#  Background oriented schlieren technique with
fast Fourier demodulation
for measuring large density-gradient fields of fluids

Takaaki Shimazaki Department of Mechanical Systems Engineering, Tokyo
University of Agriculture and Technology, Koganei Campus 6-204, 2-24-16
Nakacho, Koganei, Tokyo, Japan  Sayaka Ichihara  Department of Mechanical
Systems Engineering, Tokyo University of Agriculture and Technology, Koganei
Campus 6-204, 2-24-16 Nakacho, Koganei, Tokyo, Japan  Yoshiyuki Tagawa
Department of Mechanical Systems Engineering, Tokyo University of Agriculture
and Technology, Koganei Campus 6-204, 2-24-16 Nakacho, Koganei, Tokyo, Japan
Institute of Global Innovation Research, Tokyo University of Agriculture and
Technology, Koganei Campus 6-204, 2-24-16 Nakacho, Koganei, Tokyo, Japan

##### Abstract

In order to measure the large density-gradient fields of fluids such as
underwater shock waves, we employed a fast Fourier demodulation called Fast
Checkerboard Demodulation (FCD) method in Background Oriented Schlieren (BOS)
technique. BOS is a simple image-based measurement technique that detects the
apparent displacement (local distortion) of a background image caused by the
density gradient of the fluid in front of the background. The cross-
correlation particle image velocimetry (PIV) method, which is commonly
employed in the BOS technique for detecting the apparent displacement, uses a
random-dot background, whereas FCD uses a periodic pattern as the background
(e.g., checkerboard pattern, lattice grid pattern (grid scale)) to detect the
apparent displacement as the phase change of the pattern in the Fourier space.
In this study, we measured the apparent displacement, which is proportional to
density gradient of fluid, of an underwater shock wave using FCD and PIV. The
results showed that FCD can measure a displacement gradient of up to 2.5 times
larger than that which PIV can measure. Furthermore, we systematically
investigated the measurement limit of FCD-BOS by changing several parameters
of the periodic patterns, such as the grid size. Also, we explored the related
parameters in the Fourier space to understand the limitations. It is worth
noting that FCD-BOS with lattice grid pattern (grid scale) can measure the
apparent displacement as accurately as that with a custom-made pattern,
indicating that large density-gradient fields of fluids can be measured using
a simple setup with a commercially available (inexpensive) pattern.

##  1 Introduction

Background Oriented Schlieren (BOS) technique is a non-contact method for
measuring the density field in a fluid [1, 2, 3, 4, 5]. Compared to
conventional Schlieren imagin and optical interferometry, the measurement
system is quite simple: it consists of only a background, camera, and light
source. BOS technique has been applied to various density fields, such as gas
flows [6, 7, 8, 9, 10], shock waves [4, 5, 11, 12], flames [13, 14, 15, 16,
17], and liquid-gas interfaces [18, 19]. Furthermore, it has also been applied
to various length scales, from micrometers [2, 11, 12, 20] to meters [5, 21].
Moreover, complex 3D density fields have been also measured using multi-
directional BOS imaging [22, 15, 23].

Figure 1 shows the illustration of BOS technique. BOS technique quantifies the
density-gradient field in front of a background by comparing two background
images with and without the density gradient which are captured using a
camera. The background image with density gradient looks distorted as compared
to the background image without the density gradient because the refractive
index of a fluid changes with density. Such distortion is quantified as
”apparent displacement”. It is extremely important for the BOS technique to
accurately measure the apparent displacement between the background images
[24, 25, 26]. A random-dot background is commonly used to obtain apparent
displacement of images through analysis by digital image correlation (DIC),
which is known as particle image velocimetry (PIV) in the fluid dynamics
research community [27, 28, 29, 30] or optical flow [31, 32, 33]. However,
neither PIV nor optical flow is capable of detecting displacements of large
density gradients because they assume parallel displacement of the background
image, while in reality, random dots experience a shape distortion due to
large density gradients as illustrated in Figure 2(a).

Figure 1: Illustration of BOS technique. The black and red lines represent
light paths (optical path lengths) with and without density gradient,
respectively. Figure 2: (a) (Left) A background image of dot pattern without
apparent displacement (Reference image). (Right) A background image with
apparent displacement (Distorted image). Black squares are not only translated
but also deformed, causing problems to the displacement detection by PIV. (b)
Simple situation under a lateral apparent displacements $u_{1}$ and $u_{2}$.
The magnitude of the dot deformation is represented by the displacement
gradient $\partial u/\partial x$.

Let us consider a simple situation in which a square
${\rm\Delta}x\times{\rm\Delta}x$ on a background is distorted in one direction
(Figure 2(b)). Because the magnitudes of the displacements $u_{1}$ on the left
side of the square and $u_{2}$ on the right side of the square are different,
the size of the square after distortion is expressed as
${\rm\Delta}x^{\prime}={\rm\Delta}x+u_{2}-u_{1}$, and the rate of shape
distortion is
$({\rm\Delta}x^{\prime}-{\rm\Delta}x)/{\rm\Delta}x=(u_{1}-u_{2})/{\rm\Delta}x\approx\partial
u/\partial x$. Thus, the magnitude of the distortion depends on the
displacement gradient $\partial u/\partial x$, i.e., the second-order spatial
differential of the density field of the fluid. When the displacement gradient
is too large, PIV cannot properly acquire the apparent displacement [34].
Hence, there is a limitation in the existing PIV-based BOS measurements for
underwater shock waves and flames [35] due to their large displacement
gradients.

