KONA Powder and Particle Journal No. 41 (2024) 221–228/https://doi.org/10.14356/kona.2024002

Original Research Paper

Particle Size Measurement Using a Phase Retrieval Holography
System with a GPU-Equipped SBC †Yohsuke Tanaka 1* and Dai Nakai 2
1 Faculty of Mechanical Engineering, Kyoto Institute of Technology, Japan

2 Graduate School of Mechanical Engineering, Kyoto Institute of Technology, Japan

We have developed a phase retrieval holography system using a single-board
computer (SBC) with a graphics processing unit (GPU) for particle size measurement.
The system comprises two cameras connected to the SBC with a GPU (Jetson
NanoTM, NVIDIA®), a diode-pumped solid-state green laser, and a beam splitter.
The GPU enables us to reconstruct holograms in real-time and measure particle
size. The system can record the shapes and positions of particles falling in a static
flow in a three-dimensional volume as two holograms generating an interference
pattern. Two holograms solve the twin image problem that arises because of the lackof phase information using phase retrieval holography. We also present the requirement of this system for experimentally recording andnumerically reconstructing holograms of particles. Finally, we compare the particle size distribution obtained by the system to that ofconventional two-dimensional image measurement.
Keywords: particle size measurement, phase retrieval holography, single-board computer, graphics processing unit

1. Introduction

Digital transformation (Stolterman and Fors, 2004) has
changed society as a whole, as evidenced by the spread oftelework and other online activities due to the COVID-19
pandemic. The use of Internet of Things (IoT) (Mattern and
Floerkemeier, 2010) has also been advancing. In particular,
Malik et al. (2021) reported that the use of Industrial IoT
in industry has brought significant improvements and profitability in process visualization using the Raspberry Pi,
an inexpensive single-board computers (SBC). In powder
production, big data aggregated from each process are used
for trouble detection, quality control, and determination of
production conditions through artificial intelligence technology (Fujita et al., 2020).
As the processing speed of computers has improved, theamount of information obtained by optical measurement,
which is used in powder production, has increased frompoint measurement with phase Doppler anemometry (Durstet al., 1981) to image measurement (Lichti and Bart, 2018).
In addition, holography, invented by Gabor (1948), hasbeen used for measuring particles in three-dimensional
space. Gabor holography can be realized easily with a lightsource and a complementary metal oxide semiconductor(CMOS) camera and is insensitive to vibration because thereference and object waves are on the same optical axis.
†*

Received 20 August 2022; Accepted 19 October 2022
J-STAGE Advance published online 28 January 2023
Corresponding author: Yohsuke Tanaka;
Add: Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan
E-mail: tyohsuke@kit.ac.jp
TEL: +81-75-724-7355 FAX: +81-75-724-7355

Gabor holography is a three-dimensional microparticle
measurement with a deep focal depth, involving size
(Nayak et al., 2021), position (Memmolo et al., 2015), andvelocity (Katz and Sheng, 2010). However, only the lightintensity of the complex amplitude is recorded by the camera, which causes the twin image problem that degrades
the contrast of reconstructed particles (Latychevskaia andFink, 2007). To solve this problem, phase retrieval holography using two or more cameras has been proposed (Liuand Scott, 1987; Zhang et al., 2003). The present authors
have used this method to measure microparticles (Tanaka
et al., 2016), microbubbles (Kubonishi et al., 2018), andmicrodroplets (Nakatani et al., 2019; Tanaka et al., 2019b).
However, practical usage of Gabor holography and phaseretrieval holography is limited by the huge execution timefor volumetric reconstruction.
Parallel processing by a graphics processing unit (GPU)accelerates holography processing (Tian et al., 2010) to bemuch faster than that by a central processing unit (CPU).
In particle measurements, the CPU to GPU execution time
ratio has been shown to be higher than 100 times (Tanaka
et al., 2019a). Since 2014, NVIDIA® has released GPUequipped SBCs for AI and IoT applications, and in 2019 it
released the Jetson NanoTM as an inexpensive SBC.
This paper proposes particle size measurement using aphase retrieval holography system with a GPU-equipped
SBC with theoretical and fabrication considerations. We
also confirm the effectiveness of the system by comparing
particle size distributions measured by it and conventional
two-dimensional image measurement.

