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Citation: Chen, X.; Zhang, J.; Xi, J. 3D
Metrology Using One Camera with
Rotating Anamorphic Lenses. Sensors
2022,22, 8407. https://doi.org/
10.3390/s22218407
Academic Editors: Shichang Du,
Yiping Shao and Delin Huang
Received: 5 October 2022
Accepted: 31 October 2022
Published: 1 November 2022
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sensors
Article
3D Metrology Using One Camera with Rotating
Anamorphic Lenses
Xiaobo Chen 1, * , Jinkai Zhang 2, * and Juntong Xi 1
1School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
2School of Mechanical Engineering, Jinan University, Jinan 250022, China
*Correspondence: xiaoboc@sjtu.edu.cn (X.C.); me_zhangjk@ujn.edu.cn (J.Z.)
Abstract:
In this paper, a novel 3D metrology method using one camera with rotating anamorphic
lenses is presented based on the characteristics of double optical centers for anamorphic imaging.
When the anamorphic lens rotates
−
90
◦
around its optical axis, the 3D data of the measured object
can be reconstructed from the two anamorphic images captured before and after the anamorphic
rotation. The anamorphic lens imaging model and a polynomial anamorphic distortion model are
firstly proposed. Then, a 3D reconstruction model using one camera with rotating anamorphic
lenses is presented. Experiments were carried out to validate the proposed method and evaluate
its measurement accuracy. Compared with stereo vision, the main advantage of the proposed 3D
metrology approach is the simplicity of point matching, which makes it suitable for developing
compact sensors for fast 3D measurements, such as car navigation applications.
Keywords: 3D reconstruction; anamorphic lens; anamorphic stereo vision
1. Introduction
In modern industry, 3D metrology is an important technology, based on various
methods. Optical metrology and non-optical metrology are the two categories of 3D metrol-
ogy [
1
]. Non-optical 3D metrology methods may use a coordinate measuring machine
(CMM), a scanning electron microscope, and a scanning probe microscope [
2
,
3
], whereas
optical 3D metrology methods may employ chromatic confocal microscopy, point auto-
focus instruments, focus variation instruments, phase-shifting interferometry, coherence
scanning interferometry, imaging confocal microscopy, and stereo vision. An overview
of such methods was presented in detail [
4
–
6
]. Each 3D metrology method has its own
properties and differs in application areas, measurement accuracy, scale, efficiency, cost, etc.
Optical metrology procedures are often fast, precise, and non-destructive of the measured
objects. Stereo vision, which uses two cameras with spherical lenses, is a frequently used
optical metrology approach [
7
–
9
]. The measuring precision of stereo vision can range from
several microns to several millimeters, depending on the measured area and the spherical
lenses used. The most challenging aspect of stereo vision involves the extraction of the
corresponding points from stereo images. Projectors are widely used to generate structured
light to achieve high-quality corresponding points [
10
]. Stereo vision has been widely used
in many areas, such as industry metrology, agriculture, daily living, etc. [11,12]
Anamorphic lenses are increasingly employed in the film industry to capture wide
images for broad screens [
13
–
15
]. Anamorphic lenses are distinguished by two characteris-
tics: double focal lengths and double optical centers [
16
–
18
]. Double focal lengths means
the focal lengths differ greatly in the tangential and sagittal planes. This characteristic of
anamorphic lenses is often used in the film industry, as it allows images to be compressed
horizontally. Metrology, physics, and some optical imaging applications also use anamor-
phic lenses [
19
–
22
]. Double optical centers means there are two optical centers, one in the
tangential plane and the other in the sagittal plane. The distance between the two optical
Sensors 2022,22, 8407. https://doi.org/10.3390/s22218407 https://www.mdpi.com/journal/sensors
Sensors 2022,22, 8407 2 of 17
centers is the anamorphic distance, which is an intrinsic quantity for anamorphic lenses.
The anamorphic distance will provide anamorphic images with more 3D information,
making anamorphic lenses considerably more suitable for 3D metrology. H. Durko used
anamorphic lenses for measuring parts with large aspect ratios [
23
]. F. Blais performed an
exploratory calibration of anamorphic lenses based on a pinhole imaging model [
24
]. In
our previous work, a high-precision anamorphic lens calibration method was proposed
with 3D and 2D calibration targets [16,25].
