14th International Symposium on Particle Image Velocimetry – ISPIV 2021 

August 1–5, 2021 

 

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Physics-based universal outlier detector for flow 

statistics  

E. Saredi1*, A. Sciacchitano1 and F. Scarano1 

1Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands 
* e.saredi@tudelft.nl 

 

Abstract 

Outlier detection for PIV velocity fields is still nowadays an active field of research. In the last 

decades, several image pre-processing and processing algorithms have been developed aiming at 

increasing the dynamic velocity range of PIV measurements and reducing the measurement 

uncertainty. Nevertheless, PIV velocity fields are still often characterised by the presence of 

outliers, which potentially hamper the correct interpretation of the flow physics and negatively 

affect the evaluation of the flow statistics. The outlier detection strategies presented in literature 

are mainly based on the statistical analysis of the velocity vector with its immediate neighbour. 

Most of these algorithms have been demonstrated to be effective for instantaneous flow fields, 

where the errors associated with the outliers are order of magnitude larger than the expected 

measurement uncertainty. However, these approaches are not as effective for the flow statistics, 

where the outliers yield errors of the same order of the measurement uncertainty. To overcome 

this limitation, this paper proposed an outlier detection approach based on the agreement of the 

flow statistics to the constitutive equations, more specifically to the turbulent kinetic energy (TKE) 

transport equation. The focus is posed on the ratio between the local advection terms of TKE and a 

robust estimation of the TKE production along the local streamline. It is demonstrated that, in 

presence of outliers, the proposed principle yields a clear separation between the correct and the 

erroneous vectors. In order to assess the performance of the proposed principle, three different test 

cases are considered. For all of them, the results are compared with a reference outlier detection 

methodology, namely the universal outlier detection method proposed by Westerweel and 

Scarano (2005). The proposed turbulence transport-based approach exhibits higher performance in 

terms of percentage of outliers correctly identified in the flow statistics.  

1 Introduction 

In Particle Image Velocimetry (PIV), outliers are spurious vectors that exhibit large unphysical 

variations in magnitude and direction from neighbouring valid vectors (Westerweel, 1994). Several 

factors can lead to the appearance of outliers. Among them, unwanted light reflections, lack of 

seeding and inadequate processing methodology are the main ones (Hart, 2000; Lazar et al., 2004; 

Sciacchitano, 2019; among others). Several methods for outlier detection have been presented in 

the literature, with the focus on the analysis of instantaneous flow fields. The most common 

approach relies on the analysis of the difference between a vector and its neighbours, either in the 

temporal or in the spatial domain. In this category of outlier detection algorithms, the approach 

proposed by Westerweel and Scarano (2005) is nowadays widely applied. The latter approach 

builds upon the median filter presented by Westerweel (1994) and achieves a larger degree of 

mailto:e.saredi@tudelft.nl


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generality by introducing a normalisation of the median residual with respect to an estimate of the 

local velocity variation. The authors proved the applicability of their normalised median test in a 

wide range Reynolds numbers from 10-1 to 107. Duncan et al. (2010) extended this approach for 

unstructured data as those obtained by Particle Tracking Velocimetry (PTV) or adaptive 

interrogation algorithms. These methodologies proved to be effective for isolated outliers, whereas 

they suffer from under-detection of outliers when a cluster of false vectors is present in the data. 

This phenomenon is common when low-seeding areas, if cross-correlation based algorithms are 

employed, or strong reflections on objects are present in the raw images (Masullo and Theunissen, 

2016). In order to overcome this limitation, Masullo and Theunissen (2016) presented a new 

method based on the combination of a first spatial coherence test and a modified Gaussian-

weighted distance-based averaging median. The introduction of the former parametrises the extent 

of the region inside which the vectors are tested. This methodology showed a gain in the 

achievable spatial resolution and outlier detection probability, at a cost of an increased 

computational burden.  

In addition to the spatial domain, also the time domain has been exploited to identify erroneous 

vector in PIV data. Proper orthogonal decomposition (POD) has been considered as a filter to 

detect outlier (Wang et al., 2015; Raiola et al., 2015). In these cases, the low-order POD modes are 

used in order to reconstruct the velocity field at each time step and the difference between the 

original velocity and the reconstructed field is used as flag for the erroneous vector. This approach 

has shown a higher effectiveness for detecting clustered outliers with respect to the normalized 

median test approach.  

