# On dealing with multiple correlation peaks in PIV

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## Abstract

A novel algorithm to analyse PIV images in the presence of strong in-plane displacement gradients and reduce sub-grid filtering is proposed in this paper. Interrogation windows subjected to strong in-plane displacement gradients often produce correlation maps presenting multiple peaks. Standard multi-grid procedures discard such ambiguous correlation windows using a signal to noise (SNR) filter. The proposed algorithm improves the standard multi-grid algorithm allowing the detection of splintered peaks in a correlation map through an automatic threshold, producing multiple displacement vectors for each correlation area. Vector locations are chosen by translating images according to the peak displacements and by selecting the areas with the strongest match. The method is assessed on synthetic images of a boundary layer of varying intensity and a sinusoidal displacement field of changing wavelength. An experimental case of a flow exhibiting strong velocity gradients is also provided to show the improvements brought by this technique.

## 1 Introduction

Standard analysis of PIV images generally involves the division of sequential recordings into smaller interrogation areas and estimating the displacement of recorded particle images captured within those areas by means of cross-correlation. Particle images moving together in the same direction will produce a single strong peak in the cross-correlation map, allowing the retrieval of a representative displacement vector. For this analysis to be conducive, local in-plane displacement gradients must be negligible and an upper limit of 0.05 pixels/pixel is often suggested in literature (Keane and Adrian 1992). When strong in-plane displacement gradients are present, interrogation areas are populated by particle clusters moving in different directions. As a consequence, the correlation peak will broaden and deform as the gradients increase, until the peak splinters into multiple peaks (Fig. 1) (Scarano 2002). The unique displacement vector obtained from the correlation map is then no longer representative of the underlying flow field. Non-uniform displacement fields and deformations in the particle image pattern therefore constitute a strong source of error for PIV (Huang et al. 1993) and must be treated carefully. While smaller interrogation window sizes are preferential to reduce the effect of local velocity gradients, such reduced windows contain fewer particle images thereby increasing the measurement uncertainty and limiting the maximum displacement measurable (Adrian 1991). Both contradicting requirement on the sizing of the correlation windows contribute to a limitation in the achievable dynamic velocity range of PIV (Adrian 1999).

One of the prevailing solutions adopted to increase attainable spatial resolution and dynamic velocity range is the iterative multi-grid approach proposed by Scarano and Riethmuller (2000). The technique suggested involves the analysis of images with wide interrogation areas first, used as “predictor” of the displacement field. Velocity predictors are adopted to deform consecutive image snapshots with the aim of reducing the relative particle images displacement. This step is typically combined with a reduction of the window size and a displacement correction using the velocity predictors. The initial window size should be large enough to capture the largest displacement in the flow field while the final size should be appropriately sized to achieve the desired spatial resolution, ensuring a sufficient amount of particle images are captured. The benefit of the multi-grid algorithm is an increase in spatial resolution and an improved velocity dynamic range, which is the reason of its widespread use and success. However, in the presence of splintered peaks in the first iteration of the analysis, a reliable estimation of the flow field is impossible. This gives rise to an unrepresentative displacement predictor, which influences the remainder of the iterative process.

*et al.*on the other hand devised an intricate multi-pulse separation technique, where pulse separations across several exposures are adjusted to increase dynamic range (Persoons and ODonovan 2010). The majority of PIV applications continue to rely on standard double-pulsed systems however, imposing a unique separation time between recordings. As such, a solution to the problematic of multiple correlation peaks appearing must be sought from an algorithmic perspective.

