Accepted as poster presentation for ICPR 2014, Stockholm, Sweden on August 24~28, 2014. [Revised Version] Title: New geometric interpretation and analytic solution for quadrilateral reconstruction Author: Joo-Haeng Lee Affiliation: Human-Robot Interaction Research Team, ETRI, KOREA Abstract: A new geometric framework, called generalized coupled line camera (GCLC), is proposed to derive an analytic solution to reconstruct an unknown scene quadrilateral and the relevant projective structure from a single or multiple image quadrilaterals. We extend the previous approach developed for rectangle to handle arbitrary scene quadrilaterals. First, we generalize a single line camera by removing the centering constraint that the principal axis should bisect a scene line. Then, we couple a pair of generalized line cameras to model a frustum with a quadrilateral base. Finally, we show that the scene quadrilateral and the center of projection can be analytically reconstructed from a single view when prior knowledge on the quadrilateral is available. A completely unknown quadrilateral can be reconstructed from four views through non-linear optimization. We also describe a improved method to handle an off-centered case by geometrically inferring a centered proxy quadrilateral, which accelerates a reconstruction process without relying on homography. The proposed method is easy to implement since each step is expressed as a simple analytic equation. We present the experimental results on real and synthetic examples. [Submitted Version] Title: Generalized Coupled Line Cameras and Application in Quadrilateral Reconstruction Abstract: Coupled line camera (CLC) provides a geometric framework to derive an analytic solution to reconstruct an unknown scene rectangle and the relevant projective structure from a single image quadrilateral. We extend this approach as generalized coupled line camera (GCLC) to handle a scene quadrilateral. First, we generalize a single line camera by removing the centering constraint that the principal axis should bisect a scene line. Then, we couple a pair of generalized line cameras to model a frustum with a quadrilateral base. Finally, we show that the scene quadrilateral and the center of projection can be analytically reconstructed from a single view when prior knowledge on the quadrilateral is available. ...

1. New Geometric Interpretation and Analytic Solution
for Quadrilateral Reconstruction
Joo-Haeng Lee
Convergence Technology Research Lab
ETRI
Daejeon, 305–777, KOREA
Abstract—A new geometric framework, called generalized
coupled line camera (GCLC), is proposed to derive an analytic
solution to reconstruct an unknown scene quadrilateral and the
relevant projective structure from a single or multiple image
quadrilaterals. We extend the previous approach developed for
rectangle to handle arbitrary scene quadrilaterals. First, we gen-
eralize a single line camera by removing the centering constraint
that the principal axis should bisect a scene line. Then, we couple
a pair of generalized line cameras to model a frustum with a
quadrilateral base. Finally, we show that the scene quadrilateral
and the center of projection can be analytically reconstructed
from a single view when prior knowledge on the quadrilat-
eral is available. A completely unknown quadrilateral can be
reconstructed from four views through non-linear optimization.
We also describe a improved method to handle an off-centered
case by geometrically inferring a centered proxy quadrilateral,
which accelerates a reconstruction process without relying on
homography. The proposed method is easy to implement since
each step is expressed as a simple analytic equation. We present
the experimental results on real and synthetic examples.
I. INTRODUCTION
A new geometric framework, called generalized coupled
line camera (GCLC), is proposed to derive an analytic solution
to reconstruct an unknown scene quadrilateral and the relevant
projective structure from a single or multiple image quadri-
laterals. We extend the previous approach, called coupled
line camera (CLC), which models a rectangular frustum of a
pinhole camera using two line cameras [1], [2]. (A line camera
in our context does not refer to a capturing device such as a
line-scan camera. Rather, our geometric configuration is more
related to modeling approaches based on linear elements for
camera calibration [3] or multi-perspective image [4].)
Under CLC configuration, geometric relation among the
base rectangle, the image quadrilateral and the optical center
can be comprehensively described as simple equations of a
compact parameter set. Hence, given a single image quadri-
lateral, we can uniquely identify the frustum by reconstructing
the base rectangle and optical center using a closed-form
solution. The solution also contains a determinant that tells if
a image quadrilateral is the projection of any rectangle prior
to reconstruction.