The advent of Shake-The-Box (STB, Schanz et al., 2016) has established a new state-of-the-art to 

perform volumetric measurements. Particles are tracked individually across frames and the 

temporal information is used then to improve the accuracy of the estimation of the particle 

position at the various time-steps. The result of this operation is an ensemble of particle tracks 

inside the measurement volume. Several techniques have been presented to obtain the 

instantaneous velocity field 𝐮 or the time-averaged velocity field  �̅� on a Cartesian grid, such as 

ensemble averaging (Agüera et al. , 2016) or functional binning (Godbersen and Schröder, 2020). 

The aforementioned outlier detection strategies are typically effective for the instantaneous 

velocity when the latter is expressed on a Cartesian grid. In contrast, when the ensemble-average 

operation is conducted to determine the time-average velocity field, the magnitude of the outliers 

often decreases significantly because of their unsteady nature. As a consequence, the outliers’ 

magnitude can become of the same order of that of the correct vectors, thus decreasing the 

effectiveness of the aforementioned outlier detection methods and potentially leaving a large 

percentage of invalid vectors undetected.  

Considering the constitutive equations is another possible approach to the outlier detection 

problem. Song et al. (1999) made use of the continuity equation calculated on a Delaunay 

tessellation in order to detect spurious vectors. Due to the presence of outliers, unphysical nonzero 

values of the velocity divergence appear in the incompressible flow field. The outliers can be 

identified based as regions of high velocity divergence magnitude; however, for this purpose, the 

definition of a user-defined threshold is necessary, so as to limit the occurrence of false positive (as 

deductible from the data presented by Azijli and Dwight, 2015). 

Inspired by the two works above, in the present work a robust physics-based approach is 

proposed that employs the turbulent kinetic energy transport equation to detect the outliers in the 

flow statistics. 



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2 Physics-based outlier detection principle 

This work proposes an outlier detection approach for flow statistics. The measured velocity is 

expressed via Reynolds decomposition in terms of mean and fluctuating component: 

𝑢𝑖 = �̅�𝑖 + 𝑢′𝑖      (1) 

where the subscript i indicates the component of the velocity (𝑖 = 1,2,3 in a three-dimensional 

flow). 

In the following discussion, an outlier is defined as a velocity vector whose time-average value or 

its fluctuations depart from the exact value to an extent exceeding the measurement uncertainty. 

From the fluctuating component of equation 1, 𝑢′𝑖, the turbulent kinetic energy (TKE), expressed as 

𝑘 =  1/2(𝑢1
′ 2̅̅ ̅̅̅ + 𝑢2

′ 2̅̅ ̅̅ ̅ + 𝑢3
′ 2̅̅ ̅̅ ̅), can be obtained. Its behaviour is described by the associated transport 

equation (Hinze, 1975): 

 

𝐷𝑘

𝐷𝑡
= 𝑃 + 𝒯 − 𝜀      (2) 

where:  

𝐷𝑘

𝐷𝑡
=

𝜕𝑘

𝜕𝑡
+ 𝐴 is the material derivative of k,    (3) 

𝐴 = �̅�𝑖
𝜕𝑘

𝜕𝑥𝑖
 is the transport by advection,   (4) 

𝑃 = −𝑢𝑖′𝑢𝑗′̅̅ ̅̅ ̅̅ ̅ 𝜕𝑢𝑖̅̅ ̅

𝜕𝑥𝑗
 is the production,   (5) 

𝒯 contains all the transport terms of turbulence,  (6) 

𝜀 = 𝜈 
𝜕𝑢𝑖

′

𝜕𝑥𝑗

𝜕𝑢𝑖
′

𝜕𝑥𝑗

̅̅ ̅̅ ̅̅ ̅̅
 is the pseudo-dissipation.   (7)  

Many studies have focused the attention on the description of the relative importance of the 

different terms of the TKE equation. In Ikhennicheu et al. (2020) a list of the most recent works is 

reported where the TKE budget has been experimentally evaluated. In this work, turbulent shear 

flows are considered, such as jets, mixing layers, channel flows and boundary layers. For this kind 

of flows, the turbulent quantities change slowly together with the mean flow; the production and 

the dissipation of the turbulent kinetic energy are in balance, i.e. 𝑃 ≈ 𝜀 (Pope, 2000). Furthermore, 

equation 2 can be further simplified since in many flows the turbulent transport of 𝑘 is negligible 

(Nieuwstadt et al., 2016). From this, it is possible to conclude that 𝑃 acts as upper bound of the 

variation of TKE along a streamline, since, neglecting the transport term, a non-null dissipation can 

yield only to a reduction of TKE.  

