A Region-Aided Color Geometric Snake - Biomedical Image Analysis - page 549

of C are handled automatically and accuracy and stability are achieved using
numerically stable computations.
The internal curvature and external pressure terms of the RAGS formulation
in (10.17) can be easily transferred to a level set representation:
C t = g ( |∇ I | ) κ N → φ t = g ( |∇ I | ) κ |∇ φ |
C t = g ( |∇ I | ) α N → φ t = g ( |∇ I | ) α |∇ φ |

,

(10.18)

The other external forces in (10.17) are static vector fields derived from
image data which do not change as the active contour deforms. Static force
fields are defined on the spatial positions rather than the active contour itself.
Since N is the inward normal, the level set representation of the inward unit
normal is given by

N =− ∇ φ

(10.19)

|∇ φ | .

Then, we have

1
|∇ φ |

F · N =− ( F ·∇ φ ) . (10.20)

Combining (10.18) with (10.20) where F takes on the static force fields, the level
set representation of RAGS is given by

R ·∇ φ,

φ t = g ( |∇ I | )( κ + α ) |∇ φ |+∇ g ( |∇ I | ) ·∇ φ − β (10.21)

where g ( · ) is the stopping function as before. The expression for the curvature
of the zero level set assigned to the interface itself is given by

κ = div ∇ φ
|∇ φ |

y x

= φ xx φ

− 2 φ y φ x φ xy + φ yy φ

(10.22)

( φ

x y ) 3 / 2

+ φ

10.5 Numerical Solutions

The numerical solution for region force diffusion is discussed in detail in
Section 10.5.1, but the detailed numerical solutions for RAGS level set repre-
sentation are only presented in Appendix A as they are not critical to the under-
standing of the concepts underlying RAGS. In fact, the whole of this section can
be skipped without loss of continuity.