# Real-time and On-site Aerodynamics using Stereoscopic PIV and Deep Optical
Flow Learning

Mohamed Elrefaie  Steffen Hüttig  Mariia Gladkova  Timo Gericke  Daniel
Cremers  and Christian Breitsamter  Chair of Aerodynamics and Fluid Mechanics,
Technical University of Munich, Garching, Germany  Volkswagen AG, Wolfsburg,
Germany  Chair of Computer Vision and Artificial Intelligence, Technical
University of Munich, Garching, Germany  Corresponding authors:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We introduce Recurrent All-Pairs Field Transforms for Stereoscopic Particle
Image Velocimetry (RAFT-StereoPIV). Our approach leverages deep optical flow
learning to analyze time-resolved and double-frame particle images from on-
site measurements, particularly from the ’Ring of Fire,’ as well as from wind
tunnel measurements for real-time aerodynamic analysis. A multi-fidelity
dataset comprising both Reynolds-Averaged Navier-Stokes (RANS) and Direct
Numerical Simulation (DNS) was used to train our model. RAFT-StereoPIV
outperforms all PIV state-of-the-art deep learning models on benchmark
datasets, with a 68 $\%$ error reduction on the validation dataset, Problem
Class 2, and a 47 $\%$ error reduction on the unseen test dataset, Problem
Class 1, demonstrating its robustness and generalizability. In comparison to
the most recent works in the field of deep learning for PIV, where the main
focus was the methodology development and the application was limited to
either 2D flow cases or simple experimental data, we extend deep learning-
based PIV for industrial applications and 3D flow field estimation. As we
apply the trained network to three-dimensional highly turbulent PIV data, we
are able to obtain flow estimates that maintain spatial resolution of the
input image sequence. In contrast, the traditional methods produce the flow
field of $\sim$16$\times$ lower resolution. We believe that this study brings
the field of experimental fluid dynamics one step closer to the long-term goal
of having experimental measurement systems that can be used for real-time flow
field estimation.

###### keywords:

Deep Optical Flow Learning , Car Aerodynamics , Stereoscopic PIV , Ring of
Fire , Wind Tunnel Measurements

##  1 Introduction

Particle Image Velocimetry (PIV) is a crucial method in experimental fluid
mechanics for identifying velocity components in complex flow fields Adrian
(2005). The flow medium is initially seeded with tiny particles. Then, a laser
sheet is used to illuminate the flow domain so that the particles are visible
and a camera records a sequence of images. Eventually, the velocity vectors of
the flow can be estimated by analyzing the recorded images. The most commonly
used techniques for flow field estimation from particle images are the cross-
correlation and the optical flow methods Westerweel (1997). Given a sequence
of particle images, the cross-correlation method produces a spatially sparse
displacement field. In contrast, optical flow methods can provide a dense
displacement field Adrian (2005).

More recently, PIV processing has been approached using deep learning
techniques Rabault et al. (2017), Lee et al. (2017), Cai et al. (2019), Cai et
al. (2020), Zhang and Piggott (2020), Lagemann et al. (2021a), Lagemann et al.
(2021b), Yu et al. (2021), Gao et al. (2021), Lagemann et al. (2022), Yu et
al. (2023), and Han and Wang (2023) demonstrating remarkable success in planar
PIV evaluation. However, the complexity of real-world PIV measurements, such
as those encountered in industrial applications, demands a more sophisticated
approach than planar PIV can provide. In addition, as experimental setups
become more intricate and the volume of analyzed data grows, the
aforementioned traditional techniques can become computationally more
intensive and time-consuming, thus for volumetric flow field quantification
Lagrangian Particle Tracking is widely used.

For instance, in highly complex experiments such as the Ring of Fire (RoF)
Hüttig et al. (2023), Hüttig et al. (2024), or wind tunnel measurements Ladwig
et al. (2023) where thousands of images need to be processed to analyze the
three-dimensional flow, traditional stereoscopic PIV methods may prove
insufficient in terms of efficiency and resolution.

The considerable amount of data generated by the measurement systems
necessitates extensive processing times for traditional methods, potentially
demanding days or even weeks. To address the inherent limitations of
conventional methods in assessing three-dimensional flow, we introduce an
automated processing framework for PIV image analysis by incorporating deep
learning techniques into the pipeline. This approach ultimately demonstrates
the advantages of high automation, resulting in a significant reduction of
processing time when compared to non-automated systems. Additionally, this
method achieves superior spatial resolution, greatly enhancing the efficiency
of the experimental measurements. Additionally, although the training phase
for the model’s parameters can be resource-intensive, it can be executed
offline. Once the deep learning model has been adequately trained, the time
required for the estimation process of each image pair is approximately 40
seconds111For $1024\times 1024\,\mathrm{px}^{2}$ double-frame images on one
A100 Nvidia GPU.. This learning approach facilitates real-time and on-site
estimation and, thus, high efficiency while preserving acceptable accuracy
levels suitable for practical applications.

To summarize, our contributions are the following:

  * 1.

We introduce RAFT-StereoPIV to accommodate stereoscopic PIV, showcasing its
applicability and potential for a diverse range of real-world scenarios in
fluid mechanics (RoF and wind tunnel measurements).

  * 2.

We benchmark our deep learning model on the validation dataset Problem Class 2
Lagemann et al. (2021b) (14.4 thousand fluid flow test cases) and the unseen
test dataset Problem Class 1 Cai et al. (2019) (10 thousand fluid flow test
cases). We obtain a significant decrease in error rates compared to the state-
of-the-art: a 68 $\%$ reduction in error rates on the validation dataset and a
47 $\%$ reduction on the unseen test dataset.

  * 3.

We conduct extensive hyperparameters tuning to optimize the training of our
deep learning model. Furthermore, we introduce a smaller version RAFT-
StereoPIV-Small, significantly reducing the original model size from 5.3
million parameters to 2.2 million, which corresponds to a decrease of
approximately 59$\%$, thereby enhancing efficiency while maintaining robust
performance.

To the best of the authors’ knowledge, our work is the first successful
application of deep learning for experimental aerodynamics based on
stereoscopic PIV.

The rest of the paper is organized as follows: in Section 2, we briefly
describe relevant methods for deep learning-based PIV processing and the
measurement systems. We provide an overview of our approach for flow field
estimation using deep learning for on-site and real-time car aerodynamics in
Section 3. Finally, we cover the implementation details and present the
results on both synthetic data and real-world measurements (Section 4).
Limitations and future work are discussed in Section 5.

##  2 Related Work

###  2.1 Deep Learning for PIV

Analysis of apparent motion from images by means of neural networks has been
extensively researched in the computer vision community and pioneered in
Dosovitskiy et al. (2015). Building on such foundational contributions, the
recent works of Cai et al. (2019) and Cai et al. (2020) have inspired a number
of follow-up works on applying deep learning methods to PIV data. The authors
developed supervised deep learning models based on convolutional neural
networks (CNNs) for estimating dense motion that achieved performance
comparable to the state-of-the-art methods for PIV processing at that time.
One of the important contributions is the synthetic dataset of fluid flow
images, which contains a large number of particle images and the corresponding
ground-truth fluid motions. The dataset provided by Cai et al. (2019),
however, does not allow for generalization to complex flow fields due to the
large gap between synthetic and real-world experimental data. Motivated by the
limitations of the aforementioned dataset, Zhang and Piggott (2020) and
Lagemann et al. (2021a) proposed unsupervised deep learning models for PIV,
where the loss function is composed of different loss terms: photometric loss
between two consecutive image frames, consistency loss in bidirectional flow
estimates, and spatial smoothness. Their models produce results that are
competitive with both traditional PIV approaches and currently used supervised
learning-based methods and even surpass them in some challenging flow
situations. However, the unsupervised models could not achieve the same
accuracy as those of the supervised ones. A data-driven but slightly different
approach was done in a recent article from Wang et al. (2022), where a
physics-informed neural network was used to reconstruct dense velocity fields
from sparse experimental PIV or Particle Tracking Velocimetry (PTV) data. The
potential of such physics-informed neural networks is that they are not only
data-driven but also incorporate physical laws in the training.

Drawing inspiration from the recent optical flow architecture known as
recurrent all-pairs field transforms (RAFT) Teed and Deng (2020), numerous
studies have built upon this network Lagemann et al. (2021b), Yu et al.
(2021), Lagemann et al. (2022), Yu et al. (2023), and Han and Wang (2023).
They achieved remarkable results in the field of PIV for fluid dynamics,
thereby highlighting its significant impact and widespread adoption within the
research community. The exceptional performance of such optical flow
architecture can be attributed to an extensive training dataset, substantial
computational power, and considerable model size, emphasizing accuracy over
model efficiency and computational performance. Consequently, training such a
model presents challenges. Utilizing four Nvidia Titan X GPUs, the training
would typically require approximately 12 days.

In two related papers, Liang et al. (2022) and Liang et al. (2023) present
novel deep learning approaches to PTV for efficient and accurate flow field
measurements. DeepPTV, the first method, uses a deep neural network to learn
complex fluid flow motion from consecutive particle sets, aggregating local
spatial geometry information from neighboring particles. The second method,
GotFlow3D, introduces a recurrent graph optimal transport-based deep neural
network for 3D flow motion learning from double-frame particle sets. This
approach constructs two graphs in geometric and feature space, fusing
intrinsic and extrinsic features learned from a graph neural network to
iteratively and adaptively guide recurrent flow learning. Both methods exhibit
state-of-the-art performance in accuracy, robustness, and generalization,
offering deeper insights into the complex dynamics of various physical and
biological systems.

