Three-Dimensional Pose Estimation and Segmentation Methods - 3D Computer Vision: Efficient Methods and Applications - page 102

method is based on the minimisation of the Hausdorff distance between edges ob-
served in the image and those inferred from the model projected into the image,
taking into account mutual occlusions of different parts of the articulated object.
For minimisation of the error function, the 'extended kernel particle filter' approach
is introduced by Stößel ( 2007 ) and employed as a stochastic optimisation technique.

2.2.2 Three-Dimensional Active Contours

This section is adopted from d'Angelo et al. ( 2004 ), who describe a parametric
active contour framework for recovering the three-dimensional contours of rota-
tionally symmetric objects such as tubes and cables. The proposed algorithm is a
three-dimensional ziplock ribbon active contour algorithm based on multiple views.

2.2.2.1 Active Contours

In the snake approach by Kass et al. ( 1988 ), the basic snake is a parametric func-
tion p representing a contour curve or model:

p = v (s) for s ∈[ 0 ,l ] , (2.3)

where p is a contour point for a certain value of the length parameter s . An energy
function E C is minimised over the contour v (s) according to

l

E snake v (s) ds.

E C = (2.4)

0

The snake energy E snake is separated into four terms:
E snake v (s) = αE cont v (s) + βE curv v (s) + γE ext v (s) + δE con v (s) . (2.5)

The 'internal energy' E int =

+

βE curv ( v (s)) regularises the problem
by favouring a continuous and smooth contour. The 'external energy' E ext depends
on the image at the curve point v (s) and thus links the contour with the image. Here
we use the negative gradient magnitude of the image as the external energy, which
then becomes E ext =−∇ I( v (s))

αE cont ( v (s))

. E con is used in the original snake approach by
Kass et al. ( 1988 ) to introduce constraints, e.g. linking of points of the active contour
to other contours or springs. Balloon snake techniques (Cohen, 1991 )use E con to
'inflate' the active contour in order to compensate for the shrinking induced by the
internal energy E int . The weight factors α , β , γ , and δ can be chosen according to
the application.
The dependence of the snake model on its parameterisation may lead to nu-
merical instabilities and self-intersection problems when it is applied to complex
segmentation tasks. Such problems are avoided by implicit active contour mod-
els (Caselles et al., 1995 , 1997 ). Furthermore, a contour is not necessarily a single
curve, and modifications to extract ribbon structures consisting of parallel lines, like