SINGLE-LENS STEREOSCOPIC ARRANGEMENT
WITH OVERLAPPING OF THE VIEWS
Thierry FOURNEL, Corinne FOURNIER, Christophe VINCENT
Laboratoire Traitement du Signal et Instrumentation
UMR CNRS 5516 - Université J. Monnet
23, rue du Dr P. Michelon, 42023 Saint-Etienne cedex 2, France
E-mail : fournel@univ-st-etienne.fr
ABSTRACT
The single-camera stereoscopic arrangement proposed in this paper allows to overlap one another the left
and right views of an object field viewed from two symmetrical incidences. An affine camera records the
view transmitted through a semi-silvered mirror and the view reflected by an additional mirror. The
geometry of the Affine Reflection Transmission Arrangement is described. Its ability to measure a 3D
displacement is tested on a target pattern.
KEYWORDS
Affine lens, catadioptric stereo system, computational stereovision, single-camera stereoscopy
1. INTRODUCTION
Stereoscopic systems involving a single lens and a single sensor can be found in different
domains like Artificial Vision (Nene and Nayar, 1998 ; Mouaddib and Pegard, 1995) or Particle
Image Velocimetry (Prasad, 2000). As the camera is unmoved, catadioptric configurations are
used. By using a perspective camera the reflecting surfaces that perform projection of the scene
from a single viewpoint are planar or conic (Nayar and Baker, 1997).
Here we focus on arrangements recording stereo pairs from symmetrical incidences and using
fully-astigmatic mirrors i.e. planar mirrors.
The arrangement given in (Arroyo and Greated, 1991) (respectively in (Grant et al., 1991)) is a
single-camera version of the translation arrangement (respectively the rotational arrangement). In
the previous arrangements mirrors are located symmetrically according to the optical axis.
Therefore as the left view is reflected successively on two mirrors and recorded on the right side,
the left and right views are adjacent on the sensor. So, each view covers only half the sensor.
Y
Y
Z
O
Z
(a)
O
(b)
Figure 1. Single-camera version of:
(a) the translation arrangement, (b) the rotational arrangement
In this paper we propose a catadioptric stereo arrangement using a single lens and a single sensor
to view the object plane from two symmetrical incidences and to overlap the views one another.
If the object is motionless, a shuttering allows to separate the views in time and to work at the full
resolution. If the object is in motion, the views have to be recorded simultaneously. To separate
the views, a color coding added to a color camera can be used. Two color filters can respectively
be located in front of the mirror and the semi-silvered mirror.
2. DESCRIPTION OF THE ARRANGEMENT
An arm is mounted in front of the lens in order to reflect a second view. This arm is composed of
a planar mirror and a semi-silvered mirror. Located on the optical axis of the camera, the semisilvered mirror allows to record at once the direct view and the view reflected by the planar
mirror (Figure 2). The planar mirror is located on the axis symmetrical to the optical axis
according to the perpendicular (OZ) to the object plane. The two mirrors are oriented so that the
reflected image seems to be viewed from a virtual camera under the same angle as the real
camera that points towards the object plane. The location of the virtual camera is obtained from
the real camera by two successive symmetries. The symmetry according to the semi-silvered
mirror plane gives an image of the real camera (drawn with dashed lines on figure 2). From this
image, the symmetry according to the mirror plane gives the virtual camera.
Object coordinate
system OXYZ
Y
Z
Semi-silvered
mirror
O
Affine lens
Mirror
Virtual lens
Figure 2. Catadioptric single-camera arrangement to view according to symmetrical incidences
and to overlap the views one another on the sensor
Figure 3. The pair of the die faces 3 and 5 is viewed from the real camera
and the pair of the faces 2 and 3 is viewed from the virtual camera
Figure 3 shows the views of a die recorded with the arrangement given in figure 2. The incidence
angles are -20° and +20°.
3. GEOMETRY OF THE ARRANGEMENT
3.1 REFLECTION \ TRANSMISSION ARRANGEMENT
Let us note d1 the distance between the origin O of the object coordinate system OXYZ and the
center M1 of the semi-silvered mirror (on the optical axis), α the angle between the optical axis
and the Z axis and β the angle between the optical axis and the perpendicular of the semi-silvered
mirror.
The position of the mirror can be deduced from the three parameters d1, α, β of the catadioptric
centered system : the mirror (centered at point M2) is in the plane passing through the line of
intersection between the object plane and the semi-silvered mirror plane (centered at point M1).
The next paragraph gives a non analytical proof of it.
