Special Issue of International Journal of Computer Applications (0975 – 8887)
on Software Engineering, Databases and Expert Systems – SEDEXS, September 2012
Camera Self-Calibration by an Equilateral Triangle
A.Baataoui, I.El Batteoui
A.Saaidi Et K.Satori
LIIAN, Department of Mathematics and informatics
Faculty of Sciences Dhar-Mahraz P.O.Box 1796
Atlas-Fes, Morocco
LIIAN, Department of Mathematics and informatics
Faculty of Sciences Dhar-Mahraz P.O.Box 1796
Atlas-Fes, Morocco
ABSTRACT
In this article, we present a technique for self-calibration
of a CCD camera with constant focal distance using a
planar scene. The particularity of our technique is the use
of triangle equilateral which two vertices are defined from
the matches detected in images taken by the camera.
Using these vertices to estimate all the projection matrices
which are operated with homographies between the
images to estimate the intrinsic parameters of the camera.
Experimental results show the robustness of our
algorithms in terms of stability and convergence.
Finally, three images are sufficient to determine the
intrinsic parameters of camera by solving a system of
nonlinear equation that needs initialization and
optimization of a cost function associated to the
parameters sought by the bais of 'Levenberg-Marquardt
algorithm [15].
The paper is organized as follows: The second part
presents the model of camera and equilateral triangle. The
projection of the triangle is described in third part. Selfcalibration of cameras presented in fourth part. The
experiments are presented in the fifth and concluding part
in the sixth game.
Keywords: Self-Calibration, Equilatéral Triangle,
2. CAMERA AND EQUILATERAL
TRIANGLE
Absolute Conic , Homography.
A.
1. INTRODUCTION
We consider the pinhole camera model to transform a
point in the scene in his image, this model is defined by
the perspective projection matrix P of 34 given by the
The calibration of a camera is to determine the intrinsic
and extrinsic parameters using a known object called
calibration pattern [8, 9, 10, 18]. The grid may be threedimensional.(calibration 3D) or plane (calibration 2D).
This constraint is not always present in the applications of
computer vision, which gives rise to new methods called
self-calibration which allow to calculate the intrinsic and
extrinsic parameters without any prior knowledge on the
stage. The latter method uses 3D scenes [1, 2, 3, 21, 22,
23] or planar scenes [4, 5, 6, 7, 9] to calculate the camera
settings automatically, but they face many problems, such
as the majority of methods encounters the problem of
systems to solve non-linear, which must be initialized
while adding constraints on the model of self-calibration,
for example in [16] a constraint is added to the movement
of camera that undergoes a translation and a small
rotation, this constraint permits to estimate the
homography of the plane at infinity and to calculate
parameters of the camera.
In this paper we are interested on the camera selfcalibration with fewer constraints by the use of the object
of equilateral triangle any scene 2D and movement rigid
camera in space. Our technique is to estimate the
homography matrix between two images by the RANSAC
algorithm [11] based on points of interest detected by
Harris [12] which are matched by the correlation function
ZNCC [13, 14, 19 ]. Next, this matrix will be used with
two matches (the projections of the two vertices of an
equilateral triangle) to estimate the projection matrix and
determined the system of equations.
Model of camera
following formula : P  K ( R t ) with :

( R t) is the extrinsic parameter matrix, with R
is the rotation matrix and t the translation vector
of camera in space.

K is the intrinsic parameter matrix defined by:
 f   0 u0 


K 0  f
v0 
0
0
1 

(1)
f is the focal length,  is the scale factors,
(u0 , v0 ) are the image coordinates of the principal point
ant  is the obliquity factor.
B.
Triangle équilatéral
For any two points A and B in the plane of the stage,
there is a point C so that ABC is an equilateral triangle
set by the length of its side (Figure 1).
29
Special Issue of International Journal of Computer Applications (0975 – 8887)
on Software Engineering, Databases and Expert Systems – SEDEXS, September 2012

R  seuil : near a point of interest
B. Correlation Measure
We
use
the
correlation
measure
ZNCC  Zero mean Normalised Cross Correlation  to
match the bridges of interest detected by the Harris
algorithm, this measure is characterized by the invariance
to changes in local luminance linear and defined by the
following formula :
ZNCC (qi , q j ) 
The three heights (median, bisector and
mediator) are the same measure. They are
3
simultaneously equal a
with a is length of
2
one side of an equilateral triangle.
The three angles are the same as: 60 .
3. EQUILATERAL TRIANGLE
PROJECTION
Harris Detector [12, 17]: To extract the high points in
terms of information, we used the Harris detector is based
on the following function:
(2)
n
qi and q j two points of Harris detected in the two
images i and j .
xn  I (qi  n)  I (qi )
yn  I '(q j  n)  I '(q j )
C. Homography between images
RANSAC [11] : is an algorithm that estimates geometric
entities (homography between images in our situation)
from a dataset (matched points), whose error with respect
to the entity is found above a threshold .
33 matrix
transformation linking two points belonging to image i
and j by the following equation:
with :
Homography btween images : is a
( x, y) : The coordinates of pixel in question.
A
C
B