In this study, to solve this problem fundamentally, we employed the Fast
Checkerboard Demodulation (FCD) method which was proposed by Wildeman [36],
because unlike PIV, FCD can measure the displacement even when the background
pattern is distorted. FCD uses a background with a two-dimensional periodic
pattern to detect the displacement field from the change of wavenumber in the
Fourier space. By using FCD, Wildeman [36] has successfully detected waves on
a water surface. This study takes advantage on the robustness of FCD in
detecting large apparent displacements. It is the first time that FCD method
is used in a BOS technique. In our measurement system, the background patterns
of horizontal and vertical lines (lattice grid pattern) and of a black-and-
white square pattern (checker pattern) are used. For comparison, a density-
gradient field was measured by both PIV and FCD. Also, the data measured using
hydrophone was used as the reference (correct) data. We measured underwater
shock waves since they are good examples of flow field with large displacement
gradients. We also performed numerical validation by adding the displacement
to a synthetic background image to gain a better understanding on the
experimental values and the measurement limits of FCD-BOS.

This paper is structured as follows. Principles of BOS technique and FCD
method are described in Section 2. The experimental setup and results are
explained in Section 3. We compared the apparent displacement and displacement
gradient obtained using FCD and PIV with those estimated from the pressure
measured by the hydrophone. Finally, we also analyzed the synthetic image
under ideal conditions for discussion on the measurement limit of FCD-BOS in
Section 4.

##  2 Principle

###  2.1 Background Oriented Schlieren (BOS) technique

BOS is a technique which visualizes and quantifies the density gradient by
detecting variations in the refractive index of the fluid. As shown by the
illustration in Figure 1, the measurement target is placed between a camera
and a background where the camera is focused onto the background to capture
images with and without density gradients. FCD or PIV is then used to
determine the apparent displacement by comparing both images. The apparent
displacement in $y$-direction, $v$, (Figure 1) can be related to the gradient
of the refractive index as shown in the following equation [5] :

$v=\frac{Z_{D}}{n_{0}}\int\frac{\partial n}{\partial y}dz,$ (1)

where $Z_{D}$ is the distance from the background to the center of the density
gradient, $n$ is the refractive index of the measurement target, and $n_{0}$
is the refractive index of the ambient fluid. The relationship between the
refractive index $n$ and the fluid density $\rho$ is given by the Gladstone-
Dale equation:

$n=K\rho+1,$ (2)

where $K$ is the Gladstone-Dale constant, which is $3.34\times 10^{-4}$
$\rm{m^{3}/kg}$ for water [5]. Substituting Eq. 2 into Eq. 1 leads to

$v=\frac{KZ_{D}}{K\rho_{0}+1}\int\frac{\partial\rho}{\partial y}dz,$ (3)

where $\rho_{0}$ is the density of the ambient fluid. Similarly, the apparent
displacement in $x$-direction, $u$, is

$u=\frac{KZ_{D}}{K\rho_{0}+1}\int\frac{\partial\rho}{\partial x}dz.$ (4)

In this study, the apparent displacement measured by the BOS technique and
that estimated from direct measurement of pressure using a hydrophone are
compared to validate the ability of FCD-BOS in detecting apparent
displacement. The procedure for estimating the apparent displacement from the
pressure measured by the hydrophone is explained in $\S$ 2.3

###  2.2 Fast Fourier demodulation of a periodic background

Figure 3: (a) The checkered pattern with tiles of 20
$\mathrm{\SIUnitSymbolMicro m}$ $\times$ 20 $\mathrm{\SIUnitSymbolMicro m}$.
(b) The lattice grid pattern with black lines of 5 $\mathrm{\SIUnitSymbolMicro
m}$ wide and white regions of 15 $\mathrm{\SIUnitSymbolMicro m}$ $\times$ 15
$\mathrm{\SIUnitSymbolMicro m}$. A distorted image $I$ (1500 $\times$ 750
pixels) was created by displacing the reference image $I_{0}$. $P_{0}$ and $P$
are the power spectra in the Fourier space ($k$-space) of $I_{0}$ and $I$,
respectively. $\bm{k}_{1}$ and $\bm{k}_{2}$ are the carrier peaks. The radius
of a circle in $P_{0}$ is $k_{rad}$. $k_{cx}$ is the norm of carrier peak
$k_{c}$ along $x$-axis. The synthetic displacement (true) and the displacement
detected by the FCD are shown.

