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

Super-resolution PIV using multi-frame displacement
information

Zhenxing Ouyang1, Hua Yang1∗, Yunkang Cao1, Zhen Yang1, Zhouping Yin1

1 State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of
Science and Technology, Wuhan, 430074, China

∗ huayang@hust.edu.cn

Abstract
High-resolution (HR) fluid-flow velocity information is important to reliably analyze fluid measurements
in particle image velocimetry (PIV), such as the boundary layer and turbulent flow. Efforts in PIV to en-
hance the resolution of flow fields are mainly based on single-frame information, which follows the velocity
field estimation and may influence the final reconstruction accuracy. In this study, we propose a novel
super-resolution (SR) reconstruction technology from another perspective, which consists of two parts: a
multi-frame imaging system and a Bayesian-based multi-frame SR reconstruction algorithm. First, a split-
based imaging system is developed to obtain particle image pairs with fixed displacements. Subsequently,
we present a Bayesian-based multi-frame SR (BMFSR) reconstruction algorithm to obtain an SR particle
image. Multi-frame particle images collected by the developed system are used as the input low-resolution
images for the following novel SR reconstruction algorithm. Synthetic and experimental particle images
have been tested to verify the performance of the proposed technology, and the results are compared with
the traditional and advanced reconstruction methods in PIV. The results and comparisons show that the pro-
posed technology successfully achieves good performance in obtaining finer particle images and a more
accurate velocity field.

1 Introduction
High-resolution (HR) fluid-flow velocity information is important to reliably analyze fluid measurements in
particle image velocimetry (PIV), such as the boundary layer and turbulent flow. However, fluid-flow data
are often sparse, have limited spatial resolution and are noisy in real life for various reasons, such as measur-
ing device imperfections or instability in the observed scene, sensor resolution limitations in experiments or
insufficient mesh size in computational fluid dynamics (CFD) simulations (Gao et al., 2020). Therefore, it is
necessary to improve the flow field resolution to explore many different boundaries, small-scale structures,
turbulence transitions and so on. In PIV measurement, the HR flow field can be obtained by CFD methods,
such as large eddy simulation (LES) and direct numerical simulation (DNS) (Liu et al., 2020). Neverthe-
less, simulations can only model the flow ideally and cannot fully analyze the real situation. Consequently,
it is essential and practically useful to reconstruct flow information from spatiotemporal low-resolution (LR)
data.

Super-resolution (SR) reconstruction is a classic technique for improving the resolution of an imaging
system. It can realize the resolution enhancement of a larger imaging system with HR data on a smaller
domain (Brunton et al., 2020). For decades, various SR reconstruction methods have been proposed, which
can be divided into two types: traditional methods and deep learning (DL) methods. Traditional SR re-
construction methods mainly utilize interpolation methods to improve the flow field resolution. Keys (1981)
employed bicubic interpolation based on the filter operation with low-pass characteristics to increase the res-
olution of digital images. Takehara et al. (2000) proposed an SR reconstruction method based on a Kalman
filter and χ2-testing and verified its performance on two synthetic datasets. Next, efforts (Gunes and Rist,
2007; Alfonsi et al., 2013; He and Liu, 2017) that applied proper orthogonal decomposition (POD) modes
extracted from the DNS dataset to obtain the corresponding coefficients and reconstruct the HR flow field
were made. These traditional SR reconstruction methods are based on the least-squares approximation to es-
timate the weighting factors between the LR data and HR data, which limits spatial resolution improvement
to some extent.



In the last two decades, the application of SR reconstruction technology based on DL in fluid dynam-
ics has attracted increasing attention. DL reconstruction is a data-driven technology that can provide an
effective method for generating HR flow fields quickly without iteratively solving partial differential equa-
tions (PDEs). Fukami et al. (2019a,b) adopted the convolutional neural network (CNN) and hybrid down-
sampled skip-connection multiscale models to reconstruct two-dimensional attenuated isotropic turbulence.
(Deng et al., 2019) proposed a network-based generative adversarial network-based artificial intelligence
framework to enhance the spatial resolution of complicated wake flow behind two side-by-side cylinders;
however, (Liu et al., 2020) proposed a multitime path CNN. To reduce the increase in the visual complexity
to an LR input, which is caused by data-driven upsampling approaches, various efforts (Raissi et al., 2019;
Gao et al., 2020; Dwivedi et al., 2021) have been proposed. Despite the advantages, the success of these DL
models mainly relies on a large quantity of offline HR data as labels.

