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path with just one run of the Viterbi algorithm, although some pathological
examples may take considerably longer to compute.
(a)
(b)
(c)
Fig. 10.6. Segmentation of Cyclostephanos Dubius by circular shortest path
method. (a) The original microscope image of the diatom, (b) the polar unwrapping
with circular shortest path overlaid, and (c) the corresponding segmentation contour
(from [7]).
However, despite this improvement, a major shortcoming of all methods
based on a polar to rectangular mapping is the inability to handle concave
contours and higher dimensional objects, thus severely limiting their applica-
tion domain.
10.3 Globally Optimal Geodesic Active Contours
(GOGAC)
The classic active contour or snake model proposed by Kass [88] modeled
a segmentation boundary by a series of point masses connected by springs.
This explicit view of curves as a polygon was replaced by an implicit view of
curves as the level set of some 3D surface by Osher and Sethian [129]. Level
sets offer significant advantages over traditional snakes including improved
stability and much better handling of topology (e.g., segmentation of multiple
objects with just one contour). Another advance came in the form of geodesic
active contours as proposed by Caselles et al [34]. They demonstrated the
equivalence of their energy function to the length of a geodesic (i.e., path of
least cost, path of least time) in an isotropic space. A problem with traditional
geodesic active contours is that they are a gradient descent method and thus
have all the usual problems of initialization, termination, and local minima
associated with such methods. They simply do not have the stability and
simplicity of application of globally optimal segmentation methods.
The globally optimal GOGAC method we outline here finds closed con-
tours in the image domain itself rather than unwrapping the image through
polar to rectangular transformation. Working in the image domain means that
we cannot find simple shortest paths, as this would cause a bias towards small
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