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line representation. The Pluckerian representation is the 
canonical line representation in projective geometry. 
The Pluckerian coordinates are defined as follow: Let 
P be the cartesian coordinates of an arbitrary point on a 
line D in a 3D space and 1 be the unit direction vector 
of the line D. We introduce and often use the vector H 
which is the orthogonal from the origin O to the line D. It 
is easily seen that 
PAI= HA 
or 
PAI=N=hn 
where N = HA], and n = TN is the normal to the plane 
defined by the 3D line D and the origin O, and finally 
h = || H || = || N || represents the distance of the line to 
the origin (see fig 2). Therefore, the line equation will be 
PAI=N 
The two vectors (1, N) define the Pluckerian coordinates of 
the line D. Note that H, 1, and N form a right handed 
coordinate system. n 
  
Fig.2. The vectors n, 1, and h. 
Using this line representation we need four parameters 
to represent a 3D line. Two parameters for the unit line 
direction l, and two parameters to define the vector IN or 
H which are orthogonal to the line direction 1. 
Image lines : A line d, the projection of a 3D line D 
on the image plane is called a 2D line. In the camera 
coordinate system, this line may be considered as a 3D 
line which lies on the plane z — 1. Therefore its equation 
is simply: 
min =0 
where m — (z, y, 1)? is an arbitrary point on d. 
The vector n is the normal to the plane containing the 
3D line, its image and the camera optical center. Therefore, 
this vector n is the same as the vector n — INT introduced 
in the previous paragraph. We may even use the vector 
N to represent the image line when the 3D line is given in 
camera coordinate system. Usually we have only access to 
the image lines. Therefore, we prefer in general represent 
the image lines by a unit vector n. 
If we take n = (a, 8,7)" the line equation is written 
as: : 
az +pfy+y=0 
The vector n is a unit vector and therefore, our 2D line 
representation depend only on two parameters. 
4  Ego-motion estimation 
Much work has been done on the motion estimation from 
straight lines. In the case of discrete motion, we can par- 
ticularly mention the works of Liu, Huang, Spetsakis, Ag- 
garwal, Chellapa, and Vieville [13, 12, 20, 1, 4, 25] on the 
  
monocular sequences and that of Zhang [26] on the stereo 
sequences. And in the case of continuous line motion analy- 
sis approach we may refer to the works of Faugeras, Navab, 
and Henriksen [8, 9, 19, 11]. One may easily verify that for 
each of the fundamental formulae obtained in one case (dis- 
crete or continue) one can find a similar and related one 
in the other case [15]. In this paper, we use the continu- 
ous approach. After the first steps, when we can use more 
frames this approach shows its real advantages. 
Let us take a 3D line represented by the vectors (N, 1). 
We now describe its motion (N,1). In order to gain more 
insight into the problem, we assume that the 3D line under 
consideration is attached to a rigid body whose motion 
is described by its instantaneous angular velocity, £2, and 
linear velocity V, its kinematic screw at the origin O.We 
can also suppose that the object is static and the camera 
system has such a motion description. 
We know that the velocity P of any point P attached 
to the rigid body is given by 
The normalized direction 1 satisfies a simpler differential 
equation: 
i=QAI (2) 
The vector H can be expressed as 
H = P — (PT) (3) 
therefore, 
H = P — (PTIi - (P71 + PTi)l 
Replacing P andi by their values from equations 1 and 2, 
H — V.CQOAIP- (P7 -(v?y 
we obtain: 
H-QAH-V -(V?DI (4) 
Then it’s easy to obtain N, the time derivative of N = HAI: 
HAL+HAI 
(QAH-tV-(VTID)AI-HA(QA1) 
QA(HAD)+VAI 
N 
and we obtain: 
N=QAN+VAI (5) 
Therefore, the motion of a 3D line can be defined as follows: 
Asli] 6 
where the matrix D is defined as follows: 
Qo 
»-[i 2] 
Line Motion Field Equation: What we measure from 
the images are the 2D lines represented by the unit vectors 
n and their motion fields n. Therefore, we give here the 
line motion field equation. Line motion field equation was 
first given in [19]. We used two points on the line to obtain 
that result. Here we draw the same equation from the 
above equations. we have n = TNT: therefore