In the CLC-based reconstruction, no explicit form of camera
parameters is involved since the formulation is based on pure
geometric configuration of a pinhole projection. In application,
an image quadrilateral is represented by a set of diagonal
parameters (i.e. relative lengths of partial diagonals and the
crossing angle) rather than actual pixel coordinates. If re-
quired, unknown camera parameters such as the focal length
can be computed subsequently using a standard calibration
technique [5], [6]. Generally the previous solutions require to
reconstruct the camera parameters first [7]. For example, when
we apply the IAC (image of the absolute conic) method, the
unknown focal length should be found first [5], [8].
Another interesting feature of CLC-based reconstruction is
geometric interpretation of the solution space, which leads to
an optimized analytic solution [2]. For example, given an im-
age quadrilateral, two candidate line cameras are defined over
two solution spheres. By the constraint of common principal
axis, spheres are confined to two solution circles. Finally, the
optical center is found in the intersection of two solutions
circles. We believe a similar geometric framework can be
applied in other geometric computer vision problems such as
investigating the solution space of n-view reconstruction.
In this paper, we propose generalized coupled line camera
(GCLC) that inherits the key features of CLC and models
a projective frustum with a quadrilateral base, which targets
on a prospective application of projective reconstruction of
an unknown scene quadrilateral. While keeping the same
centering constraint of CLC that the principal axis passes
through the center of quadrilaterals, we extend the model with
additional parameters to describe the lengths of all partial
diagonals. In CLC, these parameters need not be specified
since they cancel out due to equilateral partial diagonals of
a rectangle [1], [2]. The increased number of configuration
parameters in GCLC, however, hinders to formulate a closed-
form solution for single view reconstruction. We investigate
this property and propose an analytic solution that works for
single view reconstruction under special conditions, and a
method to approximate unknown diagonal parameters from
multiple views. For practical application of CLC framework,
we need to handle an off-centered case. In this paper, we also
propose an improved method composed of simpler operations
based on geometric properties, not relying on constrained
equation solving or explicit homography as in [1].
This paper is organized as follows. In Section II, we sum-
marize the previous work on CLC [1], [2]. In Section III, we
generalize CLC and describe reconstruction solution including
off-centered cases. In Section IV, we give experimental results
on synthetic and real quadrilaterals to demonstrate the perfor-
mance. Finally, we conclude with remarks on future work.

2. d
s0
s2
q0
y2 y0
u0
u2
v0v2
vm
pc
l0
l2
m0m2
(a) Camera pose when d = 1.7.
cv0v2 vm
(b) Circular trajectory of pc for varying d.
Fig. 1. An example of a canonical line camera: m0 = m2 = 1, l0 = 0.6,
l2 = 0.4, and ↵ = 0.2.
II. PRELIMINARIES OF COUPLED LINE CAMERAS
A. Line Camera
Definition 1. A line camera captures an image line uiui+2
from a scene line vivi+2 where vi = (mi, 0, 0) and vi+2 =
( mi+2, 0, 0) for positive mi and mi+2. See Figure 1a.
Definition 2. In a centered line camera, the principal axis
passes through the center vm of the scene line vivi+2:
vm = (vi + vi+2)/2. (1)
Definition 3. A canonical line camera is a centered line
camera with two constraints for simple formulation: vm =
(0, 0, 0)T
and equilateral unit division:
kvi vmk = kvik = kvi+2k = 1. (2)
For a line camera Ci, let d be the length of the principal axis
from the center of projection pc to vm. Let ✓i be the orientation
angle of the principal axis measured between vmpc and vmvi.
Definition 4. For a canonical line camera, its pose equation
is expressed as follow:
cos ✓i =
✓
li li+2
li + li+2
◆
d = ↵i d (3)
where li = kui umk is the length of partial diagonals. Let ↵i
be the line division coefficient of the canonical configuration
↵i =
li li+2
li + li+2
(4)
According to Eq.(3), we can observe the relation among ✓i,
d and ↵i. Note that when ↵i is fixed, pc is defined along a
circular trajectory or on a solution sphere of radius 0.5/|↵|.
