Three-dimensional, three-component particle imaging …ltces.dem.ist.utl.pt/lxlaser/lxlaser2008/papers/03.1_2.pdf · Three-dimensional, three-component particle imaging using two

14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008

- 1 -

Three-dimensional, three-component particle imaging using two optical

aberrations and a single camera

Nils P. van Hinsberg1, Ilia V. Roisman

1, Cam Tropea

1

1: Department of Fluid Mechanics and Aerodynamics, Technische Universität Darmstadt, Darmstadt, Germany, [email protected]

Abstract We present a method for three-component, three-dimensional flow velocity measurements using a single CCD camera. The positions of the particles in physical space are determined by using two optical aberrations, namely astigmatism and a spherical lens aberration. Due to the combination of these two aberrations the particle images have either an ellipse-like or a star-like shape, depending on their position in depth. The ratio of the major horizontal and major vertical axis encodes the depth position of the particles. The measurement principle, the calibration procedure and the data processing algorithm for particle depth determination and velocity field calculation are explained in detail. Finally, the validation of the technique is illustrated by means of a laminar uni-directional flow.

1. Introduction

In the last decades, a considerable amount of effort has been put into establishing new techniques to measure three velocity components in a flow volume (3D-3C). These techniques allow three-dimensional structures of unsteady flows, as well as their development in time to be investigated. One of the earliest of these techniques is particle tracking velocimetry, in which single particles are followed in time to determine their velocities (Racca and Dewey (1988), Virant and Dracos (1997)). The drawbacks of this method are its restricted application to flows of moderate Reynolds numbers and the use of multiple cameras to uniquely determine the position of the particles. Another technique is stereoscopic PIV, where two cameras, placed in an angle larger than 30˚ to one another, are used to image the particles inside a laser light sheet, producing velocity information in a quasi 2D plane. A limitation of this technique is the requirement of a large optical window, as the cameras are spatially separated. Scanning PIV (Brücker (1995)) is a technique, which uses thin light sheets to scan the flow volume in order to determine the velocity fields at different depth positions inside the flow. Brücker determined the out-of-plane component of the velocity by making use of the continuity equation. However, due to the limited temporal scanning speed, this technique is restricted to flows with relatively large time scales. Another common technique to measure the third velocity component is holography (Meinhart et al. (1994)). They showed that with holographic PIV the measurement of about 105 instantaneous velocity vectors in a flow volume of 100 × 100 × 100 mm3 was achievable. This technique, however, requires a camera with a high data storage possibility as well as a complicated and time consuming evaluation process.

Two relatively new techniques are tomographic PIV (Elsinga et al. (2005)) and defocusing 3-D PIV (Willert and Gharib (1992)). In tomographic PIV a reconstruction of the measurement volume by means of tomographic algorithms is performed. As in PTV, multiple cameras are needed to reconstruct the tomographic image and to determine the exact position of each particle. As it is not always possible to position multiple cameras around the experimental set-up, new 3D-3C techniques have been developed based on a single camera. Willert and Gharib (1992) introduced a defocusing technique based on a single camera equipped with a three-hole aperture due to which a single particle is imaged on the camera chip as three particles forming an equilateral triangle. The depth of the particles can be calculated from the positions and the lengths of the triangles.


- 2 -

Unfortunately, due to the small holes of the aperture, a high laser power is required in combination with large tracer particles.

Another 3C-3D velocimetry measurement technique using a single camera has been introduced by Angarita-Jaimes et al. (2006) based upon the measured wavefront scattered by the tracer particles. They described two approaches to capture the data required to measure the wavefront: multi-planar imaging using a distorted diffraction grating and an anamorphic technique. With the first technique three images are obtained per particle corresponding to the three diffraction orders. The image diameters are linearly related to the distance of the particle from the focal plane. The anamorphic technique makes use of a combination of a cylindrical lens with a spherical lens, producing for each particle a different image size in the x- and y-direction, which define the length of the major and minor axes of each single particle image, depending on the particle position in depth. They showed that the latter technique has the largest measurement range.

In this paper we present a 3D-3C time-resolved measurement technique, based on micro-PIV and making use of two optical aberrations and a single camera. The particle position in depth is encoded by means of the two optical aberrations, namely astigmatism combined with a spherical lens aberration (Born and Wolf (1999)).