Y
Z
M1
d1
α
O
α
π−2β
β
axis of the
real camera
Ω
β
α
M2
I
I’
Figure 4. Adjustment point I of the arrangement
(the viewangles are identical)
In the horizontal plane containing the optical axis (Figure 4), let us note Ω the point where the
bisectors of the triangle OM1M2 intersect and I the point where the lines (OY) and (ΩM1)
intersect. The intersection point I’ between the line (ΩM1) supporting the first mirror and the line
supporting the second one is identical to the point I. Indeed, the triangle IOΩ rectangular at O
and the triangle I’M2Ω rectangular at M2 have the same hypotenuse :
π

ΩM 1 ⋅ sin  − β 
ΩO
2
 = ΩM 2 = ΩI ′
ΩI =
=
sin (β − α ) sin (β − α ) ⋅ sin (α ) sin (α )
The point I can be used to adjust with accuracy the position of the mirror.
(1)
3.2 AFFINE NATURE OF THE CAMERA
It must be noticed that the distance dV of the virtual camera from the origin O is greater than the
distance dR of the real camera. But to have an exact overlapping of the two views, the
magnifications must be the same for both the recordings. It is the reason why, working at a finite
distance, an affine lens (also called a “telecentric” lens) which acts as an orthogonal projection
device, is used.
Moreover the distance dV rises up respectively with the angle α and with the distance OI. But
the distance dV must be in the depth of field of the real camera.
Figure 5 shows the distance dV version the viewangle α, the distance d1=OM1 being fixed to dR/2
and the working distance dR to 180 mm. The distance dV must be less than the boundary value of
the back depth of field. There is a maximum distance OI (85 mm here) for which the angle α can
reach 45°.
Distance dV (mm)
360
340
320
300
OI=115mm
Boundary of the depth of field
280
OI=85mm
260
240
OI=55mm
220
200
180
0
5
10
15
20
25
30
35
40
45
Angle α (degree)
Figure 5. Distance dV of the virtual camera from the pointed origin of the object plane
version the view angle α
4. EXPERIMENT AND RESULTS
For testing the Affine Reflection Transmission Arrangement (ARTA), a texture was shifted of
2.50 mm according to the Z axis with a translation table (10 µm accuracy) and the displacement
field measured.
The field was imaged with an affine lens of which the working distance is 230 mm. The aperture
number was equal to 22 and the back depth of field to 70 mm (respectively the front depth of
field to 60 mm). The center O of the object plane was 180 mm far from the camera. The size of
the camera field was about 54 mm in width and 40 mm in height. The conversion factor at the
recording was about 14.3 pixel/mm according to the v horizontal and u vertical axis. The viewing
angle α was about 20° (see figure 4).
Recordings were done by masking alternatively the reflected and the transmitted views.
The semi-silvered mirror used (50 mm in width) was located on the optical axis at d1=90 mm and
did not cover the whole field. Measurements of displacement were carried out in the center of the
camera field (17 mm in width x 22 mm in height) where a linear calibration was done (Fournel et
al., 2000). The mean deviation of the calibration points projected with the calibration matrix is
about 0.15 pixel for the real camera, respectively 0.70 for the virtual camera.
Each camera gives a displacement field for which measurements are done locally by subpixel
correlation on 32x32 pixel cells. Mean values are given on table 1. The real camera gives a
standard deviation according to the displacement projection (v direction) lower than that obtained
with the virtual camera.
v
v
u
u
Figure 6. Projections of the Z displacement (2.5mm) measured
on the virtual camera (left) and on the real camera (right)
Displacement projection
Virtual camera
Real camera
u mean component
-0.56
0.24
v mean component
12.99
-13.01
u standard deviation
0.80
0.85
Table 1. Mean values and standard deviations (pixel)
of the u (up to down) and v (left to right)
measured components of the displacement projections
40
35
30
25
20
X
15
X
10
Z
5
Y
0
0
10
Y
20
-10
-5
0
5
10
Z
Figure 7. The 3D displacement measured by ARTA
v standard deviation
0.54
0.07
Displacement
Mean value
Standard deviation
X component
0.01
0.04
Y component
0.01
0.02
Z component
2.50
0.05
Table 2. Mean values and standard deviations (mm) of the components of the 3D displacement
The 3D displacement field is obtained from the two last fields and from the calibration matrix.
Mean values are given on table 2. The mean displacement is very close to the actual
displacement. The highest standard deviation (50 µm) is logically relative to the Z axis.
5. CONCLUSION
The Affine Reflection Transmission Arrangement involves a single affine camera and allows the
whole overlapping of stereo views thanks to a pair of mirrors. It has a very simple design welladapted to differential measurements. The measurements obtained with standard equipment give
an accuracy of about 0.1 mm for a 2.5 mm displacement in depth. In perspective the quality of
the image obtained by reflection has to be improved and the image coding \ decoding has to be
worked.
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