In this study, a novel 3D metrology method using one camera with rotating anamor-
phic lenses is proposed based on its characteristic of double optical centers. When the
anamorphic lens rotates
−
90
◦
around its optical axis, the 3D data of the measured ob-
ject can be reconstructed from the two anamorphic images captured before and after the
anamorphic rotation. The rest of this paper is organized as follows. Section 2presents the
anamorphic lens imaging model as well as the anamorphic distortion model. Section 3.1
describes the 3D reconstruction process employing rotating anamorphic lenses. Section 3.2
describes the point matching approach, and Section 3.3 depicts the rotating anamorphic
stereo vision. Section 4describes the experiments and evaluates the proposed method’s
measurement accuracy. Finally, Section 5concludes the paper.
2. Anamorphic Imaging Model
This section presents the imaging model and the distortion model of anamorphic
lenses, which differ significantly from those based on spherical lenses. Anamorphic lenses
are typically composed of rear spherical lenses on the back and an anamorphic attach-
ment on the front. The anamorphic attachment consists of cylindrical lenses that have no
optical power in planes parallel to the cylindrical axis but have optical power in planes
perpendicular to it.
2.1. Anamorphic Imaging Model
The imaging model of anamorphic lenses differs from the traditional pinhole imag-
ing model of spherical lenses. The anamorphic distance and the anamorphic angle are
introduced as new extra intrinsic parameters [
16
]. Figure 1depicts the imaging model
of anamorphic lenses, where ad denotes the anamorphic distance, and aa denotes the
anamorphic angle. If the anamorphic lens is precisely assembled, the anamorphic angle
aa, as shown in Figure 1, can be as small as zero. The cylindrical lenses or the anamorphic
attachment determine the anamorphic distance ad, which must be precisely calibrated.
O
W
-X
W
Y
W
Z
W
are the world coordinates, O
ci
-X
ci
Y
ci
Z
c
are the camera coordinates centered
in the CCD plane, and O
p
-X
p
Y
p
Z
c
are the pixel coordinates centered in the CCD plane. O
cx
and O
cy
are the optical centers in the horizontal plane and the vertical plane, and f
x
and f
y
are the focal lengths in the two planes. The imaging model for anamorphic lenses can be
expressed as follows:
XC
YC
ZC
=R T
XW
YW
ZW
1
YI
fy=YC
ZC+ad
XI
fx=XC
ZC
XP
YP=cos(aa)sin(aa)
−sin(aa)cos(aa) XI
YI
(1)
where [X
w
;Y
w
;Z
w
] is a point in the world coordinates, [X
c
;Y
c
;Z
c
] is the point expressed in
the anamorphic coordinates. Rand Tdenote the rotational and translating matrix from the
world coordinates to the anamorphic coordinates. (X
I
,Y
I
) are the image coordinates in the
camera coordinates in the image plane, and (X
P
,Y
P
) are the image coordinates in the pixel
coordinates in the image plane.
Sensors 2022,22, 8407 3 of 17
Figure 1.
Anamorphic imaging model. (
a
) Imaging rays in the Y
c
-O
cy
-Z
c
plane; (
b
) imaging rays in
the Xc-Ocx-Zcplane.
2.2. Anamorphic Distortion Model
Anamorphic lenses have a distinct distortion model with respect to spherical lenses.
The front anamorphic attachment introduces many more aberrations [
26
]. Based on the
aberration theory and numerical experiments for anamorphic lenses, we proposed a poly-
nomial distortion model for anamorphic lenses, which allows for high-precision camera
calibration [
25
]. As shown in Equation (2), anamorphic distortions can be divided into
radial distortions (X
rad
,Y
rad
), third-order distortions (X
3
,Y
3
), and second-order distortions
(X
2
,Y
2
). (x
d
,y
d
) denotes the distorted image coordinates, and (x
c
,y
c
) denotes the undis-
torted image coordinates. [k
1
,k
2
,n
1
,n
2
,m
1
,m
2
,x
21
,x
12
,x
03
,x
20
,x
11
,x
02
,y
30
,y
21
,y
12
,y
03
,
y
20
,y
11
,y
02
] are the 19 distortion coefficients for anamorphic lenses that must be calibrated.