Fig. 1. Astigmatism (left) and spherical aberration (right)

In Fig. 1 the principal paths of rays due to these two aberrations are shown. In case of perfect astigmatism, a point P, which is not placed on the optical axis, will not be imaged as a point in the image plane. Instead of one focal plane, two focal lines appear with orthogonal orientations. Between these two focal lines, a point will be imaged as an ellipse.

When a large F-number of the lens is used during the recordings, spherical lens aberrations appear in the imaging plane. This means that rays which are parallel to the optical axis but at different distances from this axis fail to converge to the same point. Rays which enter the lens far from the optical axis are focused more tightly than those that enter closer to the axis.

In our technique, described below, we use a combination of these two aberrations. Out of the size and shape of the particle images, we then determine the three-dimensional position of the particles inside the flow volume. Although this technique is designed for 3D-3C velocimetry measurements, in this paper the validation is described for 3D-2C measurements.

2. Measurement principles

In this section the following points are discussed:

1. The experimental setup of the technique used for calibration

2. A detailed description of the measurement technique and the calibration using fluorescent particles to determine the particle shape in dependence on the position of the particle in the

P Lens

Optical axis

Lens


- 3 -

measurement volume

2.1 Experimental setup

To understand the measurement technique, a simplified schematic of the experimental setup is shown in Fig. 2. The optical setup is based on the common micro-PIV setup to investigate micro-fluidic systems. The laser pulses of a New Wave Solo PIV III pulsed Nd:YAG laser (532 nm, 30 mJ/pulse) are expanded in diameter using a Keplerian beam expander up to a diameter of 11 mm. A dichroic filter, placed under 45˚ and having a maximum reflection in the bandwidth of 535nm ± 15nm, directs the expanded laser beams upwards where they pass the investigated flow. In this way, a cylindrical volume illumination of 11 mm in diameter is created. This part of the setup consists of a round container of Plexiglas, having a diameter of 300 mm and holding a Glycerin-water mixture (70 Vol%-30 Vol%) in which 10 µm fluorescent RhodamineB particles have been suspended. We have chosen to use fluorescent particles, because of the high amount of laser reflections at the Plexiglas-liquid interfaces. The top of the liquid is covered with a Plexiglas plate, connected to a viscometer. As the viscometer rotates with a constant frequency, a laminar uni-directional flow is created, used for validation of the proposed technique. The exact position of the covering plate is observed using a CCD Sensicam camera (1280 × 1024 pix) in order to determine the depth of the fluid. The light emitted by the particles passes through the dichroic filter is split up by a lateral displacement beamsplitter in order to create a higher spatial resolution of the images, and is recorded by two CCD Sensicam cameras. Both cameras are equipped with long distance microscopes (Nikon Nikkor 105 mm lens together with a 160 mm extension tube) and a notch filter. The notch filter has a transmission in-between 539 and 1200 nm. It is used to filter out most of the green laser light reflections. The field of view of each camera is 5.6 mm × 4.5 mm with a spatial resolution of 4.4 µm/pix. For the actual validation of the technique, only one camera has been used to record the position of the particles.

Reflected

laser light

45°Dichroic filter

Target

Beam expander

Beamsplitter

Long distance microscope

Notch filter

90° Mirror

Cam 1

Viscometer

Long distance microscope

PIV Nd:YAG

Fig. 2. Schematic experimental setup


- 4 -

2.2 Optical imaging and calibration

The proposed measurement technique is aimed at capturing the 3D-3C velocity fields inside a liquid layer at various heights above the rigid bottom. In the proposed technique, astigmatism is introduced by the dichroic filter, placed at an angle of 45˚. This means that rays, coming from the particles, which pass through the filter in the meridional plane, travel another distance to the CCD chip than rays in the sagittal plane. As a result, the focal lengths in these two planes will be different, introducing a deformation of the particle in the image plane of the camera. This image deformation depends on the thickness of the filter, t, on the refraction indices of the tilted filter, n, and on the angle of the tilted filter, θ. Matsushita (1993) and Smith (1966) discussed that the astigmatic difference between the two focal lines, ∆s, is expressed as:

( )

( )θθ

θ2222

22

sinsin

sin1

−−

−=∆

nn

nts

(1)

In Fig. 3 the measurement volume depth is plotted as a function of the angle of the tilted filter and its thickness.

In the case of perfect astigmatism, particles will be imaged as horizontal or vertical lines at the focal lines, or as an ellipse if the particle is located between these two focal lines.