Despite the advancements in PTV and deep learning for 3D flow field
measurements, there remains a gap in the literature regarding flow estimation
based on stereoscopic PIV data using deep learning techniques. In addition,
much of the existing research has been focused on methodology development and
simple test cases rather than addressing the challenges posed by real-world
experimental data, which hold significant relevance for industrial
applications. In contrast, our work leverages deep learning for stereoscopic
PIV data analysis to provide practical solutions for complex industrial flow
scenarios.

###  2.2 Ring of Fire

Understanding the aerodynamics is crucial for many aspects of car design
development. This includes the drag analysis, rain, dust, and soiling
simulation, aeroacoustics, and the validation of Computational Fluid Dynamics
(CFD) methods. The majority of experimental research in car aerodynamics is
carried out in wind tunnels using pressure probes, hot-wire anemometers,
tufts, or through optical metrology methods such as PIV, PTV, or ’Shake-The-
Box’ Schanz et al. (2016). Due to the limitations imposed by the size of the
wind tunnels and the measurement techniques employed, researchers often
simplify the problem by resorting to stationary scaled models. Therefore, wind
tunnel experiments cannot reproduce the exact aerodynamics encountered during
real driving conditions like platooning, cornering, and drafting.

Ring of Fire (RoF) Sciacchitano et al. (2015), Terra et al. (2017), Terra et
al. (2018), Spoelstra et al. (2018a), Spoelstra et al. (2018b), Spoelstra et
al. (2019), Spoelstra et al. (2020), Spoelstra et al. (2021), Hüttig et al.
(2023), Hüttig et al. (2024), Spoelstra et al. (2023) is a novel quantitative
measurement technique that can provide insights into the flow topology of
full-scale, transiting objects and determine their aerodynamic drag. It is a
non-intrusive, large-scale, stereoscopic PIV measurement technique that can be
implemented for a full three-dimensional flow field description. The test
object passes through the measurement section and the flow field is captured
in the laser light section Spoelstra et al. (2019). In addition to visualizing
the flow field, the drag coefficient can be determined by evaluating the
momentum loss in a control volume Terra et al. (2017). For better flow field
visualization and accurate drag analysis, a dense and instantaneous velocity
field with high resolution as well as resolved fluctuations and turbulent
structures is essential Spoelstra et al. (2021). Having high spatial
resolution of the flow field can assist the engineers in better understanding
and evaluating the unsteady flow phenomena around the transiting object.

In collaboration with Volkswagen AG Hüttig et al. (2023), Hüttig et al. (2024)
conducted the first RoF experiment for the automotive industry. The objective
of this research was to assess the applicability and feasibility of RoF system
for cars, quantify the flow in the wake of different models and
configurations, and additionally determine the drag and its uncertainty
through the control volume approach.

One of the main motivations for the RoF measurement system is to have an on-
site flow field estimation. Moreover, it has the potential to measure dynamic
motions without the need for complex moving ground simulations in a wind
tunnel or CFD Hüttig et al. (2023), Hüttig et al. (2024). The implementation
of such measurement technique can be done under real driving conditions with
tire and ground effects and independent of the vehicle. By using high-speed
cameras with a short acquisition time of 6000 frames per second (FPS) at
resolutions of $1024\times 1024\,\mathrm{px}^{2}$, high spatial and temporal
resolution can be achieved. However, the resultant vector field is not
immediately available after the measurement, and therefore, neither drag
estimation nor flow field visualization Spoelstra et al. (2019), Hüttig et al.
(2023), Hüttig et al. (2024). This is due to the fact that preprocessing of
the measurement data takes substantial amount of manual work and processing
time when using classical methods. In contrast, our method does not need any
preprocessing and allows an automated and real-time flow field estimation as
the deep learning model can learn different preprocessing operations and
filters from the training data.

##  3 Approach: Flow Field Estimation Using Stereoscopic PIV and Deep Learning

An overview of our approach for flow field estimation using RAFT-StereoPIV is
given in Figure 1. Our method, while demonstrated here in the context of RoF
for real-time and on-site car aerodynamics, is adaptable and effective in a
wide range of stereoscopic experimental setups.

Figure 1: Overview of our approach for flow field estimation using RAFT-
StereoPIV. a) Stereoscopic configuration of Ring of Fire where the car crosses
the measurement section and particle images are processed on-site and in real-
time by RAFT-StereoPIV to output a 3D vector field. b) The Mapping function
used for 3D reconstruction from 2D Stereoscopic PIV Data. c) Inputs to RAFT-
StereoPIV and its main operations. d) Processing pipeline RAFT-StereoPIV
(adapted from Teed and Deng (2020)), which takes a video image sequence from
the stereoscopic configuration and estimates the corresponding optical flows
of the in-plane components (in $x$\- and $y$-directions) for each camera. The
3D velocity field $(U,V,W)$ can be reconstructed from the 2D optical flow pair
as discussed in 3.2.

The RAFT-StereoPIV computational framework comprises four essential steps:

  * 1.

Feature Extraction and Context Encoding: A shared feature encoder is employed
to extract per-pixel features from both input images. Utilizing the same
architecture as the feature encoder, the context encoder generates a context
map for the first image frame. Both encoders use same convolution, where the
output size is maintained equal to the input size.

  * 2.

All-pairs Correlation: This step involves computing the visual similarity
between the two frames by taking the inner product of all pairs of feature
vectors.

  * 3.

Iterative Updates: Using a convolutional gated recurrent unit (Conv-GRU)
multiple iterative updates of the estimated flow field take place.

  * 4.

3D Reconstruction: In this step, the three-dimensional velocity field can be
reconstructed from a pair of two-dimensional optical flow data.

In this approach, features and motion priors are learned by the feature and
context encoder, as well as the update operator, all of which are
differentiable and integrated into a single end-to-end trainable architecture.

As an input to our framework, we utilize a video image sequence from the
stereoscopic configuration. The deep learning model is then used to estimate
the corresponding optical flows of the in-plane components (in $x$\- and
$y$-directions) for each camera.

To obtain the optical flow out-of-plane (in $z$-direction), we utilize a
mapping function for the 3D reconstruction from 2D stereoscopic PIV data based
on geometric calibration and digital self-calibration Prasad (2000), Wieneke
(2005).

###  3.1 Experimental Setup

####  3.1.1 Ring of Fire

The RoF experiment Hüttig et al. (2023), Hüttig et al. (2024) was conducted
outdoors at the Volkswagen AG proving ground in Germany with a tunnel
structure to keep the Helium-Filled Soap Bubbles (HFSB) inside the measurement
domain and generate a sufficient seeding density. A general overview of the
setup is shown in Figure 2.

Figure 2: Close up top view of the Ring of Fire experimental setup (Hüttig et
al. (2023)).

The test object is a Volkswagen up! with a frontal surface area of 2.07
$\mathrm{m}^{2}$, resulting in a tunnel blockage ratio of 4.5$\%$. The car’s
body was coated with a matte black foil to reduce light reflections. For the
wake analysis, two Photron SA-X2 cameras were set up at the tunnel’s entrance,
positioned 7.6 m away from the light sheet. The cameras have a sensor size of
$1024\times 1024\,\mathrm{px}^{2}$, maximum acquisition frequency of 10
$\mathrm{kHz},$ 12-bit digital output, and 20 $\mu\mathrm{m}$ pixel pitch.
They were equipped with Scheimpflug-Adaptors v3 from LaVision and Nikon Nikkor
lenses with a focal length of $\mathrm{f}=60\mathrm{\leavevmode\nobreak\ mm}$
and the lens aperture was set to $\mathrm{f}_{\\#}=4.0$. The light sheet was
about 4 cm thick, created using two high-speed lasers, Quantronix Hawk Pro and
Quantronix Hawk II, and relevant light sheet optics; these were positioned on
both sides of the road in the middle of the tunnel. Measurements were acquired
at a rate of 6 kHz resulting in 6,000 images for each run.

HFSB were released into the tunnel via fourteen 20-nozzle LaVision seeding
arrays and a 204-nozzle-generator seeding rake from TU Delft, reaching a total
production of up to 18 million bubbles per second. To allow for a high
concentration of particles in the test section, two curtains were set up at
the tunnel’s entrance and exit, and one was closed during seeding, depending
on the flow conditions. The experiments were conducted during the night when
the wind was minimal. Each run recorded images covering a 33.33 $m$ distance,
about 10 $m$ ahead and 20 $m$ behind the car. The car’s cruise control was set
to 33.33 $m/s$ when crossing the laser sheet, yielding a Reynolds number of
$3.30\times$$10^{6}$ based on the car’s height.

####  3.1.2 Wind Tunnel Measurements of Ahmed Vehicle Model

Figure 3: Stereoscopic PIV setup of the 25$\%$-scaled Ahmed body.

