i need to get summarize of this paper and make ppt

1. Optics and Lasers in Engineering 107 (2018) 247–257
Contents lists available at ScienceDirect
Optics and Lasers in Engineering
journal homepage: www.elsevier.com/locate/optlaseng
Review
Review of phase measuring deflectometry
Lei Huanga,∗, Mourad Idira, Chao Zuob, Anand Asundic
a
National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, NY 11973, USA
b
Jiangsu Key Laboratory of Spectral Imaging & Intelligence Sense, Nanjing University of Science and Technology, Nanjing 210094, China
c
School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore
a r t i c l e i n f o
Keywords:
Phase measuring deflectometry
Structure light illumination
Phase retrieval
Fringe analysis
Phase shift
Wavefront reconstruction
a b s t r a c t
As a low cost, full-field three-dimensional shape measurement technique with high dynamic range, Phase Measur-
ing Deflectometry (PMD) has been studied and improved to be a simple and effective manner to inspect specular
reflecting surfaces. In this review, the fundamental principle and the basic concepts of PMD technique are in-
troduced and followed by a brief overview of its key developments since it was first proposed. In addition, the
similarities and differences compared with other related techniques are discussed to highlight the distinguishing
features of the PMD technique. Furthermore, we will address the major challenges, the existing solutions and the
remaining limitations in this technique to provide some suggestions for potential future investigations.
1. Introduction
Quantitative three-dimensional (3D) shape metrology has already
become the key technology in industrial applications for quality con-
trol, reverse engineering, precision manufacturing, and digitization of
artwork [1–7]. Due to its high speed, non-contact, and non-destructive
testing feature, optical 3D shape metrology is one of the major metrol-
ogy techniques, especially suitable for the inspection of high-quality and
valuable surfaces [6–13].
Depending on the field of view, the optical 3D shape metrology
can be classified as pointwise or line-by-line scanning techniques [14,
15] and full-field vision-based measurement techniques [9, 16, 17]. The
full-field vision-based methods include active and passive 3D vision ap-
proaches. The structured-light-illuminated 3D vision is one of the most
typical active approaches. With the structured-light illumination, the
out-of-plane depth information is coded and recorded with a digital cam-
era. As a natural pattern in an optical shop, the fringe pattern is a classi-
cal solution of structured-light illumination. After the image acquisition
of the fringe pattern(s), quantitative phase values can be retrieved with
well-developed fringe analysis algorithms [18–22]. The fringe phases es-
tablish the correspondence between camera pixels and the illumination
points. As a result, the geometry relations between system components
can be built up to enable the 3D measurement of the Surface Under Test
(SUT). Phase Measuring Deflectometry (PMD) is one of the optical 3D
shape metrology techniques based on two-dimensional (2D) fringe phase
measurement, especially for specular reflecting surfaces [13, 23–33].
In this work, we review the principle of PMD technique and the
follow-up studies after its invention, compare the similarities and differ-
∗
Corresponding author.
E-mail address: huanglei0114@gmail.com (L. Huang).
ences with other related measurement techniques, and discuss the major
challenges, the current solutions, and their remaining deficiencies.
2. Fundamentals and concepts of PMD
The fundamental principle of PMD technique is the law of reflection.
As described in Fig. 1, an angle change 𝛼 of the SUT with respect to a ref-
erence orientation will introduce a doubled angle 2𝛼 to the reflected ray.
Usually, the sight ray of a camera, or say the probe ray, is treated as the
light source in PMD for easier understandings and analysis, although the
light is actually illuminated from a Thin-Film-Transistor Liquid-Crystal
Display (TFT LCD) screen in the physical process. In this review work,
the PMD measuring process is considered as the probe rays from the
camera are reflected by a specular SUT onto the TFT LCD screen.
As illustrated in Fig. 2, there are many possible height and slope com-
binations to explain the phase point observed by a single camera probe
ray, which is called height-slope ambiguity in PMD. By tracing where
the probe rays are deflected by the SUT in a geometrically known scene,
the vectors of the surface normal, or x- and y-slopes, can be determined
with proper regularization to the inverse problem in PMD with height-
slope ambiguity. Based on the measured slopes, the quantitative surface
topographic data can be reconstructed via numerical calculations. Iter-
ative height reconstructions and slope calculations may be necessary to
achieve self-consistent shape results [34].
Generally speaking, the implementation of a measurement with PMD
include the following steps.
