refined particle position localization in digital...

17th International Symposium on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2014

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Refined Particle Position Localization in Digital Holographic Particle

Image Velocimetry

Christina Hesseling1,*, Tim Homeyer1, Joachim Peinke1, Gerd Gülker1

1: Forwind – Center for Wind Energy Research, Institute of Physics, University of Oldenburg, Oldenburg, Germany

* correspondent author: [email protected] Abstract In digital holographic particle image velocimetry (DHPIV), the light field scattered by small tracer particles is recorded holographically. From the recorded holograms the movement of these particles within the flow of interest and therefore the flow movement is deduced. In each time step one hologram is recorded and the original light field, i.e. the intensity and the phase information, can be reconstructed from each hologram with the common convolution algorithm. In order to achieve a high temporal resolution and convenient usage, digital cameras are applied. We present here the impact of the particle position in relation to the digital sensor on the reconstructed light field. Besides optical experiments recording a water flow, numerical simulations are executed. Finally, we show particle traces resulting from particle positions which have been reconstructed with a novel reconstruction algorithm. 1. Introduction In particle image velocimetry a flow seeded with tracer particles is illuminated by laser light. The scattered light field is recorded and by reconstructing the movement of the scattering particles the behaviour of the fluid flow is measured indirectly. Classical PIV utilizes pulsed laser light illuminating the particles in one light sheet. Hence, only two velocity components in a two dimensional plane (2C 2D) are retrieved. In order to expand this technique to full 3D 3C measurement capability tomographic PIV has been developed [Elsigna]. Pulsed laser light illuminates a volume and usually the scattered light recorded by four cameras is used to retrieve the particle movement. The work presented here deals with a different approach to access full 3D 3C resolution. In-line DHPIV makes use of only a single camera. Furthermore, continuous wave lasers can be applied as the strong forward light scattering generates sufficient light intensities. Hence, the temporal resolution is no more limited by the repetition rate of the laser pulses but by the frame rate of the camera facilitating the usage of high speed cameras. In spite of its advantages, DHPIV is not yet widely distributed as especially the large depth of focus increases the error in particle localisation. Additionally, the complex fields generated by particles are sampled by sensors with low resolution and small dimensions in comparison to analogue, film-based recordings. In the optical experiments presented in this work particles with diameters of (8.69±0.12)µm and a pixel pitch of 12µm are applied. This yields poorly sampled light fields, and, additionally, border effects are caused by the truncation of interference fringes [Gire], which facilitates false and multiple particle detections. Traditionally, most algorithms for the reconstruction of particle positions are based on the intensity distribution of the reconstructed volume. Digital holography offers direct access to the complex amplitude of the light field. Hence, also the phase information can be utilized. Still, up to now it has only received attention by comparatively few algorithms, for example by [Yang], [Pan] and [deJong]. In our work a novel approach using the full complex wave reconstructed from the recorded holograms is used in order to reduce errors. Furthermore, the border effects are intensively studied and are taken into account in the algorithm. The measurement method is tested with a water flow induced by a magnetic stirrer. An algorithm for the reconstruction of particle positions from the recorded holograms has been developed, which is subdivided into three steps. Firstly, the holograms are preprocessed in order to enhance the interference structures of interest in the hologram data. Secondly, the three-dimensional complex light field corresponding to each of the holograms is reconstructed. Thirdly, particle positions are detected within these fields for each time step. The detected particle positions provide the basis for particle tracking. In the reconstruction as well as in the detection step novel approaches have been utilized in order to avoid misinterpretation of random constructive interference structures (speckles) as particles and to increase the precision of the particle localization.


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2. The Experimental Setup Figure 1 shows the setup applied in the fluid flow experiments. Beam cleaning and expansion optics are used to expand laser light of a Nd:YAG laser to illuminate a glass cuvette filled with water and tracer particles with plane waves. In order to prevent scattering from the sides of the glass cuvette, a rectangular aperture is inserted in front of the glass cuvette. In favour of a minimal influence of reflected light on the measurement, the reflecting surfaces within the optical path, for example glass surfaces, are slightly tilted with respect to each other. Partly, the light is scattered by the tracer particles in the cuvette forming the object light fields while the remaining part is serving as the reference light. These waves interfere on a CMOS camera sensor generating a digital hologram. In this way more than 600 holograms per second can be recorded with this setup.

Fig. 1 Experimental setup for hologram recording: The expanded laser light illuminates a water flow induced by a

magnetic stirrer in a glass cuvette, is diffracted by polystyrene microparticles in the water, and is recorded as in-line hologram on a CMOS camera chip.

