3.2 Optimized Algorithm For Incorrect 
Position 
This step of the reconstruction algorithm 
is required when there are systematic 
errors of positioning the central axis. 
Considering the centralization is not 
known for all views, the correct relative 
positioning is found using the 
mathematical programing problem of 
maximization of volume as in the original 
algorithm but with different constraints: 
pt ( h(x-a) ) =F for any a € R 
The maximization of volume has two 
physical consequence on the resultant 
shape, first it eliminates the empty 
intersection of any two strips of two 
silhouettes, that is, 
BPO zh. (3) 
If the data contain errors of positioning, 
the choice of one F e 5 does not imply in 
Second, in some special cases it is not 
possible to find the correct 
centralization of one view using (6) so 
the maximization of volume find the 
centralization of maximum volume only but 
not with the correct centralization. This 
choice makes the probability of the 
object is contained in the volume found. 
Note that this is the only case the object 
may not be contained in the solution 
volume. 
3.3 Combination 
For each investigated plane, there is a 
set of compact intervals of all views. 
The set of compact interval of each view 
corresponds to a set contained in the 
cross section of €, these longitudinal 
stripes in the cylinder is called 
effective stripes (ES). 
S={x € R°|G(x)=z,z e LII is CI of 
ith view) 
The procedure of enumerate all the 
combinations to be tested, adopts another 
numeric basis to write the number of 
combinations tested and to identify the 
next combination. 
This new numeric basis is constructed like 
that with J : the first figure in the 
representation varies from 0 to the total 
number of compact intervals in the first 
view, the second varies from 0 to the 
total number of CI of the second view and 
so on. 
338 
For instance, let N, N,, Ne N, total 
numbers of compact intervals respectively 
for each view, the representations become: 
00.70 - 0 
10...0 - 1 
: dO 
aa... a - RX aN 
1=1 
Besides numbering the combinations tested, 
this representation also an easy way of 
classify an combination. The combination 
number 8, 8,, ...5 à, corresponds to the 
a th compact interval of first view and so 
on. 
A subroutine which can be used for this is 
called once before each test: 
COMBINATIONS: 
for i = 1 to J 
if a = N. then 
a = 0 
v 
else 
a, = à + 1 
return 
end if 
next i 
return 
3.4 Polvgons 
The Polygon algorithm follows a systematic 
path in order to identify all the vertices 
of each polygon in order. 
For defining a polygonal intersection of n 
straight lines some point should be 
considered: 
(i) not all intersection points of two of 
the lines are vertices of the polygon; 
(ii) not all lines that pass through one 
vertices of the polygon form a side of it; 
(iii) for all line that form a side of the 
polygon pass through two and only two 
vertices of the polygon. 
This aspects although obvious are essential 
for a precise definition of the polygon. 
The process starts with the parameters of 
all lines that limit the slices in plane. 
One line is chosen for being the the first 
line that will be the parameter for 
finishing the search. Another line is 
chosen and it is verified if the point of 
intersection belong to the polygon, if it 
does, other lines are tested until the 
second point of the second line is find. 
This process continue until the second 
intersection point of the first line is 
defined. 
It is important to neglect the lines that 
have two different intersection at the same 
point, that is the tangent lines.