Worldwide PIV Challenge 2003 German Aerospace Center (DLR). Busan, Korea September 19-20, 2003. www.pivchallenge.org. Image distortion PIV based on B-Splines *. *) or: YAPA = Y et A nother P IV A lgorithm. Chris Willert Institute of Propulsion Technology, DLR, 51170 Köln, Germany.
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German Aerospace Center (DLR)
September 19-20, 2003
Image distortion PIV based on B-Splines *
*) or: YAPA = Yet Another PIV Algorithm
Chris WillertInstitute of Propulsion Technology, DLR, 51170 Köln, Germany
AbstractThe algorithm used by DLR for the PIV-Challenge is a combination of pyramid-based grid refinement (multi-grid, see e.g. Willert, 1997) and full-field image deformation which is intended to give second-order accurate displacement data. At its core a standard FFT-based correlation algorithm provides local image shift data. Special emphasis was placed on striking a balance between accuracy, robustness and processing speed. The specific characteristics of this algorithm, whose flowchart is shown on the right, will be given in the following sections.
The images from Cases A and C both benefit from image enhancement prior to correlation processing. High-pass filtering and/or dynamic histogram equalization reduce background noise and thereby increase correlation peak visibility.
A typical processing pyramid may look as follows:
1st pass : 96 96win, 48 48step (50% overlap) 2nd pass : 64 64win, 32 32step (50% overlap) 3rd pass : 32 32 win, 16 16step (50% overlap) 4th pass : 16 16 win, 8 8step (50% overlap) Final pass : 16 16 win, 8 8step (50% overlap)
One unique feature here involves down-sampling the image instead of using larger interrogation windows. This down-sampling is performed by summing up the intensities of neighboring pixels without clipping the result – that is, image intensities are preserved at a reduced spatial resolution. In effect small and fast interrogation windows (typ. 3232 px) then may be re-used throughout the first iterations. Aside from speeding up the processing with smaller windows the same validation criteria (e.g. correlation peak search area, median filtering, etc.) may be used throughout.
After validation the new displacement data is projected onto the next finer resolution using bilinear interpolation. For added stability the intermediate data is smoothed with a 33 kernel. To improve convergence, the interrogation at the final resolution is once repeated.
Generally the three highest correlation peaks are located from a limited region of interest in the correlation plane. In the final processing passes, a Levenberg-Marquardt nonlinear least-squares fit to the correlation peak is performed for sub-pixel peak location using 77 values. The procedure is described by Ronneberger et al. (1998).
Processing Flow Chart
(taken from Case Bnear bottom edge)
1616 (Final Pass)
Note that images are shifted toward each other using half the displacement for each.
After each interrogation pass the displacement data is used to deform the image data by applying half the local shift to each image in opposite directions. Bi-linear interpolation is used to sample the displacement data which gives room for further improvement.Unique here is the use of B-splines (B= basis or basic) for the interpolation of the displaced images which was found to give superior performance with respect to polynomial or cardinal sinc interpolation. One side effect in high gradient particle image data is that the interpolated image may take on negative values or overshoot (=ringing). The effect increases for under-sampled images.
Interpolation function shapes:solid line = cubic B-Splinedotted: linear interpolator
Generalized interpolation:(ßn(x) = synthesis function)For linear interpolation: c(k) = image samplesand
For cubic B-spline:
Equivalent interpolant of cubic B-spline
Note : Here c(k) are coefficients, not image samples! These are calculated a-proiri for the original images using a forward-backward recursive filter which requires 2 additions and 2 multiplications per coefficient (e.g. fast computation).(Thévenanz et al., 2002)
Non-separable 2-D interpolation using 55 points (25 function evaluations)
Separable 2-D interpolation using 55 points (25=10 function evaluations)
(Figures in part from Thévenaz et al., 2000)
Processing of all three data sets was performed with the same algorithm with slight variations in image pre-processing, peak-detection ROI and validation parameters. The results provided for the PIV-Challenge were intended to strike an optimum between high-spatial resolution and high validation rates on the one hand, and reasonable noise levels and processing speed on the other. A higher spatial resolution would have been possible but at the cost of increased noise. Alternatively, the noise could have been further reduced through massively over-sampled PIV interrogation – which significantly increases processing times.
Finally, image interpolation based on B-splines was found outperform traditional techniques (i.e. polynominal or sinc-based interpolation) both in terms of speed and precision.
 C. Willert (1997), “Stereoscopic digital PIV for application in wind tunnel flows”, Measurement Science and Technology, vol. 8, pp. 1465-1479.
 O. Ronneberger, M. Raffel, J. Kompenhans (1998), “Advanced evaluation algorithms for standard and dual plane PIV”, Proc. 9th Intl. Symp. on Appl. of Laser Techniques to Fluid Mechanics, Lisbon, Pt, July 13-16.
 P. Thévenaz, T. Blu, M. Unser(2000), “Interpolation revisited”, IEEE Transaction on Medical Imaging, vol. 19, no. 7, pp. 739-758.
 A. Fincham, G. Delerce (2000), “Advanced optimization of correlation imaging velocimetry algorithms”, Experiments in Fluids, vol. 29, no. 7, pp. S13-S22.
The PIV-Challenge image data sets were entirely processed with the PIVview software package (PIVTEC GmbH, Germany). PIVTEC is a DLR out-sourcing enterprise founded in 2001 and is dedicated to making PIV-related developments of DLR commercially available.
A demo version of PIVview may be downloaded from www.pivtec.com