(1) Properly set up the camera(s) and screen(s) to ensure the field of
view(s) can cover the desired measuring volume and place the spec-
imen surface inside the measuring volume and then adjust its tip/tilt
https://doi.org/10.1016/j.optlaseng.2018.03.026
Received 15 January 2018; Received in revised form 15 March 2018; Accepted 22 March 2018
0143-8166/© 2018 Elsevier Ltd. All rights reserved.

2. L. Huang et al. Optics and Lasers in Engineering 107 (2018) 247–257
Fig. 1. The measurement principle of the PMD is based on the law of reflection.
The reflected ray doubles the angle change introduced by the specimen slopes.
Fig. 2. For the same camera probe ray and its corresponding phase point on the
screen, there are many possible solutions with different height-slope combina-
tions.
to make the camera(s) see the fringe patterns on screen(s) via the re-
flection by the SUT.
(2) Capture the fringe images of the reflected pattern from the SUT.
(3) Analyze the fringe patterns to retrieve two-directional 2D phase in-
formation, and furthermore, the x- and y-slope values.
(4) Reconstruct the height information from slope datasets which is also
known as the 2D integration process.
2.1. Setups of PMD
The main devices for a general PMD include a digital camera, a TFT
LCD monitor, a specimen stage, and a computer. Computer-generated
fringe patterns are sequentially displayed on the TFT LCD screen. The
camera captures the reflection images of the displayed fringe patterns
via a specular reflecting SUT. The SUT shape is reconstructed by solving
an inverse problem with the captured images. Owing to the height-slope
ambiguity described in Fig. 2, several different kinds of regularization
approaches with the corresponding setups are proposed to resolve this
ill-posed problem.
To measure micron-level out-of-plane deformation or discrepancy
from a pre-known shape, the monoscopic PMD illustrated in Fig. 3 is a
simple and effective solution with the initial shape regularization [25],
which assumes the shape after deformation is very close to the reference
[35, 36], or the SUT is very similar to the pre-known shape [37]. Self-
consistent height and slope results can be obtained via iterations [34].
For different requirements and corresponding regularization in ac-
tual measurements, the PMD system may adopt additional components
(e.g. screens, distance sensors, or cameras). By introducing additional
patterns in the optical path as shown in Fig. 4(a), the reflected ray can
be determined by the two or more points of intersection on the shifted
screen [26, 38–40]. In Fig. 4(b), an additional distance sensor can be
used to regularize the ill-posed problem in the monoscopic PMD by pro-
viding a reference distance [41]. Fig. 4(c) and (d) describe the height
values can be searched by minimizing the discrepancies of the SUT nor-
mal vectors calculated from two or multiple cameras [24]. These cam-
eras can be served by a single screen as Fig. 4(c) or several screens
as Fig. 4(d). In addition, some recently proposed new configurations
require other constraints, such as parallel screens and reference plane
[42].
2.2. Image acquisition
In PMD measurement, a camera captures the reflection of the pat-
terns displayed on a screen through the specimen surface. There is a
trade-off between the spatial resolution and the angular resolution in
image acquisition. If the camera is focused on the specimen surface, the
measurement gets the best spatial resolution, but the angular resolution
will not be the optimum due to the defocusing of the screen patterns. If
the camera is focused on the reflection of the screen pattern, the mea-
surement achieves the best angular resolution but a lower spatial resolu-
tion comparing to the previous case. In practice, the cameras are usually
focused on the SUT for the following practical considerations.
(1) High spatial resolution is a common requirement for 3D shape mea-
surement, if achievable.
(2) The patterns displayed on the screen in PMD are typically sinusoidal
fringes which are smooth intensity curves and the phase calculation
is not very sensitive to a small amount of out-of-focus effect;
(3) There is less influence from the pixel grids of TFT LCD screen. Owing
to the defocusing effect, the TFT LCD pixel grids almost disappear in
the camera image, which is preferable. On the other hand, when the
camera focuses on the reflection of the screen pattern, the images
will record the TFT LCD pixel grids, which introduces an additional
error source to the follow-up fringe analysis process.
2.3. Fringe analysis and slope calculation
Once the fringe patterns are captured, the fringe phases need to be
retrieved by using the well-developed fringe analysis method as one of
the most significant intermediate results. The fringe analysis includes
fringe demodulation and phase unwrapping.
