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As this method was so remarkably effective on cell images, there was little
incentive to improve the method for the Pap smear problem itself, but we still
held a desire to develop more powerful global energy minimization techniques
that could be applied to a general class of objects. In particular, the Viterbi
algorithm based method would only work for objects that were convex and
two-dimensional.
In 2002, Appleton and Sun [8] put the problem of minimizing the energy
of closed contours unwrapped onto linear trellis onto a firm mathematical ba-
sis. Then, in 2003, Appleton and Talbot [6, 10] extended and generalized the
energy minimization approach to handle the optimal segmentation of planar
concave objects as well as convex images such as cells. This extension avoided
dependence on a coarse discretization grid so that grid-bias could be removed.
The extension to 3D was achieved in late 2003 by Appleton and Talbot [9] by
converting the shortest path techniques into an equivalent continuous maxi-
mum flow/minimal surface problem.
In this chapter we briefly describe the various energy minimization seg-
mentation techniques and show how they can be applied to solve quite dicult
segmentation and reconstruction problems in diverse domains from volumetric
medical imaging to multiview reconstruction.
10.2 Cell Image Segmentation Using Dynamic
Programming
Although the use of active contours [88] is well established, it is well known
that these methods tend to suffer from local minima, initialization, and stop-
ping criteria problems [44]. Fortunately global minimum energy, or equiva-
lently shortest-path, searching methods have been found that are particularly
effective in avoiding such local minima problems due to the presence of the
many artifacts often associated with medical images [51, 66].
The energy minimization method employed was based on a suggestion in
Gunn's dissertation [70]. A circular search space is first defined within the
image, bounded by two concentric circles centralized upon the approximate
center of the nucleus found by an initial rough segmentation technique (e.g.,
converging squares algorithm). This search space is sampled to form a circular
trellis by discretizing both the circles and a grid of evenly-spaced radial lines
joining them (Figure 10.2). This circular trellis is then unwrapped in a polar
to rectangular transformation yielding a conventional linear trellis.
Every possible contour that lies upon the nodes of the search space is then
evaluated and an associated energy or cost function is calculated. As with the
snake energy formulation of (9.3), this cost is a function of both the contour's
smoothness and how closely it follows image edges. The energy [13] is defined
by:
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