Measurement Science and Technology
Meas. Sci. Technol. 27 (2016) 104004 (7pp)

doi:10.1088/0957-0233/27/10/104004

3D velocity measurement by a single
camera using Doppler phase-shiftingholography
Nao Ninomiya, Yamato Kubo, Daisuke Barada and Tomohiro Kiire
Department of Optical Engineering, Utsunomiya University, 7-1-2 Yoto, Utsunomiya 321-8585, Japan
E-mail: nino@utmu.jp
Received 3 February 2016, revised 25 July 2016
Accepted for publication 27 July 2016
Published 16 September 2016
Abstract

In order to understand the details of the flow field in micro- and nano-fluidic devices, it isnecessary to measure the 3D velocities under a microscopy. Thus, there is a strong needfor the development of a new measuring technique for 3D velocity by a single camera. One
solution is the use of holography, but it is well known that the accuracy in the depth direction
is very poor for the commonly used in-line holography. At present, the Doppler phase-shiftingholography is used for the 3D measurement of an object. This method extracts the signal of a
fixed frequency caused by the Doppler beat between the object light and the reference light. It
can measure the 3D shape precisely. Here, the frequency of the Doppler beat is determined by
the velocity difference between the object light and the reference light. This implies that thevelocity of an object can be calculated by the Doppler frequency. In this study, a Japanese 5
yen coin was traversed at a constant speed and its holography has been observed by a highspeed camera. By extracting only the first order diffraction signal at the Doppler frequency,a precise measurement of the shape and the position of a 5 yen coin has been achieved. At
the same time, the longitudinal velocity of a 5 yen coin can be measured by the Doppler
frequency. Furthermore, the lateral velocities are obtained by particle image velocimetry(PIV) method. A 5 yen coin has been traversed at different angles and its shapes and the 3D
velocities have been measured accurately. This method can be applied to the particle flows in
the micro- or nano-devices, and the 3D velocities will be measured under microscopes.
Keywords: digital holography, 3D measurement, Doppler phase-shifting, velocity
measurement(Some figures may appear in colour only in the online journal)

1. Introduction

and the careful calibration of the 3D coordinates, 2D distribution of all the three velocity components can be measured by
Stereo PIV [4] and the 3D measurement of the 3D velocities
can also be achieved by so-called Tomographic PIV [5].
In order to understand the details of the flow field in the
micro- and nano-fluidic devices, it is necessary to measure the3D velocities under a microscope. The use of confocal PIV[6, 7] is a challenge, but it is not capable of depth-wisetracking and is very expensive. Thus, the development of a
new measuring technique for the 3D velocities by a single
camera observation is strongly needed. One solution is theuse of holography [8]. The principle of the holography was

According to the development of computer and imaging
devices, the new measuring techniques based on flow visualization have been developed and utilized in many different
fields. The typical examples are so-called PIV, i.e. particle
image velocimetry [1, 2], and PTV, i.e. particle tracking velocimetry [3]. The principal setup of PIV utilizes one highresolution camera and a laser light sheet, which illuminates
the tracer particles suspended in the fluid flow. By calculating
the displacement of particles, the 2D distribution of the 2D
velocities can be obtained. By implementing multiple cameras
0957-0233/16/104004+7$33.00

1© 2016 IOP Publishing Ltd Printed in the UKN Ninomiya et al

Meas. Sci. Technol. 27 (2016) 104004

invented in the 1960s [9, 10] and the digital reconstruction of
the hologram [11, 12] together with the use of CCD camera [13]has made digital holography [14] possible. PIV was originallyinvented as a film-based technique [15] and then integrated to adigital method [16]. At the same time, it is well known that theaccuracy in the depth direction is very poor for the commonlyused in-line holography because of the contamination by thezero order beam. In order to overcome this shortcoming, the useof two sets of orthogonal holography [17] has been proposed,
but it cannot be applied to the measurement under microscopy.
Recently, several new methods to shorten the depth-of-focus
have been invented [18–21]. On the other hand, many kindsof the phase-shifting [22] are the hope to accurately calculate
the particle position. Yet, the spatial phase-shifting needs 3 or
4 cameras whose CCD pixels are carefully aligned. Otherwise,2  ×  2 phase shifting array device [23] or tilting technique [24]has to be used even though these method sacrifices spatial resolution. Whereas the temporal phase-shifting cannot be applied tothe moving objects. As for the holographic measurement under
microscopy, many studies have been conducted [22, 25] and the
PIV measurements have also been done extensively [26–28].
But, it is not easy to apply the phase-shifting to the microscopic PIV. Notwithstanding, the time tracing algorithm for thedensely distributed moving particles does not have great progress from the time-­consuming cross correlation method [29]or the traditional PTV algorithm [30, 31].
Presently, the Doppler phase-shifting holography [32]is used for the 3D velocity measurement of an object. Thismethod extracts the signal of a fixed frequency caused by the
Doppler beat between the object light and the reference light.
The frequency of the beat is determined by the velocity difference between the object light and the reference light. Thisimplies that the velocity of an object can be calculated by the
Doppler frequency. In this study, the shapes and its 3D velocities have been measured accurately for a moving Japanese
5 yen coin at different angles.