Copyright © 2024 The Authors. Published by Hosokawa Powder Technology Foundation. This is an open accessarticle under the CC BY license (https://creativecommons.org/licenses/by/4.0/).

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Fig. 1

Original Research Paper

Recording hologram in particle measurement.

where dp is the diameter of the particle, xpi and ypi are its
center position.
Because this aperture has circular symmetry, the intensity distribution is derived by performing the integration2 πd r  πd p r  
J1  p 2  2 J1 222 πr   λz p i   πd p  πd p λz p i  
I z pi  1 
sin 




πd p r
λz p i λz p i  πd p r
 4 λz p i  
λz p iλz p i2 πd r pFig. 2 Hologram intensity of a particle from Eqn. (3) for zp = 50 mm, J1 πr 2   λz p i πd p 2
dp = 62 μm, and λ = 532 nm.

I z pi  1 sin 
λz p i λz p i 

πd(3)
pr
λz p i2. Recording and reconstruction ofholograms in particle measurement πd p r   λz p i  
πd p r
λz p i2  2 J1  πd p 2   4 λz p i  



(3)

First, we focus on the principles of recording and recon(3) space using Gaborstructing particles in three-dimensional
holography. In particular, we discuss hologram recording
conditions, the twin image problem, and particle elongationin the optical direction. Moreover, we explain the principleof phase retrieval holography used in the proposed system
to suppress the twin image problem.

where J1 represents the first-order Bessel function. The
particle information is contained mostly in the first lobe
(Vikram, 1992) as shown in Fig. 2. The recording distanceof a particle is limited to the range in which the lobe can be
resolved, that is,

2.1 Recording holograms using Gabor holography

where W/2 is the half width of the CMOS elements. Additionally, the pixel pitch ∆x of the elements is also limitedby(4)the relation,dp2

 z1 

λWe consider the case in which particles illuminated by aplane wave of wavelength λ as shown in Fig. 1. The complex amplitude Ψz at a distance z is formulated as

d pW /2(4)

1.22 λ

(5)
2πz 
2.44

exp  j λ   1  b  exp j π x  ξ  2   y  η 2 dxdy
Ψz 
∬ i

jλz
λz 
2.2 Reconstruction of holograms using Gabor(5)holographyπ22
We reconstruct holograms explained in sec. 2.1 by(1)
1  bi  exp j x  ξ    y  η  dxdy
λz
introducing the convolution kernel defined as (Onural and(1)
Scott, 1987)2
where j = –1 is the imaginary unit. The diffracting aperture

dp

x 









bi for the ith particle is defined as the transmittance function, namely,1 ( x  x ) 2   y  y  2  d / 2
pi
pi
p

bi  xpi , y pi , z pi   (2)
0 ( x  xpi ) 2   y  y pi  2  d p / 2

222(2)




hz 

1
2π 2
exp  jx  y 2 
λz
 λz


(6)

Eqn. (1) can be expressed compactly using the twodimensional convolution ** as(6)2π Ψzi  exp  jzi  1  bi  **hzi
(7)
 λ 

(7)

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The hologram intensity is recorded on a hologram as(8)

I zi  Ψzi  1  bi ***h * zi  bi **hzi

where a superscript * denotes complex conjugate, and thecomplexamplitude of particles are reconstructed as(8)(9)Ψz pi  I zi **h zpi  1  bi ***h2 zpi  bi **hz pi

The second and third terms on the right-hand side ofEqn.
(9) (9) lead to the virtual and real particle images. The
virtual particle image causes the out-of-focus particle, as
shown in the intensity line profile of Fig. 3. This is known
as the twin image problem (Latychevskaia and Fink, 2007)
due to the loss of phase information when the complex amplitude of the hologram is recorded by the CMOS camera.