The standard approach adopted in many PIV algorithms to circumvent the problem of numerous peaks in the correlation map is to simply discard peaks below a certain Signal-to-Noise (SNR) threshold (typically 1.2–1.5) (Keane and Adrian 1990). SNR is defined as the ratio between the highest and the second highest peak and can be indicative of measurement quality (Xue et al. 2014), However, a SNR below the threshold does not necessarily imply unreliable peaks. Multiple correlation peaks can be of equal height and well above the noise level. This SNR-based solution thus helps to increase the robustness of the measurement by dealing with the ambiguity of multiple peaks, but does not offer a solution to the problem of displacement vector representativeness. It will be shown in this paper that complex displacement fields with high amplitude oscillations indeed remain impossible to be measured even with the adoption of a SNR threshold. A particular solution utilising correlation peak amplitudes was proposed in Roth and Katz (2001); peak-height weighted averaging of the displacements related to all the peaks in the correlation map. While this suggestion mitigates the problem when multiple peaks relate to similar displacements, it remains insufficient in case of very dissimilar displacements. When dealing with high intensity opposite velocities such as in case of shear layers, shock waves, vortices, etc., the average of the peaks will produce a nearly zero displacement, augmenting the possibility of iterative multi-grid schemes to fail. To deal with velocity gradients in a recursive local-correlation algorithm, (Hart 2000) proposed to start with a large initial window and use the identified peak in the correlation map as an estimate to limit the range of possible particle displacements in sub-regions. Although this process can, in theory, be carried out down to correlation windows containing a single particle image, the dependency on the initial displacement estimate remains. Exactly this value can be compromised by the appearance of multiple peaks in case of strong displacement gradients. A “Reverse Hierarchical process” was proposed in Rohály et al. (2002). Rather than initiating the analysis from the largest scale, the solution proposed was to build a correlation map by starting from the smallest scale and adding interrogation areas based on inter-level correlation correction and validation. Also in this case, dynamic range and spatial resolution are shown to be improved, although no mention is made in case of splintered peaks. Corresponding correlation windows can also be sized differently (Liang et al. 2002). Such an approach could potentially help mitigate the problem of multiple peaks by taking the advantages of using small and large correlation window together. However, correlation maps produced were shown to be noisier and less reliable due to the smaller regions correlated. More importantly, improvements in accuracy and measurement range were shown to rely on displacement vectors obtained in the previous iteration. Alternatively, the interrogation process follows the standard implementation and image intensities captured within correlation windows are altered. Spatial resolution in case of non-uniform velocity fields was shown to be drastically improved in Nogueira et al. (2001) by multiplying interrogation windows with a weighting function prior to correlation, thereby increasing the importance of central pixels within a correlation window compared to those on the periphery. Several shapes and sizes of weighting functions were tested in Astarita (2007), proving that the adoption of weighting functions can improve the spatial results more than simply reducing the interrogation window size. Notwithstanding the great improvements shown, the use of weighting functions was demonstrated with velocity fields of small-scale amplitudes. Weighting functions reduce the luminosity of particles on the edges of the interrogation area, thus inherently discarding the information pertaining the displacement of those particle images. Weighting functions therefore lower the probability of multiple correlation peaks appearing, but simultaneously produce velocity vectors which are not necessarily representative of the underlying displacement field captured within the interrogation areas. Irrespective of the correlation response, also the vector location must be appropriate. In Young et al. (2004), the problem of velocity gradients has also been tackled in terms of vector location. Displacement vectors are usually anchored in the geometric centre of the interrogation area. This is not necessarily representative of the fluid motion in case of non-uniform velocity fields. The solution proposed by Young et al., was to anchor the vectors to a location more representative of the underlying flow motion. This involved shifting the particle image intensities according to the identified displacement, multiplying the shifted intensity fields and anchoring the vector to the location of the maximum element in their multiplication plane. In spite of the improvements produced by this method, a single vector per correlation window was produced, limiting its application in case of multiple peaks.

The algorithms mentioned afore all highlight the problematic surrounding the appearance of multiple peaks in a correlation map when high velocity gradients are present, yet, none offer a satisfactory solution. In this paper a novel Multiple Peak PIV algorithm (MP-PIV) is proposed, capable of dealing with splintered peaks in the correlation-map. Details of the methodology are provided in Sect. 2. In short, every prominent peak is considered to be a potential displacement. The suggested methodology enables the measurement of these displacements by detecting and analysing all the peaks of the correlation map, producing multiple vectors per correlation window. This approach thus improves the measurement of displacements in case of strong in-plane gradients. The procedure is articulated in two sub-processes. First, strong significant peaks are selected and discerned from random noise. This process is carried out through an automatic thresholding method exploiting the correlation map histogram. Each selected peak then produces one or more vectors anchored to a sub-correlation grid. Vector locations are chosen by shifting the images according to each displacement and by checking the quality of the overlapping images through cross-correlation. Not only is all the valid information contained in the correlation map used to the fullest, also vector locations become more accurate allowing a better estimation of the velocity predictor. Numerical analyses through Monte-Carlo simulations are presented for a boundary layer of varying amplitude and a sinusoidal velocity field of variable frequency in Sect. 3. Together with an experimental test case (Sect. 4) all assessments indicate the proposed solution to yield a drastic improvement in PIV measurements’ accuracy and attainable resolution.

## 2 Methodology

Once correlation windows have been analysed, the displacement field is validated using a normalised median threshold (Westerweel and Scarano 2005) and corrupted vectors are replaced using the median of their neighbours. The final displacement field is then interpolated to produce a displacement predictor for the successive iterations.