xc=xd1+k1x2
d+y2
d+k2x2
d+y2
d2+m1a2x2
d+y2
d+m2a2x2
d+y2
d2
| {z }
Xrad
+x21x2
dyd+x12xdy2
d+x03y3
d
| {z }
X3
+x20x2
d+x11xdyd+x02y2
d
| {z }
X2
yc=yd1+n1x2
d+y2
d+n2x2
d+y2
d2+m1a2x2
d+y2
d+m2a2x2
d+y2
d2
| {z }
Yrad
+y30x3
d+y21x2
dyd+y12xdy2
d+y03y3
d
| {z }
Y3
+y20x2
d+y11xdyd+y02y2
d
| {z }
Y2
(2)
3. 3D Metrology Using Rotating Anamorphic Lenses
3.1. Description of 3D Metrology Using Rotating Anamorphic Lenses
As shown in Figure 2a, the vertical position is the first anamorphic position, and
the cylindrical axis of the anamorphic attachment is vertical. After a
−
90
◦
rotation, the
cylindrical axis of the anamorphic lens is horizontal, and this anamorphic position is known
as the horizontal position. The anamorphic lens can switch between the two positions by
rotating. The imaging model of anamorphic lenses shown in Section 2can be reduced to an
ideal anamorphic lens for ease of analysis, which implies that the anamorphic angle is zero,
Sensors 2022,22, 8407 4 of 17
there are no distortions, and the rotation angle is precisely
−
90
◦
without decentering. As
shown in Figure 1and Equation (1), the following equations describe the vertical position
of an ideal anamorphic lens:
xv=XC
ZCfx
yv=YC
(ZC+ad)fy
(3)
where (x
v
,y
v
) refer to the image position of the object point P
V
=(X
C
,Y
C
,Z
C
), and P
V
is
in the anamorphic coordinates in the vertical position. After the anamorphic rotation, the
coordinates of the object point in the anamorphic coordinates change as follows:
PH=RPV(4)
Figure 2.
The two anamorphic positions. (
a
) Vertical position and (
b
) horizontal position which is
achieved by a rotation of the anamorphic lens in the vertical position by −90◦.
If the anamorphic lens rotates alone its optical axis by −90◦,Rcan be expressed as:
R=
0 1 0
−100
0 0 1
(5)
Thus, when the anamorphic lens is rotated to its horizontal position, the object coordi-
nates can be calculated as PH=[YC;−XC;ZC], and we have the following equations:
xh=YC
ZCfx
yh=−XC
(ZC+ad)fy
(6)
Given Equations (3) and (6), the coordinates of the object point can be calculated as:
XC=ad xvyv
fyxh−fxyv
YC=ad xhyv
fyxh−fxyv
ZC=ad fxyv
fyxh−fxyv
(7)
Thus, given the internal parameters f
x
,f
y
, and ad, as well as the image coordinates (x
v
,
y
v
) and (x
h
,y
h
) of an object point in an ideal anamorphic lens, the 3D coordinates of this
point can be easily reconstructed using Equation (7).
Sensors 2022,22, 8407 5 of 17
3.2. Point Matching
Point matching is crucial for stereo vision but is also challenging because the two
images are captured by two different cameras. Points in different positions will have
different image points in stereo images. Epipolar constraints and projectors are commonly
adopted to achieve robust point matching, which requires a complex calculation. Point
matching limits the applications of stereo vision in areas requiring fast or real-time 3D
reconstruction, such as car navigation. Compared with stereo vision, point matching
is much simpler in our proposed 3D metrology method. A simulation is provided to
demonstrate its image point matching, as shown in Figures 3–6. The simulated anamorphic
lens has the following internal parameters: ad = 30 mm, aa = 0
◦
,f
x
= 12 mm, f
y
= 16 mm,
and no distortions. As shown in Figure 3, the object to be built is a collection of points on a
spherical surface, with their coordinates given in the vertical position of the anamorphic
lens. The simulated image points on the image planes are shown in Figure 4, where the dot
points and the circle points express the image points when the anamorphic lens is in the
vertical and horizontal positions, respectively. It is not easy to match the image points from
these two positions in the current stage. In our method, the point matching can be greatly
simplified by using a parameter of an anamorphic lens known as the anamorphic ratio AR,
which is:
AR ≈fy
fx(8)
Figure 3.