2.3.1. Fringe demodulation
Fringe demodulation allows retrieving the wrapped phase values
from the captured fringe intensity image(s). According to the required
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3. L. Huang et al. Optics and Lasers in Engineering 107 (2018) 247–257
Fig. 3. The basic monoscopic PMD simply consists of a screen and a camera, which displays and captures fringe patterns, respectively.
Fig. 4. Some other typical PMD setups: (a) monoscopic PMD with shifted screens, (b) monoscopic PMD with a point distance sensor, (c) stereoscopic PMD, and (d)
multi-camera PMD with several screens serving different cameras.
Fig. 5. Since two-directional phases are usually needed in PMD, either a crossed fringe pattern (a) or one-directional fringe patterns in the x-direction (b) and
y-direction (c) are displayed on the screen and captured by the camera(s).
number of frames, they can be classified as single-frame and multiple-
frame methods.
(1) Single-frame methods need a single fringe pattern to calculate
the phase information. They are mainly transform-based methods
[43], such as windowed Fourier transform [21, 22, 35, 36]. As
two-directional phase values are required to calculate both x-and y-
slopes, a crossed fringe pattern as shown in Fig. 5(a) is used to carry
the two-directional phases. The captured crossed fringe intensity I(x,
y) can be expressed as
𝐼(𝑥, 𝑦) = 𝑎(𝑥, 𝑦) + 𝑏𝑥(𝑥, 𝑦) ⋅ cos
[
𝜙𝑥(𝑥, 𝑦)
]
+ 𝑏𝑦 ⋅ cos
[
𝜙𝑦(𝑥, 𝑦)
]
, (1)
where x and y are the orthogonal coordinates of the screen, a(x, y) is
the background, bx(x, y) and by(x, y) are the modulations of x- and y-
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4. L. Huang et al. Optics and Lasers in Engineering 107 (2018) 247–257
Fig. 6. Wrapped phases (a) and (b) need to be unwrapped to the absolute phase values in x- (c) and y-directions (d) for the following slope calculations in PMD.
directional sinusoidal fringes. 𝜙x(x, y) and 𝜙y(x, y) are the fringe phases
in x- and y-directions. By analyzing or filtering in the frequency domain,
the two wrapped fringe phases 𝜙𝑤
𝑥 (𝑥, 𝑦) and 𝜙𝑤
𝑦 (𝑥, 𝑦) can be retrieved as
𝜙𝑤
𝑥 (𝑥, 𝑦) = arctan
Im
[
̄
𝑓𝑥(𝑥, 𝑦)
]
Re
[
̄
𝑓𝑥(𝑥, 𝑦)
] , (2)
𝜙𝑤
𝑦 (𝑥, 𝑦) = arctan
Im
[
̄
𝑓𝑦(𝑥, 𝑦)
]
Re
[
̄
𝑓𝑦(𝑥, 𝑦)
] , (3)
where ̄
𝑓𝑥(𝑥, 𝑦) and ̄
𝑓𝑦(𝑥, 𝑦) denote the filtered x- and y-directional expo-
nential fringe patterns.
(2) Multiple-frame methods use several phase shifted fringe images to
estimate the phase values in the least squares sense, e.g. the well-
known phase shifting or say phase stepping methods [18, 20]. Usu-
ally, the phase shifted fringe pattern In(x, y) is in a one-dimensional
sinusoidal waveform as
𝐼𝑛(𝑥, 𝑦) = 𝑎(𝑥, 𝑦) + 𝑏(𝑥, 𝑦) ⋅ cos
[
𝜙(𝑥, 𝑦) +
2𝑛𝜋
𝑁
]
, 𝑛 = 0, 1, 2, … , 𝑁 − 1.
(4)
where a(x, y), b(x, y), and 𝜙(x, y) are the background, modulation, and
phase of x- or y-direction fringes, respectively. The wrapped phase 𝜙w(x,
y) can be calculated as
𝜙𝑤
(𝑥, 𝑦) = − arctan
𝑁−1
∑
𝑛=0
𝐼𝑛 sin
(
2𝑛𝜋
𝑁
)
𝑁−1
∑
𝑛=0
𝐼𝑛 cos
(
2𝑛𝜋
𝑁
)
. (5)
Worthy to note, unlike the usual approaches, Ref. [44] introduced
a novel fringe demodulation method to calculate the two-directional
phase information of crossed fringe patterns with phase shifts along one
direction.