A spatial Fourier demodulation technique of a periodic background, which is
known as Fast Checkerboard Demodulation (FCD) method and was proposed by
Wildeman [36], allows the quantitative extraction of the apparent displacement
field when a periodic checkered background is used. By photographing a
background with a periodic pattern (a lattice grid pattern or a checkered
pattern in this study) and performing Fourier transform, the apparent
displacement on the background is obtained as the phase change. Reference
image $I_{0}(\bm{\xi})$ is expressed as a general 2D periodic pattern as

$I_{0}({\bm{\xi}})=\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}a_{mn}e^{i(m{\bm{k}_{1}}+n{\bm{k}_{2}})\cdot{\bm{\xi}}},$
(5)

where $\bm{\xi}$ denotes the pixel coordinates ($x$, $y$) in the image,
$\bm{k}_{1}$ and $\bm{k}_{2}$ denote the orthogonal wavenumber vectors
extracted from the reference image (called carrier peaks), and $a_{mn}$
denotes the Fourier coefficient. The distorted image $I(\bm{\xi})$ is the
result when the reference image $I_{0}$ is modulated by the apparent
displacement ${\bm{u}}({\bm{\xi}})$, as follows:

$I(\bm{\xi})=I_{0}({\bm{\xi}}-{\bm{u}}({\bm{\xi}})).$ (6)

By performing Fourier transforming on $I_{0}$ and $I$, we obtain the
respective power spectras $P_{0}$ and $P$ with peaks at
$m\bm{k}_{1}+n\bm{k}_{2}$, which is expressed using integers $m,n$, as shown
in Figure 3. By extracting the signals within a radius of $k_{rad}$ centered
from $\bm{k}_{1}$ and $\bm{k}_{2}$ in $P_{0}$ and $P$, respectively, and
performing the inverse Fourier transform, we obtain

$\displaystyle g_{1}({\bm{\xi}})=a_{c1}e^{i{\bm{k}_{1}}\cdot{\bm{\xi}}},$ (7)
$\displaystyle g_{2}({\bm{\xi}})=a_{c2}e^{i{\bm{k}_{2}}\cdot{\bm{\xi}}},$ (8)
$\displaystyle
g_{1}^{\prime}({\bm{\xi}})=a_{c1}^{\prime}e^{i{\bm{k}_{1}}\cdot({\bm{\xi}-\bm{u}(\bm{\xi})})},$
(9) $\displaystyle
g_{2}^{\prime}({\bm{\xi}})=a_{c2}^{\prime}e^{i{\bm{k}_{2}}\cdot({\bm{\xi}-\bm{u}(\bm{\xi})})},$
(10)

where $a_{c1},a_{c2},a_{c1}^{\prime}$, and $\ a_{c2}^{\prime}$ are
coefficients derived from $a_{mn}$. The phase fields
$\phi_{1}(\bm{\xi}),\phi_{2}(\bm{\xi})$ extracted from two linearly
independent carrier peaks are

$\displaystyle\phi_{1}(\bm{\xi})={\rm
Im}(\ln(g_{1}g_{1}^{\prime*}))={\bm{k}_{1}}\cdot{\bm{u}}({\bm{\xi}})$ (11)
$\displaystyle\phi_{2}(\bm{\xi})={\rm
Im}(\ln(g_{2}g_{2}^{\prime*}))={\bm{k}_{2}}\cdot{\bm{u}}({\bm{\xi}})$ (12)

or

$\left(\begin{array}[]{ccc}\phi_{1}\\\
\phi_{2}\end{array}\right)=\left(\begin{array}[]{ccc}k_{1x}&k_{1y}\\\
k_{2x}&k_{2y}\end{array}\right)\left(\begin{array}[]{ccc}u\\\
v\end{array}\right),$ (13)

where $k_{1x}$ is the $x$-component of vector $\bm{k}_{1}$, $k_{1y}$ is the
$y$-component of vector $\bm{k}_{1}$, $k_{2x}$ is the $x$-component of vector
$\bm{k}_{2}$, $k_{2y}$ is the $y$-component of vector $\bm{k}_{2}$, and
$\bm{u}(\bm{\xi})=(u,v)$. By solving Eq. 13, we obtain the displacement $u,v$
directly as

$\left(\begin{array}[]{ccc}u\\\
v\end{array}\right)=\dfrac{1}{k_{1x}k_{2y}-k_{1y}k_{2x}}\left(\begin{array}[]{ccc}k_{2y}&-k_{1y}\\\
-k_{2x}&k_{1x}\end{array}\right)\left(\begin{array}[]{ccc}\phi_{1}\\\
\phi_{2}\end{array}\right).$ (14)

To avoid aliasing and phase wrapping, following three criteria must be
satisfied for the displacement measurement [36]:

$\displaystyle k_{s}<k_{rad},$ (15) $\displaystyle
k_{c}|\bm{u}^{\prime}|<k_{rad},$ (16) $\displaystyle k_{c}|\bm{u}|<\pi,$ (17)

where $k_{s}$ is the wavenumber of the displacement, $k_{c}$ (
${k_{c}}=|\bm{k_{1}}|=|\bm{k_{2}}|$ ) is the norm of the carrier peak of the
reference image, and $\bm{u}^{\prime}$ is the displacement gradient. In this
study, we focus on the value on $x$-axis. For this reason, although Wildeman
[36] uses $k_{c}$ in the criteria, we utilized the norm of carrier peak along
$x$-axis, i.e., $k_{cx}$. Thus, the criteria correspond to Eq. 16 and Eq. 17
in this study are