Simulated object points on a spherical surface in anamorphic coordinates of the vertical position.
Figure 4.
Simulated image points on the image plane. The dot points refer to the image points when
the anamorphic lens is in the vertical position, and the circle points refer to the image points when
the anamorphic lens is in the horizontal position.
Sensors 2022,22, 8407 6 of 17
Figure 5.
Simulated image points after anamorphic ration (AR) expansion. The dot points are rectified
horizontally by AR, and the circle points are rectified vertically by AR.
Figure 6. Simulated image points tracing from the vertical position to the horizontal position.
As a result, if the dot points and circle points in Figure 4are expanded horizontally
and vertically by AR, respectively, we obtain the corresponding image points shown in
Figure 5. Because the corresponding points are nearby dot points and circle points, point
matching in rotating anamorphic lenses becomes a simple and straightforward task. In
addition, as illustrated in Figure 6, we can trace the image positions during the anamorphic
rotation from the dot points to the circle points for point matching. Point matching in
anamorphic rotating 3D metrology is much easier than stereo vision with spherical lenses,
as shown in Figures 5and 6. The deviations between the dot points and the circle points
are due to the anamorphic distance ad. In this example, the anamorphic distance of 30 mm
was much smaller than the object distance of 1500 mm. The deviation between the dot
points and the circles in the image planes will increase when the object points are imaged
using an anamorphic lens with a large anamorphic distance.
3.3. Stereo Vision with Anamorphic Lenses
Inevitably, there will be a misalignment between the optical axis of the anamorphic
lens and the rotation axis of the rotary table, which will be magnified if the object is far
away from the lens. As a result, large errors will occur if Equation (7) is applied directly
for 3D reconstruction. A rational approach for high-precision 3D metrology is to treat the
Sensors 2022,22, 8407 7 of 17
rotating anamorphic lenses as two distinct anamorphic lenses, which constitute a stereo
vision with anamorphic lenses. The main differences between stereo vision and stereo
vision with anamorphic lenses are that spherical lenses are replaced by anamorphic lenses
and the baseline distance between the two anamorphic lenses is extremely small.
For 3D metrology using rotating anamorphic lenses, we should first calibrate the
internal parameters of the anamorphic lenses according to the anamorphic imaging model
and the anamorphic distortion model. A 3D calibration target can be used to calibrate the
anamorphic lens [
16
]. As can be seen in Equation (2), the distortion model of the anamorphic
lens has 19 unknown distortion coefficients. If only one image of a 3D calibration target is
used for anamorphic lens calibration, local convergence is likely to occur for the distortion
coefficients. Commonly, the 3D calibration target cannot cover the entire imaging area,
which will lead to a poor calibration result for other imaging areas. Therefore, to achieve a
stable calibration result for the entire imaging area, mixed calibration targets are adopted,
which means 3D calibration targets and 2D calibration targets are both required [
25
]. The
initial values of the internal parameters are calibrated using the 3D calibration target, and
those parameters are refined using 2D calibration targets. After anamorphic calibration
using mixed calibration targets, the internal anamorphic lens parameters [f
x
,f
y
,u
0
,v
0
,ad,
aa] and the 19 distortion coefficients [k
1
,k
2
,n
1
,n
2
,m
1
,m
2
,x
21
,x
12
,x
03
,x
20
,x
11
,x
02
,y
30
,y
21
,
y12,y03 ,y20,y11 ,y02] can be determined.