After the fringe demodulation, the phase values are wrapped within
[-𝜋, 𝜋] as shown in Fig. 6(a) and (b) because of the four-quadrant inverse
tangent function in Eqs. (2), (3) and (5). In order to calculate slopes
by using phase values, these wrapped phases need to be unwrapped to
absolute phases.
2.3.2. Phase unwrapping
Phase unwrapping [45] extends the wrapped phase values to break
the limit of [-𝜋, 𝜋] as illustrated in Fig. 6. According to the unwrapping
domain, the phase unwrapping can be divided into the spatial phase un-
wrapping [46, 47] and temporal phase unwrapping [48–50]. The spatial
phase unwrapping determines the fringe orders based on the phase re-
lations between neighboring pixels in space, and the temporal phase
unwrapping makes pixel-independent calculation of the fringe orders
along the time axis.
The required phase values in PMD are absolute phases, which means
the fringe orders should be consistent for all measurements. Marker-
assisted spatial phase unwrapping and multi-frequency temporal phase
unwrapping are feasible to calculate the absolute phases in practice:
(1) Marker-assisted spatial phase unwrapping. The specular speci-
mens are usually smooth and continuous surfaces, such as mirrors.
The noise on phase values are generally low and the phase is easy
to unwrap in most cases. The spatial phase unwrapping methods
[46, 47] are good solutions for such a simple unwrapping task, but
additional markers may be needed to determine the reference fringe
orders to keep the consistency of fringe orders in each measurement.
(2) Multi-frequency temporal phase unwrapping. By sequentially
displaying and capturing additional fringe patterns with several de-
signed frequencies, the multi-frequency temporal phase unwrapping
methods [48–50] utilize the phase relations between frequencies to
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5. L. Huang et al. Optics and Lasers in Engineering 107 (2018) 247–257
Fig. 7. Surface normal N can be calculated from the normalized vector of probe
ray p and reflected ray r.
determine the fringe orders. Comparing to the spatial phase unwrap-
ping, it is a more general solution for the absolute phase calculation.
In most of PMD applications, multi-frequency temporal phase un-
wrapping is employed as a flexible and robust phase unwrapping
solution.
2.3.3. Slope calculation
Once the absolute phase values are retrieved, their locations on the
screen can be determined since the period of the fringe pattern on the
screen is a known parameter, if the pattern is calibrated or the screen
pixel size can be trusted. Moreover, with the geometric calibration pa-
rameters, the global coordinates of the phase-marked intersection point
m in Fig. 7 can be determined. Furthermore, the normalized vector of
the reflected ray r can then be calculated with one of the regularization
approaches according to setups in Figs. 3 and 4.
For each pixel, the normalized vector of the probe ray p is deter-
mined by the camera calibration [51, 52]. As illustrated in Fig. 7, the
SUT normal vector N can be determined by
𝐍 = 𝐫 − 𝐩 =∶
⎛
⎜
⎜
⎝
𝑁𝑥
𝑁𝑦
𝑁𝑧
⎞
⎟
⎟
⎠
, (6)
where Nx, Ny, and Nz are the x-, y-, and z-components of the surface
normal N. The specimen surface x- and y-slopes (sx, sy) are therefore
calculated as
𝑠𝑥 = −
𝑁𝑥
𝑁𝑧
, (7)
𝑠𝑦 = −
𝑁𝑦
𝑁𝑧
. (8)
Once the slopes (sx, sy) and in-plane coordinates (x, y) are calculated,
the shape can be reconstructed from these gradient data.
2.4. Shape reconstruction from slopes
As shown in Fig. 8, the height distribution z is reconstructed from
the calculated coordinates (x, y) and slopes (sx, sy). This 2D integration
process can be express as
𝑧 = 𝑓int2
(
𝑥, 𝑦, 𝑠𝑥, 𝑠𝑦
)
, (9)
where 𝑓int2(⋅) stands for a 2D integration function.
Of course, if there are other trustable information, they can be in-
cluded in the integration process as constraints to obtain a more reliable
optimization. There are mainly three classes of reconstruction methods:
(1) Zonal reconstruction. From the Fried’s, Hudgin’s, and Southwell’s
algorithms in late 1970 s and early 1980 s [53–56] to the recent de-
velopment and improvement especially for PMD technique [57–62],
these zonal methods use the finite differences of height along x-and
y-directions to establish the relations between the measured slopes
with the unknown height values. Iterative compensations [57],
higher order difference format [58], radial-basis-functions-based as-
sistance [60], splines [61], and quadrilateral geometry [61] are
investigated in order to accurately estimate the height values by
matching their derived first derivatives with the measured slopes
in the least squares sense.