See Figure 1b.
B. Coupled Line Cameras
Definition 5. Coupled line camera is a pair of line cameras,
that share the principal axis and the center of projection.
By coupling two canonical line cameras, we can represent a
projective structure with a rectangle base. See Figure 2.
Definition 6. For coupled line camera, we can derive a
coupling constraint:
=
l1
l0
=
tan 1
tan 0
=
sin ✓1 (d cos ✓0)
sin ✓0 (d cos ✓1)
(5)
v0
v1 v2
v3
f vm
G
(a) Scene rectangle G (b) 1st line camera C0 (c) 2nd line camera C1
(d) Coupling C0 and C1 (e) Projective structure
u0
u1 u2
u3
r um
Q
(f) Projection of G to Q
Fig. 2. Coupling of two canonical line cameras to represent a projective
structure with a rectangle base.
where is the coupling coefficient defined by the ratio of the
lengths, l0 and l1, of two partial diagonals of Q. See Figure 2f.
C. Projective Reconstruction
Algorithm 1 (Single View Reconstruction with CLC). The
unknown elements of projective structure, such as the scene
rectangle G and the center of projection pc, can be recon-
structed from a single image quadrilaterals Q as in the below.
First, the pose equation of Eq.(3) and the coupling constraint
of Eq.(5) can be rearranged into a system of equations:
d =
sin ✓0 cos ✓1 cos ✓0 sin ✓1
sin ✓0 sin ✓1
=
cos ✓0
↵0
=
cos ✓1
↵1
(6)
Then, the length d of the common principal axis can be
computed from the system of equations in Eq.(6) as follows:
d =
p
A0/A1 (7)
where A0 = (1 ↵1)2 2
(1 ↵0)2
and A1 = ↵2
0(1
↵1)2 2
(1 ↵0)2
↵2
1. Once d is computed, two orientation
angles, ✓0 and ✓1, can be computed using Eq.(3).
The base rectangle G can be reconstructed by computing
its unknown shape parameter, the diagonal angle :
cos = cos ⇢ sin ✓0 sin ✓1 + cos ✓0 cos ✓1 (8)
where ⇢ is the diagonal angle of the image quadrilateral Q.
Finally, the projective structure can be reconstructed by
computing the coordinates of a center of projection pc:
pc =
d (sin cos ✓0, cos ✓1 cos cos ✓0, sin ⇢ sin ✓0 sin ✓1)
sin
(9)
D. Determinant Condition
When Eq.(7) has a valid value, two conditions should be
satisfied: (1) A0 and A1 have the same sign; and (2) the length
d of the common principal axis should not exceed the diameter

3. d
s0
s2
q0
y2
y0
u0
u2
v0v2 vm
pc
l0
l2
m0m2
(a) Camera pose when d = 1.7.
cv0v2 vm
(b) Trajectory of pc when d is not fixed.
Fig. 3. An example of a generalized line camera: m0 = 1, m2 = 1.4,
l0 = 0.6, l2 = 0.4, and ↵ = 0.2.
of each solution sphere: d  min(1/k↵0k, 1/k↵1k). These
conditions can be combined into Boolean expressions:
D = D0 _ D1 (10)
D0 =
✓
1 ↵0
1 ↵1
◆
^
✓
1 
↵0
↵1
◆
(11)
D1 =
✓

1 ↵0
1 ↵1
◆
^
✓
1
↵0
↵1
◆
(12)
where ^ and _ are Boolean and and or operations, respec-
tively. Since ↵0, ↵1 and are the coefficients from a given
image quadrilateral Q, we can determine if Q is an image
of any scene rectangle before actual reconstruction. Once the
determinant D is satisfied, Algorithm 1 can be applied.