After anamorphic lens calibration, we should calibrate the relative position of the
anamorphic lens in the vertical position and the horizontal position. Two images of the
3D calibration target are captured, one for the vertical position and one for the horizontal
position. Based on the calibrated internal parameters of the anamorphic lens and the image
of the 3D calibration target, the relative position [t
x
,t
y
,t
z
,a,b,r] between the 3D calibration
target and the anamorphic lens can be easily calibrated:
T=
tx
ty
tz
(9)
R=
cos(b)cos(r)sin(a)sin(b)cos(r)−cos(a)sin(r)sin(a)sin(r) + cos(a)sin(b)cos(r)
cos(b)sin(r)cos(a)cos(r) + sin(a)sin(b)sin(r)cos(a)sin(b)sin(r)−sin(a)cos(r)
−sin(b)cos(b)sin(a)cos(a)cos(b)
(10)
where Tand Rrefer to the translation vector and the rotation matrix. For the two anamor-
phic positions, there are the following equations between the coordinates of the 3D calibra-
tion target and the two anamorphic coordinates in the two positions:
PCV =RW
CV PW+TW
CV
PCH =RW
CH PW+TW
CH
(11)
where the superscript Wrefers to the coordinates of the 3D calibration target, and the
subscripts CV and CH refer to the coordinates of the anamorphic lens in the vertical and
horizontal positions. Given Equation (11), the relationship between the vertical and the
horizontal anamorphic coordinates can be determined as follows:
PCH =RCV
CH PCV +TCV
CH (12)
where RCV
CH and TCV
CH are determined as follows:
RCV
CH =RW
CH RW
CV −1
TCV
CH =TW
CH −RCV
CH TW
CV
(13)
The horizontal position is rotated from the vertical position by
−
90
◦
along the optical
axis; thus, the relative position between the two anamorphic positions is similar to [t
x
,t
y
,t
z
,
a,b,r] = [0 mm, 0 mm, 0 mm, 0
◦
, 0
◦
, 90
◦
]. After the anamorphic lens calibration and the
Sensors 2022,22, 8407 8 of 17
stereo calibration for the rotating anamorphic lenses, the 3D reconstruction using rotating
anamorphic stereo vision is similar to that in stereo vision with spherical lenses.
4. Experiments
4.1. Experiments
An experiment was performed to validate the proposed 3D metrology method using
one camera with rotating anamorphic lenses. The design of the experiment is shown in
Figure 7. The camera was MER-504-10Gx-P from Daheng China, and its resolution was
2448
×
2448 with a pixel size of 3.45
µ
m
×
3.45
µ
m. The anamorphic lens was mounted on
the rotary table HK130-10. The servo motor was SGM7J-02A7C6E, and the servo controller
was SGD7S-1R6A00A002, both from YASKAWA, Japan. As shown in Figure 7, the 3D
calibration target was composed of two 2D planar calibration targets, and the relative
position of the two planar calibration targets was determined by the 3D calibration target
camera, which was previously calibrated. As shown in Figure 8, the anamorphic lens was
composed of a front anamorphic attachment and a 16 mm Computar spherical lens. As
shown in Figure 9, the anamorphic attachment comprised three cylindrical lens elements,
and the effective focal length in the YOZ plane was
−
2489 mm. A paraxial lens design
for anamorphic lenses can be seen in [
27
,
28
]. The lens parameters for the anamorphic
attachment are shown in Table 1.
Figure 7. Experiment of 3D metrology using rotation anamorphic lenses.
Sensors 2022,22, 8407 9 of 17
Figure 8.
Anamorphic lens composed of a front anamorphic attachment and a rear spherical lens.
The anamorphic lens was mounted on a rotary table which could rotate the anamorphic lens by
−
90
◦
along the optical axis. (a) Side view; (b) Front view.
Figure 9. Anamorphic attachment for a Computar 16 mm spherical lens in the YOZ plane.
Table 1. Parameters of the anamorphic attachment used for the anamorphic lens.
e0(mm) e1(mm) e2(mm) T11 (mm) T12 (mm) T2(mm)
16.163062 30 40 4 4 4
f(mm) R11 (mm) R12 (mm) R13 (mm) R21 (mm) R22 (mm)
16 155 156.4 −857.6 −174.8 124.5
n1V1n2V2n3V3
1.516797 64.212351 1.672702 32.17888 1.516797 64.212351
The anamorphic lens was calibrated using 3D and 2D calibration targets, and the
internal parameters are shown in Table 2. And as shown in Figure 7, the relative position
of the vertical position with respect to the horizontal position was calibrated using the
3D calibration target, and the results were [t
x
,t
y
,t
z
,a,b,r]=[
−
0.8943 mm, 0.3591 mm,
−
0.1712 mm,
−
0.5586
◦
,
−
0.8074
◦
, 90.0059
◦
]. After the anamorphic lens calibration and the
rotating anamorphic stereo calibration, we conducted the 3D metrology using the rotating
anamorphic lens. As shown in Figure 10, to achieve dense 3D points, a checkerboard
was adopted with a square’s length of 5 mm. The original anamorphic images are shown
in Figure 10. Figure 10a shows the image when the anamorphic lens was in the vertical
position, and Figure 10b shows the image when the anamorphic lens was in the horizontal
position. Figure 10b was achieved by rotating the anamorphic lens by
−
90
◦
anticlockwise
along the optical axis. Once the corresponding corners in Figure 10a,b were determined,
the 3D coordinates of the corners could be reconstructed from the rotating anamorphic
stereo vision.
