How To Calculate Piv With An Inverted Window Size And Shape

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1 Adaptive PIV with variable interrogation window size and shape Bernhard Wieneke¹, Karsten Pfeiffer¹ 1: LaVision GmbH, Göttingen, Abstract PIV processing requires a careful selection of interrogation window size to balance robustness with accuracy of the calculated displacement especially in regions of high gradients. An adaptive PIV approach is presented here which calculates the optimal local interrogation window size and shape based on flow gradients and image quality. Using elongated Gaussian weighted elliptical window the principal direction and eccentricity is computed by minimizing the differences of displacements within the window. The interrogation window size is varied according to the local flow gradient and the correlation value. For synthetic and experimental data with a shear flow this approach is compared to standard PIV processing with round or square windows showing significant improvement in accuracy and spatial resolution. 1. Introduction In standard PIV processing the user selects a constant interrogation window size such that in each window there are a sufficient number of particles, typ. 3-10, to correlate on and to provide enough stability in the usual iterative large-to-small window iterative scheme to avoid outliers. Advanced algorithms (see e.g. PIV-Challenges, Stanislas et.al., 2008) use deformed window/images to adapt to flow gradients within the window. Such a constant window size of e.g. 32x32 pixel is capable to capture certain displacement gradients, but cannot follow gradients smaller than the effective spatial resolution of the algorithm. This is especially the case in regions of strong vortices and shear flow close to surfaces. Therefore some schemes have been proposed (e.g. Scarano, 2003, Theunissen et.al, 2007, 2008, 2009) to use elongated window shapes to enhance the spatial resolution in one direction relative to the one perpendicular while maintaining the same number of pixels and vector accuracy. These schemes vary in the kind of criteria which is chosen for selecting the optimal window direction and eccentricity. At the same time the image quality is never constant ideally requiring a locally adapted window size. The number of matching particles within each window will vary randomly and will decrease significantly in regions of low seeding density and strong out-of-plane motion. A variable window size based on a combination of local particle density and/or flow gradients has been proposed (Scarano, 2003, Becker et.al., 2008, Theunissen et.al, 2009). A similar criterion is used below based on a combination of flow gradients and correlation value reflecting the image quality by the degree of similarity between the two interrogation windows. 2. Method The method presented has two parts. First the interrogation window shape is computed from the local gradients of the displacement field, and secondly the size of the interrogation window is varied according to a combination of the local correlation value and flow gradients. Starting with round interrogation windows a first displacement field is calculated from which the flow gradients are used for shape and size calculation. Further iterations refine the displacement field simultaneously with the window shape and size. Care has to be taken to ensure robustness and convergence given the extra free parameter compared to conventional PIV. It is useful in this - 1 -

2 respect to adjust the shape and size parameter slower i.e. less sensitive than the displacement field calculation itself. Shape and size adjustment are useful to add extra accuracy once a close solution has been reached, but one should not rely on elongated windows to find such a solution in the first place. Interrogation Window Shape A variable shape is implemented by weighting all pixels of both interrogation windows with a round or elongated Gaussian elliptical function. The orientation of the ellipse is such that the major axis of the ellipse is minimizing the variation of the displacements dv taken from the previous iteration over all angles α (0-180 ): dv α = Σ ( v 0 v α ) (1) where v 0 is the displacement at the center of the interrogation window, summing over all displacements v α along the line through the center at a certain angle α. The absolute difference of the displacements - effectively being an and/or combination of the first and second derivative - is considered to be more robust than the second derivative alone previously suggested (Scarano, 2003), In principle the first derivative is taken care of by the usual iterative window deformation, but the second derivative alone is quite sensitive to local noises, which lead Theunissen et.al. (2009) to conclude that a stable window shape computation is better done on (ensemble-)averaged vector fields. The present method aims at variable shape and size for each instantaneous vector field. The length of the line is chosen to be somewhat larger than the interrogation window. Here a factor of 2 has been selected. This again makes the procedure more robust since gradients are averaged over a larger range assuming that e.g. shear zones are larger than a single interrogation window. The eccentricity ε of the ellipse, i.e. the aspect ratio of major to minor axis, is computed from dvα and the value perpendicular to this direction dv α +90 : ε = (2) The eccentricity is limited to a range of 1:1 (dv α = dv α+90 ) to 4:1 (dv α << dv α+90 ) to prevent extremely long ellipsesγ is an additional parameter to change the sensitivity of the shape calculation. A value γ = 1 has been found appropriate so far; a value of 2 or 4 already provides too many 4:1 ellipses changing direction rapidly from window to window. Given interrogation windows as small as 16 2 this can lead to windows as thin as 8 x 32 pixel at 1/e weighting factor. Note that the Gaussian pixel weighting reduces the effect of particle truncation at the window borders. So far the Gaussian weighting has been implemented by taking a square window of double the size (e.g. 32x32) and applying the weighting function on all pixels with subsequent FFT processing. Shorter processing times can be reached by taking tilted windows of appropriate smaller size and thinning out overlapping windows to ensure similar overlap in all directions (Theunissen et.al., 2009). Interrogation Window Size The size of the interrogation window should be large enough to ensure enough matching particles for robustness and accuracy. Previously the local rms of the displacement field in combination with the local density of seeding particles has been used successfully for both synthetic and real data to - 2 -