Due to the relatively low laser power and the long way the rays have to travel from the camera lens to the CCD

chip, a small F-number was used for the recordings. This introduced an additional aberration, the so-called spherical lens aberration, where rays which enter the lens far from the optical axis are focused more tightly than those that enter closer to the axis. As a result of the combination of these two monochromatic aberrations, particles are imaged differently than in the case of perfect astigmatism.

Due to the astigmatism and aberration of our optical system particles located exactly at the focal lines of the sagittal or meridional planes are imaged respectively as vertical or horizontal ellipses, whereas particles in-between these to focal lines are imaged as stars of interchangeable major and minor axes. Three exemplary images of a 10 µm RhodamineB particle are shown in Fig. 4. The change in depth between the left and the right image is about 3.5 mm. In the left recording, the particles have an ellipse-like shape with a small vertical and a large horizontal main axis, in the middle image, star-shape images can be seen, having a main vertical axis somewhat larger than the main horizontal axis and in the right image the particle images are ellipses again having a much larger vertical than horizontal axis.

In order to calibrate the imaging technique, we use dried RhodamineB particles, placed onto a Plexiglas plate. This plate has the same thickness as the bottom of the Plexiglas container, used for validation measurements afterwards, and is aligned orthogonally with the expanded laser beam and the long distance microscopes. To simulate a change in height of the particles on the Plexiglas plate,

theta [deg]

t [m

m]

10 20 30 40 50 60 70 80 901

2

3

4

5

6

7

0

1

2

3

4

5

Meas. vol.depth [mm]

Fig. 3. The dependence of the measurement volume depth on

the angle θ and the thickness t of the dichroic filter (n = 1.52)


- 5 -

the long distance microscopes are translated along a total path of 4 mm with increments of 10 µm. At each step, one image of the particles is taken.

Fig. 4. Dependence of the particle image shape on its position in the measurement volume

(left: at 0.5 mm with respect to the reference depth; centre: at 2.7 mm depth; right: at 4 mm depth)

3. Data processing

3.1 Particle image reconstruction

The subroutine for fitting the particle images is the most essential part of the pre-processing of the data for the proposed measurement technique, because the particle position in depth is directly linked to the particle image shape. In Fig. 5, on the left hand side, an original particle image is shown.

Fig. 5. Original particle image recording (left), segmentation after thresholding (middle) and reconstructed

particle image (right)

After having recorded the particle images, a mean background intensity field is determined and subtracted. After background subtraction, two additional thresholds are set, the first one to remove most of the remaining background noise, the second one to determine the pixels which might belong to the particle image (Fig 5, middle) by making use of a binarisation. As can be seen in the middle image, part of the background noise is still present. This noise is removed from the binary image by locating all connected objects that have less than a certain number of pixels and assigning them to the background. In the next step, the remaining pixels are connected to one another, so to form a reconstructed particle image (Fig. 5, right hand side).

In this way, each image is scanned on particles and a number is assigned to each particle for the next step in the data processing.

3.2 Particle depth through mean axes ratio

The combination of astigmatism and spherical lens aberration causes a wave front aberration which results in distorted particle images, as was explained in section 2.2. The emitted light intensity of each particle depends on the particle size, the position of the particle inside the measurement


- 6 -

volume as well as on the laser beam quality. This light intensity influences the width and height of each particle image. We have chosen the ratio between the main horizontal and main vertical axes a measure determining the particle distance from the bottom, since this ratio almost does not depend on the emitted light intensity of a particle.

By scanning the measurement volume along a total path of 4 mm with increments of 10 µm and determining the main axis ratio of each particle image, we receive the calibration curve given in Fig. 6. It can be seen in this figure, that the curve consists of three linear parts, denoted 1 to 3. These

linear parts represent an exponential dependence of the axes ratio from the distance, since the plot shown in Fig. 6 is log-normal. Part 1 corresponds to the depths where the particle images are ellipses extended in the horizontal direction (Fig. 4, left hand side), and part 3 to the depths where the particles are observed as ellipses extended in the vertical direction (Fig. 4, right hand side). Both curves have the same slope (i.e. the same

change in axes ratio), because the particles are imaged in the same way. Part 2 of the calibration curve corresponds to the particles which are observed as star-like images. The distance between the maximum and the minimum value of part 2 (highlighted by arrows in Fig. 6) is the astigmatic difference between the two focal lines, ∆s, and has a value of about 1.3 mm. This value is close to the theoretical value, calculated by equation 1, of 1.1 mm for the dichroic filter used (t = 4.1 mm, θ = 45˚ and n = 1.52).