### 2.1 Multiple peak detection

#### 2.1.1 Triangle thresholding

*Max pdf*) and the maximum correlation peak amplitude (

*Max corr*) is considered. When dealing with most commonly correlation coefficients,

*Max corr*will equal one. For each bin of the histogram, the distance between its value and the

*max-max*line is evaluated. This distance is plotted in red in Fig. 4 above the triangle. The bin location corresponding to the maximum distance, incremented by a normalized offset (typically 0.1), is considered as a threshold to discern good peaks from correlation noise, i.e. the first peak-selection is done on the basis of correlation peak amplitude.

#### 2.1.2 Peak refinement by Signal-to-Noise ratio

An accustomed measure of the correlation peak’s reliability and validation heuristic of the obtained particle image displacement estimate is the common ratio between the highest and second highest peak amplitude. In MP-PIV the authors revisit this definition since a SNR close to one actually implies both correlation peaks to be of equal magnitude and to potentially contain particle image displacement information provided they are both above the correlation noise. The afore triangle threshold presents an automatic means of discerning strong peaks from the correlation noise based on the peak heights. However, in case of noisy correlation maps (Fig. 3a), displacement peaks might be hard to detect or not even exist, potentially compromising the triangle threshold. For this reason, a secondary peak refinement is implemented.

*N*candidate peaks selected by the triangle threshold (e.g. the five highest peaks in Fig. 5), and denoting the set of sorted correlation peak values in descending order as \(P=\{P_1,P_2,\ldots ,P_N\}\), the SNR of the \(i\mathrm{th}\) peak is defined as

*P*(hence of higher amplitude) for which the \(\mathrm{SNR}_i\) is higher than a user-defined threshold \(\sigma _{{\mathrm{thr}}}\). Figure 5 shows an example of this filter for 50 peaks. The first five peaks are considered on the basis of the triangle threshold. With a typical SNR threshold of 1.3, the third peak is the first for which the \(\mathrm{SNR}_i\) is above the threshold. Accordingly, only the first three peaks are retained for further consideration as these peaks are supposed to be sufficiently distinguishable from correlation noise. Once the selection of correlation peaks is refined, displacement vectors are collocated on a sub-correlation grid, as described in the next section.

### 2.2 Vector allocation

^{1}

## 3 Numerical assessment

Improvements achieved with MP-PIV by resolving splintered peaks have been quantified through Monte-Carlo analyses. Assessments have been performed on synthetic images involving two different velocity fields: a boundary layer of increasing free-stream velocity, tested for different seeding densities, and a sinusoidal field of varying frequency, tested for different amplitudes. Synthetic images were generated following the approach described in Lecordier and Westerweel (2004) using Gaussian shaped tracer images, with diameters drawn from a normal probability \(\mathcal {N}(\mu ,\sigma ^2)=\mathcal {N}(3{\mathrm{\ pixel}},1{\mathrm{\ pixel}}^2)\). The mean intensity of particles was 50% of the maximum gray scale and the standard deviation was 18%. Intensity profiles were integrated over pixel elements adopting a pixel fill-ratio of unity and discretised in 16 bits. The synthetic images were not altered with artificial noise , such that observed differences in performance could be ascribed solely to algorithmic variations..

Summary of the algorithms used in the numerical assessment

Symbols | Peaks | Weighting | SNR filter | Interrogation area |
---|---|---|---|---|

MGRID | Single | No | No | Same size |

MGRIDF | Single | No | Yes | Same size |

LFC | Single | Yes | No | Same size |

DSIW | Single | No | Yes | Different size |

MP-PIV | Multiple | No | Yes | Same size |

### 3.1 Boundary layer

*u*,

*v*) related to a laminar boundary layer on flat plate was simulated by resolving Prandtl’s equations (Prandtl 1938) in the vicinity of a wall, i.e. Blasius’ solution;

*S*;

In Fig. 8b, the curves for a seeding density of 0.1 ppp show a similar behaviour to the case of lower seeding density (Fig. 8a), with the biggest difference being the change in error with DSIW. For this seeding density, the reliability of DSIW has improved, yielding error levels on par with the other methodologies tested. It is worth noting the difference between MGRID and MGRIDF. The mere implementation of a SNR threshold in MGRIDF allows a reduction of the error when dealing with less severe gradients, confirming the detrimental effect of multiple correlation peaks if left untreated. However, at higher gradients imposing a SNR threshold inhibits the detection of a valid correlation peak among multiple peaks of (near-)equal magnitude, increasing the overall error. Best results are obtained by MP-PIV with lowest error levels throughout the entire range of gradients tested. LFC, which aims to enhance spatial resolution, produces error levels which are slightly lower than the other techniques tested, but only in the lower range of gradients (between 0 and 0.25 pixels/pixel). The adoption of a weighting window is well-documented to increase the spatial resolution of the measurement (Astarita 2007; Nogueira et al. 1999). Current results show though that weighting can be harmful if the linear size of the window is not increased accordingly. Particles belonging to the periphery of the window will contribute less to the correlation peak, producing worse results in case of high amplitude displacements. This observation is confirmed by the error levels produced by LFC in case of high gradients. Figure 8a–c show that the error generated by LFC for gradients higher than 0.5 pixels/pixel can be more than twice the error produced by the other techniques.