3. Filtering of Holograms As the particles are in motion, they yield interference structures moving in time. Therefore, these structures change position and shape in a time series of holograms. Besides these moving structures, the hologram sequences contain constant interference structures, which are caused by dirt or reflecting glass surfaces for instance. Especially the protective window of the CMOS camera sensor causes distinct interference structures. So the holograms are filtered by subtracting a normalized average of all holograms in a time series from each of the recorded holograms. A demonstration of this processing step in holographic microscopy can be found in [Ooms]. Figure 2 displays the effect of this procedure showing one originally recorded hologram and the same hologram after this filtering process, both being normalized to 256 gray values. Clearly, the circular interference structures expected as particle holograms are enhanced in comparison to previously dominant interference structures generated by temporally constant objects.

Fig. 2 Originally recorded hologram (left) and the respective filtered hologram (right).

5 cm

Nd:YAG 532 nm

20x 0.4 CMOS

PC

neutral density filter

shutter pinhole 20µm

microscope objective 5 cm

lens f=50cm

rectangular aperture

Glass cuvette filled with water and polystyrene particles


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4. Reconstruction of the Complex 3D Light Field From the filtered holograms the three-dimensional complex light fields are reconstructed. For this purpose the common convolution approach [Kreis] is utilized in the Fresnel approximation [Goodman]: The convolution of the kernel

𝐾 𝑧 = !!!!⋅ 𝑒𝑥𝑝 ! ! (!!!!!)

!⋅!,

omitting the factor 𝑒!"#, with the hologram 𝐻 is the complex amplitude of the recorded wave field in the reconstruction distance 𝑧

𝐴 𝑧 = 𝐾 𝑧 ⨂𝐻 where 𝜆 describes the wavelength of the used laser light and 𝑥 and 𝑦 span the coordinate plane perpendicular to the optical axis of the system. In favour of their legibility, the dependences on 𝑥 and 𝑦 have been omitted. The origin of the 𝑥𝑦-plane is located on the optical axis. Hence, in the direction of the optical axis, i.e. in the 𝑧-direction, the reconstructed complex light field is sampled according to the chosen reconstruction distance while in the transverse plane, i.e. in the 𝑥𝑦-plane, the sampling of the light field is predefined by the resolution of the camera sensor. Still, also the resolvable information in the z-direction depends on the spatial resolution of the sensor. The longitudinal depth of focus of a reconstructed particle intensity image is approximated by !

!! with Ω as the numerical aperture of the recording system [Hinsch]. In the setup used in

the present study, the numerical aperture is restricted by the spatial resolution of the camera sensor. On the one hand the pixel pitch of the sensor 𝑑!! yields a Nyquist limit to the recordable spatial frequency of !

!!!!.

As explained in detail by [Zhang], the dimensions of the camera sensor can be used to restrict the spatial frequencies which reach the sensor to the Nyquist limit by using the sensor as a spatial low pass filter. For a particle on the optical axis, the sensor size limits the frequency to 𝑓 = !!⋅!!!

!!" where 𝑛! is the number of

pixels in the respective dimension. This is utilized in the system of the present study where no Fourier filter is introduced. 1024×1024 pixels of a camera sensor with a pixel pitch of 12µm are used. Therefore, the minimum recording distance is set to approximately 28cm. In total, this yields an expected depth of focus of the intensity distribution surrounding a particle of approximately 1mm. Furthermore, the accidental interference of several particles can yield constructive interference phenomena in the reconstructed light field. This means that in positions where no particles have been during the recording process, intensity maxima, called speckle, can appear. The prospect of this work is a reconstruction with an improved longitudinal localization and an exclusion of speckle in the particle detection process. For this purpose, besides the intensity information of the reconstructed light field, also the phase information is employed. As will be shown in the next section, the phase information has a characteristic shape, which has been demonstrated by [Yang]. We additionally demonstrate and explain distinct alterations of the shape of the intensity and phase patterns with the transverse location of the respective particle in relation to the camera sensor. 5. Impact of the particle position on the reconstructed complex light field Figures 3 and 4 show the intensity and phase distributions surrounding positions of two reconstructed particles. The coordinates are defined by a coordinate system starting in the upper left corner of the respective recorded holograms and are given in pixel units. The horizontal coordinate is referred to by 𝑥 and the vertical coordinate by 𝑦. The geometric centre of the holograms is therefore in position 𝑥, 𝑦 =(512.5,512.5) while the hologram edges correspond to the values 1 and 1024. The experimental data shown here has been retrieved from a water flow recorded by the setup sketched in figure 1. The particle positions have been retrieved by means of the algorithm which is paraphrased in section 6. Due to changes of the refractive indices within the setup, the reconstructed volume is compressed. Therefore, the depth coordinate (𝑧) retrieved from this experiment is also compressed [Hesseling]. Figures 3 and 4 show the 𝑥𝑧- and the 𝑦𝑧-planes through the phase and the intensity fields reconstructed around two retrieved particle positions. The particle in figure 3 is localized comparatively close to the optical axis of the system while the particle in figure 4 is slightly shifted in the 𝑥-coordinate and distinctly shifted in the 𝑦-coordinate closer to the edge of the sensor. Figure 3 shows a highly symmetric phase and intensity pattern which corresponds to measurements made by [Yang] of a particle on a glass slide and respective numerical simulations, both referring to locations on the optical axis.