In order to explain the proposed outlier detection principle, it is here considered a time-average 

velocity field around an airfoil described by  �̅�, 𝐮′ and 𝑘, whose streamline are schematically 

represented in figure 1. The airfoil is set at high angle of attack, which causes the separation of the 

flow along the suction side of the airfoil. The edges of the separated region are characterized by a 

high shear and then, as consequence, a high turbulence production. In this case, the presence of 

four regions of outliers is hypothesized: region A, in front of the airfoil; regions B and C, caused by 

the shadow created by the light refraction at leading and trailing edges of the (transparent) airfoil, 

and region D, located in the shear region. Along any time-average streamline of the flow, the 

evolution of 𝑘 must comply with equation 2. 



14th International Symposium on Particle Image Velocimetry – ISPIV 2021 

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Figure 1: Schematic representation of the outlier detection principle based on turbulent transport. 

Based on the above considerations, the normalized admissible TKE transport 𝜌𝑇𝑇 is defined which 

accounts for the variation of the turbulent kinetic energy along a streamline, relative to the TKE 

production: 

𝜌𝑇𝑇 =
|𝐴|

|𝑃|
< 𝑇ℎ𝑟              (8) 

Values of 𝜌𝑇𝑇 larger than a threshold Thr (theoretically close to unity, in practice larger than 1 to 

account for the noise in the measurements) are physically inadmissible and therefore correspond 

to the presence of outliers. In order to increase the robustness of the evaluation of 𝑃 with respect to 

the presence of spurious measurement data, the production term in equation 8 is substituted by 

𝑃𝑚, that represents the median of its neighbourhood. Furthermore, to avoid elevated values of 𝜌𝑇𝑇 

in regions where 𝑃 tends to zero, a minimum normalization level 𝛾𝑇𝑇 is assumed that represents 

the uncertainty on the determination of the numerator of equation 8, similarly to what proposed 

by Westerweel and Scarano (2005). With these two modifications,  equation 8 becomes: 

𝜌𝑇𝑇 =
|𝐴|

|𝑃𝑚|+𝛾𝑇𝑇
< 𝑇ℎ𝑟      (9) 

Equation 9 is proposed in this work as a means to identify outliers in statistical data from PIV or 

LPT. In regions of valid vectors, the normalised ratio 𝜌𝑇𝑇 remains below the threshold, meaning 

that the variation of the kinetic energy due to the advection along a streamline is smaller than the 

kinetic energy production (multiplied by the parameter 𝑇ℎ𝑟). Instead, values of 𝜌𝑇𝑇 larger than 𝑇ℎ𝑟 

indicate a variation of the turbulent kinetic energy greater than its production, which is physically 

inadmissible. 

Assuming an uncertainty level on the measured velocity of the order of 5% of Uꝏ, the term TT can 

be estimated: 

𝛾𝑇𝑇 =
(0.05𝑈∞)3

Δ
      (10) 



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being Δ the vector pitch of the velocity field. In the next section, the approach is assessed using 

three different datasets comprising both planar and volumetric PIV measurements. 

 3 Experimental assessment  

In order to assess the proposed principle, three different experimental datasets have been 

considered: the 2D velocity field around a two-dimensional wing with NACA 0012 section, the 

near-wake of a truncated cylinder and the near wake of the Ahmed body. Figure 2 shows three 

instantaneous raw images of the three considered datasets. The three datasets have been acquired 

with different systems: in the 2D-wing case, a single-camera two-component PIV system has been 

used (Adatrao et al, 2021); the images for the near-wake of the cylinder have been acquired with 

four cameras placed at a large tomographic aperture (Schneiders et al., 2016); finally, the near-

wake of the Ahmed body has been investigated using the Robotic Volumetric PIV system (Jux et 

al., 2018), which is composed by four cameras placed at a low tomographic angle (Saredi et al., 

2020).  