where aO and aR represent the amplitudes, φO and φR are thephase angles and ωO and ωR are the angular velocities of theobject light and the reference light. If the object travels at
velocity υO and the reference mirror at υR, the angular velocities are shifted by the Doppler effect of light as follows:⎛ 1 + υO(t )/c ⎞2⎛2υ (t ) ⎞
ω O (t ) = ω 0 ⎜⎟ ≅ ω 0⎜1 + O ⎟
(3)
⎝
c ⎠
⎝ 1 − υO(t )/c ⎠⎛ 1 + υ R(t )/c ⎞2⎛2υ (t ) ⎞
ω R (t ) = ω 0 ⎜⎟ ≅ ω 0⎜1 + R ⎟
(4)
⎝
−
1
υ
t
c
c ⎠
(
)/
⎝
⎠
R

where ω0 is the angular frequency of the light source and c isthe speed of light. Consequently, the superposition intensityof the holograms detected by the image sensor is expressed
as follows:
I (x, y, t ) = EO(x, y, t )+ ER(t ) 2
= a2O(x, y ) + a2R + 2a O(x, y )aR cos[∆φ(x, y ) − ∆ω(t )t ]
(5)
here,

∆ φ (x , y ) = φ O (x , y , t ) − φ R∆ ω (t ) = ω O (t ) − ω R (t )(6)The Fourier transformation of equation (5) in the time domain isFt(I )(x, y, ωb ) = [a2O(x, y ) + a2R]δ (ωb ) = a O(x, y )aRA1(ωb )exp{i[∆φ(x, y ) + B1(ωb )]}+ a O(x, y )aRA2 (ωb )exp{−i[∆φ(x, y ) + B2(ωb )]}(7)

If the angular frequency of the object light is higher than thatof the reference light, the  +1st order diffraction is obtainedin the second term. If the spectrum of each term can be split,
the phase difference Δφ(x, y) between the object light and thereference light is easily obtained.
The discrete Fourier transformation of the image recordedby the image sensor of the finite size is obtained as:

2. Fundamental techniques

DFT[I ](x, y, νb, m )Digital holography observes the interference of the reference
light and the observed light scattered on the surface of an
object. Then, the 3D shape of an object can be reproduced by
calculating the diffraction by a computer. But, in the commonlyused in-line holography, the diffraction is contaminated by the
0th order object light and the accuracy in the depth direction isdeteriorated. In order to improve this drawback, several phaseshifting methods have been proposed, which can calculate only
the 1st order diffraction and thus the acc­uracy in the longitudinal direction is guaranteed. Presently, Doppler phase-shiftingholography is chosen because it is capable of unsteady measurement and does not need multiple cameras. Figure 1 showsthe experimental setup of a Doppler phase-shifting holography.
The complex intensities of the object light and of the reference light are expressed as follows:N−1= ∑ I (x, y, tn )exp(−i2πtnνb, m ) m = 0, 1, 2, …, N − 1
n=0

(8)

If the  +1st order diffraction spectrum of a fixed frequency
is extracted from equation (8), the complex amplitude of an
object can be calculated very accurately.
3. 3D velocity measurement
Originally, the Doppler phase-shifting was invented to measure the object profile accurately [32]. Yet, as the beat frequency of the Doppler beat is proportional to the velocity
difference between the object light and the reference light,
the longitudinal velocity of an object can be measured by thebeat frequency of the Doppler beat. In order to prove this,Ninomiya et al [33] measured the longitudinal velocity of aconcave mirror travelling at a constant speed by a Doppler
phase-shifting holography.

EO(x, y, t ) = a O(x, y ) exp{i[φO(x, y ) − ωO(t )t ] }(1)ER(t ) = aR exp{i[φ R − ω R(t )t ]}(2)2N Ninomiya et al

Meas. Sci. Technol. 27 (2016) 104004

Figure 1. Experimental setup for Doppler phase-shifting digital holography.

Figure 2. Experimental setup for velocity measurement.

