A super-resolution approach for uncertainty estimation of PIV measurements
Andrea Sciacchitano
1,*, Bernhard Wieneke
2, Fulvio Scarano
1.
1: Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands 2: LaVision GmbH, Göttingen, Germany
*correspondent author: a.sciacchitano@tudelft.nl
Abstract A super-resolution approach is proposed for the a posteriori uncertainty estimation of PIV measurements. The measured velocity field is employed to determine the displacement of individual particle images. A disparity set is built from the residual distance between paired particle images of successive recordings. Within each interrogation window, the disparity set is treated with a statistical analysis to infer the measurement uncertainty: the mean disparity is ascribed to bias errors due to poor particle image sampling or spatial modulation effect; the dispersion of the set is related to precision errors, mainly due to random noise in the recordings and to errors in the PIV interrogation.
The performance of the estimator is first assessed via Monte Carlo simulation on a uniform flow field with varying out-of-plane displacement. The uncertainty is accurately estimated in optimal imaging condition, while it is underestimated when the imaging conditions are suboptimal.
The experimental assessment is conducted on a water jet experiment. For evaluating the performance of the estimator, the actual measurement error is computed as the difference between measured and a reference displacement field; the latter is built with an advanced processing algorithm that exploits the time redundancy of highly oversampled data to reduce the error of one order of magnitude. The capability of the super-resolution technique to quantify the uncertainty within 0.1 px accuracy is proven.
1. Introduction
Digital particle image velocimetry (PIV) is nowadays an established and reliable flow diagnostic tool capable of measuring velocity fields in two- and three-dimensional domains. Several works have been focused on PIV measurement errors. Huang et al. (1997) distinguished two major forms of errors in digital PIV, namely the mean-bias and the RMS (or precision) errors. The bias errors are mainly related to spatial modulation effects and to the particle image size. Modulation effects are dominant when the interrogation is conducted at low spatial resolution; furthermore, for particle image diameter of the order of 1 pixel, the measured displacement is biased towards the closest integer value, producing the effect commonly referred to as peak locking (Westerweel, 1997, among others). Precision errors are random errors mainly due to noise introduced during the recording process and numerical errors associated to the interrogation of PIV images. The noise in the recording process causes uncertainty of the pixel values describing the particles, resulting in uncertainty of the particle locations and displacements. The typical magnitude of precision errors is of the order of a fraction of a pixel.
A further source of uncertainty in PIV data is the insufficient number of particle image pairs due to low seeding density or out-of-plane motion, which is responsible of spurious vectors. The error associated to spurious vectors is typically orders of magnitude larger than the typical precision error; recognizing incorrect vectors is referred to as data validation (Westerweel, 1994, Westerweel and Scarano, 2005).
The uncertainty of PIV measurements is commonly estimated through a-priori approaches based on theoretical modeling or Monte Carlo simulations. Theoretical models have been formulated to evaluate the effects of several error sources (sub-pixel displacement, velocity gradients, particle-image diameter and image intensity among others) on the measurement precision (Westerweel, 2000, Westerweel, 2008). However the mentioned results are based on simplified interrogation methods (discrete window shift, Westerweel et al. 1997), which also include sources of error that are eliminated by using advanced algorithms employing window deformation (Scarano and Riethmuller, 2000). Monte Carlo simulations are widely employed to assess the performance of interrogation algorithms making use of simple models for the particles motion or more sophisticated simulation of turbulent flows (Lecordier et al. 2001). Despite the mentioned efforts, the a-priori estimate of the precision error based on numerical simulations largely underestimates the errors encountered in real experiments. As a result, the investigator is left with a number of procedures to evaluate empirically the measurement error a-posteriori and optimize the parameters of the image analysis based on it.
A very common approach, also reported in many experimental studies, is that of varying the size of the interrogation window by steps. At any reduction, the measurement spatial resolution increases at the cost of an increase of the measurement noise. When the noisy fluctuations exceed an acceptable level, the measurement is considered noise-dominated and a slightly larger interrogation window is selected for the data analysis. The result of the above procedure is still largely dependent upon the “perception” of the noise from the investigator and cannot be considered as a general approach to uncertainty estimation.
