COL 10(3), 031201(2012) CHINESE OPTICS LETTERS March 10, 2012
Linear sinusoidal phase-shifting method resistant to
non-sinusoidal phase error
Haihua Cui (www°°°uuu)
∗
, Wenhe Liao (©©©ÚÚÚ), Ning Dai ( www), and Xiaosheng Cheng (§§§>>>)
Jiangsu Key Laboratory of Digital Medical Equipment Technology, Nanjing University
of Aeronautic and Astronautic, Nanjing 210016, China
∗
Corresp onding author: cuihh@nuaa.edu.cn
Received June 29, 2011; accepted September 5, 2011; posted online November 7, 2011
Non-sinusoidal phase error is common in structured light three-dimensional (3D) shape measurement sys-
tem, thus we perform theoretical and experimental analyses of such error. The number of non-sinusoidal
waveform errors in a 2π phase period is the same as the number of steps of the phase-shifting algorithm; no
errors occur within the one-phase period. Based on our findings, a new structured light method, the linear
sinusoidal phase-shifting method (LSPS), that is resistant to non-sinusoidal phase error is proposed. Ex-
p eriments show that the non-sinusoidal waveform error is reduced to an almost negligible level (0.001 rad)
using the proposed LSPS.
OCIS codes: 120.0120, 120.2650, 100.0100, 150.0150.
doi: 10.3788/COL201210.031201.
Optical non-contact three-dimensional (3D) shape mea-
surement techniques have b een developed to obtain 3D
contours. With the recent advancement in digital display
technology, 3D shape measurement based on digital pro-
jection units has been rapidly expanding
[1−2]
. However,
the challenge remains to be in area of developing a sys-
tem with an off-the-shelf projector, such as liquid crys-
tal display (LCD) and digital mirror device (DMD), for
high-quality 3D shape measurement. One of the major
issues is the nonlinear response of the projection engines
of projectors
[1−10]
.
Projector gamma calibration is usually needed to per-
form high-quality 3D shape measurement using a digital
fringe projection and phase-shifting method
[2−10]
. This is
because the commonly used commercial video projector
is a nonlinear device purposely designed to compensate
for human vision
[7]
. A variety of techniques have been
developed for nonlinear phase errors
[2−12]
. These include
the two-step triangular-pattern phase-shifting algorithm
and the error compensation method
[5−6]
, which can re-
duce periodic measurement errors due to gamma nonlin-
earity as well as projector and camera defocus. In this
letter, the detailed mathematical model for sinusoidal
phase shifting was developed to predict the effects of
non-unitary gamma on phase-measuring profilometry
[3]
.
Zhang et al. developed a compensation method that
needs to calibrate an error look-up table (LUT) and is
based on the assumption of periodic phase error
[9−10]
. In
general, the abovementioned methods need an additional
compensatory step to reduce phase error. A new method
with defocusing binary structured patterns can be used
to eliminate nonlinear gamma
[7]
; however, controlling
the proper defocusing degree to achieve high accuracy
is difficult using this method
[8]
. Guo et al. proposed a
gamma correction method using a simple one-parameter
gamma function technique by statistically analyzing the
fringe images
[11]
. These techniques significantly reduce
the phase error caused by nonlinear gamma. Neverthe-
less, these compensation methods still have residual error
value that cannot be ignored in accurate measurement.
In addition, the actual gamma of the projector is very
complicated and is not the only factor that causes non-
sinusoidal waveforms. Our experiments show that the
nonlinear gamma of the projector changes over pixels,
and must be comp ensated one by one. All these prob-
lems hinder its applications, especially for precision mea-
surement. Hence, a technique that is resistant to non-
sinusoidal waveforms would be a significant development
in 3D shape measurement.
In this letter, we present the linear sinusoidal phase-
shifting method (LSPS), a novel coding method that
combines the advantages of nonlinear waveform error re-
sistance and the high resolution attained by the sinu-
soidal phase-shifting methods. Compared with the tradi-
tional phase-shifting method, the proposed method is far
less sensitive to the projector nonlinear gamma. The idea
originated from the following observations. Firstly, in a
2π phase period, the number of non-sinusoidal waveforms
is the same as the number of the phase steps. Secondly,
a few zero-error points occur in the 2π phase period.
Finally, although the non-sinusoidal waveforms have pe-
riodicity, the distances between each cycle and the error
amplitude are different. The last observation indicates
that it is difficult to completely eliminate the phase error
using the passively compensated algorithms
[8−10]
, which
are based on the same periodicity assumption. The first
two observations imply that the error can be completely
eliminated if the distance between each cycle is reduced
to zero, and if the zero phase error pixels are selected.
If this hypothesis is true, then a novel and robust phase
coding method can be developed without nonlinear pro-
jector gamma calibration. Experiments are presented to
verify the performance of the proposed technique.
Sinusoidal phase-shifting methods are widely used in
optical metrology because of their measurement accu-
racy. In this letter, a four-step phase-shifting algorithm,
which requires four phase-shifted images, is used. The
intensities of the four images with a phase shift of π/ 2
are
I
p
j
(x, y) = I
0
(x, y) + I
00
(x, y) cos[ϕ(x, y) + j × π/2],
j = 0, 1, 2, 3. (1)
1671-7694/2012/031201(4) 031201-1
c
° 2012 Chinese Optics Letters
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