By a change of 1 pixel in one of the main axes’ lengths, the axes ratio changes by an amount of

2105 −× , resulting in a theoretical depth resolution of 131 µm in part 1 and 3, and of 14 µm in part 2.

3.3 Change in particle shape for computation of velocity fields

After each particle has been assigned a number and its depth has been determined, the two-dimensional centre of mass is calculated and a square segment of the image, which contains the particle image, is extracted. To change the large distorted particle images into small round particle images useable for computation of the velocity fields, we make use of the data processing algorithm for the separation of anamorphic images described by Angarita-Jaimes et al (2006). They performed a one-dimensional integration of the intensity field of the particle, which means that the two-dimensional image is projected onto each of its main axes. These projections are then used to produce new two-dimensional particle images, by rotating the projections about the particle’s centre of mass, in this way producing an x-image and a y-image, as is seen in Fig. 7. The 3 images on the left show the original recorded distorted particle images, whereas the images in the middle are the x-image and y-image after projection. The asymmetry in intensity between the left and the right side of the original particle image with respect to its centre of mass is maintained by using a weighting factor. This means for the x-array, that each pixel in the synthesized image has a weighted intensity value by using weighted and interpolated intensity data from the left and right side of the x-array relative to the centre position. The same is done for the y-array, by weighting and interpolating the

1

2

3

Fig. 6. Calibration curve of the particle height inside the

measurement volume versus the main axis ratio (in

logarithmic scale)


- 7 -

upper and lower side of the y-array. The synthesized particle image is created by adding the extracted x-image and y-image together, decreasing its size to a 9×9 pixel particle and placing its centre on the centre of mass of the original distorted particle image. In this way, all the distorted particles in each recording are changed into round particles, which have the same intensity distribution as the original particles. This intensity distribution can be used for sub-pixel particle displacement calculations.

3.4 Computation of velocity fields

The proposed technique is developed as a 3-dimensional, 3-component velocity measurement technique. However in this paper, the computation and the presentation of the velocity field are described only by using a 3-dimensional, 2-component multilayer data processing algorithm.

In section 3.2 we have described the algorithm which assigns to each of the particles in the recorded images a depth inside the measurement volume. To compute the velocity fields, the measurement volume is split up into different layers having the same range in axes ratio. For each of these layers, all images are scanned, during which only the particles are taken which fall inside this depth range and deleting all the other particles. In this way, new pairs of particle images are generated with a low density of seeding particles. Due to the low particle density the number of signals is not enough to obtain valid velocity vectors from the instantaneous cross-correlation functions in the PIV processing algorithm, which can potentially lead to erroneous velocity vectors in each instantaneous vector field. For standard PIV algorithms, these erroneous vectors are normally identified and removed by interpolation of the neighbouring vectors (Westerweel (1993), Meinhart et al. (1994)), but due to the low signal of the present recordings, this interpolation does not result in satisfactory instantaneous and mean vector fields.

To overcome this problem, we have applied a modified version of the WIDIM PIV-algorithm (Scarano (2002)), based on a window-deformation and iterative multigrid cross-correlation. We first produce instantaneous averaged recordings by adding 20 particle recordings together to make sure that there are enough particle pairs for most of the interrogation areas to obtain a clear signal to noise ratio and correct correlation peak identification. The number of particle recordings per averaged recording is fixed at 20 to avoid an overlapping of the single particles and thereby reducing the signal to noise ratio. However, even after averaging over multiple images, there are

Fig. 7. The anamorphic images of tracer particles at different depth positions inside the measurement volume

and its projections. Left: the original particle images; middle: the extracted x-images and y-images;

right: the combined new particle images. All images are 61×61 pixels.


- 8 -

still interrogation areas present with low signal to noise ratios, therefore introducing relatively high errors in the velocity measurements and resulting in outliers in the mean velocity field and in the standard deviations.

Finally, we make use of the average correlation method described by Meinhart et al. (2000). For each instantaneous averaged recording the cross-correlation functions are calculated, after which the cross-correlations fields of all averaged recordings are added together to produce an averaged correlation field, out of which the locations of the signal peaks are determined. Hence, all the interrogation areas having low signal to noise ratios contribute only little to the average correlation function, whereas high signal to noise ratios produce a much higher signal peak. This averaged vector field is then used for the next iteration step in the PIV algorithm. In this way, the probability of erroneous vectors in the mean velocity field is greatly reduced.