All the models mentioned above, whether traditional or deep learning-based SR reconstruction methods,
only utilize the spatial information on the LR fluid field based on single-frame data and reconstruct follow-
ing the estimation of the velocity field. However, the information provided by only a two successive images
is limited. Moreover, the LR data that are used to generate the weighting parameters between the LR data
and HR data are not obtained simultaneously, resulting in the pertinence of these LR data being low. Fur-
thermore, the LR data used in the methods mentioned before are generally not the image itself, which may
influence reconstruction accuracy. Thus, in this study, a novel SR reconstruction technology is presented
from another perspective: SR particle image reconstruction using multi-frame displacement information. It
is proposed based on the idea of simultaneously collecting two LR particle images with a fixed displace-
ment and then generating the corresponding SR particle image with a novel SR reconstruction algorithm.
As a result, this novel reconstruction technology consists of two parts: a multi-frame imaging system and
a Bayesian-based multi-frame SR (BMFSR) reconstruction algorithm. First, a split-based imaging system
is developed to obtain pairs of particle images with fixed displacements. Subsequently, an algorithm named
BMFSR is proposed to obtain an SR particle image. Multi-frame particle images collected by the developed
system are used as the input LR images for the following novel SR reconstruction algorithm. In this way,
more correlated particle images can be collected simultaneously, and more LR spatial information can be
used. Finally, we determine whether the novel technology can reconstruct better results.

The rest of this paper is structured as follows: Sect. 2 introduces the basic scheme of the novel SR tech-
nology, including the multi-frame imaging system and the Bayesian-based multi-frame SR reconstruction
algorithm. Subsequently, experimental evaluations on synthetic images and real data are demonstrated in
Sect. 3. Finally, conclusions are discussed in Sect. 4.

2 The proposed super-resolution reconstruction technology
The proposed super-resolution reconstruction technology is a novel SR particle image reconstruction method
using multi-frame displacement information. This technology contains two main parts: a multi-frame imag-
ing system to collect double LR particle images simultaneously with a fixed displacement and a Bayesian-
based multi-frame SR (BMFSR) reconstruction algorithm to generate the corresponding HR particle image.
Finally, the HR velocity field can be obtained with the corresponding two successive reconstructed HR
particle images with an interval time.

2.1 Multi-frame imaging system
As mentioned above, traditional and deep learning SR reconstruction methods generate HR flow fields based
on LR single-frame data. Additionally, the LR data used are obtained from a series of two successive particle
frames. Different pairs of two successive frames are collected at different times, resulting in limited spatial
correlation between them, which limits the spatial information and may influence the final reconstruction
accuracy. Hence, in this study, we design a multi-frame imaging system to collect two particle images
simultaneously with a fixed displacement time to improve spatial relativity. The schematic and photographs
of this system are shown in Fig.1.

As shown in Fig.1 (a), this multi-frame imaging system consists of three parts: a spectral region, control
unit and transmit unit. The particle image intensity is defined as I and the collecting time is t. To obtain
the two relative particle images I1

t and I2
t at the same time t with a fixed displacement, in the spectral

region, a splitter is designed to split the incident light into two identical parts collected by two high-rate
sensors, sensor 1 and sensor 2, which are fixed with a preset constant displacement, and the displacement is
generated with a contact translation stage (shown in Fig.1 (c)). The particle image collection is controlled



by the control unit, which mainly consists of three parts, namely, a field programmable gate array (FPGA)
controlling the exposure of the two sensors, double data rate RAM (DDR) buffering the image data and a
solid-state disk (SSD) storing all particle images collected within a certain time. After collection, particle
images I1

t and I2
t are transmitted to a personal computer through the transmit unit to be estimated with the

proposed BMFSR reconstruction algorithm. The outer and inner photographs of this multi-frame imaging
system are shown in Fig.1 (b) and (c), respectively.