M= 
C
A
2 I
x
2
B
(4)
I (qi ) and I '(q j ) are means of pixel luminance on a
window centered respectively in qi and q j .
A. Interest points
E ( x, y )  ( x. y ).M ( x, y )t ,
 xn2  yn2
with :
a few properties of the equilateral triangle:

n
n
Figure 1. Equilateral triangle

 xn yn
 I
y
2
2
w
q j : Hij qi
(5)
2
w
w( x, y ) 
1
2 2
C (
 I
xy
with qi and q j are respectively the projections of the
)w
same point in the scene in the picture i and j .
( x2  y 2 )
2 2
e

H ij is the homography matrix between two images i
The matrix M characterizes the local behavior of the
function E . To detect points of interest, we evaluated the
following measure:
 R  Det ( M )  k  Trace 2 ( M )

2
 with : Det ( M )  AB  C
Trace( M )  A  B and k  0.04


R  0 : at the vicinity of an edge

R  0 : in a homogeneous region
(3)
and j , estimated by the knowledge of at least four
matches by applying the RANSAC algorithm.
D. Projection matrix of the equilateral triangle
Projection Matrixes : to estimate the projection matrix
we will use two references : reference Affine
and
reference
Euclidean
( B, X a ,Ya , Z a )
( B, X e ,Ye , Ze ) with Z e and Z a are perpendicular to the
plane of the triangle ABC .
Table 1 presents the coordinates of the vertices of an
equilateral triangle in two references
Affine and
Euclidean.
30
Special Issue of International Journal of Computer Applications (0975 – 8887)
on Software Engineering, Databases and Expert Systems – SEDEXS, September 2012
Table 1: Homogeneous Coordinates Of Vertices Of The
Triangle In The Two References Affine And Euclidean
Point
B
Affine plan
Euclidean plan
A
Q1  (0, 1, 1)T
a
3
Q '1  ( ,
a, 1)T
2 2
B
Q2  (0, 0, 1)T
Q '2  (0, 0, 1)T
C
Q3  (1, 0, 1)T
Q '3  (a, 0, 1)T
(6)
scale factor, by:
(7)
of camera in space to project the scene in the image i and
a


0
a
2




3
a 0  is the passage matrix, between the
S  0
2


0
1
0




reference Affine and Euclidean, of vertices of the triangle
as:
 S Qd
Lj
Image j
A
B
B
C
C
Figure 2. Projection of triangle ABC in the two images
i and j by the two matrixes Li and L j
1 0


T 
H i : KRi  0 1 Ri ti 
The
matrix
is
the
0 0



homography that permits to project the plan of the scene
in the image i , therefor formul (7) becomes:
Li : Hi S
Ri , ti represent, respectively, orientation and position
Qd'
Li
A
T
With 1 d 3 , qid  (uid , vid ) is a point in the 'image
i represents the projection of a vertices of the equilateral
triangle and Li a 33 matrix that can be defined, up to a
1 0


T 
Li : KRi  0 1 Ri ti  S
0 0



C
Image i
The triangle ABC , expressed in the Affine plan, is
projected in the image i by a matrix Li (Figure 2) as:
(uid , vid , 1)T : Li Qd
Planar scene
A
(8)
In practice there is not an automatic method to
determine the triangle in the image i . But we can always
determine which two vertices of the triangle can be
deduced four linear equations from the equation (6).
herefore we need other equations to calculate matrix Li .
(9)
And for the image j we can write:
Lj : H j S
(10)
From equations (9) and (10) we deduce that:
L j : Hij Li
(11)
with H ij is the homography betwin image i and j as:
Hij : H j Hi1
The matrix H ij
(12)
is determined by the
method
explained in the part C. in the two images i and j the
projections of the two vertices are given by:
(uid , vid , 1)T : Li Qd
(13)
(u jd , v jd , 1)T : L j Qd
(14)
with 1 d 3 . We devlop the two relations (11) and (14)
we find that:
(u jd , v jd , 1)T : Hij Li Qd
(15)
31
Special Issue of International Journal of Computer Applications (0975 – 8887)
on Software Engineering, Databases and Expert Systems – SEDEXS, September 2012
Equations (13) and (15) are sufficient to determine the
matrix Li , for this we use a method of singular value
Cost function : To solve the system (21). We minimize
the following cost function:
min
4. CAMERA SELF-CALIBRATION
with
with




