A Region-Aided Color Geometric Snake - Biomedical Image Analysis - page 547

into two types, internal forces and external forces. The internal forces impose
regularity on the curve and control the elasticity and rigidity of the snake. The
external forces pull the snake toward salient image features such as object
boundaries. Thus, the internal and external forces in (10.8) can be written as
F int = g ( |∇ I | ) κ N
F ext = g ( |∇ I | ) c N −∇ g ( |∇ I | ) ,

(10.14)

where g ( · ) is the stopping function as before. The first term of the external forces
is a constant shrink or expand force in the normal direction of the snake. It can
be separated from other external forces in the sense that it is not spatially static
in the image domain as other external forces and needs different numerical
schemes. However, considering the previous definition of snake forces and that
the constant force alone can push the snake toward boundaries, we keep it in
the external term.
The diffused region force is a feature driven force and spatially static. So we
can add the diffused region force to the external term:
F int = g ( |∇ I | ) κ N
F ext = α g ( |∇ I | ) N + β

R −∇ g ( |∇ I | ) ,

(10.15)

where R is the region force vector field obtained in (10.10) and α is a new constant
incorporating c . Constants α and β act as a trade-off between gradient forces
and region forces. In practice, β is a constant from 0 to 1 for most nonhighly
textured images. If good segmentation results are available, β should be set close
to 1.
The snake evolves under all the internal and external forces. However, only
the forces in the normal direction of the evolving contours can change the geom-
etry. The forces tangential to the contours can only change the parameterization
of the contours. Thus, a geometric snake evolving under internal and external
forces can be interpolated as

C t = [( F int + F ext ) · N ] N. (10.16)

Finally, by substituting (10.15) into (10.16), the region-aided geometric snake
formulation becomes

C t = [ g ( |∇ I | )( κ + α ) −∇ g ( |∇ I | ) · N + β

R · N ] N.

(10.17)