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Fig. 3 Wrapped phase (upper) and normalized intensity (lower) values in the 𝑥𝑧 and 𝑦𝑧 planes surrounding a

central reconstructed particle position from an experimentally recorded hologram.

Fig. 4 Wrapped phase (upper) and normalized intensity (lower) values in the 𝑥𝑧 and 𝑦𝑧 planes surrounding a

reconstructed particle position close to the left edge of a hologram and horizontally close to its centre. In spite of this, our experiments reveal a distinct effect of the transverse location of a particle on the respective reconstructed light field. This is demonstrated in figure 4. While the left part in figure 4 only shows a slight deformation of the phase and the intensity cone, an asymmetry of the phase and intensity cones in the 𝑦𝑧-plane is clearly visible. In order to distinguish this from other effects, particle holograms are simulated in different transverse locations and are used for the reconstruction of complex light fields. A circular disc with a radius corresponding to the radius of the experimentally employed particles is used to simulate a particle. The transverse resolution of the simulated light field has a pixel pitch corresponding to 0.5µm. Therefore, the matrix generated in the simulation process has the dimensions of 24576×24576 pixels. Similar to the calculation done by [Yang] and in accordance with Babinet’s principle the object wave is simulated as a superposition of a wave diffracted by a circular aperture of appropriate size and a plane wave. The corresponding wave is propagated according to Fresnel-Kirchhoff diffraction theory. The central disc of the resulting hologram in 30cm distance to the diffracting particle is shifted to the transverse position of interest, and the resolution of the hologram is downsampled to the resolution of the camera sensor. Finally, the resulting hologram is reconstructed similarly to the experimentally recorded holograms. For the two transverse locations shown in figures 3 and 4, this process yields the data shown in figures 5 and 6. These patterns are compared to the data shown in figures 3 and 4. The intensity as well as phase structures seen in the experiment are clearly visible in the simulated data fields. This is the case, although, the experimental data have been retrieved from particles moving in a water flow and the respective holograms


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have been recorded after the light field has passed several layers of changing refractive indices while the refracting surfaces are slightly tilted. A fluctuating phase value around the transverse centre of the simulated hologram, which is not visible in the experimental data, is attributed to truncation errors in the simulation, which is partly executed in single precision. The high symmetry of figure 3 can be found in figure 5 again while the deformation of the intensity as well as the phase symmetry in figure 4 is clearly visible in figure 6. So this phenomenon is traced back to the transverse location of the particle which means that interference fringes are truncated by the edges of the sensor. It is clearly not caused by other effects, like other particles surrounding the particle of interest, for instance. Therefore, depending on the particle position, modified particle characteristics are expected in the reconstructed light field. This insight is utilized in the particle detection algorithm which is paraphrased in the following section. Furthermore, characteristics expected from a particle position which are independent of its transverse location are identified as well: A higher intensity value than in the surrounding volume, and on the axis parallel to the optical axis through its transverse position a smaller phase change than in the surrounding volume. Only close to the 𝑧-position of the particle, the phase value is expected to jump from approximately –π to π.

Fig. 5 Wrapped phase (upper) and normalized intensity (lower) values in the 𝑥𝑧- and 𝑦𝑧-planes surrounding a

reconstructed particle position near the hologram centre at (𝑥, 𝑦) = (519,511) pixel from a simulated hologram.

Fig. 6 Wrapped phase (upper) and normalized intensity (lower) values in the 𝑥𝑧- and 𝑦𝑧-planes surrounding a

reconstructed particle position close to the left edge at 𝑥, 𝑦 = (427,86) pixel from a simulated hologram.

