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roads in aerial images (Fua and Leclerc, 1990 ) or blood vessels in angiographic im-
ages (Hinz et al., 2001 ), have been proposed in the literature. Snakes can also be
used as a segmentation tool in an interactive manner, where a human operator pro-
vides a rough estimate of the initial contour and can move the snake such that local
minima of the energy function are avoided (Kass et al., 1988 ).
For the method described in this section, the greedy active contours approach in-
troduced by Williams and Shah ( 1992 ) is used as the basis for the three-dimensional
snake framework. The contour is modelled by a polyline consisting of n points, and
finite differences are used to approximate the energy terms E cont and E curv at each
point p s , s
=
1 ,...,n according to
E cont v (s)
p s 1 h
p s
(2.6)
E curc v (s)
p s 1
2 p s +
p s + 1 ,
where h is the mean distance between the polyline points. The greedy minimisa-
tion algorithm is an iterative algorithm which selects the point of minimal energy
inside a local neighbourhood. The greedy optimisation is applied separately to each
point, from the first point at s
l . The energy E C ( v (s))
is calculated for each candidate point p in a neighbourhood grid H ∈ R
=
0 to the last point at s
=
d , where d
is the dimensionality of the curve. The point p min of minimum energy inside H is
selected as the new curve point at v (s) . This procedure is repeated until all points
have reached a stable position or a maximum number of iterations has been reached.
Since the greedy optimisation algorithm does not necessarily find a global min-
imum, it needs a good initialisation to segment the correct contours. Especially in
segmenting non-rigid objects, however, providing a suitable initialisation along the
whole contour might not always be feasible. In such cases, the ziplock snake al-
gorithm introduced by Neuenschwander et al. ( 1997 ) is used. Ziplock snakes are
initialised by the end points of the contour segment to be extracted and the tangents
of the curve at these points. The contour consists of active parts, which are subject
to the full energy term E snake , and inactive parts, which are influenced only by the
internal energy E int . The 'force boundaries' between the active and inactive parts of
the snake start close to the end points of the contour segments and move towards
each other in the course of the optimisation.
2.2.2.2 Three-Dimensional Multiple-View Active Contours
If volumetric images are available, the image energy E ext of a three-dimensional
contour can be calculated directly from the volumetric data. In industrial quality
inspection applications, volumetric data are not available, but it is usually possible
to obtain images acquired from multiple viewpoints. In that case E ext can be calcu-
lated by projecting the contour into the image planes of the cameras (Fig. 2.3 a). The
camera system is calibrated with the method of Krüger et al. ( 2004 ) (cf. Sect. 1.4 ).
An arbitrary number N of images ( i
1 ,...,N ) can be used for this projection.
The intrinsic and extrinsic parameters of each camera are assumed to be known.
=
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