Camera Calibration from a Single Image based on

Coupled Line Cameras and Rectangle Constraint

Joo-Haeng LeeRobot and Cognitive Systems Research Dept., ETRI

joohaeng@etri.re.kr

Abstract

Given a single image of a scene rectangle of an un-

known aspect ratio and size, we present a method to

reconstruct the projective structure and to find camera

parameters including focal length, position, and orien-

tation. First, we solve the special case when the center

of a scene rectangle is projected to the image center. We

formulate this problem with coupled line cameras and

present the analytic solution for it. Then, by prefixing

a simple preprocessing step, we solve the general case

without the centering constraint. We also provides a

determinant to tell if an image quadrilateral is a pro-

jection of a rectangle. We demonstrate the performance

of the proposed method with synthetic and real data.

1. Introduction

Camera calibration is one of the most classical topicsin computer vision research. We have an extensive listof related works providing mature solutions. In this pa-per, we are interested in a special problem to calibrate acamera from a single image of an unknown scene rect-angle. We do not assume any prior knowledge on cor-respondences between scene and image points. Due tothe limited information, a simple camera model is used:the focal length is the only unknown internal parameter.

The problem in this paper has a different nature withclassical computer vision problems. In the PnP prob-lem, we find transformation matrix between the sceneand the camera frames with prior knowledge on thecorrespondences between the scene and image pointsas well as the internal camera parameters [1]. In cam-era resection, we find the projection matrix from knowncorrespondences between the scene and image pointswithout prior knowledge of camera parameters [2].Camera self-calibration does not rely on a known Eu-clidean structure, but requires multiple images fromcamera motion [3].

Several approaches are based on geometric prop-erties of a rectangle or a parallelogram. Wu et al.

proposed a calibration method based on rectangles ofknown aspect ratio [4]. Li et al. designed a rectanglelandmark to localize a mobile robot with an approxi-mate rectangle constraint, which does not give an ana-lytic solution [5]. Kim and Kweon propose a method toestimate intrinsic camera parameters from two or morerectangle of unknown aspect ratio [6]. A parallelogramconstraint can be used for calibration, which generallyrequires more than two scene parallelograms projectedin multiple-view images as in [7, 8, 9].

Our contribution can be summarized as follows.Based on a geometric configuration of coupled linecameras, which models a simple camera of an unknownfocal length, we give an analytic solution to reconstructa complete projective structure from a single image ofan unknown rectangle in the scene: no prior knowledgeis required on the aspect ratio and correspondences.Then, the reconstruction result can be utilized in findingthe internal and external parameters of a camera: focallength, rotation and translation. The proposed solutionalso provides a simple determinant to tell if an imagequadrilateral is a projection of a scene rectangle.

In a general framework for plane-based camera cal-ibration, camera parameters can be found first usingthe image of the absolute conic (IAC) and its relationwith projective features such as vanishing points [2, 10].Then, a scene geometry can be reconstructed using anon-linear optimization on geometric constrains suchorthogonality, which cannot be formulated as a closed-form in general.

2. Problem Formulation

2.1. Line Camera

A line camera is a conceptual camera, which followsthe same projection model of a standard pin-hole cam-era. (See Fig 1a.) Let v0v2 be a line segment in the

21st International Conference on Pattern Recognition (ICPR 2012)November 11-15, 2012. Tsukuba, Japan

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d s0s2!0

"2 "0

v0v2 m

pc

l0l2

(a) Line camera

cv0v2 m

(b) Trajectory of the center of projection

Figure 1: A configuration of a line camera

scene, which will be projected as a line u0u2 in the linecamera C0. Especially, we are interested in the posi-tion pc and the orientation !0 of C0 when the principalaxis passes through the center vm of v0v2 and the centerum = (0, 0, 1) of image line.

To simplify the formulation, we assume a canon-

ical configuration where !vmvi! = 1, and vm isplaced at the origin of the world coordinate system:vm = (0, 0, 0). For derivation, we define followings:d = !pcvm!, li = !umui!, "i = !vmpcvi, andsi = !pcvi!.

In this configuration, we can derive the following re-lation:

l2l0

=d" cos !0d+ cos !0

=d0d1

(1)

where d0 = d " cos !0 = s0 cos"0 and d1 = d +cos !0 = s2 cos"2. We can derive the relation between!0 and d from (1):

cos !0 = d (l0 " l2)/(l0 + l2) = d #0 (2)

where #i = (li " li+2)/(li + li+2), which is solelyderived from a image line uiui+2. Note that !0 and dare sufficient parameters to determine the exact positionof pc in 2D. When #0 is fixed, pc is defined on a certainsphere as in Fig 1b. Once !i and d are known, additionalparameters can be also determined: tan"i = sin !i/dand si = sin !i/ sin"i.

