the position of the particle relative to the focal plane. In this respect, mask n°4 seems to be the better of the 3 geometries. However, more particles are detected when using mask n°1 than with masks n°3 or 4 as shown in figure 6.

While the defocalisation technique worked in the controlled environment, the results in 5 cannot be used to calibrate a real study because the exact position of the laser sheet within the resin block is unknown. The position of the focal point relative to the particles will also influence their apparent size. The following section will thus detail the calibration method.
<figure><img src="image_8.png" /><figcaption>Figure 6: Number of particle identified when using mask n°1, 3 and 4, relative to the distance to the focal plane.</figcaption></figure>

## 3 Calibration on particle in airflow

In this paper, the defocalisation technique is applied to a pulsated airflow seeded with oil droplets. The airflow is controlled using a Bronkhorst F-210AV-50K-AAD-33V Flowmeter and is maintained at 12 L/min. It is then seeded with oil droplets using the seeder developed by Durox et al. (Durox et al., 1999). The airflow can then be pulsed using a MOOG valve (R16K01MONSM2) piloted with a function generator, although the flow isn't pulsed for the calibration. The flow is injected through a burner with a 13.5 mm diameter outlet.

The oil droplets are illuminated by a laser sheet created using a 532nm Nd:YAG laser (Edgewave InnoSlab). This laser can be pulsed at up to 10kHz, but here we are shooting at 1kHz because the laser sheet is very thin during the calibration, thus more energy per pulse is useful to see the droplets. With this configuration, the laser shoots at 22.99 mJ/pulse.

The illuminated particles are recorded with a fastcam Photron SAX2 highspeed camera, capturing the images at 1kHz. The focal point of the camera is placed behind the burner to observe particle images of 6 to 15 px diameter wide. To ensure that particles are more easily detected, mask n°1 is placed over the lens of the camera. As the mask reduces the illuminance received, the camera is moved to receive more light thanks to Mie scattering. The optimal angle has not been investigated yet in this study, thus the camera is placed at a 38° angle from the laser sheet, as shown in figure 7.

The laser sheet is created with the same plano-cylindrical lens as in the previous section. For the calibration specifically, the laser sheet should be thin to have little variance for the apparent size of particles within one measurement. This is done using a thin slit placed between the lens and the burner. Several obstacles are also present to block parasitic reflections from the burner. The setup is represented in figure 7.
<figure><img src="image_9.png" /><figcaption>Figure 8: Example of calibration image. Very few particles are actually visible on each frame, so signal is summed over multiple frames to obtain this picture.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 7: Experimental setup during the calibration.</figcaption></figure>

The calibration is performed by moving the thin slit perpendicularly to the laser sheet, recording the particles illuminated by the thin laser sheet, and measuring their apparent diameter with the same algorithm as for the previous section. The thin slit is moved by 200 µm increments with micrometric precision with a micro-controller.

The thin laser sheet highlights that the particle-images appear smaller on the left of the sheet than on the right, which is expected as they are closer to the camera due to the angle displayed on 7. Thus, the position of the particles on the third dimensional axis would be a function of both the apparent size and the horizontal position on the image. The measurements are done on planes 200 µm apart from each other, and the corresponding apparent size (AS) of particles is then interpolated on the domain represented in full colour in figure 9.

Outside of this domain, the apparent size is estimated from the trend lines obtained when plotting AS as a function of the position of the particle for each plane.

$$AS \approx 7.9722 - 0.4000 \times z + 0.0611 \times x \quad (1)$$

Where AS is the apparent size of the particle in px, x is its horizontal position on the image in px, and z is its distance to the center of the burner in mm. The x and z axis refer to the position of the particle in the frame of reference of the camera, as displayed in figure 7. The domain where this estimation is used is represented by the hatched area in figure 9.

Thus, the apparent size of the particles detected on the calibration pictures is plotted relative to the laser sheet position and the particle position on the image in figure 9. This estimation seems coherent
<figure><img src="image_11.png" /><figcaption>Figure 9: Apparent size of particles identified during calibration, with their position on the image and the position of the laser sheet relative to the center of the burner.</figcaption></figure>

with the interpolated results.

Using these results, the position of the particle on the z-axis is deduced from its position on the x-axis and the apparent size of the detected particle. This will be showcased on a few particles in the following section.

# 4 Application of the defocalisation method to 3D PTV

In this section, the defocalisation method is used to follow particles in 3 dimensions thanks to the previously performed calibration. The experimental set-up remains mostly unchanged save for the 3 following points.

- The thin slit is removed so we can observe a larger slice of particles within the airflow. Because the laser sheet is larger, we can afford to reduce the amount of energy per pulse. Thus, the laser is shot at 2 kHz, with 21.35 mJ/pulse.