6. Detection and Tracking of Particle Positions The particle detection algorithm developed in the course of this work uses three-dimensional spatial correlation for the detection of particle positions. Three-dimensional simulated reference patterns are correlated with the light field reconstructed from a hologram, and the correlation values are used to detect particle positions. We have demonstrated that, depending on the transverse location of a diffracting particle, the respective reconstructed light field changes its characteristics significantly. Therefore, the algorithm employs different reference patterns in different transverse locations of an expected particle position. Figures 3 to 6 demonstrate that two characteristics can be expected from a transverse particle position

0 0


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independent from its location: The reconstructed intensity is higher than in the surrounding volume and the phase value on the axis parallel to the optical axis through the centre of the particle is less fluctuating than in the surrounding area. We refer to this value as the absolute phase gradient in 𝑧 -direction. These characteristics are used for a first rough estimate of the transverse, i.e. the 𝑥𝑦-, position of a suspected particle position. Then, a reference pattern is generated which corresponds to a pattern expected from a particle in this transverse location. Prior to the execution of this particle detection routine, look-up tables of reference patterns are stored in memory. These data are simulated similarly to the data shown in figures 5 and 6. The size of a reference pattern is 9 pixels in each transverse dimension and 49 planes in length while the distance between two planes corresponds to 50 µm. Therefore, each reference pattern corresponds to 108µm×108µm×2450µm. In the calculation of the correlation values, only a cylinder in each pattern surrounding the axis in the suspected transverse particle position is used. All reference patterns are generated for a simulated particle in a fixed distance of 30cm to the camera chip, which means that only the 𝑥𝑦 position is modified. Due to the cost of the calculation, reference patterns are not calculated for all pixel positions of the camera sensor but on a predefined grid and are interpolated accordingly whenever a position is to be investigated which does not correspond to this predefined grid. As our experiments and simulations have also shown a distinct effect of the particle position in comparison to the pixel grid on the reconstructed optical field, particle positions are not only simulated in centres of pixels on this pixel grid but also on pixel edges and crossings. This means that the 𝑥𝑦-position of a particle is located either in the centre of a pixel, on a horizontal or vertical pixel edge or in the centre of four pixels, which changes the sampling of the hologram and therefore the respective reconstructed image. For this reason, four different reference patterns are generated for each rough estimate of a transverse location of a particle and are correlated with the three-dimensional intensity and phase fields in the whole depth in the area surrounding this transverse location. Finally, the reference field and respective transverse and longitudinal particle coordinate with the highest correlation value is retrieved. In the correlation process, windowing is avoided. Therefore, a larger depth range is reconstructed from each of the holograms than corresponding to the flow of interest and particles close to the sides of the camera area are not detected. In order to ensure that no speckles are retrieved in this process, further validation criteria have been introduced in the algorithm. They refer, for example, to the distance between the detected depth coordinate and the nearest jump in the phase values in the detected 𝑥𝑦 position. Finally, the detected particle positions are processed by a tracking code written by [Blair] following the algorithm developed by [Crocker], and particle trajectories can be visualized with polynomial fits. One example based on the particle positions reconstructed from 30 holograms can be seen in figure 7. We cannot expect to see a swirl induced by the magnetic stirrer as a time interval of approximately only 0.05 seconds is reconstructed and, additionally, the cuvette is angular. On the left side of the figure, particle trajectories which are projected to the 𝑥𝑦-plane are shown. We find sensible trajectories which start or end close to the sensor edges which we attribute to the novel approach of tailoring the reference patterns to the transverse location of a particle. The precise location of the particles in the flow is unknown. Still, a first estimate of the error which is made in the localization of the particle positions, is the root mean square error of the traced particle positions in each trajectory to the polynomial fits generated for each trace. In the 𝑥𝑦 plane, this error is less than 60µm in each dimension while in the 𝑧-direction an error of approximately 300µm was retrieved in this way. Further validation experiments of this algorithm are in progress.

Fig. 7 Particle trajectories tracked from detected particle positions from 30 holograms. The final position of each

trajectory is marked with a circle, left: projection to the transverse plane, right: volumetric representation.

x/mm

y/mm

z/mm


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7. Acknowledgements The authors thank you for having shown interest in our research by reading this text and the Nieder-sächsische Ministerium für Wissenschaft und Kultur (Ministry for Science and Culture of Lower Saxony) for financial support.

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velocimetry. Exp Fluids 41:933-947. • [Goodman] Goodman JW (2005) Introduction to Fourier Optics, 3rd ed. Ben Roberts. • [Gire] Gire J, Denis L, Fournier C, Thiébaut E, Soulez F, Ducottet C (2008) Digital holography of

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46(31):7652–7661. • [Kreis] Kreis, TM (2005) Handbook of Holographic Interferometry, Optical and Digital Methods. Wiley-

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refined particle position localization in digital...

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