International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004 
  
& © 
Fig. 3: Cylinder fitting experiment (a) Point Cloud (b-d) Images 
with back-projected model in yellow, point measurements in 
red, and sub-sampled point cloud in white 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
Parame Image Point 
ter 1 2 3 Cloud Bot 
1 X 52.269 | 10.326 9.923 0.838 0.762 
2 Y 163.45 14.168 | 13.629 1.335 0.862 
3 Z 12.740 3.489 3.467 119.90 1.654 
4 t0 0.056 1.0E-2 | LOEO2 | 228E-3 | 2.0E-3 
5 tl 0.065 2.1E-2 2.1E2 | 3.96E-3 | 2.9E-3 
6 t2 5.283 2.538 2.534 0.359 0.237 
7| Length | 10.591 3 121 3.120 oo 1.857 
8 | Radius 169.57 16.166 15.522 0.634 0.565 
  
Table 1: Standard deviation for Cylinder fitting experiment 
Each point measurement in the image gives us a ray in 3D. 
Given a set of images with measured points we want to estimate 
those values for CSG parameters that result in the minimum 
distance between all these rays and the estimated CSG model. 
Alternately, the ray to body distance can be calculated in image 
space. There we have to compute the distance in pixels between 
an image measurement and the closest back-projected contour 
of the CSG model. The back projection must have a mechanism 
for hidden-line removal, so that the effects of self and external 
occlusions are taken into account. We follow the second 
approach, and use ACIS (2004), which a commercial geometric 
modelling engine to compute the hidden line projection of the 
model in the image. For an example see Fig. 2(b). 
Thus the fitting problem reduces to the estimation of those 
values of the CSG parameters, which minimize the sum of the 
squares of the orthogonal distance of the image measurements 
from the back-projected edges of the model in the image i.e., 
N 
min Y N^? [m,, H(1,.7,....,7,,)] (8) 
i=l 
Where Y defines the shortest distance of a given measurement 
m, in an image to the closest edge of the back projected CSG 
model H , which has M shape and pose parameters given by 
T,,T,,.-., Ty - There are Nimage measurements given by 
My, Ms... My - 
The problem of minimizing V?is also a non-linear least 
squares problem and is similar to that of minimizing 
Q discussed in the above section. We need partial derivatives 
QW- oy Qv 
T eO ritu, 
Although analytic expressions for the estimation of the partial 
derivatives for some of the CSG objects are given by Ermes et 
al (1999) here we estimate them numerically using finite 
differences. The final estimation uses Levenberg-Marquardt in 
combination with Singular Value Decomposition. The details 
are similar to the ones discussed in Section 3.1 . 
  
with respect to CSG parameters Le. 
4. FITTING EXPERIMENTS 
As it was said in the introduction, images and point clouds 
provide complementary sources of information, and by their 
combination we can expect better estimation accuracy. Edges of 
the object where laser scanner usually provide noisy data are 
captured best in the images. Additionally, while fitting bounded 
objects point clouds do not contain enough information about 
determining the bounds, whereas by providing the full edge 
outline images fix the bounds. For example in the case of a 
cylinder usually the closing lids on both sides are not scanned 
either because they are not visible due to the connections with 
other surrounding objects, or because it is not convenient to 
place the scanner in a position where the lids are visible. As a 
result we expect the length of the cylinder to be poorly 
determined by such a point cloud. In contrast the measurements 
in the image provide points on the edges and thus help improve 
the precision of the length estimate. 
To demonstrate the complementary nature of the information 
coming from images and point clouds we will do some fitting 
experiments on two test objects. Each object will be fitted three 
times, first using only point cloud, then using only image 
measurements and finally a combination of both. The point 
clouds we will use were captured using a Cyrax scanner. We 
assume standard deviation of 5mm for each point. The images 
were captured using a Nikon CoolPix camera having a 
resolution of 5 mega pixels and using a fixed focal length of 
7.34 mm. The standard deviation for image measurements is 
taken to be 1 pixel. 
4.1 Cylinder fitting 
The arrangement we used for the first experiment is shown in 
Fig. 3. A cylinder is scanned from the front, and images are 
taken from three different positions. We sce back-projected 
hidden lines in yellow, points measured on edges in red, while 
the sub-sampled point cloud is shown in white. A cylinder is 
represented by 8 parameters, 3 for the position, 3 for the axis, 
one for the radius and one for the length. In Table 1 we see the 
standard deviations obtained for different parameters by doing 
fitting to point clouds, images and to a combination of both. For 
images we did fitting separately using one, two and three 
images, while in case of both all of the three images were used. 
As expected in the case of using only point cloud the length of 
cylinder is not determined because in the absence of points on 
upper and lower lids there is not enough information in the 
point cloud for its determination. Because we use singular value 
decomposition the length parameter is taken out of the 
estimation during matrix inversion and thus its value remains 
fixed on the initial starting point this results in standard 
deviation of ee for length. 
   
  
   
  
  
  
    
  
  
  
  
  
  
  
  
    
  
  
  
   
   
   
  
  
    
    
   
     
  
   
  
  
  
  
  
   
   
    
   
   
   
   
  
   
   
  
  
  
  
  
   
  
   
  
   
    
   
   
   
     
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