Sensors 2022,22, 8407 10 of 17
Table 2. Internal parameters of the calibrated anamorphic lens.
fx(mm) fy(mm) u0(pixel) v0(pixel) ad (mm)
12.0520 16.1026 1.2790 ×1031.0023 ×10326.5516
aa (◦)k1k2n1n2
−0.5789 0.0304 −2.7151 ×10−50.0183 −3.3372 ×10−5
m1m2x21 x12 x03
−0.0155 1.277e-5 9.1986 ×10−5−0.0139 1.8928 ×10−5
y30 y21 y12 y03 x20
1.6727 ×10−50.011 1.0611 ×10−4−0.0023 1.3802 ×10−4
x11 x02 y20 y11 y02
−4.7071 ×10−55.8709 ×10−59.3657 ×10−5−3.9944 ×10−4−2.9851 ×10−4
Figure 10.
Anamorphic images. (
a
) Image when the anamorphic lens was in the vertical position,
and (b) image when the anamorphic lens was in the horizontal position.
One possible method to determine the corresponding corners between Figure 10a,b
is to trace the corners while rotating the anamorphic lens, similar to Figure 6. In order to
trace the corners correctly, the rotating speed must be limited. Another method is to rectify
the two anamorphic images by the anamorphic ratio AR, as shown in Equation (6), after
which the corresponding points are very close, as in Figure 5. Figure 11a was obtained by
expanding Figure 10a horizontally by the AR. Rotating Figure 10b by
−
90
◦
anticlockwise
and expanding the height of the rotated image by the AR, we obtained Figure 11b. As
can be seen in Figure 11a,b, the two images are very similar after AR rectification, and the
corresponding points are nearby, for the overlapping areas in Figure 11a,b.
Figure 11.
Anamorphic images after anamorphic ratio (AR) rectification. (
a
) Rectified image when
the anamorphic lens was in the vertical position, and (
b
) rectified image when the anamorphic lens
was in the horizontal position.
The chessboard corners in Figure 11a were extracted and represented as red points
in Figure 12; the corners in Figure 11b were represented by green circles in Figure 12. The
red points and green circles in Figure 12a are in their original positions. It can be seen
that the corresponding corners were not adjacent and were separated from one another.
This is because there was a misalignment between the optical axis and the rotation axis of
the rotary table. This misalignment was further amplified by the large object distance. If
Sensors 2022,22, 8407 11 of 17
Equation (7) is directly applied for 3D reconstruction, there will be significant errors. Thus,
the rotating anamorphic stereo vision was adopted for the 3D reconstruction. As can be
seen in Figure 11, the two images are similar, the only difference between Figure 11a,b
being a decentering. If we shift Figure 11b by (d
u
,d
v
)= (105.0792 pixels,
−
28.8297 pixels),
the two images will overlap. In Figure 12b, the corresponding points are very close, as
shown in Figure 5.
Figure 12.
Corners after anamorphic ratio (AR) rectification. The red points indicate the corners
in Figure 11a, and the green points indicate the corners in Figure 11b. (
a
) Image points in their
original position, (
b
) green points shift their positions entirely, after which the two images almost
overlap. This pixel decentering was due to the deviation between the optical axis and the axis of the
rotary table.
Given the internal parameters of the anamorphic lens, the relative positions between
the vertical anamorphic lens and the horizontal anamorphic lens, and the corresponding
image points, the 3D coordinates of the corners could be reconstructed using the rotating
anamorphic stereo vision. As shown in Figures 13 and 14, the 3D positions of the corners
in checkerboards were constructed. Figure 13 shows the images of the checkerboards
when the anamorphic lens was in the vertical position, and Figure 14 shows the images
of the calibration targets when the anamorphic lens was in the horizontal position. When
constructing the eight objects, the first two objects were 3D objects, and the other six objects
were 2D objects. The constructed 3D points are shown in Figure 15, while not all corners
in the checkerboard were reconstructed. Some points were no constructed for they lay in
the anamorphic gas where the construction accuracy was very low. The anamorphic gas is
discussed in detail in Section 4.2.2. As can be seen in Figure 15, the reconstruction accuracy
was low compared with that of the stereo vision. The accuracy of this metrology method is
also discussed in Section 4.2.