(2) Modal reconstruction. These reconstruction methods are based on
analytical models. By taking the first derivatives in x- and y-direction
of the analytical expressions, the model coefficients can be approx-
imated by fitting the measured slopes with analytical slopes. The
commonly used models include polynomials (e.g. Zernike [63–66],
Chebyshev [66], Legendre [67] and B-spline [64, 68]) and sinusoidal
curves (e.g. Fourier transform [69] and cosine transform [70]). Once,
the model coefficients are estimated from the slope fitting, the height
distribution can be calculated by using the coefficients as a weight
onto their corresponding modes.
(3) Piecewise reconstruction. The shape reconstruction method by us-
ing radial basis functions [71, 72] works subset by subset (usually
40 × 40 pixels or less) due to its large memory cost. A stitching pro-
cess is required if the dataset is larger than a single subset. A zonal
method can assist in stitching the height pieces [73].
3. Background and recent development
As the pioneers in profiling specular surface, Petz et al. conducted the
studies on measuring specular reflecting surface by imaging of two grat-
Fig. 8. The x-slope (a) and y-slope (b) can be used to reconstruct the height (c) via a 2D integration process.
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6. L. Huang et al. Optics and Lasers in Engineering 107 (2018) 247–257
ing planes which were physically moved in parallel [26, 27]. The year
of 2004 was a big year for PMD technique. Knauer et al. [24] published
their work on measuring specular free-form surfaces with deflectometry
based on fringe phase measurement, named phase measuring deflec-
tometry for the first time. In this work, many technical aspects were
addressed, including the fundamental principle, physical limit, system
calibration, and stereo solution for ambiguity. Bothe et al. [25] pre-
sented their research on specular surface measurement by using fringe
reflection technique from the phenomena of the reflection to the princi-
ple of the technique, and to the practical demonstration. Since then, as
a novel technique for free-form specular shape measurement, PMD has
been widely spread and studied by many research groups for different
applications.
(1) Optical inspection. Häusler et al. proposed a microscopic PMD sys-
tem with nanometer sensitivity for local surface features [74]. Tang
et al. utilized their modified PMD method to measure the 3D shape
of aspheric mirrors [37]. Balzer and Werling gave an excellent re-
view on the shape measurement from specular reflection from prob-
lem modeling to hardware and strategies selections [28]. Faber et al.
compared several options in solving the practical problem of para-
sitic reflection from the rear side of a transparent specimen [75]. Su
et al. proposed Software Configurable Optical Test System (SCOTS)
which has been used to measure large optics such as the primary
mirrors of telescopes [76, 77], and it demonstrated the first trial
to measure synchrotron mirrors with a full-field deflectometry sys-
tem [78–80]. In astronomy applications, Sironi et al. employed the
PMD technique to evaluate the free-form telescope mirrors [81, 82].
Häusler et al. discussed PMD technique from the physicist’s point of
view and the information theoretical point of view and compared it
with interferometry [83, 84]. Faber et al. gave a comparative assess-
ment of the strengths and weaknesses of PMD comparing to interfer-
ometry [85]. Liu et al. proposed Direct PMD (DPMD) which utilized
the geometric relations of parallel planes to directly calculate the
height from phase information and this approach can measure dis-
continuous specular objects [42, 86].
(2) Fast and dynamic measurements. Instead of using fringe orienta-
tion in one direction, Huang et al. applied two-dimensional crossed
fringe patterns in fringe reflection technique to measure dynamic de-
formation on specular surfaces [35]. With the advanced fringe pro-
cessing algorithms, both x- and y-slopes can be determined from a
single image acquisition [36]. Phase shifting algorithm for the two-
dimensional crossed fringe images was developed by Liu et al. for
applications allowing only a one-dimensional physical translation of
the crossed fringe pattern [44]. Liang et al. applied the single-shot
PMD technique to measure the corneal topography [87].
(3) Deformation measurement. Because the PMD technique is ex-
tremely sensitive to the out-of-plane deformations and its tolerance
to system calibration error is higher when measuring relative de-
formation, it becomes more and more attractive to apply the PMD
technique for deformation or defect measurement [36, 88, 89]. For
instance, the PMD technique has been used to measure influence
functions of a deformable mirror to perform closed-loop feedback
and control [90].