$\displaystyle k_{cx}|\bm{u}^{\prime}|<k_{rad},$ (18) $\displaystyle
k_{cx}|\bm{u}|<\pi,$ (19)

respectively. Eq. 15 is a criterion that varies with the displacement and the
background pattern, while Eq. 18 and Eq. 19 are criteria which vary only with
the background pattern. For example, in the $k$-space at $P_{0}$ in Figure 3,
$k_{cx}$ = $k_{1x}$ = $k_{rad}$ for the checkered pattern and $k_{cx}$ =
$k_{2x}$ = 2$k_{rad}$ for the lattice grid pattern. If the criterion in Eq.15
is not satisfied, aliasing artifacts will occur; if the criterion in Eq.18 is
not satisfied, overlap will occur; and if the criterion in Eq.19 is not
satisfied, phase wrapping will occur [36].

###  2.3 Estimation of the apparent displacement from the pressure measured by
the hydrophone

The apparent displacement was estimated from the pressure measured by the
hydrophone for comparison with FCD and PIV. First, the density $\rho$ was
obtained from the pressure using Tait equation:

$\rho=\left(\frac{p+B}{p_{0}+B}\right)^{\frac{1}{\alpha}}\rho_{0},$ (20)

where the hydrostatic density $\rho_{0}$ is 998 kg/m3, $B$ and $\alpha$ are
314 MPa and 7.15, respectively. Then, the refractive index gradient $\partial
n/\partial r$ was calculated by differentiating the refractive index $n$,
which was obtained from Eq. 2 using the density $\rho$. Because the shape and
pressure of a laser-induced underwater shock wave are axisymmetric in the
direction of the laser beam [37, 38], the axisymmetric field of the refractive
index gradient $\partial n/\partial x$, as shown in Figure 4, was
reconstructed. Finally based on Eq. 1, the apparent displacement $u$ was
obtained by integrating $\partial n/\partial r$ along $z$-axis as shown in
Fig. 4.

Figure 4: Procedure of converting the pressure of underwater shock wave to the
“apparent displacement” $u$. (a) Schematic of the BOS and hydrophone
measurements of underwater shock waves in this study. The shock wave is
axisymmetric about the direction of the laser beam (dotted straight line). (b)
Constructed refractive index gradient field $\partial n/\partial x$. To obtain
$u$, integration of the $\partial n/\partial x$ field was performed toward the
$z$-direction in the pink area of (a).

##  3 Experiments

###  3.1 Experimental set-up

Figure 5: (a) Experimental setup. (b) Background patterns. In the case of the
random dot pattern, the dot size is 8 µm $\times$ 8 µm. In the case of the
checkered pattern, the area of each tile is 20 $\mathrm{\SIUnitSymbolMicro m}$
$\times$ 20 $\mathrm{\SIUnitSymbolMicro m}$ or 40 $\mathrm{\SIUnitSymbolMicro
m}$ $\times$ 40 $\mathrm{\SIUnitSymbolMicro m}$. In the case of the lattice
grid pattern, the width of the black line is 5 $\mathrm{\SIUnitSymbolMicro
m}$, and the area of the white region is 15 $\mathrm{\SIUnitSymbolMicro m}$
$\times$ 15 $\mathrm{\SIUnitSymbolMicro m}$.

A schematic of the experimental setup is shown in Figure 5. Three types of
background patterns were used: random dots, checkered, and lattice grid. To
generate the underwater shock wave, a pulsed laser (Nd: YAG laser Nano S PIV,
Litron Lasers Ltd., wavelength: 532 nm, pulse width: 6 ns) was focused through
an objective lens (SLMPLN 20$\times$, Olympus, magnification: 20X, NA value:
0.25) into a point inside a container (100 mm $\times$ 100 mm $\times$ 100 mm)
filled with ultrapure water. The background was placed behind the shock wave
(distance $Z_{D}$ about 0.7 - 2.5 mm). A high-resolution CMOS camera (EOS 80D,
Canon, resolution: 4,000 $\times$ 6,000 pixels, spatial resolution: 0.6 - 1.5
$\mathrm{\SIUnitSymbolMicro m}$/pixel in this set-up) was used to capture the
background images with and without the underwater shock wave. The exposure
time of the camera was set at approximately 0.5 s, and the timing of filming
was adjusted by changing the timing of the light source. A delay generator
(Model 575, BNC) was utilized to synchronize the pulsed laser and laser light
sources (SI-LUX 640, Specialized Imaging Ltd., wavelength: 640 nm, pulse
width: 10-20 ns). The shock wave pressure was measured using a hydrophone
(Muller-Platte Needle Probe, Muller Instruments, Germany, effective diameter
$<$ 0.25 mm).

To determine the apparent displacement on the background, an open-source PIV
code (PIVlab, MATLAB, Thielicke and Stamhuis [39]) and an open-source FCD code
[36] were used. A dot occupies about 5.5 - 11.3 pixel on the image. The PIV
setting has a 50% overlap with the fast Fourier transform (FFT) multipass
interrogation (from 64 to 8 pixels).

