ih "WW 
L9 e RA N 
In order to solve this problem , is 
necessary to define inverse mapping P so 
that: 
P*(P,(F))=P*(h )=F for i=1 ton 
As the left inverse transform of Pi does 
not exist, there is infinite set of 
functions F which satisfy (1). This set * 
is, 
#={F | P,(F) = h for i=l ton } (3) 
The modification of purpose is not 
sufficient to solve our problem although, 
using this restricted reconstruction a 
special class of subset of IR^ can be 
exactly reconstructed for silhouettes take 
from some special directions. 
In practical the solution set can be 
defined in another way that leads to the 
algorithm: 
S - n * 
where 
F ={FeQN | FH =n) 
Once the problem is intrinsically ill 
posed concerning to uniqueness, we have to 
choose one element of F. As the choice is 
arbitrary, any function could be used, 
however, the method chooses the function 
which has the largest support. The new 
problem changes to: 
Let M(F) be the measure of the support of 
F, that is M (supp(F)) = volume (supp(F)). 
Find S" supp(F, ) that satisfies: 
M(s,) = max M(supp(F)) 
Fes 
Before start solving the problem, a 
further analysis in the new formulation 
should be helpful. In fact, the solution 
St © is the whole region where it is 
possible to find a body. 
Note that the solution of maximum volume 
is always convex in planes parallel to the 
image rays. 
Regarding to the conclusion above and 
assuming the data of projections, hi i=l,J 
is consistent, So is the intersection of 
the cones contained in 2, showed in figure 
2. If the distance of the camera from © 
is large enough to consider that the rays 
coming into camera are parallel, s, can be 
find ss intersection of cylinders. 
337 
3. ALGORITHM 
3.1 General Overview 
From the problem discussed over, 
reconstruction of forms from shadows 
consists in finding the intersection of 
cones or cylinders whose bases sre the 
silhouettes. This interception occurs in 
space at the center of the circle 
corresponding to the experimental layout. 
In this paper the approximation of 
cylinders is used. 
The algorithm is prepared to identify more 
than one object, what mean that more than 
one cylinder for each view is intercepted. 
As cylinder approximation is used, the 
intersections of the solids can be 
numerically done in intervals in the 
direction of the axis. This method makes 
easier finding the intersections because 
instead of solids, we use slices of planes 
which intercept forming polygons. 
In each plane, we can find more than one 
polygon even if there is only one object. 
This happens when there are non convex 
silhouettes, figure 3. 
The steps of the algorithm are: 
i) Pre-processing of projection data, that 
is, finding the binary limit (BL) that 
differ, in each projection, the existence 
or not of an object. 
ii) Reading data of a height interval of 
each view, this data corresponds to a 
horizontal line in digitized image. 
iii) Determining compact intervals (CI), 
line segments that are over the binary 
limit, that is, I = [z,,2,] is an compact 
interval if Vz e I, £ (2) > BL. Compact 
intervals of one view do not depends on 
others views. 
iv) Finding the intersections of the 
compact interval of different views. 
The fourth step is the reconstruction 
itself and include three specific steps: 
COMBINATION, POLYGON and POLYHEDRAL. 
COMBINATION is the routine that makes all 
the possible combination of compact 
intervals in order to find a non empty 
polygon, POLYGON is the routine which do 
the verify if it is empty or not and 
POLYHEDRAL construct the lateral polygons 
covering the volume.