1 Introduction
1.1 Existing noise filtering methods
2 Proposed ‘E15’ noise filtering method
svd
command) to obtain \(\varvec{\tilde{\mathbf{u}}}_k\), \({\tilde{s}}_k\), and \(\varvec{\tilde{\mathbf{v}}}_k\).2.1 Illustrative example
2.2 Outline
3 Analytic examples and discussion
3.1 Analytic minimal working example
3.1.1 Estimates of the clean singular values
3.1.2 Root mean square error (rmse)
3.1.3 Reconstruction loss
3.1.4 Estimate of reconstruction loss
3.1.5 Figure of merit for reconstruction accuracy
3.1.6 Comparison to literature methods
3.2 Effect of the distribution of singular values
PIV processing approach 
\(\epsilon {\times }10^5\)

\({\bar{\epsilon }}{\times }10^5\)

\(r_{\min }\)

\({\bar{r}}_{\min }\)

\(r_B\)

\(r_F\)

\(r_{{\tilde{F}}}\)

\(k_2\)

\(\varDelta _{r_{\min }} {\times }10^3\)

\(\varDelta _{{\bar{r}}_{\min }} {\times }10^3\)

\(\varDelta _{r_B} {\times }10^3\)

\(\varDelta _{r_F} {\times }10^3\)

\(\varDelta _{r_{{\tilde{F}}}} {\times }10^3\)


\(\circ \,\, 32\) px \(75\% \rightarrow 8\) px \(0\%\)  6.13  6.08  13  13  13  14  32  21  1.16  1.16  1.30  1.41  3.22 
\(\circ \,\, 32\) px \(75\% \rightarrow 16\) px \(50\%\)  3.22  3.19  15  15  13  25  27  42  0.75  0.75  0.77  1.12  1.19 
\(\circ \,\, 32\) px \(75\% \rightarrow 32\) px \(75\%\)  1.72  1.72  19  19  15  50  59  49  0.47  0.47  0.48  0.80  0.88 
\(\circ \,\, 64\) px \(50\% \rightarrow 8\) px \(0\%\)  6.06  6.08  13  13  13  14  21  14  1.10  1.10  1.23  1.33  2.03 
\(\circ \,\, 64\) px \(50\% \rightarrow 16\) px \(50\%\)  3.06  3.01  15  15  13  24  56  35  0.35  0.35  0.37  0.67  1.55 
\(\circ \,\, 64\) px \(50\% \rightarrow 32\) px \(75\%\)  1.43  1.41  17  21  13  56  58  54  0.27  0.28  0.28  0.57  0.58 
\(\varvec{\circ }\, \mathbf {64}\)
px
\(\mathbf {50\% \rightarrow 3\times (32}\)
px
\(\mathbf {75\%)}\)
 1.90  1.85 
15

16

15

30

52

29
 0.16  0.16  0.17  0.39  0.68 
\(\square \, 32\) px \(75\% \rightarrow 8\) px \(0\%\)  6.58  6.48  13  13  13  16  19  14  2.87  2.87  3.04  3.38  3.72 
\(\square \, 32\) px \(75\% \rightarrow 16\) px \(50\%\)  3.77  3.74  15  13  13  27  34  36  1.04  1.04  1.07  1.60  1.88 
\(\square \, 32\) px \(75\% \rightarrow 32\) px \(75\%\)  2.09  2.30  378  17  13  49  90  41  1.07  1.24  1.27  1.59  2.01 
\(\square \, 64\) px \(50\% \rightarrow 8\) px \(0\%\)  6.51  6.40  13  12  13  18  24  14  2.78  2.80  2.94  3.50  4.16 
\(\square \, 64\) px \(50\% \rightarrow 16\) px \(50\%\)  3.61  3.56  15  13  13  24  27  25  0.52  0.52  0.54  0.93  1.05 
\(\square \, 64\) px \(50\% \rightarrow 32\) px \(75\%\)  1.71  1.71  16  17  13  50  52  60  0.45  0.45  0.46  0.76  0.78 
\(\square \, 64\) px \(50\% \rightarrow 3\times (32\) px \(75\%)\)  2.14  2.10  15  15  15  27  32  29  0.20  0.20  0.22  0.43  0.51 
4 Synthetic PIV: cylinder flow
4.1 Generation of datasets
4.2 Results overview

The error estimation procedure of “Appendix 1” produces an accurate estimate \({\bar{\epsilon }}= 1.85 \times 10^{5}\) of the actual rms error \(\epsilon = 1.90 \times 10^{5}\), which is the rms difference between the clean and noisy PIV data.

The Brindise method yields rank and loss similar to those of the present approach, whereas the Raiola methods yield much higher rank and loss.

The estimated loss \({\bar{\varDelta }}_{{\bar{r}}_{\min }} = 3.0\times 10^{4}\) from (22) is within a factor of 2 of the actual \({\varDelta }_{{\bar{r}}_{\min }} = 1.6\times 10^{4}\).

Here, the reconstruction merit (23) is \(M = 91\%\).