Despite the multiple studies regarding PIV accuracy and errors, a-posteriori techniques to estimate the PIV measurement uncertainty have not received significant attention. In contrast, the topic of uncertainty estimation is being recognized as very important in the experimental fluid mechanics community as PIV plays a major role today as a tool to evaluate the accuracy of computational fluid dynamics (CFD) and is used more frequently in the analysis of industrial systems. The “PIV uncertainty workshop” held in Las Vegas in 2011 is only one of the events that demonstrates such attention.
The present work investigates an a-posteriori methodology for estimating the uncertainty of PIV measurements. The work focuses solely on precision and spatial modulation errors, assuming that all spurious vectors have been removed by data validation techniques. Also, the errors due to temporal modulation (truncation errors) are not given attention in the present framework. The technique is based on the simple principle that the error obtained from the position disparity of each individual particle pair combines within the interrogation window with a known statistical model. Therefore the proposed technique discusses how such individual contributions can be evaluated and how the estimation of the uncertainty for the displacement vector associated to a given window is extracted. The individual contributions are evaluated through a super-resolution technique. The concept of super-resolution has been introduced by Keane at al. (1995) to indicate interrogation techniques that improve the spatial resolution beyond that determined by the interrogation window size. Conventional super-resolution approaches are based on the tracking of particle images or features centroid (Keane et al. 1995, Cowen and Monismith, 1997); their results are characterized by lower robustness than correlation-based algorithms (Stanislas et al. 2008). In particular for high seeding density as commonly applied in planar PIV measurements (~0.1 particles/pixel), the detection and tracking of individual particles becomes unfeasible. The work is conducted using Monte Carlo simulation and experimental images from a well-controlled experiment performed with time-resolved PIV. For this experiment, reference
data is available which is obtained using the time redundancy of highly oversampled data and has precision approximately one order of magnitude higher than the inquired data set (pyramid correlation algorithm, Sciacchitano et al. 2012); thus the uncertainty estimation can be directly compared with the measurement errors.
2. Working principle of the technique
Conventional PIV processing algorithms provide the velocity field V = {Vx Vy} at discrete
positions X; no explicit information on the measurement uncertainty is given except for the correlation signal-to-noise ratio, which is more closely related to the measurement confidence level than the error. The present approach makes use of the super-resolution concept to evaluate the measurement uncertainty of the displacement vector (viz. velocity) obtained from cross-correlation. The individual contributions of the particles are then extracted and form the basis to evaluate the dispersion of the error. The latter is used to infer the measurement uncertainty under specific hypotheses on the statistical behavior of the error. The approach has a general validity and can be applied to any correlation based processing algorithm (single pass correlation, Keane and Adrian, 1992; discrete window offset, Westerweel et al. 1997; image deformation, Huang et al., 1993, Scarano and Riethmuller, 2000). The description of the approach is presented hereafter; the procedure can be split into four parts: a) image deformation, b) particles image detection, c) disparity vector computation, d) statistical analysis.
a) Image deformation
The inputs of the technique are the two images I1 and I2 of a pair, defined at time t1 and t2 = t1 + Δt respectively, and the velocity field V, measured with a conventional cross-correlation
based algorithm (e.g. window deformation iterative multigrid) and defined at t1 + Δt/2. The
velocity field is employed to symmetrically deform images I1 and I2 to the intermediate time
instant t1 + Δt/2 according to Scarano (2002), yielding the deformed images ∞I1and I∞2:
∞
( )
( )
( )
∞( )
( )
( )
1 1 2 2 , , , , 2 2 , , , , 2 2 x y x y t t I x y I x V x y y V x y t t I x y I x V x y y V x y ⎧ = ⎡ − Δ − Δ ⎤ ⎪ ⎢ ⎥ ⎪ ⎣ ⎦ ⎨ Δ Δ ⎡ ⎤ ⎪ = + + ⎢ ⎥ ⎪ ⎣ ⎦ ⎩ .It is important to remark that the two deformed images should match to a large degree. The procedure of image deformation requires the pixel intensity to be evaluated at sub-pixel positions; several interpolation schemes can be adopted, such as bi-cubic spline and Whittaker interpolation (Scarano and Riethmuller, 2000, Astarita and Cardone, 2005).