To compare with the displacements estimated by the hydrophone measurement, the
displacements obtained by the BOS in the range of $\pm$0.125 mm in the
$y$-direction from the center-line of the hydrophone were averaged along the
$x$-axis (see Figure 6).

Figure 6: A typical background image with a shock wave (left). $x$-components
of the displacement field detected by FCD (right). The analyzed area was
determined by considering the effective diameter of the hydrophone.

Note that the positions of the shock wave measured by the BOS and hydrophone
in $x$-direction are slightly different, as shown in Figure 6. The pressure is
inversely proportional to the distance of the shock wave propagation in this
experimental range [40]. Thus, for direct comparison with BOS results, the
shock wave pressure measured by hydrophone was corrected using following
equation:

$p_{image}=\frac{r_{hyd}}{r_{image}}p_{hyd},$ (21)

where $p_{image}$ is the pressure at the time of imaging, $p_{hyd}$ is the
pressure measured by the hydrophone, $r_{image}$ is the radius of the shock
wave at the time of imaging, and $r_{hyd}$ is the distance between the center
of the shock wave and the tip of the hydrophone.

In Figure 7, the results of the FCD and the displacement calculated from the
pressure measured by hydrophone are shown.

Figure 7: The apparent displacement in the $x$-direction and shock wave
pressure as a function of the distance from the center of the shock wave.

In following sections, we focus on the largest apparent displacement at the
distance near 2.7 mm (first peak). Note that a second peak (distance around
1.2 mm) appears in the estimated displacement because a part of the shock wave
was reflected on the background. The second peak in the image analysis does
not appear because the integral of $\partial n/\partial x$ is very small.

###  3.2 Experimental results

Figure 8: The maximum displacement estimated from the pressure measured with
hydrophone $u_{max}$(hydrophone) and the maximum displacement detected by
PIV/FCD $u_{max}$(PIV/FCD). Error bars indicate the standard deviations.
Figure 9: The error which is the difference between the maximum displacement
estimated from the pressure $u_{max}$(hydrophone) and the maximum displacement
detected by PIV/FCD $u_{max}$(PIV/FCD) against the maximum displacement
gradient of the displacement estimated from the pressure measured by
hydrophone $(\partial u/\partial x)_{max}$. Error bars indicate the standard
deviations.

The maximum displacement detected by PIV or FCD, $u_{max}$(PIV/FCD) and
maximum displacements calculated from the pressure, $u_{max}$(hydrophone), are
shown in Figure 8. For $u_{max}$(hydrophone) $>$ 4 $\mathrm{\SIUnitSymbolMicro
m}$, the maximum displacement detected by PIV is about 4
$\mathrm{\SIUnitSymbolMicro m}$, which is significantly lower than that
estimated from the measurements of the hydrophone. Such underestimation is due
to a significant distortion of the shape of the dots when
$u_{max}$(hydrophone) $>$ 4 $\mathrm{\SIUnitSymbolMicro m}$, making it
difficult for PIV to accurately detect the displacement. In contrast, the
displacements obtained by the FCD reasonably agrees with the estimated
displacement.

The error of the PIV and FCD methods is defined as the difference between the
maximum displacement detected using each method and that estimated from the
pressure measured using hydrophone. In Figure 9, the errors are plotted
against the maximum displacement gradient $(\partial u/\partial x)_{max}$,
which represents the magnitude of the distortion. As shown in the figure, PIV
underestimates the displacement when the maximum displacement gradient is
greater than 0.20. This is because the displacement detected by the cross-
correlation method does not match the original displacement owing to the large
distortion of the dots as discussed in Section 1. On the other hand, FCD can
reasonably detect the maximum displacement for the displacement gradient
$\partial u/\partial x<0.5$, which is approximately 2.5 times larger than
0.20, the measurement range of PIV-BOS. Among the FCD methods, the error of 20
$\mathrm{\SIUnitSymbolMicro m}$ checkers is less than the error of 40
$\mathrm{\SIUnitSymbolMicro m}$ checkers and 20 $\mathrm{\SIUnitSymbolMicro
m}$ lattice grid. Besides, it is also observed that 20
$\mathrm{\SIUnitSymbolMicro m}$ lattice grid, the error shifts from
overestimation to underestimation at the displacement gradient between 0.3 and
0.4. To analyze the reasons for such trends, we utilized synthetic images to
investigate the applicable range for displacement detection of FCD with
various background patterns. Synthetic images were used because the images
captured from the experiment might be affected by random errors such as image
blurring and slight misalignment of the experimental setup. The analysis are
discussed in Chapter 4.