2.3 Suppression of twin image problem using
phase retrieval holography

We use phase retrieval holography, which requires twoholograms separated by a distance Δz to obtain the phase
information as shown in Fig. 4 (Liu and Scott, 1987; Zhang
et al., 2003). This iterative phase retrieval algorithm isbased on the Gerchberg–Saxton algorithm (Gerchberg and
Saxton, 1972) using two holograms as intensity constraint

Original Research Paperconditions. The distance Δz between the two hologramswith good convergence of the iterative process is given by
Tanaka et al. (2016)

Δz 

z1
m  1, 2,, m  Nm

(10)

This method has an iterative process with four steps asshown in Fig. 5(a).
(10)
Step 1. Reconstruct the complex amplitude of the first
hologram to the second hologram with the distance:
Ψz 2k  exp  jΔzΨ z1k **hΔz  Ψz 2k exp  j 2 (11)

Step 2. Replace the absolute value of Ψz 2k with

I 2:

(11)
Ψz 2k  I 2 exp  j 2 

(12)
(52)
(51)
Step 3. Back-reconstruct the amplitude of step 2 to thefirsthologram:(12)Ψz1k  exp  jΔzΨz 2k **hΔz  Ψ z1k exp  j 2 (13)

Step 4. Replace the absolute value of Ψz1k with I1 :
(13)
(14)
Ψz1k  I1 exp  j 1
(54)
(53)

(14)

Fig. 3

Reconstruction of a hologram in particle measurement.

Fig. 4Phase retrieval holography using two holograms.

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Fig. 5Phase retrieval holography: (a) iterative process and (b) reconstruction of a hologram with retrieved phase information.

where ϕ1 and ϕ2 are the retrieved phases at the first and second hologram, respectively. The iterative loop exits whenthe mode of the reconstructed image at zp3 becomes the
maximum value (Tanaka et al., 2016). The virtual particle
image has been removed by phase retrieval holography, as
shown in the intensity line profile of Fig. 5(b) comparedwith Fig. 3.

2.4 Particle elongation in the optical axis direction

In addition to the twin image problem, another problem
is that the reconstructed particle is elongated in the opticalaxis direction. To understand this problem, the complex
amplitude reconstructed from the hologram formed by one
particle at (xp, yp) = (0,0) on the z axis can be considered asa circular aperture on a screen (Yang et al., 2006), that is,ψ z a 0, 0, z  1  exp  j (15)where ϕ is πrp2/(λz). On the other hand, the amplitude ψzoof (15)an opaque disk is derived by using Babinet’s principle(ψza + ψzo = 0) (Born et al., 1970) as
ψ z o 0, 0, z  1  cos    jsin  

whereupon we obtain the intensity as2(16) πrp 2 
 πrp 2 
I z  ψ z o  1  cos  sin 2 
 λz 
 λz 

(16)(17)

Fig. 6 shows that the intensity oscillates, with the frequency increasing toward the position of the particle. In(17) the oscillations are limited owing to the finite
practice,
sensor size, as shown in the numerical plots in Fig. 6.
Particle elongation is defined by a zero-valued position

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Original Research Paper

that oscillates symmetrically about z = 0 in the following
equation (Nakatani et al., 2019):

ΔL 

dp2
n  1, 2, , n  N 
2 2n  1 λ(18)Particle elongation can be expressed as dp2/(6λ) by substituting n = 2 into Eqn. (18) under the conditions of Fig. 2.
(18)
Furthermore, taking this elongation into account, the recording range of Eqn. (4) can be extended as
dp2
6λ z1 

d pW /2(19)

1.22 λ3. Holographic particle size measurement(19)
system using a GPU-equipped singleboard computer