With increasing seeding density error levels generally reduce. At a seeding density of 0.3 ppp (Fig. 8c) DSIW produces results which are slightly better than MGRID, MGRIDF and LFC, especially in case gradients are higher than 0.5 pixels/pixel. However, also in this case, MP-PIV obtains the lowest error levels across almost the entire range of gradients tested, confirming it being the most suitable methodology for complex velocity fields.

### 3.2 Sinusoid test

*A*of 3, 5 and 7 pixels were imposed. Measured displacement fields were fitted with a sinusoid and the reconstructed amplitude \(\hat{A}\) was normalised with the imposed

*A*producing an evolution in normalised amplitude \(\hat{A}/A\) with normalised wavelength \(\lambda ^*\). For all sinusoids tested, the seeding density was kept at a constant value of 0.1 ppp. Images were analysed once again with initial window sizes of 65 pixels and iteratively halved to 17 pixels. A mutual window overlap of 75% was set to obtain a final vector spacing of 4 pixels.

For a sinusoid of 3 pixels of amplitude, results are presented in Fig. 9a. The plots show that only for small amplitudes the frequency response of cross-correlation resembles the expected behaviour of a moving average (Scarano 2003; Theunissen 2012). MP-PIV performs slightly better compared to the other approaches, except for LFC. For this specific velocity field, the spatial resolution obtained using LFC is superior to the other techniques tested, with values of normalized amplitude higher than 0.5 up to \(\lambda ^*\approx 1.1\). This increased spatial resolution is to be expected as the advantages of using a weighting functions have already been shown in the literature (Astarita 2007). From a practical perspective however, smaller amplitudes demand reduced temporal separation between snapshots, which as advocated is not always optimal. When the sinusoid amplitude is increased from 3 to 5 pixels (Fig. 9b), LFC begins to show the same deficiencies as presented in Sect. 3.1 with an increased amplitude modulation for \(\lambda ^*<0.6\) compared to the other analysis approaches. At \(A=5 \mathrm{pixels}\), MP-PIV behaves slightly better than the other methods tested, offering a marginally higher spatial resolution in the range \(\lambda ^*=0.3\)–0.7. Increasing the sinusoid amplitude even further (Fig. 9c) to \(A=7 \mathrm{pixels}\), the frequency response of all the algorithms becomes completely different from the standard moving average assumption, confirming the strong non-linearity of the PIV algorithm (Theunissen 2012). When a sinusoid of 7 pixels amplitude is analysed, MP-PIV presents the best spatial resolution among all the algorithms tested, with LFC showing the worst spatial resolution for the same reason as described in Sect. 3.1.

### 3.3 Computational effort

An absolute measurement of the computational effort for MP-PIV is impossible to provide since it is entirely dependent on the complexity of the flow. In case of simple velocity fields (i.e. flow gradients are such that correlation maps contain a single peak), the computational overload is only constituted by the vector positioning process. All the parameters being equal to a standard PIV algorithm (i.e. correlation window size, vector spacing, number of iterations, interpolation technique, etc.), the additional overload for MP-PIV slows down the analysis by a factor of 1.5. However, such a comparison is unfair. In fact, MP-PIV produces a denser displacement field from the first iteration and could potentially reduce the number of iterations required for the predictor-corrector to converge, saving computational time compared to a standard PIV algorithm.

Figure 10 presents the computational time for MP-PIV and other techniques described in Table 1 for a pair of synthetic images of a boundary layer as described in Sect. 3.1. As the amplitude of the boundary layer increases, the flow becomes more complex and the computational time of MP-PIV rises as advocated. MP-PIV has been tested for three different seeding densities, 0.03, 0.1 and 0.3 ppp (particles per pixel) to show the dependency of the computational time on the amount of particle images. As the seeding becomes denser, the underlying flow is better sampled and correlation maps will contain more information, enabling an increased number of correlation windows to be elaborated by MP-PIV and rendering a more accurate flow field (cf. Fig. 8).