An Ar-ion laser of 514 nm is used as the light source,which is introduced to a pinhole and expanded by the spatial
filter of f  =  5 mm and then made parallel by a collimator lensof f  =  100 mm. A concave mirror of the radius of curvatureR  =  30 m is used as a target object in figure 1 and traversed
at υO  =  100 µm s−1 by the voice coil motor. The images ofinterference patterns are captured on a high-speed camera of1024  ×  1024 pixels at frame rate of 2000 fps and shutter speedof 1/70 000 s. The size of each pixel is 10  ×  10 µm. The timeseries of intensity fluctuations of the central pixel of 512 consecutive images are transformed into Fourier domain throughDFT to obtain its spectrum. It should be noted that as any partof the concave mirror is travelling at the same speed, choiceof any pixel may give the similar spectrum. A clear peak of
the Doppler beat is found at 386.7 Hz, which corresponds toυO  =  99.4 µm s−1 with the error of 0.6% to the given longitudinal velocity [33]. Thus, it has been proved that a Doppler
phase-shifting holography can be used for the measurementof the longitudinal velocity. Moreover, the surface profile ofthe concave mirror was reproduced from the phase data of thedigital holography and the rms error to the R  =  30 mm profile
is only 3.27 nm [33]. This is the evidence that Doppler phaseshifting technique is capable of measuring the 3D position andthe longitudinal velocity simultaneously.
3

N Ninomiya et al

Meas. Sci. Technol. 27 (2016) 104004

Figure 3. Beat frequency spectrum obtained from 512 images. (a) 400–911 frame. (b) 2400–2911 frame.

Figure 4. Displacement of Japanese 5 yen coin measured by 2D PIV. (Left: superimposed image of two sets together with PIV settings,
right top: magnified image, right bottom: measured velocity vectors.)

υO  =  413.4 µm s−1, which corresponds to υOz  =  100.0 µm s−1
and υOx  =  400.0 µm s−1. Figure 3 shows the measured spectrum obtained from 512 frames at two different timings of thetraverse. In order to obtain the beat frequency, the time series
of intensity fluctuation is needed. Presently, the time trace ofthe central pixel intensity for 512 frames is used. Moreover,for the later PIV measurement, two sets of 512 frames, i.e.
400–911 frames and 2400–2911 frames, are used whose timeinterval between sets is 1.000 s. Both of them have strong
peaks at (a) 396.2 Hz and (b) 399.7 Hz, which correspondto the longitudinal velocities of (a) υOz  =  101.8 µm s−1 and(b) υOz  =  102.7 µm s−1 by equation (3), and both agree wellAs the fundamental measurement technique has been
proved to work in the previous study [33], the actual measurement of the 3D velocity is carried out by the experimental
setup shown in figure 2. The Japanese 5 yen coin, whose
diameter is about 22 mm, is used as a target object and is traversed at a constant speed at a specific angle. As the reflection
from the 5 yen coin is weaker than the reference light, the 70%
ND filter is placed in front of the reference mirror in order tomatch the light intensities.
The images are recorded by a high-speed camera of1024  ×  1024 pixels at frame rate of 2000 fps. The angleof the moving stage is set at θ  =  76 degree and it travels at
4

N Ninomiya et al

Meas. Sci. Technol. 27 (2016) 104004

Figure 5. Displacement of Japanese 5 yen coin measured by 2D PIV with shifted images. (Left: superimposed image of two sets together
with PIV settings, right top: magnified image, right bottom: measured velocity vectors.)

with the given velocity. It should be noted that the choice ofa pixel and a starting frame for the spectrum extraction affectthe beat frequency only slightly.
By integrating over these 512 frames, an interference image
that contains only the frequency component of this Doppler
beat frequency can be extracted. By using this interference
image, which is free from the zero order beam, an accurate
reconstruction of an object can be done. Figure 4 shows thereconstructed images of the 5 yen coin at two different timings. The reconstruction is done manually by searching theappropriate depth position where the rms intensity takes its
maximum. The former image is colored by red and the latterby blue. It is obvious that the 5 yen coin travels in the x direction and its movement is calculated by the correlation methodcommonly used in the PIV algorithm. The expanded plot ofthe velocity vectors shows the transverse movement at a fixed
speed. By the combination of the longitudinal velocity measurement by the Doppler frequency and the transverse velocity
measurement by PIV method, 3D velocity measurement of an
object by a single camera has be achieved.
But as the 5 yen coin is also moving in the transverse
direction, the time series intensity used to obtain the spectrum
shown in figure 3 is not an exact trace of the light emitted
from a certain point of the 5 yen coin. Nevertheless, as thetransverse movement of the 5 yen coin has been measured by
PIV as shown in figure 4, the temporal movement of a fixed
point can be interpolated from the results obtained in figure 4.
Presently, each frame of the original interference image isshifted back by equation (9) so that it looks as though there isno transverse movement in order to trace the exact time seriesof the light emitted from a fixed point of the 5 yen coin.