E. Off-Centered Case
CLC assumes the principal axis passes through the centers
of the image quadrilateral Q and the scene rectangle G. When
handling an off-centered quadrilateral Qg, a centered proxy
quadrilateral Q should be found first by solving equations
that formulate edge parallelism between Q and Qg, centering
constraint of Q, and a vanishing line derived from Qg [1].
Once Q is found, the centered proxy rectangle G can be
reconstructed using Algorithm 1. Since the inferred Q does
not guarantee congruency to Qg, the target scene rectangle
Gg should be reconstructed using a homography H between
Q and G: Gg = HQg.
In this paper, we propose a new method to handle an off-
centered case. First, we derive a centered proxy quadrilateral
Q that is perspectively congruent to Qg. Then, we show that
the target scene rectangle Gg can be geometrically derived
without relying on homography. See Section III-E.
III. GENERALIZATION OF COUPLED LINE CAMERAS
As a main contribution of this paper, we generalize a line
camera to support a non-canonical configuration. Then, we
show that a pair of generalized line cameras can be coupled to
represent a projective structure with a quadrilateral base other
than a rectangle. Finally, we describe how we can reconstruct
a projective structure from a single view with a sufficient prior
knowledge to constrain the solution space. We also describe
how to handle off-centered cases.
v0
v1
v2 v3
f
vm
G
(a) Scene quad. G (b) 1st line camera C0 (c) 2nd line camera C1
(d) Coupling C0 and C1 (e) Projective Structure
u0
u1 u2
u3
r um
Q
(f) Projection of G to Q
Fig. 4. Coupling of two generalied line cameras to represent a projective
structure with a quadrilateral base. A generalized line camera Ci is assigned
for each diagonal of a scene quadrilateral G. actual values of diagonal
parameters.
A. Generalized Line Camera
Definition 7. In a general configuration of a line camera,
the principal axis may not bisect the scene line: we may not
consider the centering constraints of Eqs.(1)-(2). See Figure 3
where m0 6= m2.
Accordingly, the pose equation of a canonical line camera in
Eq.(3) should be generalized with two additional parameters,
m0 and m2. Assuming m0 > 0 and m2 > 0, the following
geometric relation holds:
li : li+2 = mi sin ✓0
d
d ˆdi
: mi+2 sin ✓0
d
d + ˆdi+2
(13)
where ˆd0 = m0 cos ✓0 and ˆd2 = m2 cos ✓0.
Definition 8. The generalized pose equation can be derived
from Eq.(13):
cos ✓i =
✓
mi+2li mili+2
mimi+2 (li + li+2)
◆
d = ↵g,i d (14)
where ↵g,i is the generalized division coefficient
↵g,i =
mi+2li mili+2
mimi+2 (li + li+2)
. (15)
For a fixed ↵g,i, the center of projection pc is defined over a
circular trajectory as in Figure 3b, or on a solution sphere [2].
B. Coupling Generalized Line Cameras
By coupling two generalized line cameras, we can represent
a projective structure with a quadrilateral base G with vertices:
v0 = m0(1, 0), v1 = m1(cos , sin ), v2 = m2/m0v0, and
v3 = m3/m1v1 where mi’s are the relative lengths of partial
diagonals or diagonal parameters of G. See Figure 4.
Definition 9. A generalized coupling constraint g is defined
as follows:
g =
l1
l0
=
m1 sin ✓1
m0 sin ✓0
(d m0 cos ✓0)
(d m1 cos ✓1)
(16)

4. C. Projective Reconstruction
Using a trigonometric identity and the pose equation of
Eq.(14), we can derive the equation for 2
g by squaring both
the sides of Eq.(16):
sin2
✓i = 1 cos2
✓i = 1 ↵2
g,i d2
(17)
2
g =
m2
1(1 m0↵g,0)2
(1 ↵2
g,1d2
)
m2
0(1 m1↵g,1)2(1 ↵2
g,0d2)
(18)
From Eq.(18), the length d of the common principal axis
can be expressed with GCLC parameters:
d =
s
Ag,0
Ag,1
(19)
where Ag,0 = m2
0(1 m1↵g,1)2 2
g m2
1(1 m0↵g,0)2
and
Ag,1 = m2
0↵2
g,0(1 m1↵g,1)2 2
g m2
1(1 m0↵g,0)2
↵2
g,1.