The experiments were conducted in the closed-loop wind tunnel at Bochum
University of Applied Sciences Ladwig et al. (2023) and Gericke et al. (2023).
This atmospheric tunnel, designed as a Göttingen-type, features a nozzle exit
with a diameter of 0.5 meters. The semi-closed test section has a length of
1.2 meters. The mesh screen of the wind tunnel was removed for this
measurement campaign to prevent the accumulation of HFSB. The turbulence
intensity was recorded at or below 3.5$\%$ for this setup. A flow rectifier
with a 20 mm mesh size was placed 100 mm upstream from the mesh’s intended
position in the wind tunnel’s settling chamber. A flat plate with an
elliptical nose, a length of 800 mm, and a width of 300 mm is centered 200 mm
after the nozzle’s exit. On top of the flat plate, a 25$\%$-scaled model of
the Ahmed body reference Ahmed et al. (1984) with a 25∘ slant angle was
mounted 215 mm behind the plate’s leading edge. The dimensions of the Ahmed
body are $261\times 97\times 72$ mm in length, width and height, respectively.
To reduce noise from laser light reflection, all elements, including the plate
and the model, were painted black. Under the Ahmed body, a NACA0020 profile
was incorporated to cover the plastic tubes connecting the pressure taps and
Scanivalve modules. The stereoscopic PIV setup consists of two Imager CX12
Cameras from LaVision (4080 $\times$ 2984 px2; 2.7 µm pixel size, 12-bit)
equipped with Scheimpflug adaptors v3 and Nikon Nikkor f=60 mm with an
aperture of f#=2.8. Illumination of the DEHS particles (mean diameter
$\approx$ 1 µm) was realized by a PIV-Laser Quantel Evergreen 200 (repetition
rate 15 Hz, 200 mJ) and corresponding light sheet optics. Control and
triggering of the cameras and the laser was performed via DaVis 10.2 and a
programmable timing unit. The experimental setup is shown in Figure 3.

###  3.2 3D Reconstruction from 2D Stereoscopic PIV Data

Figure 4: Out-of-plane velocity component reconstruction.

Since the deep learning model was trained only on 2D-planar particle images
and the corresponding optical flows are in-plane components (in $x$\- and
$y$-directions), optical flow out-of-plane (in $z$-direction) can not be
directly estimated using the pretrained network. Based on a non-symmetric
stereoscopic arrangement, accurately reconstructing the out-of-plane velocity
component is possible because the identical object is captured by two cameras
simultaneously, but from different perspectives Prasad (2000). Consequently,
true 3D velocity field $(U,V,W)$ can be reconstructed from a pair of 2D
displacements $(\Delta x_{1},\Delta y_{1})$ and $(\Delta x_{2},\Delta y_{2})$
as seen from left and right camera respectively by knowing the precise
position of the cameras with respect to the measurement plane. The 3D flow
field estimation is only possible within the area covered by both cameras from
the stereoscopic configuration. However, the overlap area is a trapezoidal
region of the light sheet due to perspective distortion. Accordingly, the
overlap area should be maximized by accurate alignment and calibration Willert
(1997), Wieneke (2005).

A mapping function is, therefore, required to estimate the projected
displacement fields. This mapping function is obtained from a calibration
process that defines an accurate relation between the object and the image
space. First, geometric calibration of the system is performed using the
pinhole model from Prasad (2000). A calibration plate was aligned with the
laser pointing axis and perpendicular to the driving direction of the car. For
this, a calibration grid with known dimensions, which consists of markings
distributed over two levels of depth, is placed inside the light sheet. The
pinhole approach approximates the camera’s positions with respect to each
point on the calibration grid by means of triangulation. For each camera, the
second-order mapping function, as employed by Willert (1997), consists of
estimating twelve coefficients. A minimum of six calibration points is
necessary to determine the coefficients based on the least-squares method.
Wieneke (2005) further proposes to perform a self-calibration procedure on
5-50 image pairs showing particles in motion. A disparity map between the
image planes is then created, using the ensemble-averaged cross-correlation.
The calibration error can be reduced even more by shrinking and deforming the
interrogation windows of the disparity map, based on the previous self-
calibration procedure.

The in-plane velocity components measured by the left camera can be described
by Prasad (2000):

$u_{1}=\frac{\Delta x_{1}}{M\cdot\Delta t}\quad\text{and}\quad
v_{1}=\frac{\Delta y_{1}}{M\cdot\Delta t},$ (1)

with $\Delta x_{1}$ and $\Delta y_{1}$ being the displacements as seen from
the left camera, $M$ being the magnification factor, and $\Delta t$ serving as
the change in time. Similarly, the in-plane velocity components $u_{2}$ and
$v_{2}$ measured by the right camera can be calculated using Eq. 1. The
velocity components can be reconstructed according to the following equations
Willert (1997), Prasad (2000):

$U=\frac{u_{1}\tan\alpha_{2}+u_{2}\tan\alpha_{1}}{\tan\alpha_{2}+\tan\alpha_{1}},$
(2)

$V=\frac{v_{1}\tan\beta_{2}+v_{2}\tan\beta_{1}}{\tan\beta_{2}+\tan\beta_{1}},$
(3)

$W=\frac{u_{1}-u_{2}}{\tan\alpha_{2}+\tan\alpha_{1}},$ (4)

where $\alpha_{1/2}$, given in Figure 4, represent the angles between cameras
1 and 2 and the z-axis in XZ-plane respectively, $\beta_{1/2}$ are the angles
between the cameras and z-axis in the YZ-plane side view.

###  3.3 The Deep Learning Model

Our network architecture is based on RAFT Teed and Deng (2020), Lagemann et
al. (2021b), Yu et al. (2021), Lagemann et al. (2022), Yu et al. (2023), and
Han and Wang (2023).

A shared convolution-based feature encoder network is applied to extract
features from grayscale particle image-pair $I^{t}$ and $I^{t+\Delta t}$. The
feature encoder is fed with low-resolution images ($32\times
32\,\mathrm{px}^{2}$) and maps them to a dense feature map with the same
resolution:

$\mathbf{w}^{(I^{t})}=g_{\theta}\left(I^{t}\right),\quad\mathbf{w}^{(I^{t+\Delta
t})}=g_{\theta}\left(I^{t+\Delta t}\right):$ (5) $\mathbb{R}^{32\times
32}\rightarrow\mathbb{R}^{32\times 32\times 256}$

Additionally, a context network, which has the same architecture as the
feature encoder network is applied to the first input image $I^{t}$ to obtain
a context feature map that will be fed later to an update block. Both the
feature and context encoders in Figure 5 consist of two convolutional layers
and six residual layers, using ReLU as an activation function and instance
normalization for normalization. We also introduce RAFT-StereoPIV-Small
variant, a smaller model, where the residual units are replaced with
bottleneck residual units, which are designed to reduce computational
complexity and the number of parameters. This alternative model is
considerably more compact, with each of its feature and context encoders
containing only about 80,000 parameters. This presents a significant contrast
to the larger variant of the model, where each encoder has approximately 1
million parameters, demonstrating the efficiency and reduced computational
burden of the RAFT-StereoPIV-Small model.

The dot product is then used as a measure of visual similarity between all
pairs of the feature maps $\mathbf{w}^{(I^{t})}$ and $\mathbf{w}^{(I^{t+\Delta
t})}$ obtained from the feature encoder. This results in a full correlation
volume between all pairs, which can be described as a single matrix
multiplication:

$\mathbf{C}\left(\mathbf{w}^{(I^{t})},\mathbf{w}^{(I^{t+\Delta
t})}\right)\in\mathbb{R}^{32\times 32\times 32\times 32}$
$C_{ijkl}=\sum_{h}\mathbf{w}^{(I^{t})}_{ijh}\cdot\mathbf{w}^{(I^{t+\Delta
t})}_{klh}$ (6)

After computing the correlation volume, a 4-layer correlation pyramid
$\left\\{\mathbf{C}_{1},\mathbf{C}_{2},\mathbf{C}_{3},\mathbf{C}_{4}\right\\}$
is constructed by average pooling the last two dimensions of the correlation
volume with kernel sizes 1, 2, 4, and 8 and equivalent stride. The pooling
operation acts as a feature selection step. That is, it selects the best
features that describe the particle images. Furthermore, a correlation feature
map is obtained using a lookup operation by indexing from the correlation
pyramid. This indexing is performed within a local neighborhood
$\mathcal{N}\left(\mathbf{x}^{\prime}\right)_{r}$, given by Eq. 7, where each
pixel $\mathbf{x}=(u,v)$ in $I^{t}$ is mapped to
$\mathbf{x}^{\prime}=\left(u+f^{1}(u),v+f^{2}(v)\right)$ with $f^{1}$ and
$f^{2}$ the current estimated optical flow.

$\mathcal{N}\left(\mathbf{x}^{\prime}\right)_{r}=\left\\{\mathbf{x}^{\prime}+\mathbf{dx}\mid\mathbf{dx}\in\mathbb{Z}^{2},\|\mathbf{d}\mathbf{x}\|_{\infty}\leq
r\right\\},$ (7) $\text{where }r=4\,\text{px}$

The grid $\mathcal{N}\left(\mathbf{x}^{\prime}/2^{k}\right)_{r}$ is used to
index the correlation volume at level $\mathbf{C}_{k}$. In the last step, an
update operator is used, which is intended to resemble the steps of a first-
order descent optimization algorithm. As a result, tied weights across depth
with bounded activation functions are used to enhance training stability and
encourage convergence to a fixed point.

Figure 5: Feature and context encoder architectures of RAFT-StereoPIV and
RAFT-StereoPIV-Small models.

An integral part of the iterative update operator is a gated activation unit
based on the GRU cell, with convolutions used in place of fully connected
layers. The architectures of the iterative update for the large and the small
models are given in Figure 6. The larger variant has approximately 3.1 million
parameters, while the smaller model’s update block contains about 2 million
parameters.

Figure 6: Update block architecture of RAFT-StereoPIV and RAFT-StereoPIV-Small
models.