- The airflow is now pulsed with a sinusoidal function at 78 Hz. This frequency was chosen empirically as an hot wire anemometry revealed that it would resonate with the burner to result in a sinusoidally pulsating airflow.

- A second camera (a fastcam photron SA5) is added to perform a 2D PIV to measure the velocity field within the flow. The velocity field is needed when following the particles to predict their position on the next frame. An example of velocity field calculated is displayed in figure 10. It should be noted that the px/mm ratio would be different on both cameras, and needs to be considered during the calculations, alongside the angle formed by the position of the two cameras. Both cameras are piloted to start the recording simultaneously, although it was noticed that the SAX2 starts 1.4 ms after the SA5. This gap was taken into account during calibration.
<figure><img src="image_12.png" /><figcaption>Figure 10: Left: Example of velocity field calculated by PIV; Right: 2D path of example particles</figcaption></figure>

To showcase the potential of the defocalisation technique, 3 different particle trajectories are followed here: one at the center of the airflow, and two in the whirls. Their 2D position on the image are represented in figure 10. The left particle (in green on the figure) is followed over 29 frames while the other two are followed over 40 frames (blue and red respectively). The whirls are visible from the position of the red and green particles. It should be noted that these three particles are not present at the same frames, and are only represented together for clarity sake.

The apparent size of the particles followed is also calculated alongside their position and plotted in figure 11. This figure shows noticeable jumps in the apparent size. The jumps are especially noticeable during the first 10 frames for the left example particle and around frame 10 for the center example particle. For the results displayed, the sub-pixel measurements were calculated with parabolic interpolation but a Gaussian interpolation was also tested for similar results, thus this might not be an interpolation issue. We are currently investigating this issue.

It should be noted that because of the angle between the laser sheet and the camera, particles appear larger on the right of the image than on the left. The fact that the left particle displays little variation of apparent size in figure 11 compared to the other two despite being within a whirl.

<figure><img src="image_13.png" /><figcaption>Figure 11: Apparent diameter of the three particles followed. Left: Obtained from correlation with pattern. Right: Obtained from window deformation</figcaption></figure>

In order to better quantify the slight changes in the apparent size between each frame, a specific approach is proposed in this paper. The images of each particle at different time steps are cropped. The
<figure><img src="image_14.png" /><figcaption>Figure 12: Trajectories of the 3 particles followed calculated with window deformation method (red). The origin of the axes is placed at the center of the burner.</figcaption></figure>

algorithm developed will try to find the best parameters to transform the first image into the second one. Window deformation techniques are widely used in image processing and this procedure does not require long CPU time; The three parameters to optimize are the horizontal and vertical position, while the third one will be the magnification (which can be below or above 1). The three optimal parameters allow to get a different view on the trajectory of the three particles. The variation of their apparent sizes can be seen in (figure 11). As seen, the results presented are smoother than the one obtained using correlation with patterns. Notably, the apparent size of the particle on the left doesn't show the expected whirl movement, whereas the right one does exhibit such a movement. A closer inspection of left particle revealed that its apparent size on the image doesn't seem to decrease, so this result is coherent with the images captured.

From the window deformation technique, the position of the particle in 3D is plotted once again. Figure 12 shows that the window deformation does not suffer from the z-axis jumps observed with the pattern correlation. The path obtained with window deformation method is coherent with what can be observed in the images: the whirls can be seen on the path for the particles in both whirls. The particle in the left whirl is shown to move away from the center of the burner as its apparent size increases as it is followed. The one at the center can be seen to deviate after around 30 frames, which is coherent with the 2D path observed in figure 10.

# 5 Conclusions

In this article, a method of 3D PTV using defocalisation is displayed. Masks are placed over the camera's lens to avoid the superposition of particle images by changing the shape of the particle images into empty circles.

Several geometries of masks were tested. The geometry of the mask proved to be important as it affects the precision of depth measurement, and the amount of particles detected. Among the four geometries tested, mask n°2 was obfuscating the image too much to be usable, mask n°1 resulted in the most particle detected, and mask n°4 in the highest depth precision. A definitive best geometry has not been decided, although mask n°1 was chosen for the PTV for a better particle tracking.

The study was conducted on oil droplets within a pulsed airflow. A calibration was performed in order to quantify the relation between the size of particle images and their position in 3D space. Using this calibration, the 3D position of oil droplets within an airflow pulsed at 78Hz was calculated and the
3D trajectories of a few droplets were plotted.

However, the apparent size measured from correlating the images with patterns showed occasional jumps that are not representative of real particle displacement. This happened for both parabolic and gaussian sub-pixel interpolation. An approach by window deformation has been used instead for a more coherent result.

This particular study only highlighted the 3D path of three selected particles within the flow, but the method could be applied to the other particles detected.

## Acknowledgements

Put acknowledgments here.

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