Bernhard Wieneke

PIV uncertainty quantification from correlation statistics

The uncertainty of a PIV displacement field is estimated using a generic post-processing method based on statistical analysis of the correlation process using differences in the intensity pattern in the two images. First the second image is dewarped back onto the first one using the computed displacement field which provides two almost perfectly matching images. Differences are analyzed regarding the effect of shifting the peak of the correlation function. A relationship is derived between the standard deviation of intensity differences in each interrogation window and the expected asymmetry of the correlation peak, which is then converted to the uncertainty of a displacement vector. This procedure is tested with synthetic data for various types of noise and experimental conditions (pixel noise, out-of-plane motion, seeding density, particle image size, etc) and is shown to provide an accurate estimate of the true error.

PIV uncertainty quantification by image matching

A novel method is presented to quantify the uncertainty of PIV data. The approach is a posteriori, i.e. the unknown actual error of the measured velocity field is estimated using the velocity field itself as input along with the original images. The principle of the method relies on the concept of super-resolution: the image pair is matched according to the cross-correlation analysis and the residual distance between matched particle image pairs (particle disparity vector) due to incomplete match between the two exposures is measured. The ensemble of disparity vectors within the interrogation window is analyzed statistically. The dispersion of the disparity vector returns the estimate of the random error, whereas the mean value of the disparity indicates the occurrence of a systematic error. The validity of the working principle is first demonstrated via Monte Carlo simulations. Two different interrogation algorithms are considered, namely the cross-correlation with discrete window offset and the multi-pass with window deformation. In the simulated recordings, the effects of particle image displacement, its gradient, out-of-plane motion, seeding density and particle image diameter are considered. In all cases good agreement is retrieved, indicating that the error estimator is able to follow the trend of the actual error with satisfactory precision. Experiments where time-resolved PIV data are available are used to prove the concept under realistic measurement conditions. In this case the ‘exact’ velocity field is unknown; however a high accuracy estimate is obtained with an advanced interrogation algorithm that exploits the redundant information of highly temporally oversampled data (pyramid correlation, Sciacchitano etxa0al (2012 Exp. Fluids 53 1087–105)). The image-matching estimator returns the instantaneous distribution of the estimated velocity measurement error. The spatial distribution compares very well with that of the actual error with maxima in the highly sheared regions and in the 3D turbulent regions. The high level of correlation between the estimated error and the actual error indicates that this new approach can be utilized to directly infer the measurement uncertainty from PIV data. A procedure is shown where the results of the error estimation are employed to minimize the measurement uncertainty by selecting the optimal interrogation window size.

Iterative reconstruction of volumetric particle distribution

For tracking the motion of illuminated particles in space and time several volumetric flow measurement techniques are available like 3D-particle tracking velocimetry (3D-PTV) recording images from typically three to four viewing directions. For higher seeding densities and the same experimental setup, tomographic PIV (Tomo-PIV) reconstructs voxel intensities using an iterative tomographic reconstruction algorithm (e.g. multiplicative algebraic reconstruction technique, MART) followed by cross-correlation of sub-volumes computing instantaneous 3D flow fields on a regular grid. A novel hybrid algorithm is proposed here that similar to MART iteratively reconstructs 3D-particle locations by comparing the recorded images with the projections calculated from the particle distribution in the volume. But like 3D-PTV, particles are represented by 3D-positions instead of voxel-based intensity blobs as in MART. Detailed knowledge of the optical transfer function and the particle image shape is mandatory, which may differ for different positions in the volume and for each camera. Using synthetic data it is shown that this method is capable of reconstructing densely seeded flows up to about 0.05 ppp with similar accuracy as Tomo-PIV. Finally the method is validated with experimental data.

Collaborative framework for PIV uncertainty quantification: comparative assessment of methods

A posteriori uncertainty quantification of particle image velocimetry (PIV) data is essential to obtain accurate estimates of the uncertainty associated with a given experiment. This is particularly relevant when measurements are used to validate computational models or in design and decision processes. In spite of the importance of the subject, the first PIV uncertainty quantification (PIV-UQ) methods have been developed only in the last three years. The present work is a comparative assessment of four approaches recently proposed in the literature: the uncertainty surface method (Timmins et al 2012), the particle disparity approach (Sciacchitano et al 2013), the peak ratio criterion (Charonko and Vlachos 2013) and the correlation statistics method (Wieneke 2015). The analysis is based upon experiments conducted for this specific purpose, where several measurement techniques are employed simultaneously. The performances of the above approaches are surveyed across different measurement conditions and flow regimes.

PIV uncertainty propagation

This paper discusses the propagation of the instantaneous uncertainty of PIV measurements to statistical and instantaneous quantities of interest derived from the velocity field. The expression of the uncertainty of vorticity, velocity divergence, mean value and Reynolds stresses is derived. It is shown that the uncertainty of vorticity and velocity divergence requires the knowledge of the spatial correlation between the error of the x and y particle image displacement, which depends upon the measurement spatial resolution. The uncertainty of statistical quantities is often dominated by the random uncertainty due to the finite sample size and decreases with the square root of the effective number of independent samples. Monte Carlo simulations are conducted to assess the accuracy of the uncertainty propagation formulae. Furthermore, three experimental assessments are carried out. In the first experiment, a turntable is used to simulate a rigid rotation flow field. The estimated uncertainty of the vorticity is compared with the actual vorticity error root-mean-square, with differences between the two quantities within 5-10% for different interrogation window sizes and overlap factors. A turbulent jet flow is investigated in the second experimental assessment. The reference velocity, which is used to compute the reference value of the instantaneous flow properties of interest, is obtained with an auxiliary PIV system, which features a higher dynamic range than the measurement system. Finally, the uncertainty quantification of statistical quantities is assessed via PIV measurements in a cavity flow. The comparison between estimated uncertainty and actual error demonstrates the accuracy of the proposed uncertainty propagation methodology.