## 4 Experimental application

Images were analysed using the same algorithms adopted in the previous section. The window size was reduced from 256 to 16 pixels in five predictor-corrector iterations and a mutual overlap of 75% between correlation windows was imposed. Images were interpolated using quintic spline (Astarita and Cardone 2005) and velocity fields were validated using the normalized median threshold (Westerweel and Scarano 2005). A mask was automatically generated to ignore pixels belonging to the disk^{2}.

The average velocity field is shown, together with the displacement magnitude, in Fig. 12. With a jet-like flow issuing from the pores, velocity gradients in the order of \(\frac{\partial u}{\partial y}\equiv 0.6\,\, \mathrm{pixels/pixel}\) are encountered near the geometrical pore boundaries. Horizontal gradients \(\frac{\partial u}{\partial x}\) can be as high as 1.1 pixels/pixel. Velocity profiles for the horizontal displacement are extracted along the red dashed lines and are presented in Figs. 13 and 14. The first remarkable result is the behaviour observed with DSIW. Due to the low seeding density of the images, estimated to be approximately 0.05 ppp, DSIW produces correlation maps which are far too noisy to produce a reliable displacement field. As a result, velocity profiles for DSIW, presented in Figs. 13 and 14 differ completely from the other techniques tested. This behaviour was already shown in the numerical assessment (Fig. 8a) and is confirmed by this experiment. The use of DSIW is therefore strongly discouraged in case of images presenting low seeding density.

### 4.1 Residual displacements

The higher displacements measured by MP-PIV are not necessarily an indication of better measurements. In order to investigate the quality in case of experimental images, a post-processing analysis which the authors refer to as residual displacements is proposed. For each pair of analysed snapshots the final displacement field is used to deform the initial images, as per standard predictor-corrector iterations. Deformed images are then re-analysed, independently, with three different window sizes: 17, 33 and 65 pixels. The vector spacing is kept constant to 33 pixels and the norm of the displacement, for each window size, is stored. The maximum displacement among the three windows tested is then selected and referred to as the residual displacement (RD). This analysis of RD provides a robust indicator of quality of the measurement for experimental images; a displacement vector is correct when the PIV images, deformed with that vector, are perfectly overlapping in the vector’s neighbourhood, in which case RD = 0. The detrimental effect of erroneous and unrepresentative displacement estimates will echo throughout the iterative process and may appear as large displacement inaccuracies in the final result. These are detectable (large RD) with only the larger window sizes.

The vertical profiles for RD are finally shown in Fig. 17. The plots present once again a generally higher value of RD in the vicinity of the jets (\(x/D_{{\mathrm{disc}}}\approx 0.05\)) where gradients are the highest and standard PIV algorithms fail to measure the displacements. MGRID and MGRIDF produce a RD of about 4 pixels, LFC values are around 8 pixels and DSIW is completely out of line with RD in excess of 18 pixels. Results are the best for MP-PIV, with levels of RD below 2 pixels across the entire profile.

## 5 Conclusions

Correlation maps in PIV image analyses can present multiple peaks when strong gradients in particle image displacement are encountered within interrogation windows. A method to detect relevant correlation peaks and generate multiple vectors per interrogation region was proposed. Strong peaks are automatically discerned from the correlation noise using a threshold based on the histogram of the correlation map. Selected peaks are analysed in sequence to understand which areas of the interrogation window contribute to which identified correlation peak. This task is fulfilled through a secondary grid of sub-interrogation areas, each half the size of the parent correlation window. The geometrical centres of these sub-windows serve as anchor locations for subsequent displacement vectors. For each identified correlation peak, sub-interrogation windows are translated according to the corresponding displacement. The most suitable displacement for each sub window is then found through a secondary cross-correlation based on, among all the tried displacements pertaining the identified peaks, the strongest SNR.

The algorithm has been assessed with Monte-Carlo simulations performed with two synthetic flow fields: a boundary layer of varying intensity and a one dimensional sinusoidal field of varying frequency and amplitude. Results indicated the proposed algorithm, MP-PIV, was able to outperform existing PIV methodologies by allowing a lower measurement error in the presence of strong velocity gradients, attaining errors which are as low as half the alternatives tested. Moreover, results for the sinusoid test showed an improved spatial resolution in case of high amplitudes.

Finally, an experimental test case offered a further opportunity to explore the advantages of MP-PIV in case of real images. Images for a porous disc were analysed with different image processing methodologies and residual displacements were presented and compared. Velocity profiles and residual displacements confirmed the results obtained for the synthetic test cases with MP-PIV allowing a better measurement of image displacements in the flow area most affected by velocity gradients.

## Footnotes

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