The applied Conv-GRU consists of two main gates: the update and the reset
gate. It takes the flow, the correlation, and the hidden state $h_{t-1}$ and
yields a new hidden state $h_{t}$, which is then passed through two
convolutional layers and finally outputs the flow update $\Delta\mathbf{f}$.
The output of the update gate $z_{t}$ for the time step $t$ is calculated as:

$z_{t}=\sigma\left(\operatorname{Conv}_{3\times
3}\left(\left[h_{t-1},x_{t}\right],W_{z}\right)\right),$ (8)

where $x_{t}$ denotes the concatenation of flow, correlation, and context
features, $h_{t-1}$ the hidden state information of the previous units,
$\sigma$ the sigmoid activation function, and $W_{z}$ the weight matrix of the
corresponding unit. The update gate gives the model the ability to choose how
much data from previous iterations will be included in the predictions made in
the future. The reset gate $r_{t}$ is the second important part of the Conv-
GRU and its output is computed in the same manner as the update gate but with
a different weight matrix:

$r_{t}=\sigma\left(\operatorname{Conv}_{3\times
3}\left(\left[h_{t-1},x_{t}\right],W_{r}\right)\right)$ (9)

Based on the output of the reset gate $r_{t}$, a new memory content
$\tilde{h_{t}}$ is computed using the tanh activation function and the weight
matrix $W_{h}$ in order to store the relevant information:

$\tilde{h_{t}}=\tanh\left(\operatorname{Conv}_{3\times
3}\left(\left[r_{t}\odot h_{t-1},x_{t}\right],W_{h}\right)\right)$ (10)

where $\odot$ denotes the Hadamard (element-wise) product operator. Finally,
the hidden state vector $h_{t}$ is obtained using the output of the update
gate, the memory content, and the hidden state vector at the previous time
step:

$h_{t}=\left(1-z_{t}\right)\odot h_{t-1}+z_{t}\odot\tilde{h_{t}}$ (11)

It is worth mentioning that each local update incorporates knowledge of both
small and large displacements since correlation volumes are constructed for
all pairs at both high and low resolution. A total number of 12 iterative
updates are used for the optical flow estimation. The hidden dimension is 128
for RAFT-StereoPIV and 96 for RAFT-StereoPIV-Small. In the RAFT-StereoPIV
model, two convolutional GRU update blocks with 1$\times$5 and 5$\times$1
filters are used, whereas RAFT-StereoPIV-Small model utilizes a single GRU
with 3$\times$3 filters.

Both models output optical flow of the in-plane directions $x$ and $y$. The
training of the networks is supervised on the $l_{1}$ loss with exponentially
increasing weights between the predicted $\mathbf{f}_{i}$ and ground truth
flow $\mathbf{f}_{gt}$ over the full sequence of predictions,
$\left\\{\mathbf{f}_{1},\ldots,\mathbf{f}_{N}\right\\}$:

$\mathcal{L}=\sum_{i=1}^{N}\gamma^{N-i}\left\|\mathbf{f}_{gt}-\mathbf{f}_{i}\right\|_{1},$
(12) $\text{where }\gamma=0.8\quad\text{and}\quad N=12$

Averaged endpoint error (AEE) is used as an evaluation metric and it
represents the average of the L1-norm of the estimated and the ground-truth
optical flow over all pixels:

$\mathrm{AEE}=\left\|\mathbf{f}_{\mathrm{es},N}-\mathbf{f}_{\mathrm{gt}}\right\|_{1}$
(13)

##  4 Evaluation

In this section, we will thoroughly assess the performance and characteristics
of the proposed method. To this end, we will utilize different datasets, each
offering unique challenges and properties, and conduct a series of
experiments.

###  4.1 Datasets

####  4.1.1 Open-source Synthetic PIV Datasets

Figure 7: Particle images and the corresponding optical flow from the open-
source synthetic PIV dataset Problem Class 2 used to train RAFT-StereoPIV and
RAFT-StereoPIV-Small.

##### Problem Class 1

The Problem Class 1 dataset, introduced by Cai et al. (2019), consists of
diverse fluid motion structures that have been generated using CFD. This
dataset includes simulations of various flow cases, such as Cylinder, which
represents flow over a circular cylinder, in addition to those sourced from
the Johns Hopkins turbulence databases Lee and Moser (2015). Moreover, the
dataset contains cases involving direct numerical simulations (DNS) of
turbulence and sea surface flow driven by a Surface Quasi-geostrophic (SQG)
model, as originally presented in Carlier (2005) and Resseguier et al. (2017),
respectively. The dataset is distinguished by its high particle density,
particle peak intensity ranging from 200 to 255 counts within an eight-bit
grayscale, and maximum particle displacement of $\pm$10 px. These
characteristics produce images that closely resemble those obtained in ideal
experimental conditions.

##### Problem Class 2

The public Problem Class 2 dataset Lagemann et al. (2021b) is based on the
same ground-truth flow fields of the synthetic dataset of Problem Class 1,
however, with reduced particle density, intensity, and overall signal-to-noise
ratio (SNR) in order to model image conditions similar to the experimental
data. The maximum in-plane displacement was increased from $\pm$ 10 px to
$\pm$ 24 px. The particle images have a mean SNR value of approximately 3.6,
ranging from $[0.1,70]$ and Gaussian noise with a mean $\mu_{n}=0.01$ and a
variance $\sigma_{n}=0.075$ based on the normalized pixel values.

Problem Class 2 dataset consists of 15000 particle image-pairs with their
corresponding ground truth flow field in $x-$ and $y-$direction and is split
into 12000 ($80\,\%$) for training and 3000 ($20\,\%$) for validation. The
input data used for training our models RAFT-StereoPIV and RAFT-StereoPIV-
Small is smaller chunks from Problem Class 2 of size $32\times
32\,\mathrm{px}^{2}$ and therefore comprises 100,000 samples for training and
15,000 for validation. Some examples of the particle images and the
corresponding optical flow from the open-source synthetic PIV dataset used to
train the deep learning models are shown in Figure 7.

A comparison between the characteristics of the two datasets Problem Class 1
and Problem Class 2 is given in Table 1.

| Problem Class 1 | Problem Class 2
---|---|---
Particles Diameter | $2\mathrm{px}\lesssim d_{p}\lesssim 6\mathrm{px}$ | $0.22\mathrm{px}\leq d_{p}\lesssim 5\mathrm{px}$
Particles Density | $N_{ppp}\gtrsim 0.025$ | $0.005\leq N_{ppp}\leq 0.1$
Background Noise | $\mathrm{SNR}>10$ | $1\leq\mathrm{SNR}\leq 16$
Max Displacement | $\mathbf{V}_{\max}\leq 10\mathrm{px}$ | $\mathbf{V}_{\max}\leq 16\mathrm{px}$
Out-of-Plane Motion | $w/\Delta z=0$ | $w/\Delta z\leq 0.\overline{0}$
Table 1: Overview of the public benchmark datasets and their characteristics.

We implemented a significant optimization to Problem Class 2 dataset by
transitioning the image storage format from float32 to uint8. This
modification resulted in a substantial reduction of the dataset’s size from
26.52 GB to 8.22 GB, thereby enhancing data efficiency and processing speed.

####  4.1.2 Synthetic Dataset of Varying Particle Image Conditions

To evaluate the performance of RAFT-StereoPIV under varying particle image
conditions, we conduct a systematic analysis using synthetic PIV images with a
range of parameters. In this study, we focus exclusively on varied image
parameters, considering factors such as particle diameter, particle seeding
density, particle out-of-plane motion, and the introduction of Gaussian noise
to the images. This comprehensive examination enables us to assess the
robustness and generalizability of the RAFT-StereoPIV model across various
scenarios.

Synthetic images offer considerable advantages, as they allow for precise
control over a wide range of image parameters, which is often unattainable in
experiments due to inherent uncertainties such as local density and
temperature fluctuations, laser and imaging optics-related uncertainties, or
pulse-to-pulse variations in laser power and energy distribution. Furthermore,
independent manipulation of image parameters enables the examination of their
specific effects, which is frequently challenging in measurements, as
different parameters may be interdependent or influenced by a common cause.
Consequently, the analysis of inference results from these synthetic images
yields crucial insights into the most significant image parameters, enhancing
our understanding of their impact on the overall system.

####  4.1.3 Ring of Fire and Wind Tunnel Experimental Data

Figure 8: Ring of Fire experimental data after applying the mapping function.
Here, we visualize challenging nature of the data, where red zoomed-in region
indicates the low seeding density of the particles. The yellow patch shows
large variation in the particles’ size. The green one shows both low seeding
density and variation in the particles’ size.

The particle images from RoF and the wind tunnel measurements are imperfect
and the flow field estimation is challenging due to several factors: highly
turbulent three-dimensional flow within the light sheet results in high
uncertainty, non-uniform distribution of seeding particles, and different flow
directions with local gradients within and perpendicular to the laser sheet
plane as seen in Figure 8. Compared to synthetic images used for training the
models (see Figure 7), we can see that the experimental data is more
challenging and provides larger uncertainties and error sources.

