2.4 Particle image analysis
2.4.2 Signal-peak detection and displacement determination
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and decreases with increasing displacement, no matter in which direction or if the displacement of the particle images is not constant. In order to optimise the performance of the experiment, the number of paired particle images is of primary impor-tance as only these add to the signal strength. In addition the number of correct pairings with respect to incorrect combinations has to be optimised for increasing the signal-to-noise ratio and thus the detectability of the signal. This can be achieved either by changing the seeding concentration, the magnification of the imaging system or simply by increasing the light-sheet thickness, see equation (2.7). In-plane loss-of-pairs, caused by particles entering or leaving the measurement volume during the illuminations, due to the motion of the fluid, can be re-duced by decreasing ì or eliminated when the search window contains the corresponding particle image pattern selected with the template. This can be achieved either by correlating differently sized samples as described before or by using window-shifting in connection with multi-pass interrogation techniques [106, 109]. Using this approach two equally sized sam-ples, separated by the local particle image shift are cross-correlated after the local shift has been determined within a first evaluation. Since the correlation windows are symmetrically shifted with respect to each other and relative to the exact measurement position, it becomes necessary to determine where the footprint of the vector should be located (centre or corner).
This is especially important when different evaluation methods are compared because the par-ticle pattern employed for the evaluation may be different for the same pixel coordinates of the displacement vector [97]. Out-of-plane loss-of-pairs can be compensated by shifting the light-sheet in the direction of the mean particle displacement, as proposed in [52], or by using multi-light-sheet arrangements when the out-of-plane motion is not uniform, see chapter 4.
The effect of in-plane gradients can be reduced by increasing the magnification of the imaging system, by decreasing the time-separation between the two illuminations or by using prop-erly shaped interrogation windows whose linear dimension is reduced in the direction of the gradient.
2.4.2 Signal-peak detection and displacement determination
In order to increase the accuracy in determining the location of the displacement peak from
b
#dce pixel to sub-pixel accuracy, an analytical function is fitted to the highest correlation peak by using the adjacent correlation values [109]. Usually two one-dimensional Gaussian fits along the two coordinates through the highest correlation coefficient are applied. The moti-vation for this particular function is based on the fact that under ideal imaging conditions the
2 Particle Image Velocimetry
shape of the signal peak is of Gaussian shape, like a diffraction limited image of a particle, and close to 3 pixels in each spatial direction for realistic applications of PIV.
f
indicates the exact location of the maximum andD
ã&ï k
are coefficients of no direct interest.
Using this expression for the main and the adjacent correlation values and the fact that the first derivative of this expression must be zero, the position can be estimated with sub-pixel accuracy. Unfortunately, differences from this ideal Gaussian shape are normal due to the discretisa-tion, electronic noise, velocity gradients and optical aberrations and introduce artefacts which reduce the performance of this fit and lead to systematic measurement errors. The peak-locking effect for example which is introduced by under-sampling the particle images may be amplified by using this peak-fit due to the characteristic variation of the RMS error as a function of the sub-pixel location of the correlation peak.
Better results can be achieved by fitting a two dimensional Gaussian function to a larger number of points by using the iterative Levenberg-Marquardt method as described in [88]. The weighting of the values should be according to the Fisher transform in order to compensate the non-normal distribution of the correlation-coefficient error (the error is zero for values equal 1 and increases with decreasing amplitude). This peak-finder also works properly for non-Gaussian shaped correlation peaks and is less sensitive to sub-pixel displacements compared with the three point Gaussian peak-fit.
Figure 2.13 shows the distribution of the measured T -displacements as a function of the sub-pixel shifts along the other coordinate calculated from simulated images with a size of
z # }a
&3#
z }
pixel . The evaluation has been performed by using the 2nd order accurate based multi-pass interrogation procedure with three-point Gaussian peak-fit and the FFT-based free-shape cross-correlation along with the two dimensional Gaussian peak-fit. As the distribution of the measured velocity is nearly independent of the sub-pixel location when the two-dimensional Gaussian peak is applied, only the graph with the largest deviation from the true displacement has been plotted for comparison. The enormous variation of the measured displacement as a function of the sub-pixel shift in
ñ
-direction should be noted when the three-point Gaussian peak-fit is applied, see figure 2.13 left. It results in a strong peak-locking effect compared with the two-dimensional Gaussian peak fit analysis despite of the large particle image diameter (z c
}
pixel).