Eq.(19) states that d can be computed from known diagonal
parameters, mi and li, of a single pair of scene and image
quadrilaterals, not relying on their diagonal angles, and ⇢.
Algorithm 2 (Single View Reconstruction with GCLC). Once
the length d of the common principal axis has been found
using Eq.(19) with prior knowledge on diagonal parameters,
we can compute the orientation angles, ✓0 and ✓1, using the
pose equation of Eq.(14). Then, the diagonal angle of a scene
quadrilateral and the center of projection pc can be computed
using Eqs.(8) and (9), respectively. ⇤
If we have no prior knowledge on diagonal parameters mi
of G, we can infer them using multiple image quadrilaterals
Qj from different views. By setting m0 = 1, the number of
unknown diagonal parameters of G is reduced to three: m1,
m2 and m3. For each Qj, the crossing angle j of Eq.(8) is
expressed with m1, m2 and m3, and coefficients derived from
known diagonal parameters li,j of Qj. Since the reconstructed
j’s should be identical regardless of views, the following
identity should hold: cos j = cos j+1. Hence, if we have
four different views, we can formulate three equations of three
unknowns, m1, m2 and m3:
cos 0 = cos 1 = cos 2 = cos 3 (20)
The number of views are varying according to the degree of
freedom in diagonal parameters.
Although an analytic solution for Eq.(20) is not found
yet, the problem can be formulated as minimization of the
following objective function:
fobj =
n 1X
j=0
k cos j cos j+1k2
(21)
where n is the number of views. Generally, Eq.(21) can be
solved using a numerical nonlinear optimization method [9].
Since optimization may get stuck in a local minima, we may
check the validity using determinant of Eq. 24.
Algorithm 3 (n-View Reconstruction with GCLC). When
Algorithm 2 cannot be applied due to lack of knowledge
on the scene rectangle G, but we have multiple image
(a) Reference: Gg and Qg (b) Inferring a centered Q in blue
(c) Reconstruction of G and Gg (d) Congruency of G and Gg
Fig. 5. Reconstruction of a synthetic quadrilateral Gg from an off-centered
quadrilateral Qg: m0 = 1, m1 = 0.75, m2 = 1.35, m3 = 1.4 and =
1.35. Diagonal parameters mi and the vanishing line is given.
quadrilaterals Qj from n different views, we can find the
unknown mi’s by minimizing the objective function of
Eq.(21). Then, we can apply Algorithm 2 for one of the
views to reconstruct the projective structure. ⇤
The number of views required in Algorithm 3 depends on
the number of unknown mi’s. For a general quadrilateral of
three unknown mi’s except m0 = 1, at least 4 views are
required according to Eq.(20). For a parallelogram with known
m0 = m2 = 1 and unknown m1 = m3, at least 2 views are
required to find m1. See Section IV for real examples.
D. Determinant Condition
Similarly as in Section II-D, we can derive, from Eqs.(14)
and (19), a condition Dg that can determine if Q is projection
of a centered scene quadrilateral G with known mi’s.
Dg = Dg,0 _ Dg,1 (22)
Dg,0 =
⇣
m1(1 m0↵g,0)
m0(1 m1↵g,1)
⌘
^
⇣
1 
↵g,0
↵g,1
⌘
(23)
Dg,1 =
⇣

m1(1 m0↵g,0)
m0(1 m1↵g,1)
⌘
^
⇣
1
↵g,0
↵g,1
⌘
(24)
E. Off-Centered Case
Let an off-centered image quadrilateral Qg be projection
of a scene quadrilateral Gg, which is also off-centered and
unknown yet. See Fig. 5a. To apply Algorithms 2 and 3, we
provide a method to find a centered proxy quadrilateral Q that
is an image of a centered scene quadrilateral G. Specially, G is
guaranteed to be congruent to Gg through parallel translation
by t. We also show that the translation vector t can be
computed in image space. Hence, we do not need to compute
homography H between G and Q to reconstruct Gg as in CLC.