##  4 Numerical approach: measurement limit of FCD-BOS

###  4.1 Numerical condition

In order to determine the applicable range of displacement detection of FCD,
we used synthetic images (image size: 1500 $\times$ 750 pixels, spatial
resolution: 1 $\mathrm{\SIUnitSymbolMicro m}$/pixel) that imitate three types
of background patterns: random dots, lattice grid, and checkered. The
synthetic images were modulated with synthesized apparent displacement $u(r)$,
i.e., “true displacement”, based on the Gaussian distribution as follows:

$u(r)=\left\\{\begin{array}[]{ll}u_{max}\exp\left(-\dfrac{(r-r_{c})^{2}}{\beta}\right)&\quad(r\geq
r_{c})\vspace{1em}\\\
u_{max}\exp\left(-\dfrac{(r-r_{c})^{2}}{9\beta}\right)&\quad(r<r_{c}),\end{array}\right.$
(22)

where $u_{max}$ varies between 1 - 11 $\mathrm{\SIUnitSymbolMicro m}$ at an
interval of 1 $\mathrm{\SIUnitSymbolMicro m}$, $r$ is the distance from the
origin indicated in Figure 10, ${r_{c}}$ is set to 400
$\mathrm{\SIUnitSymbolMicro m}$, and $\beta$ is set to $400\times 10^{-12}$
m2. This $u(r)$ resembles the displacement of a shock wave. For each interval
of $u_{max}$, the origin of $u(r)$ was issued to 20 different positions
relative to the background pattern. These 20 data were used to obtain the
averaged value and its error range.

Figure 10: Synthetic images (1500 $\times$ 750 pixels) and the $x$-direction
displacement $u$ (maximum displacement is 5 $\mathrm{\SIUnitSymbolMicro m}$ (=
5 pixels) in this figure). The radius of the circle in each $P_{0}$ is
$k_{rad}$. Symbols are explained in the caption of Figure 3.

###  4.2 Numerical results

####  4.2.1 Size and orientation of the background patterns

Figure 11: The maximum displacement gradient of the displacement given
modulated to the reference image $(\partial u/\partial x)_{max}$ and the error
which is the difference between the maximum displacement given modulated to
the reference image $u_{max}$(estimation) and the maximum displacement
detected by PIV/FCD $u_{max}$(PIV/FCD). Error bars indicate the standard
deviations estimated by using 20 sets of images. Figure 12: (a) Power spectra
$P_{d}$, which is obtained by the Fourier transformation of the true
displacement $u(r)$, for $u_{max}=$ 1 $\mathrm{\SIUnitSymbolMicro m}$. The
subfigure on the upper right shows the enlarged view. The radius of the
circle, $k_{rad}$, varies depending on the pattern. (b) The ratio of the sum
of $P_{d}$ contained within $k_{rad}$ to the total sum of $P_{d}$. Figure 13:
The measurement error of checkered patterns of different sizes and
orientations against the maximum displacement gradient. Error bars indicate
the standard deviations. Figure 14: The measurement error of lattice grid
patterns of different sizes and orientations against the maximum displacement
gradient. Error bars indicate the standard deviations. Figure 15: The
displacement fields and the wavenumber intensity of synthetic images modulated
with true displacement for $u_{max}=$ 5 $\mathrm{\SIUnitSymbolMicro m}$
measured using 12 $\mathrm{\SIUnitSymbolMicro m}$ lattice grid patterns of
different line thickness: (a) 1 $\mathrm{\SIUnitSymbolMicro m}$/11
$\mathrm{\SIUnitSymbolMicro m}$, (b) 6 $\mathrm{\SIUnitSymbolMicro m}$/6
$\mathrm{\SIUnitSymbolMicro m}$,(c) 8 $\mathrm{\SIUnitSymbolMicro m}$/4
$\mathrm{\SIUnitSymbolMicro m}$, and (d) 10 $\mathrm{\SIUnitSymbolMicro m}$/2
$\mathrm{\SIUnitSymbolMicro m}$. The lattice grid patterns are shown in the
insets.

Figure 10 shows the typical result for $u_{max}=$ 5
$\mathrm{\SIUnitSymbolMicro m}$. Overall, the results show similar trends as
the experimental results. The displacement detected by PIV is noisy, while
those detected by the FCD methods are similar to the true displacement.
However, the measurement accuracy of FCD depends on the background pattern
where the displacement fields of 20 $\mathrm{\SIUnitSymbolMicro m}$ checkers,
20 $\mathrm{\SIUnitSymbolMicro m}$ ($0^{\circ}$) lattice grid, and 20
$\mathrm{\SIUnitSymbolMicro m}$ ($45^{\circ}$) lattice grid are almost
identical to the true data, but the displacement field of 40
$\mathrm{\SIUnitSymbolMicro m}$ checkers is partially different from true
data. The reason for this is discussed in the later part of this section. For
the lattice grid patterns without rotation ($0^{\circ}$) and with a rotation
of $45^{\circ}$, $k_{cx}$ are 0.32 pixel-1 and 0.23 pixel-1, respectively
although $k_{rad}$ are 0.16 pixel-1 for both patterns. Nevertheless, since the
lattice grid patterns of both $0^{\circ}$ and $45^{\circ}$ satisfy Eq. 19,
there is no notable difference between their displacement fields.