Figure 1: Schematic (a), outer photograph (b) and inner photograph (c) of the multi-frame imaging system.

2.2 Bayesian-based multi-frame SR reconstruction algorithm
Current SR reconstruction methods in PIV measurement are mainly based on a single frame, that is, one in-
put and one output. The formulation of the single-frame-based SR reconstruction methods can be expressed
as:

argmin
ψHR

∥∥ψLR −SKψHR
∥∥2

+ηR(ψHR) (1)

where ψLR denotes the LR flow data, such as the velocity field, and ψHR is the corresponding HR data to
be reconstructed. S and K correspond to downsampling and filtering with a blur kernel, respectively. R is
a function of the HR data ψHR to constrain the first term so that the solution is unique and the parameter η
balances the first and second terms. With increasing input LR image numbers, the ideal HR flow data can
be obtained from Eq.1.

However, LR particle images utilized in the single-frame-based methods are collected at different times,
whether within one successive two-frame pair to estimate one LR velocity field or without, resulting in the
spatial information of a single-frame particle image being limited, which may influence the final recon-
struction accuracy. Hence, based on the multi-frame imaging system, we also propose a Bayesian-based
multi-frame SR (BMFSR) reconstruction method, which involves two particle frames collected at the same
time with a constant displacement into the reconstruction function and is modified from Liu and Sun (2014).
This novel SR algorithm is proposed from another perspective: SR particle image reconstruction. Let the
image intensity be I, the first collecting time be t and the second be t +∆t, where ∆t is the interval time.
Then, taking time t as an example, the new reconstruction formulation is as follows:

Jt
∗ = argmin

Jt

θ1
∥∥SKJt − I1

t

∥∥+η∥∇Jt∥+θ2
∥∥SKFwJt − I2

t

∥∥ (2)

where Jt
∗ is the object function to be minimized and Jt is the upsampling image to be reconstructed at time

t. I1
t and I2

t are the LR particle images collected by sensor 1 and sensor 2 at time t. ∇Jt is the gradient of
HR image Jt . Fw is the warping matrix corresponding to the displacements w between J1

t and J2
t . η is a

parameter that balances the influence of sparsity on the derivative filter to model the priors of HR image Jt
and is usually set to 0.02. θ1 and θ2 are parameters that control the noise and outlier when Jt is warped to
generate the adjacent frame. Note that L1-norm penalization is adopted instead of L2-norm used in Eq. 1,
because that L1-norm can achieve better stability and convergence (Seong et al., 2019).

By solving Eq. 2, the HR image IHR
t can be obtained. In the same way, the correlated HR image IHR

t+∆t
will be obtained with Eq.2 from the other two frames I1

t+∆t and I2
t+∆t . Finally, the velocity field of the flow is

estimated with an open source (Vejrazka, 2021), which is a cross-correlation algorithm, named DCC in this
paper. Bicubic interpolation is used to obtain a full-size velocity field.

With the Bayesian-based multi-frame SR reconstruction algorithm, more spatial information at one time
can be used to reconstruct the HR data, resulting in the reconstruction accuracy improvement.



2.3 Minimization and parameter setting
For the novel SR reconstruction algorithm, we can estimate the HR image by solving Eq.2 when the current
flow field w, blur kernel K and noise levels θ1 and θ2 are given. To simplify the estimation, we replace the
L1-norm with a non-quadratic penalty function ϕ(s2) =

√
s2 + τ2. The symbol τ is a prefixed small constant,

which is usually set as τ = 0.01, to ensure that ϕ is strictly convex.
This object function of Eq. 2 can be solved by the iterated reweighted least squares (IRLS) method

(Liu et al., 2009), which iteratively solves the following linear system:

[θ1KT ST P1SK+η((Jt)
T
x PS(Jt)x+(Jt)