The mean and standard deviation following from the graphs in figure 2.13 are summarised in table 2.1. It should be noted that the reliability of the three-point peak-fit is strongly limited, with respect to the two dimensional Gaussian peak-fit, due to the variation of the standard de-viation as a function of the sub-pixel location (one order of magnitude). For this reason, the analysis of the recordings acquired for the investigation presented in chapter 5 to 7, was performed with the the iterative Levenberg-Marquardt method and the displacement estima-tion with sub-pixel accuracy was performed with a two dimensional Gaussian peak-fit routine.
2.4 Particle image analysis
FIGURE 2.13: Numerical comparison between two-dimensional (solid lines) and three-point Gaus-sian peak-fit (graphs with symbols (left column) and dotted lines) with iterative Levenberg-Marquardt method for two fixed particle image displacements (top: (*)+7 pixel, bottom: (*)+-0/ pixel) as a function of the6 -shift. The graph represents the probability density functions for a set of simulated displacement fields (each 2000 by 16000 pixel in size) analysed with a 2nd order accurate multi-pass interrogation technique with3 3 pixel interrogation window and 50% overlap.
Figure 2.14 shows the distribution obtained by analysing experimental data. The displacement was achieved by transforming one image as will be outlined in the following chapter. It can
ñ
0.4 #dc#a#e b #dc#]& &c } b
TABLE 2.1: Comparison between two-dimensional (bottom row) and three-point Gaussian peak-fit for two particle image dis-placements in ) -direction (centre column (*) +
pixel, right column(*)K+
-0/
pixel) as a function of the6 -shift.
2 Particle Image Velocimetry
be seen that the functional dependence of the distributions on the displacement agrees nicely with the result shown in figure 2.13.
For completeness it should be mentioned that various techniques have been proposed in the past to optimise the evaluation procedure with regard to accuracy and spatial resolution and to overcome the limitations associated with the required image density and the tolerable gradients, for details the interested reader may consult the existing literature [22, 27, 33, 72, 97]. The excellent performance of the 3-point Gaussian fit when the location of the correlation maximum is exactly symmetrical relating to a pixel stimulated Lecordier et al [72] to make use of the accuracy by transforming the measured image in such a way that the measured velocity becomes zero after the transformation, see section 3.2.2. To increase the spatial resolution without increasing the probability that a random correlation peak will exceed the height of the displacement peak and deteriorate the velocity measurement, it is possible to calculate the correlation twice with a small spatial separation in such a way that the image displacement
1.3 1.4 1.5 1.6
FIGURE2.14: Experimental comparison between two-dimensional (solid lines) and three-point Gaus-sian peak-fit (graphs with symbols (left column) and dotted lines) with iterative Levenberg-Marquardt method for two fixed particle image displacements (top: (*)+7 pixel, bottom: (*)+-0/ pixel) as a function of the6 -shift. The graph represents the probability density functions for a set of measured displacement fields (each 1280 by 1024 pixel in size) analysed with a 2nd order accurate multi-pass interrogation technique with3 3 pixel interrogation window and 50% overlap.
2.4 Particle image analysis is identical but the noise uncorrelated [27]. Once this has been done, the signal peak can be clearly detected by calculating a correlation of the correlation (the uncorrelated noise will vanish), but again it should be taken into account that evaluation schemes based on few particle images are highly sensitive to noise that is introduced by the characteristics of digital cameras.
In order to enhance the signal quality when strong in-plane gradients are present, Huang et al [33] has proposed to transform the image in such a way that the deformation vanishes after the mapping, see also [22]. This approach works fine as long as the flow under consideration possesses only in-plane gradients (
^
T
,
^
ñ
ands
T
, s
ñ
). In case of strong out-of-plane gradients ( ^ E ) this and most of the other sophisticated methods will fail, as can be easily realized by considering a turbulent boundary layer experiment with a light-sheet parallel to the flat plate. Under these conditions the apparent particle image gradients caused by particles from different layers cannot be compensated because of arising instabilities in the image analysis.