Table 1. Velocities of Japanese 5 yen coin (76 degree).

Stage velocity (μm s−1)
Measured velocity (μm s−1)

υOz

υOx

100.0
100.7

400.0
401.2

1 Nx − 1 Ny − 1∑ ∑ F (k , l )NxNy k = 0 l = 0⎡ ⎛ kx⎡ ⎛ kxly ⎞⎤ly ⎞⎤× exp⎢2π i⎜ + ⎟⎥ ⋅ exp⎢2π i⎜ 0 + 0 ⎟⎥⎢⎣ ⎝ Nx⎢⎣ ⎝ Nx
Ny ⎠⎥⎦
Ny ⎠⎥⎦f (x − x 0, y − y0 ) =(9)Then the beat frequencies obtained from the spectrum of theshifted images are (a) 391.2 Hz and (b) 392.6 Hz, which correspond to the longitudinal velocity of (a) υOz  =  100.5 µm s−1and (b) υOz  =  100.9 µm s−1. It is quite obvious that the image
shifting improves the measurement accuracy of the longitudinal velocity. Figure 5 shows the reproduced images of the
5 yen coin with the image shifting. It can clearly be seen thatthe shape of the 5 yen coin is more obvious with the use of
the shifted images and it is quite a matter of course but thetransverse movements are hardly seen in figure 5, which is anevidence that the image shifting has been done properly.
Table 1 summarizes the result of the 3D velocity measurement by a Doppler phase-shifting holography. The voicecoil motor used for the moving stage has a positioning acc­
uracy of 0.05 µm and the nominal velocity fluctuation is lessthan 1%, which has been proved by a high-speed camera to
be much smaller. The vertical velocity is almost zero as thetraversing stage moves in the horizontal direction. Now it can5

N Ninomiya et al

Meas. Sci. Technol. 27 (2016) 104004

Figure 6. Errors in the velocity measurement in transverse and longitudinal directions. (a) υOx. (b) υOz.

be concluded that this technique is capable of 3D velocity
measurement.
Finally, similar measurements have been repeated with different angles of the traverse. The results are shown in figure 6
and it is quite obvious that this technique is valid for different
angles of the traverse with the error less than 2%.

[4] Guezennec Y G, Brodkey R S, Trigui J and Kent J C 1994
Algorithms for fully automated three-dimensional particle
tracking velocimetry Exp. Fluids 17 209–19[5] Elsinga G E, Scarano F, Wieneke B and van Oudheusden B W
2006 Tomographic particle image velocimetry Exp. Fluids
41 933–47[6] Park J S, Coi C K and Kihm K D 2004 Optically sliced microPIV using confocal laser scanning microscopy (CLSM)
Exp. Fluids 37 105–19[7] Kinoshita H, Kaneda S, Fujii T and Oshima M 2007 Threedimensional measurement and visualization of internal flowof a moving droplet using confocal micro-PIV Lab Chip
7 338–46[8] Schnars U and Jüptner W 2005 Digital Holography Digital
Hologram Recording, Numerical Reconstruction, and Related
Techniques (Berlin: Springer) (doi: 10.1007/b138284)[9] Thompson B J 1964 Diffraction by opaque and transparent
particles Opt. Eng. 2 43–0
[10] Trolinger J D, Beltz R A and Farmer W M 1969 Holographic
techniques for the study of dynamic particle fields Appl.
Opt. 8 957–61[11] Goodman J W and Lawrence R W 1967 Digital image
formation from electronically detected holograms Appl.
Phys. Lett. 11 77–9
[12] Adams M, Kreis T and Jüptner W 1997 Particle size andposition measurement with digital holography Proc. SPIE
3098 234–40[13] Schnars U and Jüptner W 1994 Direct recording of hologramsby a CCD target and numerical reconstruction Appl. Opt.
33 179–81[14] Nishihara K, Hatano S and Nagayama K 1997 New
method of obtaining particle diameter by the fast Fouriertransform pattern of the in-line hologram Opt. Eng.
36 2429–39[15] Nayler J L and Frazer B A 1921 Preliminary report upona experimental method of investigation, by the aid of
kinematographic photography, the history of eddying flow
past a model immersed in water Technical Report of theAdvisory Committee for Aeronautics for the year 1917–18
vol. 1 pp 18–25[16] Willert C E and Gharib M 1991 Digital particle image
velocimetry Exp. Fluids 10 181–93[17] Sheng J, Malkiel E and Katz J 2003 Single beam twoviews holographic particle image velocimetry Appl. Opt.
42 235–50[18] Dixon L, Cheong F C and Grier D G 2011 Holographic
deconvolution microscopy for high-resolution particle
tracking Opt. Express 19 16410–7