Figure 13.
Images for 3D construction when the anamorphic lens was in the vertical position. (
a
,
b
)
refer to images of 3D targets and (c–h) refer to the images of a 2D target.
Sensors 2022,22, 8407 13 of 17
4.2. Accuracy Analysis
As shown in Equation (14), the pixel error was the main source of error for this
metrology, though other parameters such as ad,f
x
, and f
y
are also very important. This
method’s measuring principle is based primarily on the anamorphic distance, which
changes the image position compared to spherical lenses. The anamorphic distance ad
shown in Section 4.1 was small in comparison to the object distance; thus, any small
pixel position error would have a significant impact on the measurement results. From
Equation (7), we have:
dXC=∂XC
∂xh
δxh+∂XC
∂xv
δxv+∂XC
∂yv
δyv
dYC=∂YC
∂xh
δxh+∂YC
∂yv
δyv
dZC=∂ZC
∂xh
δxh+∂ZC
∂yv
δyv
(14)
where
δ
x
h
,
δ
x
v
, and
δ
y
v
are the pixel errors. If we substitute the pixel errors in Equation (14)
with δ, the point error can be deduced from Equation (14) as follows:
∆xyz =pdXC2+dYC2+dZC2
=ad ·δ·qx2y2·f2
x·f2
y+x2·f2
x+x1y1·fx·fy+y2·f2
y
(fy·xh−fx·yv)2
(15)
where the coefficients x2y2,x2,x1y1, and y2in Equation (15) are given by:
x2y2=(xh−yv)2
x2=2y4
v
x1y1=−2y2
vx2
h+(xv+yv)xh−xvyv
y2=x4
h+(xv+yv)2x2
h−2xvyv(xv+yv)xh+x2
vy2
v
(16)
To further evaluate the measurement accuracy, we conducted two simulation experi-
ments. The simulation anamorphic lens was the same as the anamorphic lens shown in
Section 3.2:fx= 12 mm, fy= 16 mm, aa = 0◦, and ad = 30 mm.
4.2.1. Accuracy Analysis for a Point
First, we simulated an object point with the coordinates [X
C
, Y
C
, Z
C
] = [500 mm,
500 mm, 1500 mm] in anamorphic coordinates for the vertical position. The pixel coordi-
nates of the point in the vertical and the horizontal anamorphic positions were calculated
using Equations (3)–(6). Given the pixel coordinates and the calibrated parameters of the
rotating anamorphic stereo vision, the 3D coordinates of the point could be calculated. The
pixel errors in the image plane were assumed to have a Gaussian normal distribution with
a standard deviation of 0.2 pixels, which was the calibration result of the anamorphic lens.
Then, 5000 3D points were reconstructed with varying pixel errors, and the reconstructed
3D points are shown in Figure 16. The standard deviation of the reconstructed 3D point
was 17.3378 mm.
4.2.2. Accuracy Analysis for a Surface
We also simulated a set of object points in a plane. The X
C
and Y
C
coordinates
ranged from
−
1000 mm to 1000 mm, with an interval of 50 mm, and all the Z
C
were
1500 mm. The pixel errors in the image plane satisfied a Gauss normal distribution, and the
standard deviation of the pixel error was set to 0.2 pixels. The standard deviation for each
reconstructed 3D point is shown in Figure 16. As can be seen in Figure 17, the measuring
accuracy for the points away from the X
C
axis was high, while it was very low if the object
points were near the X
C
axis. The points near the X
C
axis were removed, which left a gap
in Figure 17. Thus, there was a measuring gap if rotating anamorphic lenses were used for
3D construction. We named this gap the anamorphic gap.
Sensors 2022,22, 8407 14 of 17
Figure 16.
Image showing 5000 constructed 3D points for an object point in [500 mm, 500 mm,
1500 mm] with different pixel errors. The standard deviation of the constructed 3D points was
17.3378 mm.
Figure 17.