T
y PS(Jt)y)+θ2FT

w KT ST P2SKFw]Jt = θ1KT ST P1I1
t +θ2FT

w KT ST P2I2
t
(3)

where (Jt)x and (Jt)y are the x- and y-derivative matrices of the HR image Jt . T is the transpose operator
of the corresponding matrix; for example, ST is the transpose of the downsampling S. To simplify the
equations, we use the following notation:

P1 = diag(ϕ′(
∥∥SKJt − I1

t

∥∥2
)),Ps = diag(ϕ′(∥∇Jt∥2)),P2 = diag(ϕ′(

∥∥SKFwJt − I2
t

∥∥2
)) (4)

Here, diag denotes the diagonal weight matrices. The IRLS iterates between solving Eq. 3 and Eq. 4 are
based on the current estimation to obtain the final ideal HR particle image Jt .

The solution for solving Eq.2 is based on the assumption that the flow field w, the blur kernel K and the
noise levels θ1 and θ2 are given. The flow field w can be estimated with an optical flow Drulea and Nedevschi
(2011). Hence, the unknown parameters in the generative models include K, θ1 and θ2.

We use Bayesian maximum a posteriori estimation (MAP) to find the optimal solution:

argmax
I,w,{θ1,θ2}

p(Jt ,w,K,{θ1,θ2}
∣∣{I1

t , I
2
t }) (5)

In addition, the sparsity on derivative filter responses is used to model the prior of HR image Jt , flow field
w, and the blur kernel K:

p(Jt)=
1

ZJt (η)
exp{−η∥∇Jt∥}, p(w)=

1
Zw(λ)

exp{−λ(∥∇u∥+∥∇v∥}, p(K)=
1

ZK(ξ)
exp{−ξ∥∇K∥}

(6)

where ZJt (η), Zw(λ) and ZK(ξ) are normalization constants only dependent on parameters η, λ and ξ, re-
spectively. u and v are the horizontal and vertical components of the flow velocity w.

Then, similar to Eq.2, when solving the noise levels θ1 and θ2, the HR image Jt and the blur kernel K are
given, and the Bayesian MAP estimation for the noise parameters has the following closed-form solution:

θi =
α+Nq −1

β+NĪ
, Ī =

1
N

N

∑
q=1

∥∥(Ii
t −SKFwiI)(q)

∥∥, i = 1,2 (7)

where Ī is the sufficient statistics, N is the pixels of the LR image and α and β are two constant parameters.
For the blur kernel K, the solution is similar to the HR image Jt ; the upsampling Jt and downsampling matrix
S are given. The minimization formulation is as follows:

argmin
K

θ0
∥∥SJtK− I1

t

∥∥+η∥∇K∥ (8)

which is also optimized by IRLS and not discussed in detail here.
With the iteration of the estimations of the noise controlling parameters θm

1 and θm
2 , optical flow field wm,

intermediate upsampling image Jm
t and blur kernel Km, where m is the current iteration number, the optimal

HR particle image Jt can be generated. We use 5 iterations to generate the HR particle image with η = 0.02
for the image derivative, λ = 1,000 for the optical flow field, ξ = 0.7 for the derivative of the blur kernel and
α = 1 and β = 0.1 for noise. The upsampling scale is set to be 4.



3 Experimental evaluations
In this section, we illustrate the performance of the proposed SR reconstruction technology on synthetic
and experimental images. First, we focus on the performance of the proposed BMFSR method and verify
the advantage on a synthetic vortex flow in Sect. 3.1. Subsequently, in Sect. 3.2, real experimental micro-
PIV data are tested to demonstrate the practicality of the novel SR reconstruction technology combining
the multi-frame imaging system and the BMFSR algorithm, which utilizes the multi-frame displacement
information.