In this section, we illustrate the details of the holographic
particle size measurement system using a GPU-equipped
SBC (Jetson Nano, NVIDIA). Fig. 7 shows the opticalsetup for the phase retrieval holography system with twoholograms. The laser beam (532 nm, 5 mW, Edmund®) ismagnified using an objective lens (40/0.65) and converted
to a collimated beam using a beam expander (TS Scorpii®,
8×, Edmund). After passing through the measurement
volume (6.9 mm × 6.9 mm × 40 mm) where the particles(SP20SS, Mitsubishi Chemical Corporation) fall at theirterminal velocity, the collimated light is split by a beam
splitter and recorded by two CMOS cameras (BFS-U331S4C-BD2, FLIR, processing rates: 0.4 fps (Gabor holography) and 0.5 fps (phase retrieval holography), shutter
speed: 200 μs, image size: 512 pixels × 512 pixels, cell
pitch: 3.45 μm) fitted with telecentric lenses (VS-TCT0565/S, 0.5×, WD: 65.7 mm, VS Technology®). Two cameras

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Original Research Paper

Table 1 Comparison of execution time comparison between CPU andGPU for Gabor and phase retrieval holography.

[s]Fig. 6 Theoretical and numerical distribution of light intensity ofFig. 2 through the center of the particle.

[Ratio]CPUGPU

CPU/GPU

Gabor

82.5

2.0

41.2

Phase retrieval

101.7

2.6

39.1

used image measurement to determine the sizes of particles stuck to a glass plate, as shown in Fig. 8(a). We used
image analysis software (Fiji 2.3.0: Schindelin et al., 2012)
to apply dynamic thresholding (Prewitt et al., 1966) to
the recorded image to obtain a binarized image, and wemeasured the particle size as the Feret diameter, as shownin Fig. 8(b).

4. Results

Fig. 7 Optical setup for particle size measurement using phaseretrieval holography system with a GPU-equipped SBC: (a) laser; (b)objective lens; (c) collimator lens; (d) ND filter; (e) observation box; (f)beam splitter; (e) telecentric lens; (h1) CMOS camera 1; (h2) CMOS
camera 2; (i) NVIDIA Jetson Nano as GPU-equipped SBC.

are installed at positions z1 = 10 mm and z2 = 20 mm under
the condition of m = 1. The misalignment of a rotation anda shift in the xy plane between the two cameras was calibrated numerically within one pixel by bundle adjustmentof the image distortion correction (Russ and Neal, 2016)using a calibration plate printed with random dots.
The holograms recorded by the two cameras are processed by the NVIDIA Jetson Nano with JetPack® SDK
4.6.1. Camera control and image acquisition are conductedwith the Spinnaker® SDK C++ v2.5.0.80 library. The
CUDA C/C++ 10.2 library and Thrust API are used forimage processing and hologram reconstruction calculation
using the GPU. The textbook by Shimobaba and Ito (2019)is informative, detailing GPU programming of hologram
recording and reconstruction. OpenCV 4.5.4 is used todisplay images in GUI windows during processing. The
application for bundle adjustment of the two cameras is
implemented in Julia® 1.7.2.
Table 1 gives the CPU and GPU execution times forreconstruction processes using Gabor and phase retrieval
holography. The CPU to GPU execution time ratio indicates that a GPU-equipped SBC is essential for early-timeprocessing.
For comparison with holographic measurement, we