See Section II-E and [1].
Algorithm 4 (Reconstruction from an Off-Centered Quadri-
lateral). An off-centered scene quadrilateral Gg can be recon-
structed from its image Qg by adding extra steps to the GCLC
methods presented in Section III-C. See Figure 5:

5. Qg
Q
om
ug,0 ug,1
ug,2
ug,3
um
u0
u1
u2
u3
w0
w1
wd,0
wd,1
wm
Fig. 6. Derivation of a centered proxy quadrilateral Q that is perspectively
congruent to Qg. Assume the vanishing line w0w1 is given.
1) Infer a centered proxy quadrilateral Q from Qg such
that Q is projection of a centered scene quadrilateral
G that is congruent to the target quadrilateral Gg. See
Algorithm 5;
2) Apply Algorithm 2 to Q to reconstruct the corresponding
centered quadrilateral G and the center of projection pc.
If multiple Qg,j are available, apply Algorithm 3.
3) The target scene quadrilateral Gg can be computed as
translation of G: Gg = G + t where t can be computed
from displacement s = um om between centers of Q
and Qg using Algorithm 6.
Algorithm 5 (Centered Proxy Quadrilateral). Assuming a
vanishing line w0w1 is given, we can find a centered proxy
quadrilateral Q by perspectively translating an off-centered
quadrilateral Qg. See Figure 6:
1) Find the intersection points wd,i between the vanishing
line w0w1 and each diagonal ug,iug,i+2 of Gg.
2) Find the intersection point wm between the vanishing
line w0w1 and the line of translation omum.
3) Find the intersection point u0 between the line ug,0wm
and the line omwd,0. Similarly, find u2 from ug,2wm and
omwd,0.
4) Find the intersection point u1 between the line ug,1wm
and the line omwd,1. Similarly, find u3 from ug,3wm and
omwd,1.
5) The i-th vertex of Q is ui. ⇤
Note that Algorithm 5 is composed of simple line-line inter-
sections rather than geometric constraint solving as in [1].
Algorithm 6 (Perspective-to-Euclidean Vector Transforma-
tion). With GCLC defined with known Q and G (as in Fig. 4),
we can project an image vector s to a scene vector t. First,
we perspectively decompose s along two diagonals of Q:
1) Find the intersection points us,0 between the line u0om
and the line umwd,1. Similarly, find us,1 from u1om and
umwd,0.
2) For each decomposition coefficient si of us,i, compute
the coefficient ti for vi using Eq. 26.
3) The corresponding scene vector t can be expressed as a
vector sum of two diagonal vectors, t0v0 + t1v1, of G
assuming vm = 0. See Fig. 4b. ⇤
Algorithm 6 is based on the following property of a general-
ized line camera.
um
ug,m
u0
u1
us,0
us,1
w0
w1
wd,0
wd,1
v0
vm
vg,m
v1
vt,1
vt,0
G
Gg
Q
Qg
Fig. 7. Perspective-to-Euclidean vector transformation.
d
q0
v0v2 vm vt,0vt,2
pc
l0
l2
u0
u2
us,0
us,2
m0m2
um
Fig. 8. Scaling transformation in a generalized line camera, which is
explained as a cross ratio between corresponding four points.
Using projective invariance of cross-ratio [8], the following
holds for two sets of collinear points, (vt,0, v0, vm, v2) and
(us,0, u0, um, u2), in the scene and images lines, respectively:
sili(li + li+2)
li(sili + li+2)
=
timi(mi + mi+2)
mi(timi + mi+2)
(25)
where si = kus,i umk/li and ti = kvt,i vmk/mi. (See
Fig. 8.) By solving Eq.(25) for ti, we get the following relation
between ti and si:
ti =
simi+2(li + li+2)
simi+2li + ((1 si)mi + mi+2)li+2
(26)
Hence, if a line camera is defined, a scaling factor si of image
line can be mapped to ti of the scene, and vice versa.