In order to assess the performance of the turbulence transport-based principle, its results are 

compared to the ones obtained by the universal outlier detection, hereafter referred as UOD, 

proposed by Westerweel and Scarano (2005). In their work, the authors proposed to identify 

outliers based on the value of a normalized residual 𝑟0
∗ defined as: 

𝑟0
∗ =

|𝑈0−𝑈𝑚|

𝑟𝑚+𝛾𝑈𝑂𝐷 
       (11) 

where 𝑈0 is the velocity vector considered, 𝑈𝑚 the median of its neighbours, 𝑟𝑚 the median of the 

neighbour residuals defined as 𝑟𝑖 = |𝑈𝑖 − 𝑈𝑚| and 𝛾𝑈𝑂𝐷 the minimum normalization level, set to 

0.1 pixel. In the original paper, velocity measurements were considered outliers when the 

normalized residual 𝑟0
∗ exceed the value of 2; the same threshold is used in this analysis. 

 

                 
Figure 2: Raw images of the three experimental datasets considered in this work. Left: flow around 

a 2D-wing NACA 0012; center: near-wake of a truncated cylinder; right: near-wake of the Ahmed 

body. 

The sources of outliers differ among the considered test cases. For the 2D-wing case (figure 2, left) 

the presence of the wing body and the shadow created by the refraction at the leading edge create 

outlier regions in the velocity field, as it can be seen in figure 3 (top-left), where the instantaneous 

velocity field is presented. Considering the case of the cylinder, a strong reflection on the object 

surface is visible in the raw image presented in figure 2 (center). This reflection cannot be 

eliminated with pre-processing and causes the appearance of outlier instantaneous tracks along 

the line of sight of the considered camera, as visible in the highlighted region in figure 3 (center-

left). For the Ahmed body flow, studied with a coaxial system (Schneiders et al., 2018), an excess of 

particles coupled with the background noise in the middle of the images causes the appearance of 

spurious tracks at the center of the measured volume close to the cameras (figure 3, bottom-left). 

The outliers that appear in the instantaneous fields, represented by the velocity vector on a 



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Cartesian grid in the 2D-wing case and particle tracks for the other two cases, corrupt the 

correspondent time-average and ensemble-average velocity fields, presented for all the cases in the 

right column of figure 3. For the 2D-wing case, the region correspondent to both the shadow and 

the perspective section of the wing are characterized by erroneous low velocity. This effect is 

clearly visible in the area correspondent to the shadow. For the cylinder case, the detrimental effect 

of laser reflections that generates invalid vectors inside the cylinder geometry is evident. 

Furthermore, a region of flow characterized by quasi-null streamwise velocity can be spotted in 

the area correspondent to the erroneous tracks, indicated with an ellipse in figure 3 (center-left). 

This can be explained by the fact that the erroneous particles generated by the surface reflection 

yield random tracks whose mean velocity is approximately null. Similarly, the presence of 

spurious tracks in the near-wake of the Ahmed body produces an unphysical region of 

deceleration and acceleration at X = 0.2 m in the free-stream region of the ensemble-averaged flow 

field (figure 3, bottom-right).         

 

 ure 3: Left column: Instantaneous streamwise velocity field 𝑢 represented on a Cartesian grid and 

at particle locations for the planar case and for the 3D cases, respectively. In all the cases, the areas 

interested by erroneous measurements are highlighted. Right column: time-averaged and 

ensemble-averaged velocity field  �̅� for the 2D case and the 3D cases, respectively. For the near-

wake of the Ahmed body, the region labelled A indicates the area considered correct for the 

determination of  𝜂𝑓𝑝, presented in figure 6. 