(4) Study on misalignment. Similarly, the influence of the misalign-
ment can also be determined by using PMD technique [91]. Davies
et al. utilized the PMD technique to determine the influences of 5°-
of-freedom misalignment of segmented telescope mirrors [92].
(5) Working beyond the visible light. With the success of PMD in the
visible region, it has been extended to other ranges of the wavelength
spectrum for specific applications. Sprenger et al. proposed a novel
method using Ultraviolet (UV) source in deflectometry to avoid the
parasitic reflections from the rear side of transparent specimens [93].
Su et al. explored deflectometry with Infrared (IR) source and camera
to measure rough optical surfaces [94]. Due to the lack of convenient
devices like the TFT LCD monitor in visible light to generate the
phase-shifted fringe patterns, instead of measuring the fringe phases,
the methods working with UV and IR sources calculate the intensity
peak while scanning a slit across the radiation source.
4. Related measurement techniques
There are several related techniques linked to the PMD technique.
Their similarities and differences are addressed in this work to view
these similar techniques from different angles.
4.1. Comparison with phase measuring profilometry
From the data acquisition to the image processing, the PMD is very
similar to the Phase Measuring Profilometry (PMP) [9, 18] which mea-
sures the 3D shape of diffused surfaces by using fringe projection. PMP
has a longer study history going back to early 1980 s [18]. In both
techniques, their raw data are commonly sinusoidal fringe patterns.
Their fringe analysis procedure similarly includes fringe demodulation
[19, 43] and phase unwrapping [46, 47, 50]. The fringe pattern(s) can
be demodulated by using least-squares-based phase shifting algorithms
[19] or single-frame demodulation techniques such as transform-based
methods [43]. Therefore, they share the same practical issues as well as
the existing solutions in fringe analysis, such as the nonlinear response
of the digital light device [95–98].
Although these two techniques measure the 3D shape based on the
geometry relation of the source and detectors. There is a major differ-
ence: PMP measures the diffuse surface based on the optical triangula-
tion as illustrated in Fig. 9(b), while PMD works with the specular re-
flection based on the law of reflection in Fig. 9(a). The retrieved phase
values in PMP are directly related to height data, while the phase values
in PMD are linked to both slopes and height. Therefore, proper regular-
ization and numerical integration are needed in PMD. When measuring
partially specular and diffuse surfaces, these two techniques can be com-
bined by using the PMP height data to regularize the inverse problem
of PMD [99, 100].
4.2. Comparison with pointwise scanning deflectometry
Based on the law of reflection, the PMD technique measures the full-
field SUT slopes from each measurement as shown in Fig. 10(a). Relying
on the same principle, the single-point deflectometry technique, such as
autocollimator illustrated in Fig. 10(b), only provides readings of the
intensity-weighted average x- and y-slopes in a small area in each mea-
surement.
The single-point deflectometry technique is commonly more pre-
cise than the full-field PMD technique. For instance, the measuring
precision is typically around 0.1 μrad RMS for the Long Trace Pro-
filer [101], Nanometer Optical component measuring Machine [102],
or Nano-accuracy Surface Profiler [103, 104] which are widely used
for the synchrotron mirror inspection, while the measuring precision of
full-field PMD technique is typically larger than 10 μrad RMS. Compar-
ing to the pointwise scanning deflectometry, the PMD technique is good
for full-field measurements with higher speed. Moreover, the field of
view and dynamic range of PMD systems are much easier to adjust for
different applications.
4.3. Comparison with Hartmann wavefront sensing
Turning to the system configuration and measuring process, the PMD
is similar to the Hartmann Wavefront Sensor (HWS) [105] in some as-
pects. As Su et al. [76] claimed, their SCOTS can be considered as a
reverse Hartmann test. The similarities between PMD and HWS include
that the incoming rays in HWS are similar to the camera probe rays in
PMD. By employing the fringe phase values, the screen in PMD is func-
tional as the detector in HWS as shown in Fig. 11. In both techniques,
the direct readouts in a single measurement are x- and y- slopes in a 2D
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7. L. Huang et al. Optics and Lasers in Engineering 107 (2018) 247–257
Fig. 9. PMD (a) and PMP (b) share many similar aspects in pattern generation, image acquisition, and fringe analysis.
Fig. 10. PMD measures the full-field slopes (a) while the pointwise deflectometry measures the average slopes of a small area (b).
Fig. 11. The sampling point on the PMD can vary by the specimen distance (a), while the sampling points of HWS with respect to its detector is always fixed (b).