###  4.2 Implementation Details

Our RAFT-StereoPIV and RAFT-StereoPIV-Small models are implemented in the
open-source framework Pytorch Paszke et al. (2017). In our training, we
parallelize across two Quadro GV100 GPUs. RAFT-StereoPIV has approximately 5.3
million trainable parameters. In contrast, its more streamlined counterpart,
RAFT-StereoPIV-Small, maintains a compact structure with around 2.2 million
trainable parameters. All modules are initialized from scratch with random
weights. We follow the same approach as Teed and Deng (2020) and clip
gradients to the range $[-1,1]$. During training, we use the AdamW optimizer
Loshchilov and Hutter (2019) with $\beta_{1}=0.9$, $\beta_{1}=0.999$, and
weight decay $\lambda$ = 0.01. The bigger model was trained with a batch size
of 196 for 120 epochs, while the smaller one was trained with a batch size of
250 for 60 epochs. During training, the optical flow is initialized to zero in
the update block. However, during inference on the RoF images, we apply the
warm-start initialization, where the optical flow estimated from the previous
pair of frames is used as an initialization for the next update. The results
obtained using the aforementioned initialization are better than with zero
initialization. It is worth mentioning that the convex upsampling module used
in the original RAFT implementation Teed and Deng (2020) is not used in RAFT-
StereoPIV or RAFT-StereoPIV-Small since no downsampling of the original input
during feature extraction takes place. In addition, normalization is not
included in the GRU updates, because it can be problematic for regression
tasks like optical flow Teed and Deng (2020). Furthermore, by utilizing
automatic mixed precision instead of single-precision in our PyTorch
implementation, we realized a notable enhancement in computational efficiency.
This approach led to a reduction in training time by approximately 30$\%$
streamlining our computational process.

###  4.3 Results

####  4.3.1 Comparative Analysis on Problem Class 1

Table 2 presents a comparative analysis between state-of-the-art deep optical
flow learning methodologies and our models, RAFT-StereoPIV and RAFT-StereoPIV-
Small, on the unseen test dataset, Problem Class 1. The best values from each
experiment are highlighted in bold. RAFT-StereoPIV and RAFT-StereoPIV-Small
outperform all other methods across all flow instances.

State-of-the-art deep learning methods’ averaged endpoint error for Problem
Class 1
---
Method | Cylinder | JHTDB-channel | DNS-turbulence | SQG
UnLiteFlowNet32-PIV Lagemann et al. (2021b) | 65.9 | 41.9 | 44.3 | 40.1
UnLiteFlowNet-PIV Zhang and Piggott (2020) | 7.9 | 14.5 | 22.5 | 21.6
LightPIVNet Yu et al. (2021) | 5.2 | 22.8 | 19.7 | 24.9
RAFT32-PIV Lagemann et al. (2022) | 2.1 | 3.6 | 6.8 | 5.9
RAFT-StereoPIV-Small (Ours) | 1.3 | 1.3 | 3.4 | 1.8
RAFT-StereoPIV (Ours) | $\mathbf{1.3}$ | $\mathbf{1.3}$ | $\mathbf{3.3}$ | $\mathbf{1.9}$
Table 2: Comparison between the state-of-the-art deep optical flow learning
methods and our models RAFT-StereoPIV and RAFT-StereoPIV-Small on the unseen
test dataset Problem Class 1. The unit of averaged endpoint error is
normalized to pixels per 100 pixels. The best values from each experiment are
marked in bold. RAFT-StereoPIV and RAFT-StereoPIV-Small achieve the best
performance on all the flow cases.

####  4.3.2 Systematic Investigation of the Effects of Particle Image
Parameters

Our systematic analysis of synthetic PIV images showcases the RAFT-StereoPIV
model’s adaptability and robustness. We compare RAFT-StereoPIV with RAFT32-PIV
Lagemann et al. (2022). The study reveals the following results (see Table 3):

Averaged endpoint error for synthetic images with varying particle image
conditions
---
Parameter | Value | Experiment
Cylinder | $\begin{array}[]{c}\text{JHTDB}\\\ \text{Channel}\end{array}$ | $\begin{array}[]{c}\text{DNS}\\\ \text{turbulence}\end{array}$ | SGQ
$0.5-1.25$ |  $\mathbf{0.86}$ (1.01) |  $\mathbf{0.77}$ (0.79) |  $\mathbf{1.01}$ (1.02) |  $\mathbf{0.84}$ (0.94)
| $1.25-2.5$ |  $\mathbf{0.56}$ (0.64) |  $\mathbf{0.62}$ (0.66) |  $\mathbf{0.65}$ (0.79) |  $\mathbf{0.77}$ (0.83)
$d_{p}$ | $2.5-3.75$ |  $\mathbf{0.60}$ (0.61) |  $\mathbf{0.63}$ (0.75) |  $\mathbf{0.73}$ (0.83) |  $\mathbf{0.84}$ (0.86)
| $3.75-5$ |  $\mathbf{0.61}$ (0.70) |  $\mathbf{0.58}$ (0.66) |  $\mathbf{0.77}$ (0.81) |  $\mathbf{0.94}$ (0.97)
| $5-7$ |  $\mathbf{0.73}$ (0.84) |  $\mathbf{0.79}$ (0.91) |  $\mathbf{1.02}$ (1.03) |  $\mathbf{1.02}$ (1.12)
| 0.01 |  1.38 (1.95) |  1.43 (2.14) |  2.19 (2.51) |  1.47 (2.16)
$N_{ppp}$ | 0.05 |  $\mathbf{0.60}$ (0.61) |  $\mathbf{0.63}$ (0.75) |  $\mathbf{0.73}$ (0.83) |  $\mathbf{0.84}$ (0.86)
| 0.1 |  0.44 ($\mathbf{0.32}$) |  0.38 ($\mathbf{0.33}$) |  $\mathbf{0.41}$ (0.50) |  0.69 ($\mathbf{0.60}$)
| 4.3 |  0.47 (0.45) |  0.46 (0.49) |  0.52 (0.80) |  0.62 (0.69)
SNR | 2.9 |  $\mathbf{0.60}$ (0.61) |  $\mathbf{0.63}$ (0.75) |  $\mathbf{0.73}$ (0.83) |  $\mathbf{0.84}$ (0.86)
| 2.0 |  0.75 (0.86) |  0.59 (0.73) |  0.64 (0.75) |  0.67 (0.78)
Table 3: Summary of the performance of RAFT-StereoPIV and (RAFT32-PIV Lagemann
et al. (2022)) on synthetic PIV images, specifically assessing the impact of
particle diameter, particle density, and signal-to-noise ratio on averaged
endpoint error (AEE) values. The best values from each experiment are marked
in bold.

  * 1.

The first five experiments focus on the effect of varying particle diameters
($d_{p}$), providing evidence of the RAFT-StereoPIV model’s robustness to
varying $d_{p}$ values. Contrasting with the RAFT32-PIV, which experiences a
noticeable degradation in performance when particle diameters exceed double
the size of the training distribution. The RAFT-StereoPIV model exhibits
strong robustness to fluctuations in particle diameters, tolerating a range
from 0.5 to 7 pixels, even though particle diameters larger than 5 pixels fall
outside the domain distribution.

  * 2.

The second set of experiments investigates the error sources related to
varying particle densities ($N_{ppp}$). The results indicate that higher
$N_{ppp}$ values correspond to lower AEE values for both models as more
feature information becomes available.

  * 3.

The final set of experiments examines the effect of varying signal-to-noise
ratios (SNR), encompassing both changes in noise levels and peak intensities.
For both models, the AEE’s progression is similar, revealing that higher SNR
values result in lower performance.

This thorough examination of synthetic PIV images with diverse parameters
allows us to assess the RAFT-StereoPIV model’s generalizability and robustness
across various scenarios. RAFT-StereoPIV outperforms RAFT32-PIV baseline
method and achieves state-of-the-art performance on all test cases.

####  4.3.3 Hyperparameters Analysis

Our hyperparameters analysis for the RAFT-StereoPIV model yielded several
significant findings. We discovered a trade-off between batch size and
variance in gradients, with larger batch sizes delivering better performance,
albeit limited by available GPU memory. An increase in dropout probability
enhanced the model’s performance but required an extended training duration
for optimal generalization on unseen data. Among the various normalization
methods tested, including batch, instance, group, and no normalization,
instance normalization emerged as the most robust, despite group normalization
initially showing superior performance during training and validation.
Moreover, we found that using a training dataset of 20,000 samples yielded
good results, but expanding the dataset further could still be beneficial by
serving as a regularization technique and helping reduce overfitting. Lastly,
the number of training epochs, indicating the frequency of passing the
training dataset through the network, significantly influenced the model’s
convergence and optimization. These results provide valuable insights for
optimizing the RAFT-StereoPIV model, aiding in our larger goal of
democratizing access to large-scale deep learning models by leveraging
community resources and exploring the hyperparameters space.

By fine-tuning RAFT-StereoPIV’s hyperparameters, we managed to cut down the
training time from roughly two days to 13.6 hours, resulting in a 73 $\%$
reduction in training time when using 2 Quadro GV100 GPUs, thereby greatly
enhancing the computational efficiency of the model. Most notably, our efforts
led to a substantial decrease in error rates: a 68 $\%$ reduction on the
validation dataset, Problem Class 2, which contains 14,400 fluid flow test
cases, and a 47 $\%$ reduction on the unseen test dataset, Problem Class 1,
with 10,000 fluid flow test cases. The detailed hyperparameters analysis is
given in B.

Figure 9: Ring of Fire experimental data - Development of the normalized out-
of-plane streamwise velocity component in the wake of the car at $x$ = 0.25,
0.50, 0.75, and 1.00 m. (Top) we show the results obtained from LaVision DaVis
10.2 after postprocessing and (bottom) - from the deep learning model RAFT-
StereoPIV. RAFT-StereoPIV outputs dense per-pixel optical flow field, whereas
the cross-correlation-based LaVision DaVis 10.2 outputs only sparse velocity
field with 40 $\times$ 30 vectors.

####  4.3.4 Flow Field Estimation from Ring of Fire

Figure 10: Spatial resolution comparison in the wake of the car from Ring of
Fire - Highlighting the difference in the spatial resolution between the
sparse vector field output from the cross-correlation based LaVision DaVis
10.2 (top) and the dense per-pixel optical flow field output from RAFT-
StereoPIV (bottom).