4. Conclusions
This paper presents a newly developed 3D velocity measuring
technique based on a Doppler phase-shifting holography.
It can measure the 3D velocity of an object by the single
camera observation very accurately with the error less than
2% for any angle of traverse.
This method envisions its application to particle motion
detection in the micro- or nano-devices and thus the 3D
particle velocities can be measured simultaneously by asingle camera. As this method does not need the depth-wisetracking, it can easily be coupled with conventional 2D PIV.
Nevertheless, the application of this technique to the particle
flows, the limit of particle seeding density and the depth offield that this method can be applied remain to be proved bya future study. But, it will contribute to the development of
micro- and nano-fluidic devices.
Acknowledgment
A part of this work has been supported by the JSPS KAKENHI Grant Number 25420147 and by the Center for Optical
Research and Education (CORE) at Utsunomiya University.
References[1] Keane R D and Adrian R J 1992 Theory of cross-correlation
analysis of PIV images Appl. Sci. Res. 49 191–215[2] Adrian R J 2005 Twenty years of particle image velocimetry
Exp. Fluids 39 159–69[3] Nishino K, Kasagi N and Hirata M 1989 Three-dimensional
particle tracking velocimetry based on automated digital
image processing J. Fluid Eng. 111 384–91
6

N Ninomiya et al

Meas. Sci. Technol. 27 (2016) 104004

time resolution using micro digital-holographic particletracking velocimetry Opt. Rev. 12 442–4[27] Sheng J, Malkiel E and Katz J 2006 Digital holographic
microscope for measuring three-dimensional
particle distributions and motions Appl. Opt.
45 3893–901[28] Sheng J, Malkiel E and Katz J 2008 Using digital holographic
microscopy for simultaneous measurements of 3D near wall
velocity and wall shear stress in a turbulent boundary layer
Exp. Fluids 45 1023–35[29] Pu Y and Meng H 2000 An advanced off-axis holographic
partcile image velocimetry (HPIV) system Exp. Fluids
29 184–97[30] Meng H, Pan G, Pu Y and Woodward S 2004 Holographic
particle image velocimetry: from film to digital recording
Meas. Sci. Technol. 15 673[31] Kim S and Lee S J 2009 Measurement of Dean flow in a
curved micro-tube using micro digital holographic particle
tracking velocimetry Exp. Fluids 46 255–64[32] Kikuchi Y, Barada D, Kiire T and Yatagai T 2010 Doppler
phase-shifting digital holography and its application tosurface shape measurement Opt. Lett. 35 1548–50[33] Ninomiya N, Kubo Y, Barada D and Kiire T 2015 Direct
measurement of logitudinal velocity by doppler phaseshifting holography J. Vis. 18 409–12[19] Soulez F, Denis L, Fournier C, Thiébaut E and Goepfert C
2007 Inverse-problem approach for particle digitalholography: accurate location based on local optimization
J. Opt. Soc. Am. A 24 1164–71[20] Guildenbecher D R, Gao J, Reu P L and Chen J 2012 Digitalholography recosntruction algorithm to estimate themorphology and depth of nonspherical absorbing particles
Proc. SPIE 8492 849303[21] Toloui M and Hong J 2015 High fidelity digital inline
holographic method for 3D flow measurements Opt. Express
23 27159–73[22] Zhang T and Yamaguchi I 1998 Three-dimensional
microscopy with phase-shifting digital holography Opt.
Lett. 23 1221–3[23] Awatsuji Y, Sasada M and Kubota T 2004 Parallel quasiphase-shifting digital holography Appl. Phys. Lett.
85 1069–71[24] Murata S, Harada D and Tanaka Y 2009 Spatial phase-shifting
digital holography for three-dimensional particle tracking
velocimetry Japan. J. Appl. Phys. 48 09LB01[25] Xu L, Peng X, Miao J and Asundi A K 2001 Studies of digital
microscopic holography with applications to microstructure
testing Appl. Opt. 40 5046–51[26] Satake S, Kunugi T, Sato K, Ito T and Taniguchi J 2005 Threedimensional flow tracking in a micro channel with high7