Snakes and Active Contours - Bezier and Splines in Image Processing and Machine Vision - page 185

Finally the force term can be expressed by the following term [89]:

x 2 ) 2 . E force = − k ( x 1 −

(9.8)

This force energy E force represents the energy of a spring connected between
a point x 1 on the contour and some point x 2 in the image plane. In practice,
there could be multiple force terms—one for each spring added. These forces
may be used by a human expert to direct and guide the evolution of the snake.

Special Cases and Variations on a Theme

By adjusting the values of the α and β terms in the internal energy of (9.5),
snakes of varying elasticity and stiffness can be produced. If β i is set to zero
at control point ν i , we allow the snake to become second-order discontinuous
(flexible) at ν i and develop a corner. This is analogous to folding a piece of
corrugated cardboard to make a cardboard box—the fold then behaves like a
flexible hinge between the stiff cardboard sides. This property allows snakes
to better conform to corners of objects such as car licenseplates and allows
for the possibility of embedding shape grammars into snakes.
In some applications, the contractive behavior of the membrane term is in-
convenient, as it may pull the snake away from the best solution. In such cases,
setting α to a low value or zero yields the so-called thin-plate splines ,which
behave much more like wooden splines and are best compared to Bernstein-
Bezier and B-splines.
Due to the contractive nature of the membrane term, snakes must always
be initialized outside the region of interest, so they can contract down onto the
image features like a contracting elastic band. In some situations, this behavior
may be inconvenient. For this reason, Cohen [43] proposed both inflationary
and deflationary forces normal to the surface of closed snakes to force them to
either grow or shrink as illustrated in Figure 9.4; he used the term balloons to
refer to these modified closed snakes. Balloons can be initialized either within
or outside image objects of interest. Figure 9.5 shows the application of a
balloon to the cell image segmentation problem.

Fig. 9.4. Balloons with inflationary and deflationary forces.