IV. EXPERIMENT
We give experimental results on real and synthetic exam-
ples. All the experiments were performed in Mathematica
implementations.
We applied Algorithm 4 to real-world quadrilaterals found
in web images of modern architectures. We assume each image
is independently taken by unknown cameras and not altered
(by cropping). Each input quadrilateral Qg,j is specified in
red lines in Fig. 9a and Fig 10a. To infer a centered proxy
quadrilateral Qj using Algorithm 5, we find a vanishing line
using patterns of parallel lines such as window frames [10].
Once a set of centered quadrilaterals Qj are found, we
estimate unknown diagonal parameters mi that minimize the
objective function fobj of Eq.(21). In the experiment, we
used NMinimize[] function of Mathematica for non-linear
optimization [9]. With mi known, we can reconstruct the
centered scene quadrilateral Gj which is congruent to the
target scene quadrilateral Gg,j. See Fig. 9b and Fig 10b. The
result of reconstructed 3D view frustum is omitted for the page
limit.

6. #1 #2 #3 #4
(a) Input: web images of Fountain Place in Dallas, Texas.
(b) A reconstructed quadrilateral with different textures of given
images.
Fig. 9. Reconstruction of a quadrilateral from four views using Algorithm 4.
Input
• Two images from uncalibrated cameras
#1 #2
(a) Input: web images of the Dockland in Hamburg, Germany.
Output
•Reconstructed parallelogram!
m1=2.87 (err 2.8%), phi=0.61 (err 0.7%), inc=24. 29 (err 1.2%) using Ref-#2
(b) A reconstructed parallelogram with different textures of given
images.
Fig. 10. Reconstruction of a parallelogram from two view using Algorithm 4.
For a quadrilateral case of Fig. 9, four images were used.
The optimization converges when fobj  3.7 ⇥ 10 4
with
m1 = 2.46639, m2 = 0.476389, m3 = 1.25378. The mean
of four j is 1.77297 with variance 5.9974 ⇥ 10 5
. The
optimization takes about 3 seconds in 2.6 GHz Intel Core i7.
Time for other reconstruction steps is trivial through evaluation
of analytic expressions. For a parallelogram of Fig. 10, it
converges when fobj  10 30
with m1 = 2.87419 and
= 0.606594 in 0.06 second.
We also applied Algorithm 4 to the synthetic quadrilateral
G of Fig. 4 with four different views. The optimization for mi
converges when fobj < 10 15
in 3 seconds. The mean error
of reconstructed mi is 1.2⇥10 7
. Timing is similar to the real
example of Fig. 9, but precision is much higher due to absence
of noise sources such as lens distortion or feature detection.
When added random noises of 1-pixel radius to vertices of
Qj in 1280 ⇥ 1024 image, the precision dropped with errors
6.9 ⇥ 10 3
and 4.3 ⇥ 10 3
in mi and , respectively.
V. CONCLUSION
We proposed a novel method to reconstruct a scene quadri-
lateral and projective structure based on generalized coupled
line cameras (GCLC). The method gives an analytic solution
for a single-view reconstruction when prior knowledge on
diagonal parameters is given. Otherwise, required parameters
can be approximated beforehand from multiple views through
optimization. We also provide an improved method to handle
off-centered cases by geometrically inferring a centered proxy
quadrilateral, which accelerates a 2D reconstruction process
without relying on homography or calibration. The overall
computation is quite efficient since each key step is represented
as a simple analytic equation. Experiments show a reliable
result on real images from uncalibrated cameras.
To apply the proposed method to a real-world case with an
off-centered quadrilateral, a vanishing line should be available
for each view. This condition can be easily satisfied in a
specially textured quadrilateral of artifacts [11]. Otherwise,
we need other types of prior knowledge to infer a centered
quadrilateral. For example, a predefined parametric polyhedral
model can be a good candidate [12].
Lastly, coupled line projectors (CLP) [13] is a dual of CLC.
We expect that generalized CLP can be combined with GCLC
for a projector-based augmented reality application.
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