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least 
squares adjustment. The observation equations are based 
on the parametric equations of the straight line (eq. (2)). 
The vector of unknowns is: 
Xa. — X, Y Zur Bi ln e Tes Ay Apr Anl 
where 
(h,.M, N) are the direction cosines of the n™ straight 
line; 
À, is the unknown distance between the perspective 
center of the projector and the p'^ reference plane. The 
planes are supposed to be parallel to the camera 
reference system. 
The vector of observations is: 
Lb = IX Y. Zi x^. Y^ Zu. X^, Y^. Z X',, 
Yo Z^ Xoo, Yo, 22g... XC, Yoo Zen X15, Vy, 
Zr XN ean on Zee] 
where the X",, Y",, Z", are the coordinates of the n° 
projected point over the p™ reference plane. These 
coordinates were computed using the parameters of 
space resection for each plane. 
A constraint must be introduced to eliminate correlation 
between A, and the direction parameters (l, m; nj), 
otherwise the results will not converge. The simplest 
way to establish a constraint is to fix the A, parameter. 
The direction vectors and the others À; will be related to 
A. Non normalised values of the direction vector 
parameters will be obtained. This means that: 
2 + m2 +n? = 1 (5) 
After the simultaneous estimation process each direction 
vector can be individually normalised. 
Another approach could be to establish the following 
constraint: 
Im?» zi (6) 
This option was also tested but the first one showed to 
be simpler and more efficient if the computational costs 
taken into account. 
5. SURFACE FITTING AND INTERPOLATION 
The obtained point coordinates in the object space define 
an irregular mesh of triangles. In order to generate a 
range image, a regular grid must be interpolated from the 
irregular data. Triangles could be selected using Delanay 
Triangulation but such approach is time consuming. In 
our approach, we have previously recognised each 
projected point. Using this knowledge it is more efficient 
to establish neighbourhoods and then apply an 
interpolation method considering a set of nine neighbour 
points. This interpolation scheme is based on the 
adjustment of a surface over the nine points which are 
371 
represented by the following mathematical model: 
Z - a«b.X«cY«dX?«eXY«fY? (7) 
Selection of points to be used in the adjustment can be 
carried out using the minimum distance criteria. The 
central point of a 3x3 matrix is the closest with the 
points to be interpolated (points of the regular grid). 
The interpolated regular grid can be transformed into a 
range image associating a pixel to the Z coordinate. This 
range image is suitable for automatic analysis in practical 
applications of measuring volumes, dimensions, shapes 
and deformations. 
6. RESULTS 
6.1 System Calibration 
Results obtained with the proposed mathematical model 
of system calibration using simulated data are presented. 
Inner orientation parameters of the camera were 
supposed to be known from a previous camera 
calibration with self calibrating convergent cameras. 
6.1.1 Simulated Data 
In order to verify the accuracy of the proposed sequential 
system calibration, simulated data were generated using 
the following camera parameters: 10mm focal length, 
5x4 mm? imaging area, and 10x10 um? pixel size. The 
position of the perspective center of the projector was 
supposed to be (X,=300mm, Y,=10mm, Z,= 10mm), 
with respect to the camera reference system. 
Eight control points were introduced in the projection 
plane. Fifteen points were projected onto the projection 
plane which was moved at three different positions with 
respect to the camera: 
-1050mm, -900mm, -750mm. It means that the 
separation (AZ) between each projection plane was 
150mm. The geometric configuration of the control and 
projected points and the projection planes are presented 
in the figure 3. 
Object coordinates of the projected points onto the p^ 
reference plane were obtained from the parameters of 
the projected straight lines. These parameters were 
computed from the simulated image coordinates of a grid 
located in the projector. 
Finally, image coordinates of projected and control points 
were computed using the collinearity equations. Random 
errors with a standard deviation of 3um were introduced 
in the image. This error is equivalent to 1/3 of the pixel 
size and can be easily obtained by any of the presented 
feature extraction methods. 
The simulated data consist of a set of image coordinates 
of control and projected points over three reference 
planes. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B2. Vienna 1996