Modeling Tumor Blood Vessel Dynamics - Mathematical Methods and Models in Biomedicine - page 135

which gives the Euclidean length of a curve C . It is well known that the fastest way
to reduce L is the Euclidean

curvature and n is the normal to the curve. This flow is known as the Euclidean curve
shortening flow. Importantly, the principle of curve shortening naturally promotes
smoothness during the evolution process.
Following on this same idea, a constant speed can be added to the evolution
equation, giving a curve that expands or contracts monotonically with speed

( C ) is to evolve C via the flow given by C t = κ n ,where κ

ν
relative to the shortening flow, which in this formulation now exists to provide
regularization:

C t =( ν + κ ) n (31)

In image processing, this constant speed is known as a balloon force [ 16 ] and has
the tendency to drive the curve outward until the boundary of an object is hit.
Therefore, what is required is some criteria for stopping the curve evolution based
on the underlying image, giving a final curve evolution of the form:

C t = g ( I )( ν + κ ) n , (32)

where g 0 at the boundary of the region of interest. Typically, a nonlinear

response function related to the image gradient is used to denote the edge of
an object [ 13 ]. What remains to be determined is a flexible and efficient way of
representing the boundary curve C and its evolution equations. To this end, the level
set method has gained traction in the image processing community, following the
application by Osher and Sethian [ 64 ] to curvature-dependent boundary evolution.
Importantly, the level set technique offers a number of advantages over the classical
snakes approach, including the ability to adapt naturally to topology changes (e.g.,
splitting/merging of regions) as well as handling discontinuities and cusps in the
boundary.

( I ) →

C.2 Level Set Method

2 , the level set method represents a curve C

Given an object Ω ⊂ R = ∂Ω as the zero

level set of a higher-order embedding function denoted φ , which satisfies:

⎧
⎨

φ ( x )= 0 , x ∈ ∂Ω

If φ ( x ) < 0 , x ∈ Ω \ ∂Ω (33)

⎩

2

φ ( x ) > 0 , x ∈ R \ Ω

Two important properties of the level set function are that the boundary normal
is given by n

= ∇φ /| ∇φ |

κ =

( ∇φ /| ∇φ | )

. Returning to (32), the update
equation can now be given in terms of the level set function:

and div