1.2 Cones and Cylinders From Plane Images 
Each of the n images filmed by the video 
and stored in magnetic disk are 
pre-processed into a binary image, one 
color level for the object and another for 
the background. 
The binary image is a mapping over the 
two-dimensional Euclidean space defining a 
set where the body was identified. This 
set may be convex or concave and results 
from projection of the body over a plane 
positioned straight over one direction. 
In figure 2 the experimental layout is 
showed. 
® un 
ul 
Figure 2 
The discussion of the use of cylindrical 
geometry instead of conical geometry 
depends specially on the distance from 
object an camera and the largest dimension 
of the object itself. The resultant 
errors from the use of cylindrical 
geometry can be estimated and used only 
when applicable, that is, when it is 
comparable to errors from digitalization. 
2. PROBLEM FORMULATION 
2.1 Mathematical Formulation 
Originally, the three-dimensional object 
reconstruction problem can be 
mathematically described as: 
Let us define F: R+ IR the function that 
characterizes an object in space. This 
function can refer to any property of the 
body itself or m surface property like 
color. It should be assured that this 
function can determine the position and 
the form of any object. 
Before we introduce the inverse problem 
consisting of the three-dimensional 
reconstruction, we will define the direct 
problem. 
For describing the direct problem some 
definition are missing: 
Considering that F e À c C[R*], the piece 
wise continuous function set, let P : O » 
D where: 
336 
-(hec[R^] | n : i^. m ), 
the projection mapping such that, 
P(F)-h and P(0)- 9 
where Q is the null vector in NR and o is 
~ 
the null vector in F 
Another property of a projection mapping 
is that if EF) z h for all i where 
E: 9 * R and h : a R ; 
8, c IR" and ene =¢ forixj; 
e, € IR^ and ana zd for i* j ; 
then, for F : IR^. IR such that Ax) = F,(x) 
forallxe 6 and Hx) = O0 elsewhere and 
for H : R^. IR such that Ky) - h (y) for 
all y € a and Ky) = 0 elsewhere, the 
application of the projection mapping 
results in: 
P(F) = H (1) 
The silhouette is a kind of projection 
mapping that for F : © + R where © = 
supp(F), that is, support of function F, 
P(F) =h such that h(y) =K for all y 
It can be proved from this definition of 
silhouette, that the inverse mapping such 
that, 
-1 
P Ph (h)=h 
can be found and it is the right inverse 
transforming of P. Nevertheless _ the 
projection Pa is not unique because Pa =F 
as P is not an injective trangform. Thus, 
the left inverse transform P such that 
-q - 
PPE =F 
does not exist. 
This preliminary analysis leads us to the 
conclusion that the initial purpose of 
finding F from a finite number of 
silhouettes is not feasible. 
However, finding the body function may not 
be the better way of reconstruct the body 
form. If this is our interest, finding 
the support of F, will be enough. Colors, 
texture and densities become important in 
other kinds of reconstruction. So, after 
changing the goal, the problem is: 
Given h, : IR^, IR and P, ,i-1,...,J ,Sv find 
supp(F) c 2 c IR? according to the 
following correlation: 
PAF) = h, (2) 
t 3 
Fh OO BN cB be