14th International Symposium on Particle Image Velocimetry – ISPIV 2021
August 1–5, 2021

Ultrasound PIV Uncertainty Quantification
R. Derakhshandeh1∗, S. Bhattacharya1, B. A. Meyers1, P. P. Vlachos1

1 School of Mechanical Engineering, Purdue University, West Lafayette Indiana, 47907, USA
∗ rderakhs@purdue.edu

Abstract
Ultrasound Particle image velocimetry (UPIV) is a non-invasive flow measurement technique where acoustic-
opaque flow tracers are injected into a working fluid and ensonified to create ultrasound images. These
images are processed using PIV cross-correlation based algorithms to measure the velocity field (Kim et al.,
2004). UPIV is useful for opaque flows, primarily where complex flows exist, accordingly, it is used in many
industrial and clinical research applications such as studying intracardiac flow (Crase et al., 2007). Further-
more, the measurement provides suitable temporal and spatial resolutions for improved diagnostic metrics.
Mentioned applications and the sensitive diagnostic industrial and clinical decisions made based on these
measurements intensifies the importance of characterizing the UPIV measurement accuracy and associated
uncertainty. However, quantifying UPIV measurement uncertainty is non-trivial due to the complexity of
possible uncertainty sources, their combination, and propagation through the measurement chain.

The formation of a particle image by ultrasound significantly differs from optical imaging, introducing
unique aspects to image quality that must be considered. Particle images are formed across several ultra-
sound scan lines, yielding an elliptical particle image shape. Furthermore, the particle’s reflected pressure
wave is converted to a digital signal that undergoes signal modulation, and this process forms a non-Gaussian
point spread function (PSF) along the scan line direction. Additionally, clusters of tracers produce a single,
bright image intensity and speckle image pattern. Compared to conventional PIV images, UPIV incurs sig-
nificantly higher image noise due to lack of filtration for the ultrasound reflection of the non-tracer obstacles.

(a) (b)

Figure 1: Comparison of a conventional PIV image(a), and an Echo PIV image(b)(Crase et al., 2007)

Although existing 2D PIV uncertainty methods can potentially be applied to UPIV, the image formation
aspects introduced above must be taken into account to quantify UPIV measurement uncertainty. Therefore,
in this work, we modified the methodology of three direct uncertainty estimation methods, Image Matching
(IM) (Sciacchitano et al., 2013)Moment of Correlation (MC) (Bhattacharya et al., 2018), and Correlation
Statistics (CS) (Wieneke, 2015), in order to increase their sensitivity to error sources specific to UPIV. In the
following paragraphs, these improvements are explained in detail.

In the IM method, the standard uncertainty is defined as the standard deviation of the disparity error
histogram. The disparity error relates to the residual mismatch between paired particle image positions. In
regular PIV, the position is estimated using a 3-point Gaussian fit. However, UPIV scans lead to significantly



stretched elliptical particle image patterns with aspect ratios 1 to 4, causing similar pixel intensities in the 3-
point neighborhood of the center. Thus, a 3-point fit often fails or is inaccurate. To overcome this limitation,
we instead use a least-square elliptic Gaussian fit for particle image position estimation and disparity values
to calculate the IM uncertainty.

For the MC method, the uncertainty is estimated directly from the cross-correlation plane by extracting
the probability density function (PDF) of the displacements. The standard algorithm assumes a Gaussian
PDF exists and convolves it with another circular Gaussian kernel to broaden the peak region. This leads
to a more reliable PDF standard deviation estimate. The MC uncertainty calculation uses the sum of the
covariance matrices for Gaussian convolution, which simplifies to the sum of the variance terms for a circular
Gaussian kernel, expressed as Px =

√
C2

x −D2 and Py =
√

C2
y −D2. Here, Px and Py are the PDF’s major

and minor axis, Cx and Cy are the convolved PDF peak diameters, and D is the circular Gaussian kernel
diameter. The kernel diameter is dynamically estimated from the cross-correlation peak’s width. Since in
a UPIV measurement, the cross-correlation peak is non-circular, the convolution kernel is also defined as
a non-circular Gaussian function. Thus, a theoretical framework for a generically rotated elliptic Gaussian
kernel convolution is deduced. Accordingly, the elliptical Gaussian kernel major axis (Dx), the minor axis
(Dy), and the orientation angle (α) are calculated by equations (1), (2), and (3) respectively.

P2
x =

(C2
x +C2

y )− (D2
x +D2

y)+
√
(C2

x −C2
y )

2 − (D2
x −D2

y)
2 −2(C2

x −C2
y )(D2

x −D2
y)cosα

2
(1)

P2
x =

(C2
x +C2

y )− (D2
x +D2

y)−
√

(C2
x −C2

y )
2 − (D2

x −D2
y)

2 −2(C2
x −C2

y )(D2
x −D2

y)cosα

2
(2)

α =
(C2

x cos2(β)+C2
y sin2(β)−D2

x)− (Py/Px)
2(C2

x sin2(β)+C2
y cos2(β)−D2

y)

P2
x − (P4

y /P2
x )

(3)

In equation (3), β is the orientation of convolved PDF peak obtained from the least square fitting.
The CS method models the uncertainty through the cross-correlation peak’s asymmetry attributed to

the covariance sum of the pixel-wise intensity difference between two matching interrogation windows. In
general, this sum is computed over a neighborhood region proportional to the particle image size. For UPIV
measurements, the CS uncertainty calculation is modified to dynamically set the covariance sum region in
each direction proportional to the estimated autocorrelation diameter.

The performance of the modified IM, MC, and CS methods are tested using synthetic ultrasound images
(Meunier and Bertrand, 1995) that represent a range of error sources, including particle image diameter,
noise, seeding density, displacement, and flow shear rate. For the baseline condition, 512× 512 images
with 3× 9 pixel diameter particles are generated with a seeding density of 0.03 particles per pixel with
Multiplicative (σ2 = 0.01) and additive (σ2 = 0.01,µ = 0) noise and constant displacements of 2.15 pixels
in both the x- and y-directions. Also, to consider for the ghost particles, random particles has been added to
the images; the number of these random particles is 5% of the total particles. The relevant parameters for
the Monte Carlo image simulations are presented in the Table 1. The parameters were varied independently
to test each method’s sensitivity to different error sources.

Table 1: Monte Carlo test matrix

Diameter Y-direction 3 pixels
Diameter X-direction 3:1:12 pixels

Displacement Y-direction 0:0.05:3 pixels
Displacement X-direction Same as it is in Y-direction

Shear Y-direction 0
Shear X-direction 0:0.05:015 pixels/frame/pixel

Density 0.01:0.005:0.05
Additive noise 0:0.01:0.1

Multiplicative noise 0:0.01:0.1

Figure 2 shows the RMS error and the RMS of the predicted uncertainty for each method as a function of
particle image aspect ratio (AR). The particle image aspect ratio varies from 1 to 4 by varying the x-direction



image diameter from 3 to 12 pixels, and the y-direction image diameter is kept constant at 3 pixels. Each
point in the plot represents the RMS value over 12800 measurements. For ideal prediction, the RMS of the
error distribution should match the RMS of the uncertainty distribution (Sciacchitano et al., 2015).

The current results show close agreement between the predicted and expected IM uncertainty values for
up to 1.5 AR. MC uncertainty predictions are within 10% of the RMS error between AR values 1 and 2.
All three methods’ performance deteriorates for AR values greater than 3, signifying further development
may be required to account for the specific particle image characteristics in UPIV. Further development and
validation will be done to improve each method’s performance to the different error sources.

Figure 2: Comparison of the uncertainty methods results over different particle aspect ratio

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