4. Validation To validate the proposed measurement technique, we use a laminar uni-directional flow between two disks having a diameter of 300 mm, of which the upper one is rotating with a constant frequency. The liquid is a Glycerin-water mixture (70Vol%-30 Vol%) in which 10 µm fluorescent RhodamineB particles have been suspended. One camera captures the side side view, recording the exact position of the covering plate to determine the depth of the fluid, the other camera captures the view from below to record the particle displacements inside the measurement volume (Fig. 2). The recorded measurement volume has dimensions of 5.6×4.5×2.1 mm, resulting in a minimum in-plane spatial resolution of 4.4 µm/pix, and is positioned at a distance of 99 mm from the rotating axis. The out-of-plane theoretical spatial resolution is 131 µm, as was derived in section 3.2. The upper disk rotates with a constant frequency of 0.83 Hz, resulting in a maximum azimuthal velocity in the measurement volume of 524 mm/s. The mean velocity profile of the liquid in the measurement volume can be well approximated by:

zh

zru 3.249=

Ω=ϕ (2)

since the radial coordinate r of the illuminated volume is much larger that the layer thickness h. In (2) the vertical coordinate z equals 0 mm at the bottom of the liquid layer and 2.1 mm at the surface of the rotating disk. To get a reliable mean velocity field, 3000 particle image pairs of the volume are recorded with a time delay between the pulses of 600 µs at a frequency of 3 Hz, resulting in 150 instantaneous averaged recordings per investigated layer inside the measurement volume. These averaged recordings are then processed by the data processing software described in section 3, resulting in velocity fields at different depths inside the volume. The images have been analyzed with a window size of 64×64 pixels and an overlap factor of 50%, resulting in an in-plane spatial resolution of 141 µm. We have placed the layers in such a way, that each one covers a range in main axes ratio of 0.5, going from 0.1 to 3.6. This coincides with the physical depths in the measurement volume and the theoretical mean velocities (eq. 2) given in Table 1. In Fig. 8 the mean velocity fields are shown for two depths. The graph at the left is placed at a depth range of 0.85 to 1.02 mm, meaning the velocity is integrated over a depth of 170 µm. For the right image, the velocity field is integrated over a range of 120 µm, in between 0.73 and 0.85 mm depth. In both images, the flow is from the top to the bottom. Despite the low particle density inside the flow, we receive a very uniform velocity field, as a result of the average correlation method.


- 9 -

Table 1: Analysed ranges of main axes ratio and corresponding physical depths and theoretical velocities

Main axes ratio [-] Physical depth in the measurement volume [mm]

Theoretical mean velocity [mm/s]

0.1 – 0.6 2.10 – 1.29 422.6

0.6 – 1.1 1.29 – 1.02 287.9

1.1 – 1.6 1.02 – 0.85 231.8

1.6 – 2.1 0.85 – 0.73 196.9

2.1 – 2.6 0.73 – 0.63 169.5

2.6 – 3.1 0.63 – 0.56 148.3

3.1 – 3.6 0.56 – 0.49 130.9

The mean average velocity for the image at the left and at the right equals 205 ± 14 mm/s and 170 ± 17 mm/s respectively. This corresponds with a divergence of 13% in comparison with the theoretical values given in Table 2, as plotted in Fig. 9. The large difference can be explained by the choice of the aspect ratio. To receive preliminary results we have chosen a threshold for the data

processing which is constant for the whole image. Due to the uneven distribution of the background noise over the recorded particle images, certain particles will be given a higher aspect ratio, resulting in a lower depth determination of these particles; hence, resulting in a lower local particle

displacement and flow velocity than the theoretical values. This can be overcome by giving each particle in a particle image its own local threshold by optimizing its background intensity level.

The turbulence intensity fields for both analyzed depths are plotted in Fig. 10. For most of the

1 2 3 4 5X [mm]

0.5

1

1.5

2

2.5

3

3.5

4

Y[m

m]

120 130 140 150 160 170 180 190 200 210 220 230 240

Utot

[mm/s]

Uref

1 2 3 4 5X [mm]

0.5

1

1.5

2

2.5

3

3.5

4

Y[m

m]

120 130 140 150 160 170 180 190 200 210 220 230 240

Utot

[mm/s]

Uref

Fig. 8. Mean velocity field at a range of depth of 0.85 – 1.02 mm (left) and at 0.73 – 0.85 mm (right)

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2100

150

200

250

300

350

400

450

Depth [mm]

Velo

city [m

m/s

]

Fig. 9. Mean velocity at different depth levels inside the measurement volume

(* theory; × experiment)