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8. L. Huang et al. Optics and Lasers in Engineering 107 (2018) 247–257
grid with coordinates (x and y), or in a simple phrase, two pieces of slope
maps. The height or wavefront is calculated from x, y, x-slope, and y-
slope. Therefore, they share some technical aspects, such as wavefront
reconstruction algorithms. In fact, at the very beginning of the PMD
study, the 2D integration techniques are learned from the earlier wave-
front reconstruction work for the HWS [106]. Moreover, looking at the
potential post-processing on the measurement results, the surface shape
in PMD and the wavefront in HWS are commonly decomposed by us-
ing Zernike, Chebyshev, or Legendre polynomials for optical aberration
analysis.
However, their differences are obvious as well. The calibration in
HWS determines the geometric relations between the detector and the
pinholes on the Hartmann mask, while the calibration in PMD deter-
mines the relative geometric relations between the camera probe rays
and the screen. The sub-aperture pinholes in HWS are fixed once the
mask is mounted with the detector, so the sampling positions will not
change with respect to the detector. As illustrated in Fig. 11, although
the probe rays from the camera are consistent in PMD, the sampling
positions still can vary along the height level of the surface under test
(SUT). This difference will introduce two major issues in PMD. First, the
slope calculation in PMD is not as straightforward as it is in HWS which
always calculates slopes with the calibrated pinhole-detector distance.
Instead, by only knowing the geometric relation between camera probe
rays and display screen in PMD, the height-slope ambiguity exists when
calculating the slope with unfixed sampling position. It requires addi-
tional reasonable hypothesis or other means to regularize this ill-posed
problem. Second, the pinhole array of HWS is commonly in a regular
geometry, which makes the wavefront reconstruction relatively easier.
Although the 2D integration techniques in PMD originate from wave-
front reconstruction in HWS, the slope data in PMD have its own fea-
tures: owing to the unfixed sampling position and off-axis perspective,
the slope samplings in world coordinates are not in a regular grid, such
as the rectangle, but more generally in a quadrilateral grid. This issue
requires additional modifications on the classical wavefront reconstruc-
tion algorithm to reduce the integration error due to the irregular grids
[62].
5. Major challenges, current solutions, and remaining
deficiencies
As mentioned above, there are many technical challenges in PMD.
After years of research and development, some of the earlier difficulties
have been overcome with the current solutions, and some deficiencies
are still remaining and require more robust and flexible solutions.
5.1. Calibration
System calibration is and will always be a mandatory and critical
procedure before any metrology instrument able to deliver a reliable
measurement. The calibration process of a PMD can usually be divided
into the geometry calibration and the screen calibration. The purpose
of geometry calibration is to determine the geometric relations of the
display screen with respect to the camera rays (based on camera cali-
bration). Reference flat mirror with markers can be used to complete
the geometry calibration [23]. Inspired by the idea of bundle adjust-
ment as an iterative self-consistency approach, Olesch et al. developed
a self-calibration procedure for arbitrary specular surfaces to improve
the global accuracy of the reconstructed height data [34]. Xiao et al.
proposed a simple and flexible approach to carry out the PMD calibra-
tion with a marker-less flat mirror and optimization with bundle ad-
justment [107]. This method does not need any markers on the cali-
bration flat mirror, which is a great advantage because the accuracy
of the marker positions directly influences the calibration result in the
traditional method. Based on Xiao’s calibration method for monoscopic
PMD, Ren et al. further developed a calibration method for the stereo-
scopic PMD [108]. The stereoscopic PMD setup is first treated as two
separate monoscopic PMD systems and calibrated one by one with it-
erative optimization, and then they are merged into one cost function
to have an overall optimization. Instead of utilizing the PMD devices
only, laser tracker with retro-reflectors has been used to calibrate the
distances to assist the PMD calibration [79]. By using the laser tracker,
the calibration procedure gets more possibilities to define different ref-
erence planes and get distance and dimensions more flexible and easier,
instead of relying on camera vision only. Zhou et al. proposed a flexible
and simple PMD calibration method based on the combination of re-
flection rays determined by the varied points on a screen and reflection
images of a plane mirror without fiducials placed at three different loca-
tions [109]. In some strategies with additional geometry assumptions,
such as the parallel planes in DPMD [42, 86], particular alignment and
calibration techniques are needed to reduce the systematic error. How-
ever, since the low-frequency slope errors are much easier to propa-
gate into the height data as low-frequency profile errors via integration,
even with the existing PMD calibration approaches, there is still an open
question on how to easily determine the geometry relations with better
accuracy to fully utilize the high sensitivity of the slope measurement.