To explain the working principle of the technique, the distinction between the ideal case (perfect images with no noise) and real case is remarked hereafter. In the ideal case, the particle images of a pair superimpose perfectly in the deformed images ∞I and ∞1 I (see figure 1, first row); therefore, 2
the cross-correlation function exhibits a displacement peak centered at the origin (null relative displacement) with a peak width proportional to the particle image diameter dτ. As discussed by
Adrian, 1991, the uncertainty of the displacement measurement is proportional to the particle image diameter dτ and to the uncertainty of the algorithm employed to locate the particle centroid.
Figure 1. First row: image deformation in the ideal case. a) Particle images of ∞I1; b) Particle images of ∞I2; c) Superposition of ∞I1and I∞2: the green particles of ∞I1and the red particles of I∞2superimpose perfectly, yielding the particles displayed in yellow; d) Correlation function between ∞I1 and ∞I2; e) Profile of the correlation function C at y = 0. Second row: image deformation in the real case. f) Particle images of I∞1; g) Particle images of ∞I2; h) Superposition of ∞I1and I∞2: the particle images do not superimpose perfectly (yellow: particles correctly superimposed; green: particles of I∞1without the pairing particle in I∞2; red: particles of ∞I2without the pairing particle in ∞I1); i) Correlation function between I∞1 and ∞I2; j) Profile of the correlation function at y = 0: the dashed line represents the profile computed in the ideal case.
In the real case (figure 1, second row), the image deformation techniques only yield an approximation of the overall particle motion and individual particle images will not match perfectly. As a consequence, the correlation between different particle image pairs yields a peak at positions scattered around the origin of the correlation space. Each relative displacement obtained from a particle image pair can now be regarded as the elemental contribution to the actual cross-correlation. In the linear approximation, the correlation peak for the entire window equals the sum of all individual peaks from particle pairs. Clearly, the dispersion of the data causes the window cross-correlation peak to broaden as shown in the figure 1.j. Moreover, the correlation peak may not be centered at the origin of the correlation space, yielding a non-null displacement. Furthermore, the peak magnitude drops as a result of the signal broadening. The position of the maximum in the broadened signal is more sensitive to the noisy contributions resulting in a higher measurement uncertainty. In this case, the uncertainty of the displacement measurement is proportional to the dispersion of the residual displacements of the particles.
b) Particle images detection
After the image deformation, the super-resolution analysis is applied to the individual particle images identified inside the interrogation window. To detect the particle images, the intensity product image Π =I I∞∞1 2 is considered; the peaks of correspond to particle images correctly paired (figure 2). Since the cross-correlation is a quadratic operator, those maxima have the largest contribution in building the correlation peak. Let us define the peaks image the binary image composed by the peaks of :
σ ~ dτ dτ a) b) c) d) e) σ ~ relative displacement dispersion f) g) h) i) j)
( )
, 1 if( )
, is a relative maximum 0 otherwise i j i j ⎧⎪ Π ϕ = ⎨ ⎪⎩ .Each point (i, j) where is non-null indicates a particle image pair; the peak of the corresponding particle images is detected in I∞1 and I∞2 in a neighborhood of search radius r (typically 2 pixels), centered in (i, j).
Figure 2. a) Deformed image ∞I1; b) deformed image ∞I2; c) intensity product image ; d) peaks image . c) Disparity set computation
Once the particle images have been detected as explained in section b), their position is computed with subpixel accuracy through a conventional 3-point Gaussian fitting (Westerweel, 1997) of the intensity peak, centered in the locations corresponding to the relative maxima of I∞1and ∞I2. The Gaussian fitting provides two particle position sets X1={x11, x12, …, x1N} and X2={x21, x22, …, x2N}, where the superscript indexes the deformed image and N is the number of particle pairs in
an interrogation window.
Figure 3. Particle images of I∞1 (hollow squares) and ∞I2
(full circles) and disparity vectors.