Relations between the maximum displacement gradient and error are shown in
Figure 11. PIV underestimates the apparent displacement as the displacement
gradient increases, similar to the experimental results shown in Figure 9. On
the other hand, the measurement error of FCD (20 $\mathrm{\SIUnitSymbolMicro
m}$ checkers) is sufficiently low for $(\partial u/\partial x)_{max}<0.5$
while that of FCD (20 $\mathrm{\SIUnitSymbolMicro m}$ lattice grid) has non-
trivial values for $(\partial u/\partial x)_{max}>0.45$. Since the value of
$k_{cx}$ for the lattice grid pattern (20 $\mathrm{\SIUnitSymbolMicro m}$) is
0.32 pixel-1, based on Eq. 19, phase wrapping occurs when the displacement is
larger than 9.8 $\mathrm{\SIUnitSymbolMicro m}$, which corresponds to a
displacement gradient of 0.42. Thus, FCD (20 $\mathrm{\SIUnitSymbolMicro m}$
lattice grid) underestimates the displacement when $(\partial u/\partial
x)_{max}$ $>$ 0.45. The FCD (40 $\mathrm{\SIUnitSymbolMicro m}$ checkers)
gradually underestimates the displacement as the displacement gradient
increases because of the resulting aliasing artifacts as described by Eq. 15.

To investigate further how the criterion in Eq.15 offsets the results in our
condition, the power spectra $P_{d}$, which is obtained by the Fourier
transformation of the true displacement $u(r)$, is analyzed. Since the
analysis results are similar for every displacement between $u_{max}$= 1 and
11 $\mathrm{\SIUnitSymbolMicro m}$, here only the results for the synthetic
image of $u_{max}$= 1 $\mathrm{\SIUnitSymbolMicro m}$ are discussed. Figure
12(a) shows the $P_{d}$ for $u_{max}$= 1 $\mathrm{\SIUnitSymbolMicro m}$,
where the red dashed circle in the zoomed-in figure shows the boundary of
$k_{rad}=$ 0.26 pixel-1. Since Eq.15 shows that the measurement accuracy
correlates to a ratio of the sum of $P_{d}$ contained within $k_{rad}$ to the
total sum of $P_{d}$, the ratio is plotted as a function of $k_{rad}$ in
Figure 12(b). In the same figure, dashed vertical lines are plotted for the
$k_{rad}$ values of the patterns of different sizes, i.e., $k_{rad}=$ 0.08
pixel-1 (40 $\mathrm{\SIUnitSymbolMicro m}$ checkers and lattice grid), 0.16
pixel-1 (20 $\mathrm{\SIUnitSymbolMicro m}$ checkers and lattice grid), and
0.26 pixel-1 (12 $\mathrm{\SIUnitSymbolMicro m}$ checkers and lattice grid).
The ratios for $k_{rad}=$ 0.08 pixel-1, 0.16 pixel-1, and 0.26 pixel-1 are
65%, 87%, and 95%, respectively. Since ratio for 40
$\mathrm{\SIUnitSymbolMicro m}$ checkers is lower that that for 20
$\mathrm{\SIUnitSymbolMicro m}$ checkers, the resulting aliasing artifacts in
40 $\mathrm{\SIUnitSymbolMicro m}$ checkers have caused more noise in the
detected displacements than those of 20 $\mathrm{\SIUnitSymbolMicro m}$
checkers, as shown in Figure 10. Besides, they have also caused the
underestimation of the displacement by 40 $\mathrm{\SIUnitSymbolMicro m}$ as
shown in Figure 11. Such results show that the measurement accuracy can be
further improved when the ratio is increased through increasing the values of
$k_{rad}$ using the patterns of smaller size.

Here, we focus on $k_{rad}=$ 0.26 pixel-1 (12 $\mathrm{\SIUnitSymbolMicro m}$
checkers and lattice grid) which correspond to a ratio of 95%. For that, the
measurement accuracy of 12 $\mathrm{\SIUnitSymbolMicro m}$ checkers and
lattice grid were evaluated based on the respective measurement errors shown
in Figure 13 and Figure 14.

In Figure 13, when the maximum displacement gradient is less than 0.35, 12
$\mathrm{\SIUnitSymbolMicro m}$ checkers has the smallest error. Similarly, in
Figure 14, when the maximum value of the gradient is less than 0.25, 12
$\mathrm{\SIUnitSymbolMicro m}$ lattice grid has the smallest error. However,
in Figure 13, the errors of 12 $\mathrm{\SIUnitSymbolMicro m}$ and 12
$\mathrm{\SIUnitSymbolMicro m}$ 45∘ checkers deviate from one another when
$(\partial u/\partial x)_{max}>$ 0.35. Similarly, in Figure 14, the errors of
12 $\mathrm{\SIUnitSymbolMicro m}$ and 12 $\mathrm{\SIUnitSymbolMicro m}$ 45∘
lattice grids also deviate from one another when $(\partial u/\partial
x)_{max}>$ 0.25. The reason for these deviations can be understood based on
Eq. 19. For checkered pattern of 12 $\mathrm{\SIUnitSymbolMicro m}$
$45^{\circ}$ ($k_{cx}$ = 0.37 pixel-1), the maximum measurable displacement is
8.5 $\mathrm{\SIUnitSymbolMicro m}$, which corresponds to $(\partial
u/\partial x)_{max}$= 0.37, while for that of 12 $\mathrm{\SIUnitSymbolMicro
m}$ $0^{\circ}$ ($k_{cx}$ = 0.26 pixel-1), the maximum measurable displacement
is 12 $\mathrm{\SIUnitSymbolMicro m}$, which corresponds to $(\partial
u/\partial x)_{max}$= 0.51. Thus, as shown in Figure 13, the measurement
errors of 12 $\mathrm{\SIUnitSymbolMicro m}$ 45∘ checkers show non-trivial
values for $(\partial u/\partial x)_{max}>$ 0.35, owing to phase wrapping.

