State of the Art of Level Set Methods in Segmentation and Registration of Medical Imaging Modalities - Biomedical Image Analysis - page 54

where g is a positive decreasing function. Segmentation is achieved via mini-
mization of this energy functional equivalent to the computation of geodesics in
a Riemannian space according to a metric that weights the Euclidian length of
the curve with the term g ( |∇ I ( C ( p )) | ).
Minimization of the functional is performed via derivation of the Euler-
Lagrange system:

∂ C

∂ t = g ( |∇ I | ) κ N − ( ∇ g ( |∇ I | ) . N ) N ,

(2.14)

where κ is the Euclidian curvature of the curve C and N is the unit normal
vector to the curve. Implementation with a level set framework is performed by
embedding the curve C into a level set function φ . Using the following property
on the curvature term:

∇ φ
|∇ φ |

κ = di v ,

(2.15)

and the following equivalence of relationships between a curve C and its asso-
ciated level set function φ :

∂ C
∂ t = α N
∂φ
∂ t = α |∇ φ | ,

(2.16)

the level set formulation is expressed as:

g ( |∇ I | ) di v ∇ φ

|∇ φ |

∂φ
∂ t =|∇ φ |

+∇ g ( |∇ I | ) ∇ φ
|∇ φ |

(2.17)

g ( |∇ I | ) ∇ φ
|∇ φ |

=|∇ φ | di v

To improve convergence speed and allow the detection of non-convex objects,
the authors also introduced a modification of the initial formulation with the
introduction of a constant inflation term ν g ( |∇ I | ) |∇ φ | leading to the following
functional:

di v g ( |∇ I | ) ∇ φ

|∇ φ |

+ ν g ( |∇ I | )

∂φ
∂ t = |∇ φ |

(2.18)

= g ( | ∇ I | )( κ + ν ) | ∇ φ | +∇ g ( | ∇ I | ) ∇ φ.

Applications of the geodesic deformable model to medical imaging have been
tested by both groups of pioneering authors. Yezzi et al. tested their geodesic
deformable model in [17] on 2D images for cardiac MRI, breast ultrasound with