Hereafter, we follow a standard method to quantitatively evaluate the experimental results by computing
the peak signal-to-noise ratio (PSNR), the root mean square error (RMSE) and the average angle error
(AAE) over N pixels of the image:

PSNR=10·log10
2552

1
N

N
∑

n=1
|Jt(n)−J(n)|2

, RMSE=

√
1
N

N

∑
n=1

|wes
n −wtr

n |
2, AAE=

1
N

N

∑
n=1

arccos(
wes

n ·wtr
n

|wes
n |·|wtr

n |
) (9)

where J and Jt are the ground truth and the reconstructed HR particle images, respectively. wtr and wes

denote the ground truth and the estimated velocity fields.

3.1 Test on synthetic vortex flow
First, we investigate the performance of the proposed BMFSR method on a synthetic vortex flow. Two pairs
of HR particle images with 512 × 512 pixels are generated, whereby 7,864 particles with a mean diameter
of 6 pixels and size variation of 0.3 pixels are randomly distributed. No out-of-plane motion or noise is
considered, and the sheet thickness is set to 0.333. The displacement between these two pairs is set to 0.25
pixels and considered to be the background HR particle images of sensor 1 and sensor 2. In this experiment,
a synthetic velocity field is a Lamb-Ossen Vortex, similar to (Carlier and Wieneke, 2005). The exact solution
of the Navier-Stokes equation is:

u(r,θ) =
Γ0

2πr
(1− e

−r2

r0
2 ) (10)

where Γ0 is the initial circulation and r0 is the core radius. In this study, Γ0 = 2000, r0 = 50 pixels. LR
particle images are generated by downsampling the HR particle images above with a downsampling scale
of 4. During dowmsampling, blur and noise are added to simulate the real scene.

The HR particle images of the background reconstructed by the traditional bicubic interpolation and
reconstructed by the proposed BMFSR are shown in Fig.2. As shown, the image reconstructed by BMFSR
is more approximate to the background particle image. In addition, we compare the reconstruction quality
with the evaluation indices PSNR, RMSE and AAE, and the results are shown in Fig.3. Compared to bicubic
interpolation, the PSNR of our proposed method BMFSR is the highest, whether for I1

t or I2
t . In Table 2,

Bicubic-DCC denotes that the DCC results are set to be the background truth (Ours-DCC is the same)
because both bicubic interpolation and our method estimate the velocity with an open source DCC and it
has its inherent estimation error. The bold denotes the best and the underlined denotes the second here.
In Table 2, it can be seen that the proposed method BMFSR performs better than the traditional bicubic
interpolation method relative to whether the ground truth or the result of the basic DCC algorithm.

We present the energy maps and velocity distributions of different methods in Fig. 4. The comparison
result suggests that the proposed BMFSR method outperforms the traditional bicubic interpolation method.
In particular, at the bottom right, the bicubic result is worse, which may be caused by the blur and noise of
the downsampling operator in generating LR particle images. However, the proposed BMFSR method can
decrease the influence of blur and noise to some extent.

It can be concluded that the proposed BMFSR reconstruction algorithm is well adapted to SR reconstruc-
tion and shows great advantages in particle image reconstruction and HR velocity estimation compared to
the traditional bicubic interpolation method. Then, the overall performance of the proposed super-resolution
reconstruction technology is verified in a real experiment.



(a) (b) (c)

Figure 2: HR particle images of (a) the ground truth; (b) the traditional bicubic method; (c) the proposed
BMFSR method.

Figure 3: Results of different methods on the reconstructed particle image PSNRs and the estimated velocity
errors.

(a) (b)

(c) (d)

Figure 4: Energy maps and velocity distributions of different methods: (a) the ground truth; (b) result of
DCC; (c) result of the traditional bicubic method; (d) result of the proposed BMFSR method.