In this section, we report use of the fabricated system
to measure particle size distributions in three-dimensional
space using Gabor holography and phase retrieval holography. The effectiveness of the system is confirmed bycomparing its measured distributions with that measured
conventionally by two-dimensional image measurement.
Reconstructed volumes (6.9 mm × 6.9 mm × 40 mm)
of Gabor and phase retrieval holography are shown in
Figs. 9(a) and 10(a), respectively. Particles are elongatedalong the optical axis, as shown in Eqn. (18) and Fig. 6.
Although particle size can be measured from the elongation(Nakatani et al., 2019), to reduce the computational load
for real-time processing, we measure the size by using thebinarized image with the minimum value in the optical axis
direction in the volume.
The obtained minimum-intensity image, binarized image, and intensity histogram are shown in Figs. 9(b) and9(c) and Figs. 10(b) and 10(c), respectively. As comparedin Figs. 3 and 5, phase retrieval holography, which suppresses the twin image problem, has a histogram peakat the value of zero light intensity as focused particles.
In contrast, Gabor holography, which is affected by thetwin image problem, has no peak in the histogram at thezero value. Consequently, Gabor holography has a larger
threshold value than that of phase retrieval holography,
and its binarized particle size is smaller than that of phaseretrieval holography.
The Feret diameter is measured from the binarized
images as in the image measurement of Fig. 8(b), and theparticle size distributions of Gabor and phase retrieval holography are shown in Figs. 11(a) and 11(b), respectively.
Gabor and phase retrieval holography have peaks of similar
size. However, as mentioned above, Gabor holography has
more particles below 40 μm than does phase retrieval holography because of the smaller size due to the twin image
problem.

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Original Research Paper

Fig. 8 Particle size distribution by image measurement: (a) observation of particles with telecentric lens; (b) histogram of particle size distribution(number of particles Np = 1147).

Fig. 9 Gabor holography: (a) reconstructed particles in measurement volume; (b) imposed volume and binarized image (threshold number: 23); (c)intensity histogram.

The cumulative frequency distribution functions are
generated from the particle size distribution of image
measurement, phase retrieval, and Gabor holography. The
distributions determine the sizes corresponding to the 10th,
50th, 90th percentiles, and the span, labeled d10, d50, d90,
and dspan respectively. From the values in Table 2, unlike
Gabor holography, phase retrieval holography enables
the measurement of almost the same distribution levels as
those of image measurement.

5. Conclusion

We have illustrated a phase retrieval holography systemwith a GPU-equipped SBC with theoretical and fabrication

226

considerations. This system enables real-time measurements of particle size distributions in a three-dimensional
space with almost the same accuracy as that of conventional two-dimensional image measurement. The results
show that the fabricated system has reached the stage ofpractical particle-size measurements in three-dimensional
space.

Data Availability Statement

The data on particle distributions by image measurement, Gabor holography, and phase retrieval holography is available publicly in J-STAGE Data (https://doi.
org/10.50931/data.kona.22002518).

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Original Research Paper

Fig. 10 Phase retrieval holography: (a) reconstructed particles in measurement volume; (b) imposed volume and binarized image (threshold number:8); (c) intensity histogram.

Fig. 11 Particle size distributions by (a) Gabor holography (number of particles Np = 1390) and (b) Phase retrieval holography (number of particles
Np = 1837).
Table 2 Particle size distributions of Feret diameter.

[μm]

[-]

d10

d50

d90

dspan*

Image measurement

53.9

65.4

74.3

0.3

Phase retrieval

44.1

61.7

74.3

0.5

Gabor

24.8

52.5

65.4

0.8

* dspan = (d90 – d10)/d50

Acknowledgments

Our research on the phase retrieval holography system
for measuring particle size distribution was supported by

the Hosokawa Powder Technology Foundation (Grant
Number HPTF20108).

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Authors’ Short Biographies
Yohsuke Tanaka
Yohsuke Tanaka is an Associate Professor at the Faculty of Mechanical Engineering, Kyoto
Institute of Technology. He worked for 3 years as a field engineer at Kyocera Corporation after receiving his Master’s Degree in 2002 from the Kyoto Institute of Technology. He received his Ph.D. in
Engineering from Osaka University in 2008. He received fellowships (DC2) from the Japan Society
for Promotion of Science from 2007 to 2008. His research interests include optical, flow, and soundmeasurements.

Dai Nakai
Dai Nakai received his Bachelor of Engineering from the Kyoto Institute of Technology in 2022.
He has since been researching particle measurement using holography as a Master’s student at the
Kyoto Institute of Technology. He is currently researching development of a machine learning-based
holographic collision detection system to observe microdroplet collision in precipitation processes.

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