 



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The presence of outliers in the instantaneous velocity fields does not influence only the time-

averaged field �̅� but also the fluctuating component of the velocity 𝒖′. In order to evaluate this 

effect along all the three velocity components, the turbulent kinetic energy is here considered, in 

the form of √𝑘/𝑈∞. For all the cases analysed, the outlier regions present a sharp increase of 

turbulent kinetic energy with respect to the surrounding (figure 4). It can be noted that the 

extension and the shape of the outlier regions result more visible in the flow fluctuations, 

visualised by means of the turbulent kinetic energy in figure 4, than in the time-averaged flow 

fields (figure 3-right). Nevertheless, the turbulent kinetic energy alone can hardly be used as a 

criterion for outliers detection, because its physically admissible values vary greatly among the 

experiments. 

 
Figure 4: Contour of normalized turbulent kinetic energy √𝑘/𝑈∞ for the three experimental 

datasets: 2D-wing (left), cylinder (centre) and Ahmed body (right). For the near-wake of the 

Ahmed body, the region labelled B indicates the area considered erroneous for the determination 

of  𝜂𝑑, presented in figure 6. 

In order to assess the performance of the proposed principle, the two outliers indicators, namely 

the normalized admissible TKE variation of the current turbulence transport-based approach (𝜌𝑇𝑇 

of equation 9 and the residual 𝑟0
∗ of Westerweel and Scarano (2015) are presented in figure 5 for all 

the three considered cases. When considering the universal outlier detection, it should be 

remembered that the presence of false vectors is flagged by the residual 𝑟0
∗ exceeding the value of 

2. For both the 2D case and the cylinder case, the universal outlier detection fails to detect the 

outlier region. In the former, only a small region at the leading edge of the shadow is flagged as 

erroneous. In the latter, the area affected by reflection is not detected as erroneous, with only some 

vectors at the edges of the flow domain indicated as outliers. The UOD shows better results when 

applied to the data of the near-wake of the Ahmed body. In this case, a portion of the erroneous 

region in the free-stream is correctly detected. 

The right column of figure 5 presents the spatial distribution of the ratio 𝜌𝑇𝑇 for all the considered 

cases. For all of them, 𝜌𝑇𝑇 reaches the highest value at the edges of the erroneous region, with 

𝜌𝑇𝑇 > 10, an order of magnitude difference with respect to the adjacent correct region. Considering 

the 2D-wing case, the edges of the shadow region and the front and top edges of the area occupied 

by the prospective view of the wing body are clearly detected as faulty vectors. When the cylinder 

case is considered, the edges of the erroneous region are clearly detected. It is remarked that the 

indicator  𝜌𝑇𝑇 exhibits a large separation between the erroneous region (𝜌𝑇𝑇 typically exceeding 10) 

and the correct portion of the flow (𝜌𝑇𝑇 < 1).  

When the third case is considered, the near-wake of the Ahmed body, the proposed method 

correctly detects the boundary of the erroneous region highlighted in 4 (right). Also in this case, 

the separation in terms of 𝜌𝑇𝑇 between the correct and the erroneous region is larger than one 

order of magnitude. In addition, two regions close to the Ahmed body are highlighted as 



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erroneous. Unfortunately, their validity cannot be determined since both the velocity field 

presented in figure 3 (bottom-right) and the TKE presented in figure 4 (right) do not show 

unphysical behaviour. This suggests the presence of false positives in those regions.   

 
Figure 5: (Left column) Normalized residual 𝑟0

∗ of the universal outlier detection algorithm 

proposed by Westerweel and Scarano (2005) for all the considered datasets. Top: 2D-wing; center: 

near-wake of a truncated cylinder; bottom: near-wake of an Ahmed body. (Right column) 

Normalized admissible TKE variation 𝜌𝑇𝑇 defined by equation 9 for all the considered datasets. 

Top: 2D-wing; center: near-wake of a truncated cylinder; bottom: near-wake of an Ahmed body. 

In order to quantitatively evaluate the performance of the turbulence transport-based principle in 

comparison with the reference methodology, results are presented in terms of detection and false 

positive rates 𝜂𝑑 and 𝜂𝑓𝑝, respectively. The former is defined as the number of spurious vectors 

correctly flagged as erroneous divided by the total number of outliers. The false positive rate 𝜂𝑓𝑝 is 

defined as the ratio between the number of correct vectors flagged as outliers and the total number 

of correct vectors in the considered portion of the flow. To evaluate the robustness of the two 

methodologies, the two ratios have been evaluated varying the thresholds between 0 and 4 for the 

UOD, and between 0 and 10 for the turbulence transport-based approach. To perform this analysis, 

the dataset regarding the near-wake of the Ahmed body has been considered. The calculation of 

𝜂𝑓𝑝 requires the knowledge of the number of correct vectors that are wrongly flagged as outliers. 