2 Particle Image Velocimetry
3 Stereo-scopic Particle Image Velocimetry
The conventional PIV technique described in the previous sections yields reliable results as long as the flow under investigation is two-dimensional and parallel to the light-sheet. In case of turbulent flows with a strong velocity component being normal to the light-sheet, the out-of-plane velocity component remains unknown and the in-plane components are biased due to the perspective error as indicated in figure 3.1 for a simulated flow with a pure out-of-plane velocity component.
FIGURE 3.1: Magnitude and direction of the projection-error as a func-tion of the image locafunc-tion for a constant out-of-plane displacement.
As this error is directly proportional to the viewing angle, according to figure 3.1, the object distance must be increased to minimise the error while keeping the field of view. As this approach requires long focal length lenses it is obvious that this approach is not satisfactory.
To overcome this constraint completely and to obtain the out-of-plane velocity component, a stereoscopic observation arrangement has to be applied, which will be outlined in the follow-ing sections.
3.1 Principles
Using the stereoscopic recording technique, the images of tracer particles are recorded si-multaneously from two different viewing directions, and the correct displacement (without
3 Stereo-scopic Particle Image Velocimetry
perspective error) of the particle ensembles are reconstructed by using the proper equations.
The basic recording arrangements can be classified either with respect to the camera position relative to the light-sheet or with respect to the propagation direction of the light-sheet plane according to figure 3.2. The left drawing reveals a configuration where both cameras are lo-cated on the same side of the light-sheet. As a consequence, this recording arrangement can be operated in forward/backward or ninety degree configuration when the propagation direction of the light-sheet plane is considered. An alternative arrangement is shown in the right drawing of the same figure. In this case the cameras are separated by the light-sheet plane so that the pure forward, backward and ninety degree arrangements are possible. As long as only the in-tensity of the scattered light is considered, the most efficient light-sheet camera-configuration is the purely forward scattering set-up, according to the Miescattering diagram in figure 2.2, followed by the forward/backward configuration, purely backward and finally ninety degree case. This may change when the state of polarisation has to be taken into account beside the intensity (this will be further analysed in chapter 4 and the experimental parts of this thesis).
FIGURE 3.2: Stereoscopic recording arrangements. The orientation of the principle observation ray (oblique lines) relative to the light sheet (dark plane) or the propagation direction of the light-sheet plane (indicated by the arrows) can be used to define various recording configurations with different properties. The observation direction from one light-sheet side (left drawing) allows forward/backward and ninety degree imaging and from opposite sides (right drawing) forward, backward and ninety degree.
Beside the above classification, it is common practice to differentiate between the stereo-scopic recording approaches regarding to the field distortions into translation and angular displacement methods, see figure 3.3. In case of the translation method the light-sheet plane, the main plane of the lens and the image plane are parallel in relation to each other. As a result, the magnification factor is constant across the field of view (e.g. the image of a regular grid appears undistorted) and the image analysis varies only slightly from the analysis outlined in the previous chapter, see [44] for details. The drawback, on the other hand, is the decreased performance of this arrangement for increasing stereo opening angles. This happens because of optical aberrations and the decrease of the modulation transfer function towards the edges of the field of view (see also section 4.7) and because of the limited overlap of the observation areas of both cameras when the CCD sensor is not shifted with respect to the optical axis, see [44]. For these reasons the angular displacement method is usually applied where the light-sheet plane and the main plane of the lens intersect in a common line1. In this config-uration the magnification factor varies across the field of view and typical distortions appear
1The translation imaging configuration can be seen as a special case of the angular-displacement arrangement with the line of intersection between the image plane, the main plane of the lens and the object plane at infinity.
3.1 Principles
Lens plane Image plane
Observation plane Observation plane
FIGURE3.3: Stereo-scopic imaging configurations. Left: translation method. Right: angular displace-ment method.
as indicated in figure 3.4 for both camera-light-sheet arrangements shown in figure 3.2. The size, shape and location of the dark areas indicate the image of a rectangular area in the object space as a function of the camera arrangement. The difficulties associated with this effect and possible solutions will be further analysed in section 3.2.
FIGURE3.4: Linear field distortions of a regular grid due to the oblique observation directions for two angular displacement camera arrangements. Left: both cameras are located on the same side of the light-sheet according to the left drawing in figure 3.2). Right: cameras are separated by the light sheet (right drawing in figure 3.2).