Figure 4. Generic distribution of the disparity vector.
The disparity set D, computed as the difference between X2 and X1, expresses the residual distance between the particles of ∞I1and ∞I2:
D= {d1, d2, …, d N}= X2- X1.
Figure 3 illustrates the particle images of ∞I1and I∞2 and the resulting disparity vectors di. d) Statistical analysis and uncertainty computation
In each interrogation window, the disparity set is treated in a statistical manner to infer the measurement uncertainty. The correlation peak can be regarded as formed by elemental contributions obtained from the particle image pairs; its broadening due to precision errors is proportional to the dispersion of the disparity set, computed as the standard deviation σ = {σu, σv}
of D, and scales with the inverse of the square root of N. Therefore, the uncertainty due to random errors is calculated as 𝝈 𝑁 .
The mean of the disparity set may differ from zero (see figure 4). In fact, a non-null mean disparity µ = {µu, µv}is ascribed to bias errors, which have two origins: poor particle image
sampling and spatial modulation effect. In the former case, the particle image displacement is biased towards the closest integer displacement; this error cannot be detected by the super-resolution estimator because the particle position itself is subjected to peak-locking. The latter is due to insufficient spatial resolution and can be recognized by the present approach. The combined uncertainty δ is computed as the Euclidean sum of the bias and random contributions and represents the total measurement uncertainty:
{
}
2 2 u v N δ δ ⎛ ⎞ = = + ⎜ ⎟ ⎝ ⎠ σ δ µThe entire procedure followed by the super-resolution estimator for uncertainty quantification is summarized in the scheme of figure 5.
3. Numerical assessment
A Monte Carlo simulation has been conducted to assess the super-resolution estimator. Since the actual displacement field is known, the comparison between estimated and actual uncertainty is straightforward.
Synthetic images are generated through a random distribution of 16,000 particles over an array of 400×400 pixels (seeding density of 0.1 particles per pixel). The particle images have quantization level of 8 bits and a shape which follows a Gaussian distribution. Two different imaging conditions are investigated, with mean particle image diameter of 2 px (optimal condition) and 0.5 px (suboptimal condition). The standard deviation of the diameter distribution is set to 0.2 px
1 1 1 ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎩ ⎭ M N X X 1 N ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎩ ⎭ M d d 2 1 2 ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎩ ⎭ M N X X ∞ 1 I ∞ 2 I
Figure 5. Scheme of the super-resolution estimator for uncertainty estimation.
I1 I2 Raw images
Deformed images Image deformation based on a predictor Particle images detection Particles position sets Disparity set computation Disparity set Statistical analysis Intensity product image Uncertainty Intensity product
for both cases. A Gaussian-shaped laser sheet is simulated having width dz = 30 px. The camera
read-out error is set to 5 counts, while the pixels fill factor equals 1. A uniform displacement field is reproduced, with null vertical displacement and horizontal displacement equal to 2.1 px. Tests are performed with uniform out-of-plane displacement ranging between 0% and 40% of the laser sheet thickness.
The recordings are processed with the WIDIM algorithm (Scarano and Riethmuller, 2000), using interrogation windows of 17×17 px2 and 33×33 px2, both with 50% overlap.
For brevity, the results are discussed only for the horizontal displacement component u. The actual error ε is calculated as the difference between measured and exact displacement field. The actual mean bias error µA is computed as the arithmetic mean (in time and, since the flow is
uniform, also in space) of the actual error ε; the standard deviation σA from the mean expresses
the RMS error. The actual uncertainty δA is obtained as the root sum square of µA and σA.
2 2
A A A
δ = µ σ+
The super-resolution estimator provides an estimate δ of the instantaneous uncertainty of each displacement vector. The average estimated uncertainty δE is computed as the average of δ in
time. Since the flow is uniform, the former value is averaged also in space.
In the present simulation, the main error source is the out-of-plane motion, which causes a variation of the particle images intensity level. For overlapping particle images and varying relative intensity, the correlation peak is shifted, leading to a biased displacement estimate that depends on particle image intensity, width and overlap (Nobach and Bodenschatz, 2009; see figure 6). This effect alone limits the accuracy of PIV measurements to the order of 0.1 pixels.