Standard deviation for object points in a plane. The standard deviations for object points
with small Y
C
coordinates were removed for large reconstruction errors. The standard deviation of
the pixel position was 0.2 pixels, the ad was 30 mm, and the ZCwas 1500 mm.
If we individually increase the anamorphic distance from 30 mm to 100 mm, the
measuring accuracy will increase correspondingly, as shown in Figure 18. The lens parame-
ters ad,f
x
, and f
y
are closely related because they are determined by the anamorphic lens
structure, and it is not possible to simply change one parameter independently [
25
]. We
proposed paraxial lens designs for anamorphic lenses with zero anamorphic distance [
26
],
but designing an anamorphic attachment with an extremely large anamorphic distance
appeared to be difficult.
Sensors 2022,22, 8407 15 of 17
Figure 18.
Standard deviation for object points in a plane. The standard deviations for object points
with small Y
C
coordinates were removed for large reconstruction errors. The standard deviation of
the pixel position was 0.2 pixels, the ad was 100 mm, and the ZCwas 1500 mm.
The measuring precision of this 3D metrology method appeared to be lower than
that of stereo vision with spherical lenses. The fundamental reason for this is that, in
comparison to stereo vision, the baseline distance between the two anamorphic positions
was quite small. The measuring precision will be as good as that of stereo vision with
spherical lenses if the two anamorphic lenses are sufficiently apart from each other. Because
the two images are captured with the same revolving anamorphic lens, the main advantage
of this 3D metrology is its ease of point matching. The appropriate matching points are
very close if the two images are rectified with the anamorphic ratio, as shown in Figures 5
and 12. Alternatively, as illustrated in Figure 6, it is possible to trace the matching points by
rotating the anamorphic lens. When the anamorphic lens is rotated at a rapid speed, it can
measure 3D data four times in one circle, which is ideal for a quick 3D reconstruction with
low measurement accuracy. One potential application of this would be car navigation.
5. Conclusions
A 3D metrology method using one camera with rotating anamorphic lenses is pro-
posed in this paper. The anamorphic lens can rotate by
−
90
◦
along the optical axis, captur-
ing two anamorphic images of an object before and after the rotation. This setup is similar
to that of stereo vision; therefore, it can be considered a stereo vision with anamorphic
lenses. The main differences between stereo vision and rotating anamorphic lenses are
that anamorphic lenses replace spherical lenses, and the baseline distance between the
two anamorphic lenses is very small. This feature will make rotating anamorphic lenses
a compact sensor suitable for 3D metrology in a constrained space. The feasibility of the
proposed method in 3D reconstruction is based on one internal parameter of anamorphic
lenses, i.e., the anamorphic distance, which will slightly change the image position of the
object. The corresponding points between the two anamorphic images are close if the two
anamorphic images are rectified by the anamorphic ratio, which substantially simplifies
the corresponding point matching and reduces the calculation amount. When compared to
stereo vision, the measurement accuracy of the rotating anamorphic stereo vision is lower,
especially for the image points near the X
C
axis of the anamorphic lens in the vertical posi-
tion. These characteristics make the rotating anamorphic stereo vision suitable for a fast 3D
reconstruction without a high demand for measurement accuracy, such as car navigation
applications. Further research might include high precision anamorphic lens calibration,
error compensation, point matching for rotating anamorphic lenses, and anamorphic lens
designs with a large anamorphic distance.
Sensors 2022,22, 8407 16 of 17
Author Contributions:
Conceptualization, J.Z., X.C. and J.X.; methodology, X.C. and J.Z.; software, X.C
and J.Z.; validation, J.Z.; data curation, J.Z.; writing—original draft preparation, J.Z.; writing—review and
editing, X.C. and J.Z. All authors have read and agreed to the published version of the manuscript.
Funding:
This research was funded by the National Natural Science Foundation of China (52175478),
the Defense Industrial Technology Development Program (JCKY2020203B039), the Science and
Technology Commission of Shanghai Municipality (21511102602), the Doctoral Foundation of the
University of Jinan (XBS1641), and the Joint Special Project of Shandong Province (ZR2016EL14).
Data Availability Statement:
The data underlying the results presented in this paper are not publicly
available at this time but may be obtained from the authors upon reasonable request.
Conflicts of Interest: The authors declare no conflict of interest.
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