In order to evaluate this, a volume containing only correct vectors has been selected by visual 

inspection. The planar section of the considered volume is indicated as region A in figure 3 



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(bottom-right). On the contrary, the calculation of 𝜂𝑑 requires the knowledge of the erroneous 

vectors. For this purpose, the area circled and indicate as region B in figure 4 (right) has been 

considered. Since the proposed principle aims to detect the edges of erroneous regions, only the 

vectors located at the edge of the faulty area have been considered. Once isolated this group of 

vectors, the amount of them flagged as erroneous by the UOD and by the turbulence transport-

based outlier detection has been calculated. The results obtained in terms of 𝜂𝑑 and 𝜂𝑓𝑝 by the two 

methods varying the correspondent threshold is presented in figure 6. It has to be noted that the 

optimal values of 𝜂𝑓𝑝 and 𝜂𝑑 are 0 and 1, respectively. It is expected that, when increasing the 

value of the threshold of the two algorithms, both correct detection probability and the false 

positive rate decrease. Both the methodologies presents the expected behaviour. However, for the 

UOD approach the correct-detection and false-positive curves decrease at similar rates, whereas 

for the turbulence transport-based approach the false-positive curve drops much more rapidly. As 

a consequence, when the typical threshold of 𝑟0
∗ = 2 is used for the UOD method, only 5% of the 

correct vectors are detected as outliers (5% over-prediction), but less than 50% of the outliers are 

identified (only 45% correct prediction). Conversely, using the turbulence transport-based 

approach with a threshold of 5 yields 5% over-prediction and 95% correct prediction. This result 

supports the conclusion that the median test based on the velocity field is not able to detect outliers 

when the flow statistics are considered, while the proposed turbulence transport-based approach 

is able to detect the outliers in the flow statistics. 

 

Figure 6: Comparison of the detection ratio 𝜂𝑑 and the false positive ratio 𝜂𝑓𝑝 obtained by 

universal outlier detection and the turbulence transport-based approach proposed in this paper. 

The data are referred to the near-wake of Ahmed body dataset. 

4 Conclusions 

A novel outlier detection criterion is presented that invokes the physical principle of turbulence 

transport and applies to the flow statistics evaluated by PIV or PTV. The approach relies on the 

fact that the measured flow field should satisfy the turbulent kinetic energy transport equation. 

However,  outliers often produce un imbalance between the terms of the transport equation. The 

ratio between the measured turbulent kinetic advection rate and the production term along a 

trajectory (𝜌𝑇𝑇) is introduced to define a criterion for the detection of the boundaries of regions of 

erroneous data. In order to guarantee the robustness of the method, the production term is 

substituted with the median of its neighbours. The approach has been tested on three different 

datasets, obtained with three different techniques: planar PIV, large-aperture 3D-LPT and small-



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aperture 3D-LPT The proposed turbulence transport-based outlier detection principle compares 

favourably with the state-of-the-art universal outlier detection when considering diverse 

experimental cases. In order to compare quantitively the performance of the proposed method 

with the reference, the dataset regarding the near-wake of the Ahmed body has been considered. 

Considering the two optimal threshold, 𝑟0
∗ = 2 for UOD of Westerweel and Scarano (2005) and 

𝜌𝑇𝑇 = 5 for the turbulence transport-based approach, the results of the two methods are compared 

in terms of detection ratio 𝜂𝑑 and false positive ratio 𝜂𝑓𝑝. Both of the methods yielded a similar 

over-prediction of the outliers (about 5%); however, while the algorithm from Westerweel and 

Scarano could only detect less than 50% of the outliers, the proposed turbulence transport-based 

outlier detector was capable to identify 95% of the outliers. In case of large regions of outliers, once 

the boundary of the outlier regions has been identified with the proposed methodology, standard 

contour tracing algorithms can be applied in order to isolate the of the erroneous regions.  

 

 

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