Figure 6. Intensities I1 (a) and I2 (b) in case of overlapping particle images and cross-correlation function (c): the
centerlines of the correlation map are indicated with white lines, the correlation peak is indicated with a black dot.
For non-overlapping particle images, the correlation peak is not shifted even if the relative intensity varies. However, the out-of-plane displacement may be responsible of a strong drop of the particle image intensity: as a consequence, in presence of camera read-out noise, the particle shape is not neatly defined and the correlation peak broadens, yielding larger uncertainty in the displacement estimate (Figure 7).
b) c)
Figure 7. Particle image intensity (left) and cross-correlation between the particle images (right) with and without out-of-plane displacement w.
Furthermore, the out-of-plane displacement causes loss-of-pairs due to particles that travel outside the illuminated region. According to Keane and Adrian (1992), this effect may cause a major reduction of the probability of valid displacement detection.
Figure 8. Actual and estimated uncertainty for varying out of plane displacement w; left: optimal imaging conditions (dτ = 2 px); right: suboptimal imaging conditions (dτ = 0.5 px).
In optimal imaging conditions, the measurement error is dominated by the out-of-plane displacement: the larger the out-of-plane displacement, the higher the measurement uncertainty, accordingly to Adrian and Westerweel, 2011. For moderate out-of-plane motion (w/dz ≤ 0.2), the
super-resolution estimator is able to quantify the uncertainty within 0.01 px accuracy (figure 8 left). In presence of larger displacements in z-direction, the out-of-plane loss-of-pairs reduces the number of particle pairs composing the disparity vector: in this case, the super-resolution estimator typically underestimates the uncertainty and the discrepancy between estimated and actual uncertainty rises.
When the imaging conditions are suboptimal (figure 8 right), the mean bias error is predominant with respect to the precision error: the actual uncertainty is approximately constant and above 0.2 px. The super-resolution estimator only estimates the uncertainty due to precision errors and bias errors related to spatial modulation effects; in contrast, it fails in reproducing the uncertainty
caused by peak locking errors, which is major for small particle images. This limitation of the estimator is due to the inadequacy of the three-point Gaussian fitting to accurately determine the position of particle images having diameter below 1 pixel (Westerweel, 1997). As a consequence, in suboptimal imaging conditions the uncertainty is strongly underestimated.
4. Experimental assessment on a water jet
The experimental assessment is performed using existing PIV measurements conducted in the water jet facility of the High Speed Laboratory at TU Delft. A laminar water jet is issued from a circular nozzle of 10 mm diameter (D) at vjet = 0.45 m/s (7 pixel units), yielding a diameter based
Reynolds number ReD = 5,000. The magnification factor equals 0.44. A sequence of snapshots is
recorded in continuous mode at 1.2 kHz. In the laminar jet, vortices are shed regularly at frequency of 30 Hz; therefore, each period of shedding is sampled with a set of 40 images. The thorough description of the experiment is reported in Violato and Scarano (2011). For the present investigation, it is worth mentioning that the jet is issued in the laminar regime, generating a shear layer at x/D = ± 0.5; along the shear layer, vortices are shed between y/D = 2 and 2.5, which undergo pairing in 2.7 < y/D <3.5. Further downstream, the transition to turbulent regime occurs. This experiment has been selected for the variety of flow features (high velocity gradients in the shear layer, out of plane motion in the turbulent regime, uniform flow in the laminar core, stagnant flow outside the jet; see figure 9 left), which lead to a wide range of measurement uncertainty.
The procedure to assess the estimator is described hereafter. Firstly, the snapshots are processed with the WIDIM interrogation algorithm (Scarano and Riethmuller, 2000) using 32 × 32 pixels interrogation windows with 50% overlap, yielding the displacement field (viz. velocity)
V = {uv}, which is affected by a measurement uncertainty. Furthermore, an advanced
multi-frame technique, namely the pyramid correlation (Sciacchitano et al. 2012), is employed to process the snapshots at higher spatial resolution (interrogation windows of 22 × 22 pixels) and build the instantaneous reference displacement fields Vref = {uref vref}; this technique makes use of
the time redundancy of the highly oversampled data to reduce the error of an order of magnitude. The uncertainty on V is evaluated in two independent ways:
1) through the super-resolution estimator, yielding = { u v};
2) determining the actual measurement error = V–Vref and then following the approach
described in section 3.