3 ensure e.g. smaller windows in regions of large gradients and larger windows for lower seeding densities (Theunissen et.al., 2009). Here a similar approach is used replacing absolute seeding density by the previously calculated correlation coefficient (range -1 to +1). The normalized correlation coefficient provides a measure how well the two interrogation windows have been matched by the previous vector calculation pass and incorporates different degrading effects like low seeding density, high image noise, out-of-plan motion leading to (partly) non-matching particles, and high gradients inside the interrogation window. A low correlation coefficient C should be compensated by larger interrogation windows to ensure stability and accuracy. On the other hand high local gradients should rather be tackled by smaller windows if image quality i.e. robustness permits and this is given by the second term of the local weighting factor W based on the rms of the displacement field σ rms inside the interrogation window: W = 1 / ( C γ corr σ rms γ rms ) (3) where γ rms and γ corr have been added to adjust the sensitivity of the two terms and to check which parameter has which effect. So far γ rms = γ corr = 1 has been found appropriate. Further analysis of the optimal ratio is under way. The weighting factor W is further processed to arrive at the final window size. Given e.g. a reference window size of 32 2 pixel, W is mapped to a factor between 1 and corresponding to 322 and 162 windows by eliminating first all vector positions with the 10% highest and lowest values of W and mapping a factor of linearly to the range of the lower 10% threshold to the median value of W, and mapping to the range of the median value of W to the higher 90% threshold. This effectively provides an overall average window size of A variability of factor 2 may seem low and could indeed be increased for experiments with a wide range of image quality and flow gradients. Nevertheless such a range can already provide a significant improvement in spatial resolution with a factor of 4 compared to conventional PIV e.g. 8x32 elongated windows compared to round/square 32x32 windows. 3. Numerical Simulations Using synthetic data of a shear flow the possible improvement of elongated elliptical window shapes is investigated. The flow pattern is similar to two counter-rotating cylinders of a diameter of 256 pixel in between with an effective shear zone of about 10 and 20 pixels (10-90% threshold) over which the displacement goes from -2.5 px to +2.5 px. The maximum shear rate for both cases is 12.5 % and 25 %, respectively. The image quality is Fig. 1 Circular shear zone of dv=5px within about 20 pixel width, rigid rotation inside and outside. Processed with 32x32 adaptive windows, 75% overlap, every second vector in x/y direction shown enlarged by a factor 3, windows drawn 4x smaller. Image size = 512x512 pixel. Color = shear strength

4 ideal in order to concentrate on how window size and shape determine the ability to follow small flow features with strong gradients. As shown in Fig. 1 with 32 2 windows the shape adaptation works well with elongated 4:1 windows aligned with the shear zone. No size adaption has been activated. Fig. 2 shows the difference between the measured and the true displacement for 32 2 adaptive window processing. Fig. 3 shows the rms-difference inside a range of about +-20 and +-30px, respectively, around the shear zone comparing different processing methods at 16 2, 32 2, and 64 2 window size. The adaptive window shape method is compared to standard PIV processing with a constant square and round Gaussian weighted interrogation window. For the thin shear zone (Fig.3 left) the adaptive method has 30% less error than using Fig. 2 Difference dv between true and calculated displacement for 32x32 adaptive windows. constant round 32x32 windows. For 64x64 windows the adaptive method only computes elongated windows with up to 2:1 aspect ratio since windows with 128 x 32 pixel are already so long that they do not fit well within the curved shear zone. Therefore the improvement relative to the other methods is not as large. Fig. 3 Difference between true and calculated displacements within the shear zone - 4 -

