A fast all-in-one method for automated post-processing of PIV data

Abstract

Post-processing of PIV (particle image velocimetry) data typically contains three following stages: validation of the raw data, replacement of spurious and missing vectors, and some smoothing. A robust post-processing technique that carries out these steps simultaneously is proposed. The new all-in-one method (DCT-PLS), based on a penalized least squares approach (PLS), combines the use of the discrete cosine transform (DCT) and the generalized cross-validation, thus allowing fast unsupervised smoothing of PIV data. The DCT-PLS was compared with conventional methods, including the normalized median test, for post-processing of simulated and experimental raw PIV velocity fields. The DCT-PLS was shown to be more efficient than the usual methods, especially in the presence of clustered outliers. It was also demonstrated that the DCT-PLS can easily deal with a large amount of missing data. Because the proposed algorithm works in any dimension, the DCT-PLS is also suitable for post-processing of volumetric three-component PIV data.

Figure 1

Top panel: Reduced cut-off frequency of the DCT-PLS (at −3dB) as a function of the smoothness parameter (s). The cut-off frequencies of common average filters are also represented (dashed horizontal lines). Bottom panel: Spatial frequency responses (SFR) of the DCT-PLS for two s values (0.1 and 0.5). The SFR of the 2×2 and 3×3 average filters are depicted for comparison. The dashed horizontal line corresponds to the −3dB limit.

Figure 2

Vortical cellular flow (32×32) corrupted by Gaussian noise, outliers and missing data before (left) and after (right) smoothing by the DCT-PLS. The normalized root mean squared error (NRMSE, see equation 7) was 14% here.

Figure 3

Vortical cellular flow (32×32) corrupted by a Gaussian noise of variance (0.01×V_max)² and 15% of outlying vectors. Two configurations were tested in this study: scattered (left) and clustered (right) spurious vectors.

Figure 4

Normalized root mean squared errors (NRMSE, see equation 7) between the original and the post-processed vortical flow fields, as a function of the percentage of outlying vectors, in the following situations: A and B: Gaussian noise with standard deviation (STD) 1% V_max, with scattered (A) and clustered (B) spurious vectors; C and D: Gaussian noise with standard deviation 10% V_max, with scattered (C) and clustered (D) spurious vectors. CONVL stands for the conventional method while DCT-PLS represents the DCT-based penalized least squares method.

Figure 5

Effect of clustered missing data on the post-process of the vortical flow (noise STD = 10% V_max) with the DCT-PLS: normalized root mean squared errors (between the post-processed and original velocity fields) as a function of percentage of missing vectors.

Figure 6

Effect of masks on the post-process of the vortical flow (noise STD = 10% V_max) with the DCT-PLS. Left panel: one example with a diamond-shaped mask of radius 8 pixels. Right panel: normalized root mean squared errors (between the post-processed and original velocity fields) as a function of percentage of masked pixels.

Figure 7

Experimental PIV data measured in a turbulent jet (from Westerweel et al., see text for details). A) Raw PIV velocities. B) PIV velocities post-processed with the DCT-PLS. C) Residuals (velocities in A – velocities in B). D) Contour plots of the velocity magnitudes (solid lines: using the DCT-PLS, dashed lines: using the conventional method). The background represents the velocity magnitudes obtained with the DCT-PLS (Fig. 7B).

Figure 8

Experimental PIV data of a vortex pair (from Willert et al., see text for details). A) Raw PIV velocities. B) PIV velocities post-processed with the DCT-PLS. C) Residuals (velocities in A – velocities in B). D) Contour plots of the velocity magnitudes (solid lines: using the DCT-PLS, dashed lines: using the conventional method). The background represents the velocity magnitudes obtained with the DCT-PLS (Fig. 8B).

Figure 9

Efficiency of the DCT-PLS with missing data. A) 50% of vectors have been randomly removed from the raw PIV velocities illustrated in Fig. 7A. B) The DCT-PLS reconstructed the velocity field with reasonable accuracy (compare with Fig. 7B). C) Contour plots of the velocity magnitudes (solid lines: using the DCT-PLS on the raw PIV data of Fig. 7A, dashed lines: using the DCT-PLS on the modified raw PIV data of Fig. 9A). The background represents the velocity magnitude obtained with the original raw data (Fig. 7B). D) Effect of the percentage of missing vectors on the normalized root mean squared error (NRMSE, see equation 8) between the post-processed PIV fields with missing vectors (turbulent jet, see Fig. 9B) and the post-processed PIV field without missing data (see Fig. 7B).

Figure 10

Application of the DCT-PLS approach with PIV measurements performed in a left ventricular model (IRPHE, Marseilles, France). From top to bottom: A) raw PIV data, B) PIV velocity field post-processed with the DCT-PLS, C) residuals.

Figure 11

Oversmoothing due to the DCT-PLS with a supersonic flow (courtesy of Dr. Kompenhans and Dr. Schroeder, DLR’s institute of aerodynamics and flow technology). A) Raw PIV data. B) PIV velocity field post-processed with the DCT-PLS. C) Difference between the raw and post-processed fields. D) Close-up that reveals the presence of oversmoothing along the shock wave.

Figure 12

Left panel: effect of the number of vortices on the normalized root mean squared error (NRMSE): comparison between the DCT-PLS and the conventional method (CONVL). Velocity field size = 128×128, noise standard deviation = 0.1×V_max, percentage of spurious vectors = 10%. Right panels: two velocity amplitude fields (before being corrupted) are represented for an illustrative purpose (2×2 and 8×8 vortices).

Figure 13

Automated post-processing of a 3-D corrupt mock flow (left versus right panels) with the DCT-PLS method. The bottom panels display the velocity field on a horizontal x-y plane at the middle of the volume of interest before (left) and after (right) post-processing.

Figure 14

Normalized root mean squared errors between the post-processed and original velocity fields (NRMSE, see equation 7) as a function of percentage of spurious vectors in the 3-D mock flow (see also Fig 13). Two noise standard variations were tested: 0.1 V_max and 0.01 V_max.