Since the uncertainty of the reference velocity is at least one order of magnitude lower than the uncertainty of the velocity computed with WIDIM, the latter approach provides the actual
measurement uncertainty = { u v}.
Figure 9 right depicts the estimated uncertainty on an instantaneous axial displacement field. The lower uncertainty (below 0.1 px) is found in the stagnant region outside the jet, while slightly higher values take place in the laminar core (|x|/D ≤ 0.4, y/D ≤ 4). The uncertainty becomes larger along the shear layer (x/D = ± 0.5) and in the turbulent region (y/D > 4). In the shear layer, the main source of uncertainty is the low spatial resolution responsible of the inaccurate representation of the displacement gradients; in the turbulent region, the high uncertainty is due to both the low spatial resolution and the out-of-plane motion.
Figure 9. Left: instantaneous displacement field; right: instantaneous uncertainty δv on the axial displacement component.
Figure 10. Actual and the estimated uncertainty on the axial displacement component; bottom: profile 1; top: profile 2. For clarity, one every two points is displayed in the plots.
To assess the estimator, the time average estimated uncertainty , computed averaging the instantaneous uncertainty of 500 velocity fields, is compared to the actual uncertainty . The comparison is conducted along two profiles, extracted in the laminar region (y/D = 2.5) and in the turbulent region (y/D = 6.5) respectively. Figure 10 reports the comparison for the uncertainty on the axial velocity component; similar results are obtained for the radial component.
Along both profiles, the estimated uncertainty δE follows the same trend as the actual uncertainty δA. In the laminar region (profile 1, displayed in figure 10 bottom), the uncertainty is below 0.05
pixel units in the jet core and the stagnant regions outside the jet, while it shows peaks up to 0.2 px in the shear layer. The estimated uncertainty correctly reproduces the uncertainty peaks with accuracy of 1% vjet. Along the turbulent profile (figure 10 top), the actual mean uncertainty rises
up to 0.28 pixels mainly due to truncation errors and out-of-plane motion; the super-resolution estimator reproduces the increase of uncertainty within 0.1 px accuracy (1.5% vjet).
To investigate the agreement between instantaneous uncertainty measured with the super-resolution estimator and instantaneous actual error, the time series of the two quantities are extracted from two points, one along the shear layer (P1, figure 11 left) and one in the jet core (P2,
figure 11 right). The two quantities are not supposed to be identical (even if the measurement error is null in a point, the uncertainty may be non-null), but are expected to give similar indications on the measurement accuracy. In particular, the error and uncertainty on the axial displacement component are displayed.
The plot of figure 11 left shows a similar trend between actual error and estimated instantaneous uncertainty. The signal exhibits low frequency fluctuations (in the range 10-20 Hz) of magnitude
7 px Profile 1 Profile 2
P1
P2 Profile 1
up to 0.4 px well captured by the super-resolution estimator. Those fluctuations are ascribed to shear layer oscillations occurring when a vortex is shed, which are not accurately captured by the interrogation algorithm because of the low spatial resolution. Furthermore, high frequency fluctuations of low magnitude (up to 0.1 px) are present which can be associated to random noise in the recordings.
In the jet core (point P2, figure 11 right) the flow is laminar and the measurement uncertainty is
mainly ascribed to precision errors related to random noise and PIV interrogation errors; the actual error exhibits lower magnitude (0.1 px) than in the shear layer. The super-resolution estimator indicates an uncertainty level of the same order of the actual error and accurately reproduces the peaks of uncertainty.
Figure 11. Time series of actual error v (in absolute value) and estimated uncertainty v on the axial displacement; left: point P1 along the shear layer; right: point P2 in the jet core. Notice the different scale in the vertical axis.