3.2 Test on experimental laminar boundary layer flow
To illustrate the generalization of the proposed method in the real flow field, a micro-PIV experiment was
carried out to acquire the particle images of the laminar boundary layer with a constant inflow velocity, and
the platform is shown in Fig. 5(a). Orange fluorescent particles with a diameter of 0.3 µm were dispersed
in distilled water and stored in a syringe. Then, an Era NE-1000 injection pump drove the syringe at 79
µm/s. The flow field is in a microfluidic chip, whose cross-section is 30 µm wide × 28 µm deep. The
whole flow field was illuminated using fluorescent light (Nikon IntensiLight C-HGFI) with a wavelength
of 532 nm, which was the same as the light passed by the filter in the multi-frame imaging system and
imaged by a Nikon Eclipse Ti-S inverted fluorescent microscope with a 60 × magnification lens. The field
is captured using the multi-frame imaging system with an interest region of 1,296 × 408 pixels. The flow
field that we set is a laminar flow field, which has minimum velocity at the edge of the pipe and maximum
velocity in the center of the pipe. In the experiment, we compare the performance of the traditional bicubic
interpolation reconstruction method and our proposed BMFSR method. Note that the particle image that is
used to generate the HR image of the bicubic interpolation method is only I1

t ; however, those of our proposed
BMFSR are I1

t and I2
t .

Figure 5: (a): Micro-PIV platform; (b) HR image with bicubic interpolation; (c) HR image with our pro-
posed method; (d) errors of different methods.

LR HR_bicubic Ours

Boundary layer structure: Ours>HR_bicubic>LR

None Fine Finer

Figure 6: Top: the energy maps and velocity distributions of unsteady micro-PIV flow. Down: a close up in
the region −20 ≤ Y ≤−18 pixels and 38 ≤ X ≤ 51 pixels.

Low-resolution images are directly acquired by the two sensors mentioned above. The HR particle im-
ages reconstructed from the traditional bicubic interpolation and our proposed method are shown in Fig.5(b)



and (c), respectively. In comparison, the HR particle image reconstructed with our method is clearer be-
cause of the blur kernel estimation, and the result is consistent with the synthetic experiment above. The
mainstream velocities measured are approximately 76.63 µm/s, 80.28 µm/s and 79.83 µm/s for uLR, uHR

bi and
uHR

ba , respectively; that is, the errors with the theoretical velocity are 3.00%, 1.62% and 1.05%, as shown in
Fig.5(d). Additionally, as shown in Fig.5(d), the proposed method obtains the lowest RMSE of the vertical
component, which is 0.0832. To further verify the performance of our method, we compare the energy
maps and velocity distributions of three different methods: LR uLR, HR with bicubic reconstruction uHR

bi
and ours uHR

ba in Fig.6-top. Fig.6-down shows the close up of these three velocity distributions in the region
−20 ≤ Y ≤ −18 pixels and 38 ≤ X ≤ 51 pixels. The upsampling scale factor of reconstruction here is 4.
The sampling rate in the horizontal direction of uHR

bi and uHR
ba is 4. We can observe that the distribution of

uHR
ba is finer at the boundary. Therefore, it can be concluded that the proposed technology performs better in

SR reconstruction.

4 Conclusion
In this paper, we propose a novel super-resolution (SR) reconstruction technology for particle image ve-
locimetry from the perspective of SR particle image reconstruction using multi-frame displacement infor-
mation. This technology consists of two parts: a multi-frame imaging system and a Bayesian-based multi-
frame SR (BMFSR) reconstruction algorithm. First, two correlated particle images are collected by the
multi-frame imaging system at the same time with a constant displacement. Subsequently, these two images
are utilized to generate an HR image with the proposed BMFSR method. This proposed SR reconstruction
technology has been assessed by many experiments. First, a synthetic vortex flow is used as the benchmark
to evaluate the performance of the proposed BMFSR algorithm. The results show that the proposed method
can provide accurate reconstruction results. Compared to the traditional bicubic interpolation method, it also
shows a great advantage in terms of the quality of the reconstructed particle image or the estimated high-
resolution (HR) velocity. The overall performance of the proposed SR reconstruction technology is verified
with an experimental laminar boundary layer flow. The results show that the L1 error for the mainstream
velocity and the RMSE of the vertical component of ours are both the lowest. In addition, the estimated flow
structure of the laminar boundary layer of our proposed technology is also the finest.

Acknowledgements
This work was supported by the National Natural Science Foundation of China (grant no. 51875228) and
the National Key R&D Program of China (grant no. 2020YFA0405700).

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