- 10 -

recorded flow field the turbulence intensity is below 20%, which is still quite high for a uni-directional laminar flow, but can be explained by the rotating Plexiglas disk, which was rotating under a slight angle. This resulted in a vertical motion of ± 0.15 mm, which equals 14% of the total liquid depth. For both images, an area is found where high turbulence intensity levels are measured. These high turbulence levels can only be explained by large outliers in this area. The low laser intensity in the lower left corner of the recorded particle images results in a wrong combination of particles pairs, hence, in outliers. To determine the mean velocity field, we make use of the averaging technique, after which we apply a vector validation to substitute the outliers. For the calculation of the turbulence intensity, we use the instantaneous cross correlation fields, therefore outliers will be included.

5. Conclusions and outlook

The proposed method can determine the three-component, three-dimensional velocities inside a measurement volume by using a single camera. By using two optical aberrations, namely astigmatism and a spherical lens aberration, the positions of the particles in physical space can be determined, after which either the three velocity components can be calculated or the volume can be split up into different layers and at each layer the two-dimensional velocity field can be resolved. The technique is easy to build up and the alignment and calibration is straightforward and fast.

The measurement uncertainty in the mean velocity field is approximately 13% in comparison with the theoretical values. This error can be decreased by giving each particle in a particle image its own local threshold by optimizing its background intensity level, therefore optimizing the axes ratio for each particle, and placing each particle at its right depth inside the measurement volume. This procedure will be implemented in the data processing software.

Acknowledgements This work was financed by a grant of the German Research Foundation under No. TR 194/34.

References

1 2 3 4 5X [mm]

0.5

1

1.5

2

2.5

3

3.5

4

Y[m

m]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tu [-]

1 2 3 4 5X [mm]

0.5

1

1.5

2

2.5

3

3.5

4

Y[m

m]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tu [-]

Fig. 10. Turbulence intensity field at a range of depth of 0.85 – 1.02 mm (left) and at 0.73 – 0.85 mm (right)


- 11 -

Angarita-Jaimes, N.; McGhee, E.; Chennaoui, M.; Campbell, H. I., Zhang, S.; Towers, C. E.; Greenaway, A. H.; Towers, D. P. (2006) Wavefront sensing for single view three-component three-dimensional flow velocimetry. Exp Fluids 41: 881 – 891

Born, M.; Wolf, E. (1980) Principles of Optics. Pergamon Press: 203 – 218

Brücker, C. (1995) Digital-Particle-Image-Velocimetry (DPIV) in a scanning light-sheet: 3D starting flow around a short cylinder. Exp Fluids 19: 255 – 263

Elsinga, G. E.; Scarano, F. ; Wieneke, B.; van Oudheusden, B. W. (2005) Tomographic particle image velocimetry. Proc. 6th Int Symp on Particle Image Velocimetry, Pasadena: 1:12

Matsushita, H. (1993) Color separation/synthetic optical system including two dichroic mirrors angled for correction of astigmatism. United States Patent: 5200857

Meinhart C. D.; Barnhart D. H.; Adrian R. J. (1994) An interrogation and vector validation system for holographic particle image fields. Proc 7th Int Symp Appl of Laser Techniques to Fluid Mech, Lissabon: 1.4.1 – 1.4.6

Meinhart, C. D.; Wereley, S. T.; Santiago, J. G. (2000) A PIV Algorithm for Estimating Time-Averaged Velocity Fields. Journal of Fluids Engineering 122: 285 – 289

Racca, R. G.; Dewey, J. M. (1988) A method for automatic particle tracking in a three-dimensional flow field. Exp Fluids 6: 25 – 32

Scarano, F. (2002) Iterative image deformation methods in PIV. Meas. Sci. Technol 13: R1 – R19

Smith, W. (1966) Modern Optical Engineering. McGraw-Hill: 83 – 84

Virant, M.; Dracos, T. (1997) 3D PTV and its application on Lagrangian motion. Meas. Sci. Technol 8: 1539 – 1552

Westerweel, J. (1994) Efficient detection of spurious vectors in particle image velocimetry data. Exp Fluids 16: 236 – 247

Willert, C. E.; Gharib, M. (1992) Three-dimensional particle imaging with a single camera. Exp Fluids 12: 353 – 358

Documents

Three-dimensional, three-component particle imaging …ltces.dem.ist.utl.pt/lxlaser/lxlaser2008/papers/03.1_2.pdf · Three-dimensional, three-component particle imaging using two