5. Conclusions
A super-resolution estimator has been introduced to quantify the uncertainty in planar PIV measurements. The measured velocity field is employed to determine the displacement of individual particle images and the residual distance between paired particle images is measured. In each interrogation window, a disparity vector is built from the residual distances, which is employed for a statistical analysis to provide the instantaneous uncertainty of the measured displacement.
The performance of the estimator has been assessed in presence of out-of-plane motion via Monte Carlo simulation considering different imaging conditions and interrogation window sizes. In optimal imaging conditions and up to large out-of-plane displacement, the discrepancy between quantified and actual uncertainty is below 0.05 pixel units. When the imaging conditions are suboptimal, the three-point Gaussian fitting becomes inadequate to accurately detect the particle image position and the discrepancy between estimated and actual uncertainty rises up to 0.1 px.
The estimator is also assessed on a water jet experiment. For the specific case, instantaneous reference displacement fields are built with an advanced processing technique, namely the pyramid correlation (Sciacchitano et al. 2012). Since the reference displacement fields have measurement errors one order of magnitude lower than the displacement fields computed with WIDIM, they are used to calculate the actual measurement error and uncertainty. The capability of the super-resolution estimator to quantify the uncertainty within 0.1 px accuracy has been proven in a wide gamut of flow conditions: shear layer, quiescent flow and turbulent flow. The consistency between instantaneous uncertainty and the actual error has also been shown.
6. References
Adrian RJ (1991), Particle-image techniques for experimental fluid mechanics, Annu. Rev. Fluid mech. 23:261-304.
Adrian RJ, Westwerweel J (2010), Particle image velocimetry, Cambridge University Press. Astarita T, Cardone G (2005), Analysis of interpolation schemes for image deformation methods
in PIV, Exp Fluids 38:233-243.
Cowen EA, Monismith SG (1997), A hybrid digital particle tracking velocimetry technique, Exp Fluids 22:199-211.
Huang H, Dabiri D, Gharib M (1997), On errors of digital particle image velocimetry, Meas. Sci. Technol. 8:1427-40.
Keane RD, Adrian RJ (1992), Theory of cross-correlation analysis of PIV images, Applied Scientific Research 49:191-215.
Keane RD, Adrian RJ, Zhang Y (1995), Super-resolution particle image velocimetry, Meas. Sci. Technol. 6:754-768.
Lecordier B, Demare D, Vervisch LMJ, Réveillon J, Trinité M (2001), Estimation of the accuracy of PIV treatments for turbulent flow studies by direct numerical simulation of multi-phase flow, Meas. Sci. Technol 12:1382-1391.
Nobach H, Bodenschatz (2009), Limitations of accuracy in PIV due to individual variations of particle image intensities, Exp Fluids 47:27-38.
Scarano F (2002), Iterative image deformation methods in PIV, Meas. Sci. Technol. 13:R1-R8. Scarano F, Riethmuller ML (2000), Advances in iterative multigrid PIV image processing, Exp
Fluids S51-S60.
Sciacchitano A, ScaranoF, Wieneke B (2012), Multi-frame pyramid correlation for time-resolved PIV, Exp Fluids, under review.
Stanislas M, Okamoto K, Kähler CJ, Westerweel J, Scarano F (2008), Main results of the third international PIV Challenge, Exp Fluids 45:27-71.
Violato D, Scarano F (2011), Three-dimensional evolution of flow structures in transitional circular and chevrons jets, Phys. Fluids 23, 124104.
Westerweel J (1994) Efficient detection of spurious vectors in particle image velocimetry data sets, Exp Fluids 16:236–247.
Westerweel J (1997), Fundamentals of digital particle image velocimetry, Meas. Sci. Technol. 8 1379.
Westerweel J (2000), Theoretical analysis of the measurement precision in particle image velocimetry, Exp Fluids S3-S12.
Westerweel J (2008), On velocity gradients in PIV interrogation, Exp Fluids 44:831-842.
Westerweel J, Scarano F (2005), Universal outlier detection for PIV data, Exp Fluids 39:1096-1100.