Non-uniform optical transfer functions in particle imaging: calibration and application to tomographic reconstruction

A new approach to the weighting function, which describes particle imaging in tomographic reconstruction, is introduced. Instead of assuming a spatially homogeneous mapping function of voxels to the images, a variable optical transfer function (OTF) is applied. By this method, the negative effects of optical distortions on the reconstruction can be reduced considerably. The effects of these improvements in reconstruction quality on the methods of tomographic particle imaging velocimetry, as well as 3D particle tracking are investigated. A method to calibrate the OTF to experimental circumstances is proposed as an additional step to the volume self-calibration. It is shown that this kind of calibration is able to capture the predominant particle imaging both for simulated as well as experimental data. The most common distortions of particle imaging are blurring due to a small depth of field and astigmatism due to imaging optics. The effects of both of these distortions on reconstruction and correlation quality are investigated via simulated data. In both cases, a strong influence on relevant parameters can be seen. Reconstructions using a spatially varying OTF, calibrated to the imaging conditions, show a significant improvement in reconstruction quality and the accuracy of the particle peak position, as well as in the accuracy of the gained displacement vector field when using two time steps. Evaluation of experimental data by PTV methods shows a reduction in ghost particle intensity and improvements in peak position accuracy. A computationally efficient method of applying the OTF to tomographic reconstruction is introduced.

Variational Adaptive Correlation Method for Flow Estimation

A variational approach is presented to the estimation of turbulent fluid flow from particle image sequences in experimental fluid mechanics. The approach comprises two coupled optimizations for adapting size and shape of a Gaussian correlation window at each location and for estimating the flow, respectively. The method copes with a wide range of particle densities and image noise levels without any data-specific parameter tuning. Based on a careful implementation of a multiscale nonlinear optimization technique, we demonstrate robustness of the solution over typical experimental scenarios and highest estimation accuracy for an international benchmark data set (PIV Challenge).

Peak-locking reduction for particle image velocimetry

A parametric study of the factors contributing to peak-locking, a known bias error source in particle image velocimetry (PIV), is conducted using synthetic data that are processed with a state-of-the-art PIV algorithm. The investigated parameters include: particle image diameter, image interpolation techniques, the effect of asymmetric versus symmetric window deformation, number of passes and the interrogation window size. Some of these parameters are found to have a profound effect on the magnitude of the peak-locking error. The effects for specific PIV cameras are also studied experimentally using a precision turntable to generate a known rotating velocity field. Image time series recorded using this experiment show a linear range of pixel and sub-pixel shifts ranging from 0 to ±4 pixels. Deviations in the constant vorticity field (ω z ) reveal how peak-locking can be affected systematically both by varying parameters of the detection system such as the focal distance and f-number, and also by varying the settings of the PIV analysis. A new a priori technique for reducing the bias errors associated with peak-locking in PIV is introduced using an optical diffuser to avoid undersampled particle images during the recording of the raw images. This technique is evaluated against other a priori approaches using experimental data and is shown to perform favorably. Finally, a new a posteriori anti peak-locking filter (APLF) is developed and investigated, which shows promising results for both synthetic data and real measurements for very small particle image sizes.

On Sparsity Maximization in Tomographic Particle Image Reconstruction

This work focuses on tomographic image reconstruction in experimental fluid mechanics (TomoPIV), a recently established 3D particle image velocimetry technique. Corresponding 2D image sequences (projections) and the 3D reconstruction via tomographical methods provides the basis for estimating turbulent flows and related flow patterns through image processing. TomoPIV employs undersampling to make the high-speed imaging process feasible, resulting in an ill-posed image reconstruction problem. We address the corresponding basic problems involved and point out promising optimization criteria for reconstruction based on sparsity maximization, that perform favorably in comparison to classical algebraic methods currently in use for TomoPIV.

A Variational Approach to Adaptive Correlation for Motion Estimation in Particle Image Velocimetry

In particle image velocimetry (PIV) a temporally separated image pair of a gas or liquid seeded with small particles is recorded and analysed in order to measure fluid flows therein. We investigate a variational approach to cross-correlation, a robust and well-established method to determine displacement vectors from the image data. A soft Gaussian window function replaces the usual rectangular correlation frame. We propose a criterion to adapt the window size and shape that directly formulates the goal to minimise the displacement estimation error. In order to measure motion and adapt the window shapes at the same time we combine both sub-problems into a bi-level optimisation problem and solve it via continuous multiscale methods. Experiments with synthetic and real PIV data demonstrate the ability of our approach to solve the formulated problem. Moreover window adaptation yields significantly improved results.