We evaluate the performance of RAFT-StereoPIV on the challenging time-resolved
experimental data from RoF. Since no ground-truth optical flow exists, we
benchmark our results with the cross-correlation-based commercial software
DaVis 10.2 from LaVision GmbH.

Reducing the size of the interrogation window is the simplest approach to
improve the spatial resolution when using traditional methods, although doing
so increases the uncertainty Spoelstra et al. (2021). Therefore, given the
camera resolution, a compromise between an image size large enough to capture
the entire wake of the car and an interrogation window small enough to capture
the small-scale structures must be found. At the same time, it is necessary to
maintain an appropriate SNR to reduce the number of spurious velocity vectors
as well as the uncertainty on the estimated drag. For the flow field
estimation using DaVis 10.2, we consider the following processing steps Hüttig
et al. (2023), Hüttig et al. (2024): sliding sum of correlation function,
followed by multi-pass and finally, universal outlier detection Westerweel and
Scarano (2005) for post-processing. Multi-pass comprises

  * 1.

a single pass with a 64$\times$64 interrogation window, 50 $\%$ overlap, and
square weighting

  * 2.

three passes at a window size of 48$\times$48, 75 $\%$ overlap and Gaussian
weighting.

Therefore, the smallest interrogation window that will be used for DaVis 10.2
is 48$\times$48 with 75 $\%$ overlap. This results in a highly averaged flow
field information. In comparison, the results obtained from our model RAFT-
StereoPIV are dense per-pixel optical flow field information.

Since the resolution of the images from most experimental data is much larger
compared to the training data resolution ($32\times 32\,\mathrm{px}^{2}$), we
implement a folding/unfolding scheme that generates cutouts of the input
images. As a result, multiple forward passes are performed when processing
large PIV images.

Overall, Figure 9 shows comparable flow structures at different positions in
the wake of the car for both methods, namely DaVis 10.2 and RAFT-StereoPIV.
However, sharp and more detailed flow field information is achieved with the
deep optical flow learning model RAFT-StereoPIV. In contrast to the results
from DaVis 10.2 the data handling and the flow field estimation using deep
learning, including processing the PIV images, can be completely automated and
simultaneously done on multiple GPUs. Neither preprocessing nor postprocessing
operations were applied to the images or the resultant data from RAFT-
StereoPIV.

It is observable that the predictions derived from RAFT-StereoPIV encompass
certain outliers. Employing an outlier detection, such as universal outlier
detection Westerweel and Scarano (2005), has the potential to mitigate these
outliers. However, the primary emphasis of this study is to present the
unprocessed predictions generated by the deep learning model, as this approach
facilitates the measurement of the instantaneous three-dimensional velocity
field with high resolution and enables the visualization of fine turbulent
flow structures.

Figure 10. illustrates a spatial resolution comparison in the car’s wake from
RoF experiment. This comparison highlights the contrast between the sparse
vector field output from the cross-correlation based LaVision DaVis 10.2
(shown at the top) and the detailed per-pixel optical flow field from RAFT-
StereoPIV (shown at the bottom). Applying our trained network allowed us to
maintain the spatial resolution of the original image sequence, a significant
enhancement over traditional methods that typically produce flow fields at a
resolution about 16 times lower.

####  4.3.5 Flow Field Estimation from Wind Tunnel Measurements

The PIV analysis of the flow field in the wake of the Ahmed body consists of
several steps. For the reduction of background noise and reflections a
directional minimum filter with a filter length of $L=11\,$pixels was applied,
leading to a better signal-to-noise ratio. Consequently a multi-pass PIV-
calculation was performed (1 pass: window size 96$\times$64, square weighting,
50 % overlap; 2 to 3 pass: window size 16$\times$16, Gaussian weighting, 75 %
overlap), leading to a grid spacing of 0.165 mm. Post-processing included the
universal outlier detection (Westerweel and Scarano (2005)) within a region of
5$\times$5 vectors, followed by linear interpolation to fill up empty spaced,
and finally all 100 instantaneous vector fields were averaged. Here, in
contrast to RoF we apply RAFT-StereoPIV on the preprocessed double-frame
particle images and not the raw data.

In summary, a good agreement between the flow structures obtained from DaVis
10.2 and RAFT-StereoPIV, as can be seen in Figure 11. This includes the
recirculation zone in the wake as well as the separation point. However, due
to the chosen angular stereo setup and remaining surface reflection, which
could not be completely eliminated in the pre-processing step, there exist
some discrepancies, such as outliers and artifacts.

Figure 11: Wind Tunnel Measurements - In-plane mean velocity component
$\overline{u}$ in the streamwise symmetry plane of the Ahmed body at $y$ = 0.
(Left) The results obtained from LaVision DaVis 10.2 and (right) - from the
deep learning model RAFT-StereoPIV. RAFT-StereoPIV outputs dense per-pixel
optical flow field 3000 $\times$ 3519, whereas the cross-correlation-based
LaVision DaVis 10.2 outputs only sparse velocity field with 750 $\times$ 880
vectors.

##  5 Limitations and Future Work

Deep learning in Particle Image Velocimetry shows great potential but faces
challenges such as data dependency, lack of interpretability, and high
computational demands. Future efforts in deep learning-based PIV should
explore the development of models trained to integrate specific preprocessing
filters. These would target noise reduction, outlier removal, or minimizing
reflections, thereby enhancing data quality before analysis. Additionally,
improving data augmentation, synthetic and wind tunnel data utilization,
transfer learning for broader generalization, model interpretability, and
computational efficiency remains crucial. Further advancements in real-time
processing could also broaden the scope of deep learning applications in PIV.

##  6 Conclusions

By integrating advanced deep learning techniques with both Ring of Fire and
wind tunnel measurements, we are able to achieve flow field visualization with
high spatial resolution, a key element in aerodynamic testing and vehicle
development. We achieved a new state-of-the-art on all benchmarks, in terms of
both prediction accuracy and training time. Recurrent All-Pairs Field
Transforms for Stereoscopic Particle Image Velocimetry (RAFT-StereoPIV)
surpasses all existing supervised and unsupervised learning-based approaches
in performance. In addition, it is the first flow estimation based on 3D
reconstruction from 2D stereoscopic PIV data using deep learning. Our proposed
method has the potential for real-time and on-site characterization of
automotive aerodynamics under real driving conditions, as well as in
controlled wind tunnel environments. As the vehicle passes through the laser
sheet, as in the open-road setting Ring of Fire, RAFT-StereoPIV effectively
visualizes the flow field in the vehicle’s wake. This capability provides a
detailed and dynamic insight into the aerodynamic interactions, crucial for
comprehensive analysis.

We believe that this study helps pave the way forward to enable the
development of more advanced measurement techniques for stereoscopic PIV in
general and industrial aerodynamics in particular.

#### Author Contributions

ME handled the development and implementation of the proposed method and
testing on provided data. ME, with MG, designed, trained, tuned, and evaluated
the deep learning models. ME proposed the 3D optical flow reconstruction
module and started the effort with SH and TG to evaluate this idea. SH and TG
were responsible for conducting the RoF experiment and the wind tunnel
measurements of Ahmed Vehicle Model. TG, DC, and CB were responsible for
research supervision. All co-authors meaningfully contributed to manuscript
writing and review prior to submission.