Nevertheless, it is noteworthy that because the operation of taking
the difference from the reference makes the deformation result less sen-
sitive to the imperfection of the calibration, the calibration requirement
for deformation measurements is lower than the calibration requirement
for absolute shape measurements [88]. This makes the PMD very suit-
able for the out-of-plane deformation measurement.
Screen imperfection was considered and a calibration procedure was
proposed in Refs. [23, 110]. In many PMD measurements reported in
the literature, the screen imperfection has not yet been well addressed
or carefully calibrated. With the further development in geometric cal-
ibration for PMD, at a certain point, the screen imperfection will need
to be carefully considered, or the PMD system may be more smartly set
up to minimize the influence by the screen imperfection.
5.2. Height-slope ambiguity
As mentioned earlier when introducing the setups of PMD in
Section 2.1, there are several approaches proposed to regularize the in-
verse problem, e. g. shape assumption [35–37], translated screens [26,
38, 39], distance regularization [41], and stereovision [24]. Besides,
The model-based optimization [111–114] is another way to deal with
this issue. In modal PMD (MPMD) [111], models with analytical expres-
sions are used to represent the shape of SUT. In this way, both height
and slopes are represented with a set of coefficients of modes, and the
ray tracing correspondence on the screen can be adjusted by changing
these coefficients. By optimizing these coefficients to minimize the dis-
crepancy between ray tracing and actual measurement, the slope and
height can be determined at the same time to best explain the measure-
ment data.
5.3. Rotational vectors
Theoretically, the gradient data of the SUT must be a conservative
vector field. However, due to the phase errors, and mainly the calibra-
tion errors and improper assumptions, the measured slope results con-
tain rotational vectors, instead of being a pure conservative vector field.
These rotational vectors will introduce problems to the following 2D in-
tegration process [115]. How to reduce this type of error is a challenging
task. Recently there is a method proposed by Xiao et al. for processing
the rotational vectors in deflectometry with sparse representation [116].
The MPMD can involve the screen pose to the optimization to extremely
reduce the residual rotational vectors [111]. Better initial value from
pre-calibration is always helpful to avoid a nonlinear optimization in
MPMD being stuck in a local minimum.
254

9. L. Huang et al. Optics and Lasers in Engineering 107 (2018) 247–257
5.4. Shape reconstruction from slopes (2D integration)
By learning from the wavefront reconstruction in wavefront sensing
[106] and after several years of investigation especially for PMD [57–
62, 70, 71, 73, 115, 117], the 2D integration for shape reconstruction
from slopes has been well developed.
The performance of the modal reconstruction in terms of accuracy
and speed mainly depends on how close the measured x-and y-slopes
can be fitted with a finite number of the selected modes in the chosen
mathematical model. The shape reconstruction becomes a representa-
tion by the selected modes weighted by the coefficient determined from
slope fitting process.
On the other hand, the performance of the zonal reconstruction is
mainly based on the selected format of the finite difference and the
sampling grid geometry of the slope data. A recent work on the zonal
reconstruction provides a general solution in a more common grid in
PMD with good performance in both accuracy and speed [62].
Many practical issues have been considered in the zonal and modal
reconstruction methods, e.g. the incomplete dataset with arbitrary aper-
ture [70], high accuracy [58, 61, 66], speed comparison [62, 115], and
irregular grids [62, 71]. A lot of reconstruction algorithms are ready to
be used in order to meet various requirements in accuracy and speed for
different applications.
6. Conclusions
With many years of research and development, a few aspects of the
PMD technique have been studied from its physical limitations and sys-
tem calibration to the regularization and shape reconstruction. As a high
dynamic range and low-cost measurement solution, the PMD technique
has been applied to several scientific applications for deformation, cur-
vature, and shape measurement. Accurate calibration for absolute shape
measurement still needs to be improved and it is a current limitation of
the PMD technique. Further investigations are still necessary to make
the PMD technique more flexible to calibrate, easier to use, and more
accurate in practical measurements for industrial applications.
Acknowledgment
This research used resources of the National Synchrotron Light
Source II, a U.S. Department of Energy (DOE) Office of Science User
Facility operated for the DOE Office of Science by Brookhaven National
Laboratory under Contract No. DE-SC0012704.
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