5 Constant square windows are slightly worse than round windows probably due to truncation errors. Clearly 75% overlap outperforms 50% overlap since more vector positions are available to fit the strong gradients. The adaptive method can be further improved by using 75% overlap everywhere. Right now for 4:1 elliptical windows the overlap is only 50% in the direction of the minor axis and 87.5% in the direction of the major axis. Fig. 4 PIV-Challenge 2003 case B10, color = correlation value For the thicker shear zone of about 20 pixel width the overall errors are much lower. Since the gradient is already so small there is little difference between all methods with 16x16 windows. Still with 32x32 or larger windows the advantage of shape adaptation is significant. From the curves it is also clear that one should use windows as small as possible since the effect of small windows is larger than using the window shape adaptivity. The performance of both shape and size adaptation has been checked with image 10 of Fig. 5 PIV-Challenge 2003 case B10, interrogation window size: red = 32 2, black = Fig. 6 PIV-Challenge 2003 case B10, close-up of shear region at lower image border with horizontally stretched windows (drawn factor 4 smaller), every second vector shown in x-direction, color = u-displacement in pixels. PIV-Challenge 2005 case B. This synthetic data has regions of strong out-of-plane motion with low correlation values at the top-left in Fig. 4 and at the same time a boundary shear flow at the lower border of the image. Using γ rms = γ corr = 1 and 32 2 pixel reference window size leads to largest windows with 32 2 pixel for the low correlation region and small 16 2 windows with strong 4:1 elongated windows at the boundary shear flow (Fig. 5 and 6). Without using the rms of the displacement for the size calculation (γ rms = 0) the shear flow still has elongated windows but with larger size. Without using the correlation (γ corr = 0) the region with low correlation values has smaller window sizes. Using both rms and correlation provides the best result

6 4. Experimental Results The size and shape adaption method has been tested successfully for various experimental data. Below is shown the performance of the adaptive method (Fig. 7 right) for part of the jet flow image 50 of PIV-Challenge 2003 case A. Comparing constant and adaptive 32x32 windows with a vector grid spacing of 8 pixel (75% overlap) one can notice slightly higher peak vorticity values for the adaptive method. When using 16x16 windows more details become visible but both constant and adaptive methods suffer from outliers. With 32x32 windows and 4 pixel vector spacing (87.5% overlap) the robustness is Fig. 7 PIV-Challenge 2003 case A50, processed with constant window size and shape (left) and adaptive method (right), color = vorticity. Top 32x32ov75%, middle 16x16ov75%, bottom 32x32ov87%. Left: constant round Gaussian windows, right: adaptive shape and size

7 regained except 2 small regions for the adaptive method - and small flow features become visible but at the expense of longer processing times. As shown in Fig. 7 bottom the adaptive method performs slightly better than the constant window method, visible by higher peak vorticity values and sharper flow details resolving smaller vorticies and shear zones not previously visible. Still the overall noise level remains low as can be deduced by the almost laminar flow region at the top which is without significant gradients. Here all small scale vorticity structures are due to vector noise. Ensemble Data Averaging As pointed out by Theunissen et. al. (2009) the adaptive method is particularly suited for correlation averaging (Meinhart et.al., 1999) where one is summing up the correlation maps at each vector position over all images and computing the vector field from the summed correlation map at the end. This method is often used for µpiv with laminar and only slightly varying flows. This method can easily be extended to the usual multi-pass scheme refining interrogation window size and shape. The added robustness from using multiple images simultaneously allows smaller final interrogation Fig. 8 Top 16x16ov75%, every 2 nd vector shown, color = V, middle 32x32ov75%, bottom 32x32ov75% adaptive window size and shape

8 windows than what would be possible processing single images. Fig. 8 shows a comparison of ensemble averaging over 56 images with 16x16 and 32x32 px windows with 75% overlap compared to adaptive window size (32x32 to 16x16) and shape. In the sharp bend the adaptive approach using smaller 16x16 windows can resolve some small flow feature which is not visible with standard 32x32 px processing, while in the bulk of flow the adaptive approach retains the lower error from using larger windows. Processing here with 16x16 still leaves considerable flow fluctuations due to residual noise. 5. Summary A size and shape adaption method has been developed which increases the ability of PIVprocessing to resolve small flow structures with high gradients. The spatial resolution is improving while maintaining robustness and accuracy. Using synthetic data of a thin shear flow it is shown that adaptive window shapes can significantly increase the accuracy of the displacement field. The method also performs reliably for experimental data with visible improvements. References Becker F, Wieneke B, Yuan J, Schnörr C (2008) Variational correlation approach to flow measurement with window adaptation, 14th Int.Symp. Laser Appl Fl. Mech., Lisbon, Portugal, July Meinhart CD, Wereley ST, Santiago JG (1999), A PIV Algorithm For Estimating Time-Averaged Velocity Fields, Proceedings of Optical Methods and Image Processing in Fluid Flow, 3rd ASME/JSME Fluids Engineering Conference, San Francisco. Scarano F (2003) Theory of non-isotropic spatial resolution in PIV, Exp Fluids 35: Stanislas M, Okamoto K, Kähler CJ, Westerweel J, Scarano F (2008) Main results of the third international PIV Challenge, Exp Fluids 45: Theunissen R, Scarano F, Riethmuller ML, (2007) An adaptive sampling and windowing interrogation method in PIV, Meas Sc Techn 18: Theunissen R, Scarano F and Riethmuller ML (2008) On improvement of PIV interrogation near stationary interfaces, Exp Fluids 45: Theunissen R, Scarano F, Riethmuller ML (2009) Spatially adaptive PIV interrogation based on data ensemble, Exp Fluids 48: