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A priori information about camera location and 
orientation (space resection) can be used to draw a 
polygon which probably contains the searched feature. A 
Hough transform can be limited to this subimage, 
reducing mathematical operations and generating a few 
candidate clusters. Using a filtering approach, 
parameter estimation is recursively improved for each 
new observation introduced and, the better is the 
camera location estimate, the smaller is the subimage 
to be processed. 
Kalman Filtering is a suitable method to get 
optimal estimates for parameters in robotics 
eye-in-hand vision dynamic space resection. 
Broida and Chellapa (1986) proposed an approach 
for the estimation of object motion parameters based on 
a sequence of noisy images. The movement of the body is 
modeled as a function of time and recursive filtering 
techniques are used to estimate the modeled parameters 
from the data measured in the images. In this approach 
image coordinates of two points in the body over a 
sequence of images are used as measurements. 
A similar approach is adopted by Liang, Chang and 
Hackwood (1989) to address the problem of a 
vision-based robot manipulator system. In their method 
intrinsic (interior) and extrinsic (exterior) 
orientation parameters are recursively estimated based 
on several images of a single point. 
1.3 Objectives 
This paper presents a filtering and feature-based 
approach to  space-resection (camera calibration). A 
mathematical model is developed which is based on 
straight features. This model is treated by Kalman 
filtering techniques, and it is shown that an iterative 
process between image processing and parameter 
estimation can be defined aiming a reduction on the 
computational effort. 
2. GEOMETRIC MODEL FOR LINE CORRESPONDENCE 
2.1 Object to image space transformation 
There are several possible ways to define the 
sequence of transformations that relates two reference 
systems. In this paper, object to image space 
transformation is done by translating the origin of the 
object space until it reaches the image space origin 
and, then, rotating the resulting system around the 
resulting system axes. This transformation is described 
mathematically by equation (2.1.1) where Xc, Yc, Zc are 
the coordinates of the camera perspective center (image 
space origin), R is the rotation matrix, and A is a 
scale factor. 
x X - Xc 
y = AR Y - Yc (2.1.1) 
z Z - Zc 
Let k, ¢ and w be the rotation sequence, the 
resulting rotation matrix R is given by: 
cos cosK  cos( senK+senl send cosK sen senK-cosw send cosK 
-cos® senK cos cosK-senW send senK senW cosK+cost) sen senk 
sen® -senW cos® cosw cos® 
(2.1.2) 
2.2 The interpretation plane 
The following concepts will be developed supposing 
that systematic errors in the image coordinates, such 
as symmetric radial distortion and horizontal scale 
factor, were previously eliminated and that the image 
coordinates were reduced to the principal point (image 
center). It will also be assumed that focal length is 
known and that the camera and object remain static 
during one image frame acquisition. 
  
  
The interpretation plane is defined by a straight 
line in the image and the camera perspective center in 
image space reference system. Similarly, a straight 
line in the object space and the camera perspective 
center define a plane. This plane was called the 
interpretation plane by Barnard (1983). 
Let pi = (x1, y1, f) and p2 = (x2, y2, f) be two 
distinct image points defining a line % the 
interpretation plane % associated with £ can be 
represented by its normal vector N 
f.(y, = y, 
> + > 
Ns pxp, f(x, - x,) (2.2.1) 
sh 41% 
and the interpretation plane equation is given by 
f yx * f(x -x.).y * (x, .y,7x, y, ).z = 0 (222) 
The representation based on the line described by 
two points is suitable for analytical photogrammetry, 
but not for digital photogrammetry and vision. The line 
representation in polar coordinates (0-p) has 
advantages for applications in those areas mainly if 
Hough transform is used. For a description of Hough 
transform see Gonzalez and Wintz (1987). 
Considering the parametric representation of a 2D 
straight line: 
y=ax +b (2.2.3) 
and its normal representation: 
cos0.x * sino. y - p = 0 (2.2.4) 
with: 
a = -cotan @ = tan « (2.2.5) 
b = p/sine = -p/cosæ (2.2.6) 
the equation of interpretation plane can be written as: 
f.cos0.x + f.sin6.y - pz = 0 (2.2.7) 
and the normal vector to the interpretation plane is: 
f.cosO 
> 
N = | f.sin0 (2.2.8) 
-p 
2.3 Measurement Model 
The concept of interpretation plane can be useful 
in deriving models with particular features. Liu, Huang 
and Faugeras (1990) proposed a model which enables two 
steps for the camera parameters search. Tommaselli and 
Lugnani (1988) presented a model called "equivalent 
planes model" that is based on equivalence between 
parameters of the interpretation plane in object and 
image space. The approach to be developed in this paper 
is based on this model but eliminating the parameter 
(A) which is a scale factor between parameters of the 
planes in object and image space. 
Let £ be a straight line in the object space 
with the parametric representation: 
X X1 1 
P: 1Y | = } Yi | + EC} M (2.3.1) 
Z Zi n 
where Xi, Yi, Zi are the object coordinates of a known 
point in the line; l, m, n are the directions of the 
line and t is a parameter. 
Let be the normal vector to the interpretation 
plane in the object space and defined by the vector 
product, between the direction vector n and the vector 
(PC - C) (see Fig. 2,3.1). 
Rotating the f vector in order to compensate for 
the angular differences between object, and image space 
reference systems, the normal vector N and the rotated