S ' 




0
'

(16)
a
2
3
a
2
0
a
0




.




the
matrix
Ri
is
T
orthogonal ( Ri Ri  I3 ) , So the relation (16) can be
written as follows:
LTi  Li
 S 'T S ' S 'T RiT ti 
: 

 tT R S '
tiT ti 
i i

T
with   KK

(17)
is the projection of the absolute conic
: I3 in all images.
z3i 
 the matrix contains the first
z2i 
T
two lines and columns of the matrix Li  Li , therfor from
equation (17) we find that:
T
Mi : S ' S
 ij2  z3i z2 j  z2i z3 j
n represents the number of images used. To solve the
function (22) using the Levenberg-Marquardt algorithm
[15] which requires a very important step initialization.
Initialization : to initialize the function (22), we assume
that certain conditions are satisfied on the vision system.

The principal point is to the center of the image,
therefore u0 and v0 are known.

Pixels are squared therefore   1 .

by replacing these data in the system of equations
(21), we find the following system between
image i and j :
 
1
 z1i
We note by M i  
 z3i
(22)
l 2ij  z3i z1 j  z1i z3 j
After the development of the Equation (7) we obtain:

j i 1 i 1
 ij2  z1i z2 j  z2i z1 j
A. Self-calibration equations
K 1Li : Ri S ' RiT ti

n 1
  ij2  l 2ij  ij2 
n
decomposition. the L j matrix is determined by equation
(11).
'
(18)
This formula expresses the relation between the intrinsic
camera parameters and those of a triangle. Therefore for
two images i and j , relation (18) can be written :
Mi : M j

with   f 4 f 2
(23)

T
,  is a 3n 2 matrix and  is a 3n
vector such as  and  elements are expressed in
function of u0 , v0 ,  , Li et L j .
5. EXPERIMENTATIONS
To experiment with this technique and demonstrate its
effectiveness, we took three images of 512 × 512 of two
2D scenes unknown by an camera whose intrinsic
parameters are kept constant (Figure 3).
(19)
From the relation (19) we deduce the following equalities
between image i and j :
z1 j z3i z3 j z3i z3 j
z1i

,

,

z2i z2 j z1i z1 j z2i z2 j
Image 1
(20)
These equations (20) brought us to a system of three
nonlinear equations:
 z1i z2 j  z2i z1 j  0

 z3i z1 j  z1i z3 j  0

 z3i z2 j  z2i z3 j  0
(21)
Image 2
Image 3
Figure 3.The three images of a scene unknown 2D used for
self-calibration of the camera
To detect the rich points in terms of information, we used
the Harris algorithm that eliminates the noise by a
Gaussian filter and gives better results deal with the
transformations of the image related to the rotation,
scaling, change of views, change brightness and noise
related to the sensor [20, 16, 17]. the Harris points are
shown in figure 4:
32
Special Issue of International Journal of Computer Applications (0975 – 8887)
on Software Engineering, Databases and Expert Systems – SEDEXS, September 2012
6. CONCLUSION
Image 1
In these papers, we treated the problem of camera selfcalibration plan, using only two points of interest, which
is the projection of the vertices of a triangle in each
image. These points will be used to estimate the
homography matrix between images and the projection
matrix for each pair of image, leading to the end to a
system of nonlinear equations for determining the intrinsic
parameters of camera. Our technique therefore allows the
camera self-calibration with a 2D scene known, with a
simple, reliable and robust compared to other methods.
Image 2
7. REFERENCES
Image 1
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[3]
Manolis I.A. Lourakis and R.Deriche. Camera selfcalibration using the kruppa equations and the
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[4]
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[11]
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Image 3
Figure 4. The Harris points
For matching points detected by Harris in the three
images, we used ZNCC correlation measure which is
invariant to linear changes in local luminance, these are
presented in Figure 5:
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1
Image 1
Image 3
Figure 5. The Matched points
The homographies between images and projections
matrixes of a triangle (define by two vertices from interest
points, the third vertices is not used) in all images are
calculated, in short the resolution of a non-linear equation
system permits to estimate elements of the image of the
absolute conic and to calculate the intrinsic camera
parameters.
Table 2 presents the initial result and optimal intrinsic
parameters of the camera of two scenes:
Table 2: Initial And Optimal Solution Of The
Intrinsic Camera Parameters.
Initial solution
Optimal solution
f

u0
v0
1125
1175
1
0.92
256
261
256
263
33
Special Issue of International Journal of Computer Applications (0975 – 8887)
on Software Engineering, Databases and Expert Systems – SEDEXS, September 2012
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