## References

  * Adrian (2005) Adrian, R.J., 2005.  Twenty years of particle image velocimetry.  Experiments in fluids 39, 159–169.  doi:10.1007/s00348-005-0991-7.
  * Ahmed et al. (1984) Ahmed, S.R., Ramm, G., Faltin, G., 1984.  Some salient features of the time-averaged ground vehicle wake.  SAE transactions , 473–503URL: http://www.jstor.org/stable/44434262.
  * Cai et al. (2020) Cai, S., Liang, J., Gao, Q., Xu, C., Wei, R., 2020\.  Particle image velocimetry based on a deep learning motion estimator.  IEEE Transactions on Instrumentation and Measurement 69, 3538–3554.  doi:10.1109/TIM.2019.2932649.
  * Cai et al. (2019) Cai, S., Zhou, S., Xu, C., Gao, Q., 2019.  Dense motion estimation of particle images via a convolutional neural network.  Experiments in Fluids 60, 1–16.  doi:10.1007/s00348-019-2717-2.
  * Carlier (2005) Carlier, J., 2005.  Second set of fluid mechanics image sequences.  European Project Fluid Image Analysis and Description (FLUID) , 0018–9456.
  * Dosovitskiy et al. (2015) Dosovitskiy, A., Fischer, P., Ilg, E., Häusser, P., Hazirbas, C., Golkov, V., Smagt, P.v.d., Cremers, D., Brox, T., 2015.  Flownet: Learning optical flow with convolutional networks, in: 2015 IEEE International Conference on Computer Vision (ICCV), pp. 2758–2766.  doi:10.1109/ICCV.2015.316.
  * Gao et al. (2021) Gao, Q., Lin, H., Tu, H., Zhu, H., Wei, R., Zhang, G., Shao, X., 2021.  A robust single-pixel particle image velocimetry based on fully convolutional networks with cross-correlation embedded.  Physics of Fluids 33.  doi:10.1063/5.0077146.
  * Gericke et al. (2023) Gericke, T., Erdogdu, A., Huettig, S., Ladwig, M., Lindken, R., 2023.  Reconstruction of time-averaged 3d pressure fields of an ahmed body with pressure from piv. part ii: Shake-the-box, in: 15th International Symposium on Particle Image Velocimetry–ISPIV, pp. 19–21.
  * Han and Wang (2023) Han, Y., Wang, Q., 2023.  An attention-mechanism incorporated deep recurrent optical flow network for particle image velocimetry.  Physics of Fluids 35.  doi:10.1063/5.0155124.
  * Hüttig et al. (2023) Hüttig, S., Gericke, T., Sciacchitano, A., Akkermans, R., 2023\.  Automotive on-road flow quantification with a large scale stereo-piv setup, in: 15th International Symposium on Particle Image Velocimetry – ISPIV 2023, San Diego, California, USA.
  * Hüttig et al. (2024) Hüttig, S., Kühn, M., Gericke, T., Bloem, G., Sciacchitano, A., Akkermans, R., 2024\.  On-road car aerodynamics with the ring of fire concept.  Submitted to Experiments in Fluids - arXiv preprint arXiv:xxxx.xxxxx -, –.
  * Ladwig et al. (2023) Ladwig, M., Gericke, T., Huettig, S., Lindken, R., 2023\.  Reconstruction of time-averaged 3d pressure fields of an ahmed body with pressure from piv. part i: Scanning stereo-piv, in: Symposium on Particle Image Velocimetry. San Diego State University.
  * Lagemann et al. (2021a) Lagemann, C., Klaas, M., Schröder, W., 2021a.  Unsupervised recurrent all-pairs field transforms for particle image velocimetry.  14th International Symposium on Particle Image Velocimetry doi:10.18409/ispiv.v1i1.120.
  * Lagemann et al. (2021b) Lagemann, C., Lagemann, K., Mukherjee, S., Schröder, W., 2021b.  Deep recurrent optical flow learning for particle image velocimetry data.  Nature Machine Intelligence 3, 641–651.  doi:10.1038/s42256-021-00369-0.
  * Lagemann et al. (2022) Lagemann, C., Lagemann, K., Mukherjee, S., Schroeder, W., 2022\.  Generalization of deep recurrent optical flow estimation for particle-image velocimetry data.  Measurement Science and Technology doi:10.1088/1361-6501/ac73db.
  * Lee and Moser (2015) Lee, M., Moser, R.D., 2015.  Direct numerical simulation of turbulent channel flow up to $\mathit{Re}_{{\it\tau}}\approx 5200$.  Journal of fluid mechanics 774, 395–415.  doi:10.1017/jfm.2015.268.
  * Lee et al. (2017) Lee, Y., Yang, H., Yin, Z., 2017.  PIV-DCNN: cascaded deep convolutional neural networks for particle image velocimetry.  Experiments in Fluids 58, 1–10.  doi:10.1007/s00348-017-2456-1.
  * Liang et al. (2022) Liang, J., Cai, S., Xu, C., Chen, T., Chu, J., 2022\.  DeepPTV: Particle tracking velocimetry for complex flow motion via deep neural networks.  IEEE Transactions on Instrumentation and Measurement 71, 1–16.  doi:10.1109/TIM.2021.3120127.
  * Liang et al. (2023) Liang, J., Xu, C., Cai, S., 2023.  Recurrent graph optimal transport for learning 3d flow motion in particle tracking.  Nature Machine Intelligence 5, 505–517.  doi:10.1038/s42256-023-00648-y.
  * Loshchilov and Hutter (2019) Loshchilov, I., Hutter, F., 2019\.  Decoupled weight decay regularization.  arXiv:1711.05101.
  * Paszke et al. (2017) Paszke, A., Gross, S., Chintala, S., Chanan, G., Yang, E., DeVito, Z., Lin, Z., Desmaison, A., Antiga, L., Lerer, A., 2017\.  Automatic differentiation in pytorch.  URL: https://api.semanticscholar.org/CorpusID:40027675.
  * Prasad (2000) Prasad, A., 2000.  Stereoscopic particle image velocimetry.  Experiments in Fluids 29, 103–116.  doi:10.1007/s003480000143.
  * Rabault et al. (2017) Rabault, J., Kolaas, J., Jensen, A., 2017.  Performing particle image velocimetry using artificial neural networks: a proof-of-concept.  Measurement Science and Technology 28, 125301.  doi:10.1088/1361-6501/aa8b87.
  * Resseguier et al. (2017) Resseguier, V., Mémin, E., Chapron, B., 2017.  Geophysical flows under location uncertainty, part III SQG and frontal dynamics under strong turbulence conditions.  Geophysical & Astrophysical Fluid Dynamics 111, 209–227.  doi:10.1080/03091929.2017.1312102.
  * Schanz et al. (2016) Schanz, D., Gesemann, S., Schröder, A., 2016.  Shake-the-box: Lagrangian particle tracking at high particle image densities.  Experiments in fluids 57, 1–27.  doi:10.1007/s00348-016-2157-1.
  * Sciacchitano et al. (2015) Sciacchitano, A., Caridi, G.C.A., Scarano, F., 2015.  A quantitative flow visualization technique for on-site sport aerodynamics optimization.  Procedia Engineering 112, 412–417.  doi:10.1016/j.proeng.2015.07.217.
  * Spoelstra et al. (2021) Spoelstra, A., Hirsch, M., Sciacchitano, A., Scarano, F., 2021\.  Uncertainty assessment of the ring of fire concept for on-site aerodynamic drag evaluation.  Measurement Science and Technology 32, 044004.  doi:10.1088/1361-6501/abb50d.
  * Spoelstra et al. (2020) Spoelstra, A., Mahalingesh, N., Sciacchitano, A., 2020.  Drafting effect in cycling: On-site aerodynamic investigation by the ‘Ring of Fire’.  Multidisciplinary Digital Publishing Institute Proceedings 49, 113.  doi:10.3390/proceedings2020049113.
  * Spoelstra et al. (2018a) Spoelstra, A., de Martino Norante, L., Terra, W., Sciacchitano, A., Scarano, F., 2018a.  An assessment of the ring of fire approach for indoor and outdoor on-site sports aerodynamic investigation, in: 19th international symposium on the application of laser and imaging techniques to fluid mechanics.
  * Spoelstra et al. (2019) Spoelstra, A., de Martino Norante, L., Terra, W., Sciacchitano, A., Scarano, F., 2019.  On-site cycling drag analysis with the Ring of Fire.  Experiments in Fluids 60, 1–16.  doi:10.1007/s00348-019-2737-y.
  * Spoelstra et al. (2018b) Spoelstra, A., Terra, W., Sciacchitano, A., 2018b.  The Ring of Fire for in-field sport aerodynamic investigation, in: Proceedings, MDPI. p. 221.  doi:10.3390/proceedings2060221.
  * Spoelstra et al. (2023) Spoelstra, A., Terra, W., Sciacchitano, A., 2023.  On-site aerodynamics investigation of speed skating.  Journal of Wind Engineering and Industrial Aerodynamics 239, 105457.  doi:10.1016/j.jweia.2023.105457.
  * Teed and Deng (2020) Teed, Z., Deng, J., 2020.  Raft: Recurrent all-pairs field transforms for optical flow, in: European conference on computer vision, Springer. pp. 402–419.  doi:10.1007/978-3-030-58536-5_24.
  * Terra et al. (2017) Terra, W., Sciacchitano, A., Scarano, F., 2017.  Aerodynamic drag of a transiting sphere by large-scale tomographic-piv.  Experiments in Fluids 58, 1–14.  doi:10.1007/s00348-017-2331-0.
  * Terra et al. (2018) Terra, W., Sciacchitano, A., Scarano, F., van Oudheusden, B., 2018\.  Drag resolution of a piv wake rake for transiting models.  Experiments in Fluids 59, 1–6.  doi:10.1007/s00348-018-2570-8.
  * Wang et al. (2022) Wang, H., Liu, Y., Wang, S., 2022.  Dense velocity reconstruction from particle image velocimetry/particle tracking velocimetry using a physics-informed neural network.  Physics of Fluids 34, 017116\.  doi:10.1063/5.0078143.
  * Westerweel (1997) Westerweel, J., 1997.  Fundamentals of digital particle image velocimetry.  Measurement science and technology 8, 1379.  doi:10.1088/0957-0233/8/12/002.
  * Westerweel and Scarano (2005) Westerweel, J., Scarano, F., 2005\.  Universal outlier detection for piv data.  Experiments in fluids 39, 1096–1100.  doi:10.1007/s00348-005-0016-6.
  * Wieneke (2005) Wieneke, B., 2005.  Stereo-PIV using self-calibration on particle images.  Experiments in fluids 39, 267–280.  doi:10.1007/s00348-005-0962-z.
  * Willert (1997) Willert, C., 1997.  Stereoscopic digital particle image velocimetry for application in wind tunnel flows.  Measurement science and technology 8, 1465.  doi:10.1088/0957-0233/8/12/010.
  * Yu et al. (2021) Yu, C., Bi, X., Fan, Y., Han, Y., Kuai, Y., 2021.  LightPIVNet: An effective convolutional neural network for particle image velocimetry.  IEEE Transactions on Instrumentation and Measurement 70, 1–15.  doi:10.1109/TIM.2021.3082313.
  * Yu et al. (2023) Yu, C., Fan, Y., Bi, X., Kuai, Y., Chang, Y., 2023\.  Deep dual recurrence optical flow learning for time-resolved particle image velocimetry.  Physics of Fluids doi:10.1063/5.0142604.
  * Zhang and Piggott (2020) Zhang, M., Piggott, M.D., 2020\.  Unsupervised learning of particle image velocimetry, in: International Conference on High Performance Computing, Springer. pp. 102–115.  doi:10.1007/978-3-030-59851-8_7.

#### Acknowledgements

We thank Christian Lagemann for many fruitful discussions and valuable
feedback. We also express our gratitude to Steffen Ropers for his assistance
in training the deep learning models.