2.2. Coupled Line Cameras

A standard pin-hole camera can be represented withtwo line cameras coupled by sharing the center of pro-jection. (See Fig 2.) Let a scene rectangle G havetwo diagonals v0v2 and v1v3, each of which followsthe canonical configuration in Section 2.1: !vmvi! = 1where vm = (0, 0, 0). Each diagonal vivi+2 is pro-jected to an image line uiui+2 in a line camera Ci.When two line cameras Ci share the center of projec-tion pc, two image lines uiui+2 intersect at um on thecommon principal axis, say pcvm. The four vertices ui

form a quadrilateral H , which is the projection of the

(a) Pin-hole Camera (b) Line Camera C0 (c) Line Camera C1

Figure 2: A pin-hole camera and its decomposition intocoupled line cameras.

rectangle G in a pin-hole camera with the center of pro-jection at pc. Note that the principal axis passes throughvm, um and pc.

Using this configuration of coupled line cameras, wefind the orientation !i of each line camera Ci and thelength d of the common principal axis from a givenquadrilateral H . Using the lengths of partial diagonals,li = !umui!, we can find the relation between the cou-pled cameras Ci from (1):

tan"1

tan"0

=l1l0

=sin !1(d" cos !0)

sin !0(d" cos !1)(3)

Manipulation of (2) and (3) leads to the system of non-linear equations:

d =$ sin !0 cos !1 " cos !0 sin !1

$ sin !0 " sin !1=

cos !0#0

=cos !1#1

(4)where $ = l1/l0. Using (4), the orientation !0 can berepresented with coefficients, #0, #1, and $, that aresolely derived from a quadrilateral H:

tan!02

=

!

A0 +A1 ± 2#A0A1

A1 "A0

="

D± (5)

where

A0 = B0 + 2B1, A1 = B0 " 2B1

B0 = 2(#0 " 1)2(#20 + #2

1)" 4#20(#1 " 1)2$2

B1 = (#0 " 1)2(#0 " #1)(#0 + #1)

The actual value of !0 should be chosen to make d > 0using (2). With known !0, we can compute values of!1 and d using (4). Based on the result of this section,the explicit coordinates of pc and the shape of G arereconstructed in Section 3.1.

2.3. Rectangle Determination

In our configuration of coupled line cameras, the ex-istence of !i implies that H is the projection of a canon-ical rectangle G. As the orientation angle !0 should be

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a positive real number, the expression inside the outersquare root of (5) also should be a positive real number:

D± =A0 +A1 ± 2

#A0A1

A1 "A0

> 0 (6)

The above condition guarantees that a quadrilateral His the projection of a scene rectangle G, which will befully reconstructed in Section 3.1.

3. Reconstruction and Calibration

3.1. Reconstructing Projective Structure

We define a projective structure as a frustum with arectangular base G and an apex in the center of projec-tion pc. (See Fig 2.) Since G has a canonical form andparameterized by %, its vertices can be represented asv0 = (1, 0, 0), v1 = (cos%, sin%, 0), v2 = "v0, andv3 = "v1 where % is the crossing angle of two diag-onals. Since !vi! = 1, the crossing angle % is repre-sented as

cos% = $v0, v1% (7)

Two diagonals of the quadrilateral H intersect at theimage center um on the principal axis with the crossingangle &.

To compute %, we denote the center of projection aspcc = (0, 0, 0), the image center as uc

m = (0, 0,"1),the first two vertices of H as uc

0 = (tan"0, 0,"1)and uc

1 = (cos & tan"1, sin & tan"1,"1), the centerof G as vcm = (0, 0,"d), and the vertices of G asvci = siuc

i/!uci!. Since vi = vci " vcm, % can be com-

puted using (7) with known values of &, d, si, and "i.The coordinates of pc = (x, y, z) can be found by

solving following equations: d cos !0 = x, d cos !1 =x cos%+ y sin%, and x2+ y2+ z2 = d2, which are de-rived from the projective structure. Now, the projectivestructure has been reconstructed as a frustum with thebase G(%) and the apex pc.

Note that the reconstructed G is guaranteed to be arectangle satisfying geometric constraints such as or-thogonality since the proposed method is based on acanonical configuration of coupled line camera of Sec-tion 2.

3.2. Camera Calibration

Since our reconstruction method is formulated usinga canonical configuration of coupled line cameras, thederivation based on H can be substituted with a givenimage quadrilateral Q. Now we can find a homographyH using four point correspondences between G(%) andQ. A homography is defined as H = sKW where s, K

w0w1

u3

g

u3

u1

u2

u0

g! u0

um

Qg

Q

Figure 3: Finding a centered quad Q (in blue) from aoff-centered quad Qg (in red) using vanishing points.

and W denote a scalar, a camera and a transformationmatrices, respectively. Note that we assume a simplecamera model K: the only unknown internal parameteris a focal length f . Since the model plane is at z = 0, itis straightforward to derive K and W. See [2] for details.

