3.3 Reconstruction algorithm
the j-th parallel reticular position (i, j), with m, k, and b the mass associated
to the point, the constant spring linking the point with its nearest four points of
the cross of the reticulum, and the constant spatial damper, respectively. Notice
that subscript i = j = 0 (i = n
m
+ 1 and j = n
p
+ 1) for the representation of the
four nearest points in the reticulum corresponds to i = n
m
and j = n
p
(i = j = 1),
respectively.
The two poles have to be treated separately due to their topological peculiarity.
For these points, the previous model becomes
m¨
x
nt
+ b x
nt
+ k(n
m
x
nt
-
n
m
j=1
x
1,j
) = f
nt
for the north pole, and
m¨
x
st
+ b x
st
+ k(n
m
x
st
-
n
m
j=1
x
n
p
,j
) = f
st
for the south pole, where the subscripts nt and st indicate quantities refereed to
the north and south pole, respectively.
The term f
i,j
is the external force acting on the mass associated to the sample
point (i, j), attractive with respect to the border of the visual hull, which is pro-
gressively reduced once the corresponding point comes in or out from the visual
hull:
f
i,j
=
i,j
(t
i,j
)F
a
n
i,j
,
where n
i,j
is the unit vector pointing from the current point (i, j) to the estimated
centroid of the object, that defines the direction of the force, and
i,j
(t
i,j
)F
a
is
the amplitude of the force. In particular, F
a
is the maximum force module and
i,j
(t
i,j
) (-1, 1] is a discrete numerical sequence of scale factors which is defined
as follows:
i,j
(t
i,j
) = -
i,j
(t
i,j
- 1)
where (0, 1),
i,j
(0) = 1, and t
i,j
= 0, ..., is a discrete step time incremented
every time the point (i, j) comes in or out of the visual hull.
The stability of the system, for any non-trivial initial condition of the ellipsoid,
leads the reconstruction elastic surface to shrink toward its center of mass until
the visual hull is intersected. The dynamic evolution of the system reaches the
equilibrium when the shape of the surface produces a dynamic equilibrium between
the elastic forces generated by the grid and the repulsive forces depending on the
contours of the visual hull. The result is that the initial elastic reconstruction
surface shapes itself on the unknown object.
From a performance point of view, the accuracy of the reconstruction process
depends on:
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