#### Disclaimer

The results, opinions, and conclusions expressed in this publication are not
necessarily those of Volkswagen Aktiengesellschaft.

#### Conflict of interest

The authors declare that they have no conflict of interest.

#### Data Availability Statement

This study used both open-source datasets and proprietary data from Volkswagen
AG. The proprietary data from Volkswagen AG is confidential and not publicly
available. Access was granted under a specific agreement and may be available
upon request, subject to Volkswagen AG’s approval and data sharing policies.

##  Appendix A RAFT-StereoPIV Approach

The pseudo-code for our approach, where we refer to the 4D description of the
flow field as the 3D flow information estimated sequentially over time, is
provided in Algorithm. 1.

Input: $n$ grayscale image-pairs $\\{I^{t}$, $I^{t+\Delta t}\\}$, calibration
angles $\\{\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\\}$, magnification factor
$M$
Use mapping function to map the input images: $f(I^{t}$, $I^{t+\Delta
t})=(\tilde{I}^{t},\tilde{I}^{t+\Delta t})$

1:for $i=1..n$ do $\triangleright$ iterate over $n$ number of image-pairs

2: for $j=1..2$ do $\triangleright$ iterate over two cameras

3: Estimate optical flow $\Delta x_{i,j}$, $\Delta y_{i,j}$ using the RAFT-
StereoPIV model

4: $u_{i,j}\leftarrow\frac{\Delta x_{i,j}}{M\cdot\Delta t}$ ;
$v_{i,j}\leftarrow\frac{\Delta y_{i,j}}{M\cdot\Delta t}$

5: end for

6: Reconstruct 3D velocity components from 2D velocities:

7:
$U_{i}\leftarrow\frac{u_{i,1}\tan\alpha_{2}+u_{i,2}\tan\alpha_{1}}{\tan\alpha_{2}+\tan\alpha_{1}}$,

8:
$V_{i}\leftarrow\frac{v_{i,1}\tan\beta_{2}+v_{i,2}\tan\beta_{1}}{\tan\beta_{2}+\tan\beta_{1}}$,

9: $W_{i}\leftarrow\frac{u_{i,1}-u_{i,2}}{\tan\alpha_{2}+\tan\alpha_{1}}$

10:end for

11:Ensemble averaging over $n$ image-pairs to reduce the effect of noise and
outliers:

12:$\overline{U}\leftarrow\frac{1}{n}\sum_{i=1}^{n}U_{i}$ ,
$\overline{V}\leftarrow\frac{1}{n}\sum_{i=1}^{n}V_{i}$ ,
$\overline{W}\leftarrow\frac{1}{n}\sum_{i=1}^{n}W_{i}$

Algorithm 1 4D Flow Field Estimation Using Stereoscopic PIV and Deep Learning

##  Appendix B Hyperparameters Tuning

In this study, we endeavor to facilitate the training of large-scale deep
learning models using community resources. The high computational demands
required to train those models create substantial obstacles to reproducibility
and further development, especially for researchers who have limited access to
high-performance computing infrastructure. By leveraging community resources
and systematically exploring the hyperparameters space, our approach aims to
democratize access to the state-of-the-art models, thereby fostering a more
inclusive research environment and stimulating advancements in the field. In
addition, we aim to enhance the overall performance, generalizability, and
quality of results on real-world data. To achieve this, our study involves
conducting large-scale analysis of hyperparameters to optimize the training
process of the underlying optical flow network.

To better understand the importance of specific hyperparameters of RAFT-
StereoPIV and the performance difference based on the chosen set of
hyperparameters, we ran several experiments. Results are shown in Table A1. We
report the AEE values for both the training and validation datasets, however,
during the process of selecting the optimal hyperparameter values, only the
AEE values obtained from the validation dataset are taken into account.
Sections (A)-(F) of the table test exclusively a single hyperparameter, such
as batch size, dropout probability $P_{drop}$, normalization method, initial
learning rate $\alpha_{init}$, the reduce factor $\gamma$, the size of the
training dataset $D$, and the number of training epochs $N_{epochs}$. On the
other hand, section (G) tests different combinations of these hyperparameters,
based on results obtained from the previous experiments.

Batch size: Smaller batch size means greater variance in the gradients and
thus noisy updates, however, having larger batch sizes is mostly limited by
the GPU memory available. Therefore, we tested different batch sizes
{4,16,32,64,128,192} to explore the trade-off between variance in the
gradients and memory requirements when selecting the batch size. From (A), we
observe, in accordance with expectations, larger batch sizes lead to better
model performance.

Dropout: Our findings indicate that the model’s performance improves as the
dropout probability increases. However, it is essential to note that an
extended training duration is necessitated to achieve optimal generalization
capabilities on unseen test data, allowing the model to effectively learn
robust representations from the data.

Normalization: Our experiments showed that group normalization demonstrated
superior performance on the training and validation datasets. However, when
applied to the unseen test dataset, the model performed poorly, which led us
to opt for instance normalization. One possible explanation for the poor
performance of group normalization on the test dataset could be that it is
less capable of generalizing to data with distribution shifts or variations
not present in the training and validation sets. By contrast, instance
normalization is more robust to such discrepancies, making it a more suitable
choice for this application.

Learning rate: We tested various learning rate values drawn from the
literature and closely monitored the learning curves to assess the impact on
the training dynamics and final model performance. A learning rate of 0.001
proved to be the most effective, yielding the best balance between training
speed and model accuracy. This value facilitated steady convergence without
causing the training to stall or overshoot the minimum loss.

Size of training dataset: The fundamental objective of experiment (E) was to
address the inquiry of determining the minimum training set size necessary for
achieving satisfactory model performance. Our findings suggest that a training
dataset of 20,000 samples can still yield very good results.

Number of epochs: Finally, the number of training epochs $N_{epochs}$ refers
to the number of times the training dataset is passed through the network
during training, which affects the convergence and optimization of the model.
While longer training durations can lead to overfitting, we determined that
the optimal number of epochs for our training was 120.

Experiment | Batch Size | $P_{drop}$ | Norm | $\alpha_{init}$ | $\gamma$ | $D$ | $N_{epochs}$ |  | $\begin{array}[]{c}\text{ Train }\\\ \text{ AEE }\end{array}$ | $\begin{array}[]{c}\text{ Val }\\\ \text{ AEE }\end{array}$ | $\begin{array}[]{c}\text{ Time }\\\ \text{ hrs }\end{array}$
---|---|---|---|---|---|---|---|---|---|---|---
Base | 4 | 0.0 | ’instance’ | 0.0001 | 0.2 | 104k | 80 |  | 0.1084 | 0.1111 | 46.3
| 16 |  |  |  |  |  |  |  | 0.0644 | 0.0663 | 21.2
| 32 |  |  |  |  |  |  |  | 0.0555 | 0.0577 | 17.6
(A) | 64 |  |  |  |  |  |  |  | 0.0499 | 0.0517 | 15.5
| 128 |  |  |  |  |  |  |  | 0.0474 | 0.0495 | 14.2
| 196 |  |  |  |  |  |  |  | 0.0442 | 0.0459 | 13.6
| 196 | 0.1 |  |  |  |  |  |  | 0.0379 | 0.0394 | 13.6
| 196 | 0.2 |  |  |  |  |  |  | 0.0400 | 0.0415 | 13.6
(B) | 196 | 0.3 |  |  |  |  |  |  | 0.0375 | 0.0389 | 13.6
196 | 0.4 |  |  |  |  |  |  | 0.0348 | 0.0363 | 13.6
196 |  | ’batch’ |  |  |  |  |  | diverged | diverged | 13.6
(C) | 196 |  | ’group’ |  |  |  |  |  | 0.0361 | 0.0374 | 13.6
| 196 |  | ’none’ |  |  |  |  |  | 1.6160 | 1.6371 | 13.6
| 196 |  |  | 0.01 |  |  |  |  | diverged | diverged | 13.6
| 196 |  |  | 0.001 |  |  |  |  | 0.0552 | 0.0568 | 13.6
(D) | 196 |  |  | 0.00001 |  |  |  |  | 0.1458 | 0.1512 | 13.6
196 |  |  | 0.000001 |  |  |  |  | 0.6235 | 0.6368 | 13.6
196 |  |  |  |  | 20k |  |  | 0.0349 | 0.0363 | 2.7
| 196 |  |  |  |  | 40k |  |  | 0.0340 | 0.0354 | 5.2
(E) | 196 |  |  |  |  | 60k |  |  | 0.0332 | 0.0346 | 7.8
196 |  |  |  |  | 80k |  |  | 0.0338 | 0.0352 | 10.5
196 |  |  |  |  |  | 40 |  | 0.0327 | 0.0341 | 6.8
(F) | 196 |  |  |  |  |  | 120 |  | 0.0339 | 0.0354 | 20.4
| 196 |  |  |  |  |  | 160 |  | 0.0370 | 0.0384 | 27.2
| 196 | 0.4 |  |  | 0.05 |  | 100 |  | 0.0333 | 0.0348 | 17.1
(G) | 196 | 0.4 |  |  | 0.05 |  | 120 |  | 0.0339 | 0.0352 | 20.4
| 196 | 0.5 |  |  | 0.05 |  | 160 |  | 0.0343 | 0.0357 | 27.2
Table A1: Comprehensive results of the large-scale hyperparameter exploration
experiments (RAFT-StereoPIV). Unlisted values are identical to those of the
base model. The models were trained on two Quadro GV100 GPUs. The best values
from each experiment are marked in bold.