3.3. Off-Centered Quadrilateral

In a centered case above, the centers of a scene rect-angle G and an image quadrilateral Q are both alignedat the principal axis, which is exceptional in reality. Theproposed method, however, can be readily applied tothe off-centered case by prefixing a simple preprocess-ing step.

Let Qg and Gg denote an off-centered image quadri-lateral and a corresponding scene rectangle, respec-tively. In the preprocessing step, we will find a cen-tered quadrilateral Q which will reconstruct the projec-tive structure where the projective correspondences be-tween Qg and Gg are preserved. We can show that suchQ can be geometrically derived from Qg using the prop-erties of parallel lines and vanishing points. (See Fig 3.)

We choose one vertex ugi of Qg as the initial vertex

u0 of Q, say u0 = ug0. First, we compute two vanishing

points wi from Qg: w0 as intersection of ug0u

g1 and ug

2ug3

and w1 as intersection of ug0u

g3 and ug

1ug2. Let u2 be the

opposite vertex of u0 on the diagonal passing throughu0 and um: u2 = a(um " u0) + u0 where a is an un-known scalar defining u2. Then, we can make symbolicdefinitions of the other vertices of Q: u1 as intersectionof two lines u0w1 and u2w0, and u3 as intersection ofu0w0 and u2w1.

Note that the above definitions of ui guarantees thateach pair of opposite edges are parallel before projec-tive distortion. The symbolic definitions of ui abovecan be combined in one constraint: the intersection ofu0u2 and u1u3 equals to the image center um, whichis formulated as a single equation of one unknown a.Since we can find the value of a, all the vertices ui ofthe centered image quadrilateral Q can be computed ac-

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(a) Reference Gr and Qg (b) Centered quad. Q

(c) Reconstruction (d) Rectifcation

Figure 4: Experiment with a synthetic rectangle Gr.

(a) Reference Gr and Qg (b) Centered quad. Q

(c) Reconstruction (d) Rectifcation

Figure 5: Experiment for a near-infinite vanishing point.

cordingly.

As a remark, we need to specially handle singularcases where two vanishing points w0 and w1 are not de-fined [10]. However, experimental results show that theproposed method is numerically stable for near-singularcases.

4. Experiments

We demonstrate the experimental results for bothsynthetic and real data. For experiment, the proposedmethod was implemented in Mathematica, and a Log-itech C910 webcam was used for real data with auto-focus enabled and 1280& 720 resolution.

4.1. Synthetic Data

We tested with a synthetic scene containing a ref-erence rectangle Gr (with random values of crossingangle %r, size, position and orientation) and a corre-sponding image quadrilateral Qg generated by a knowncamera in Fig 4a.

Figure 6: Relative errors in reconstruction (!vi " vm!and %) and calibration (pc and f ) when added noisesin a diameter of dmax pixels around each vertex of Qg .Result is the average of 100 experiments for dmax.

First, we find a centered quadrilateral Q using van-ishing points defined by Qg in Fig 4b. Then, the pro-jective structure is reconstructed as a frustum with thecentered scene rectangle G and the center of projec-tion pc in Fig 4c. Then, the homography H betweenG and Q are derived, which is further decomposed toprojection K and transformation W matrices as in Sec-tion 3.2. Using the homography H and the given off-centered quadrilateral Qg , we can reconstruct the targetscene rectangle Gg(%g) in Fig 4d.

In this example, the overall error can be measuredbetween Gr and Gg , which is found to be negligible:e! = !%g " %r! < 10!14 and ||vgi v

gi+2 " vri v

ri+2|| <

10!13. (Note that these error metrics are sufficient sincethe geometric constraint such as orthogonality is alwayssatisfied as in Section 3.1.) Then, the focal length canbe correctly reconstructed: f = 1280 in this example.

Our method handles an off-centered quadrilateralbased on its vanishing points in the image plane, whichmay be numerically unstable according to the paral-lelism in geometry. In Fig 5a, Gr has been placed togenerate a near-infinite vanishing points of Q in theimage plane as in Fig 5b. To test robustness near asingular case, we added a small quantity of rotation,say !%r = 10!n where n > 1, to the singular an-gle. According to the experiment, the proposed methodis very stable even when the configuration approachesthe singular case. For example in Fig 5, when !%r =10!10, we encounter a near-infinite vanishing point at(1.56807 & 1013, 1547.54). However, the error is al-most negligible: e! < 10!5 .

To test robustness, we added random noises to eachvertex of reference quadrilateral Qg in a diameter ofdmax pixels. (See Fig 6.) In a practical situation whendmax = 1, overall errors are less than 2%. Generally,reconstruction errors are smaller than calibration errors.

4.2. Real Data

We also tested with real images. In Fig 7a, the off-centered quadrilateral Qg is the image of an ISO A4

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