As for the results for lattice grid patterns shown in Figure 14, phase
wrapping occurs for 12 $\mathrm{\SIUnitSymbolMicro m}$ $0^{\circ}$ and 12
$\mathrm{\SIUnitSymbolMicro m}$ $45^{\circ}$ when $(\partial u/\partial
x)_{max}>0.26$ and $0.34$, respectively. Based on Eq. 19, for the lattice grid
pattern of 12 $\mathrm{\SIUnitSymbolMicro m}$ $45^{\circ}$ ($k_{cx}$ = 0.37
pixel-1), the maximum measurable displacement is 8.5
$\mathrm{\SIUnitSymbolMicro m}$, which corresponds to $(\partial u/\partial
x)_{max}$= 0.37, while for that of 12 $\mathrm{\SIUnitSymbolMicro m}$
$0^{\circ}$ ($k_{cx}$ = 0.52 pixel-1), the maximum measurable displacement is
6.0 $\mathrm{\SIUnitSymbolMicro m}$, which corresponds to $(\partial
u/\partial x)_{max}$= 0.26. Therefore, phase wrapping started occurring at a
smaller value of $(\partial u/\partial x)_{max}$ for 12
$\mathrm{\SIUnitSymbolMicro m}$ of $0^{\circ}$. As for the lattice grids of 12
$\mathrm{\SIUnitSymbolMicro m}$ (2$\mathrm{\SIUnitSymbolMicro
m}$/10$\mathrm{\SIUnitSymbolMicro m}$) $45^{\circ}$ and 12
$\mathrm{\SIUnitSymbolMicro m}$ (6$\mathrm{\SIUnitSymbolMicro
m}$/6$\mathrm{\SIUnitSymbolMicro m}$) $45^{\circ}$, phase wrapping occurred at
the same value of $(\partial u/\partial x)_{max}$ because $k_{cx}$ is
identical.

In summary, the limiting factors for FCD can be understood from the parameters
in Fourier space as shown in Eq. 15, Eq. 18, and Eq. 19. It is worth noting
from a practical view point that the results showed that FCD of lattice grid
patterns can measure displacement as accurate as that of checkered patterns,
although it is slightly more restricted.

####  4.2.2 Line thickness of lattice grid pattern

For practical purpose, we investigated the influence of line thickness of the
lattice grid pattern. The results of the different line thicknesses of 12
$\mathrm{\SIUnitSymbolMicro m}$ lattice grid patterns for the maximum
displacement $u_{max}$ = 5 $\mathrm{\SIUnitSymbolMicro m}$ are shown in Figure
15. In these cases, all FCD criteria (Eq. 15, Eq. 18, and Eq. 19) are
satisfied. While $k_{cx}$ and $k_{rad}$ are the same for all cases, the
wavenumber intensity $P$ of the synthetic images depends on the line
thickness. If the lines are significantly thicker than the white regions, as
shown in Figure 15(d), the $u$ field is different from the true displacement
described in Eq. 22. Since $u$ fields in Figure 15(a)(b) are identical to true
displacements, it is recommended to use lattice grid patterns with lines that
are thinner than the white regions for FCD-BOS measurements.

##  5 Conclusion

We applied Fast Checkerboard Demodulation (FCD) method for Background Oriented
Schlieren (BOS) technique. In general, BOS tecnique uses a random-dot
background to detect the displacement. However, to measure large density-
gradient fluid fields such as shock waves, the dotted pattern distorts causing
the displacement detection by cross-correlation PIV method fails to work
properly. From the experimental results, FCD is superior to PIV in terms of
detecting large displacement gradients and the maximum measurable displacement
gradient of FCD is 2.5 times larger than that of PIV. Through analysis using
synthetic data, the limiting factors of the FCD can be understood from the
parameters in Fourier space as shown in Eq.15, Eq.18, and Eq.19. The results
of the analysis shows that line thickness influences the limiting factors of
the FCD of lattice grid patterns. On a final note, FCD-BOS can be easily
constructed and customized for various thermal fluids since it just consists
of a commercially available grid scale, still camera and light source. Based
on measurement limitation of FCD-BOS reported in this paper, users of FCD-BOS
could select proper background with suitable parameters such as the line width
of lattice grid patterns and region size of checker patterns.

## Acknowledgement

This work was supported by the Kawai Foundation for Sound Technology $\&$
Music, KAKENHI Grant-in-Aid for Scientific Research (A), Grant Number
20H00222/20H00223, and JST, PRESTO Grant Number JPMJPR21O5, Japan. The authors
would like to thank Masaharu Kameda (Professor, Tokyo University of
Agriculture and Technology) and Jingzu Yee (PhD candidate, Tokyo University of
Agriculture and Technology) for their valuable discussions and suggestions.

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