Biomedical Engineering Reference
In-Depth Information
This can be viewed as a vector-valued partial differential equation balancing in-
ternal and external forces at equilibrium given by
F
int
+
F
ext
=0
,
(5)
where
F
int
represents the internal force due to stretching and bending factors,
given by
α
∂
β
∂
2
τ
(
s
)
∂s
2
.
∂
2
∂s
2
∂
∂s
τ
(
s
)
∂s
F
int
=
−
(6)
The first term in Eq. (6) is the stretching force derived from the surface tension,
while the second term represents the bending force. The external forces couple the
contour to the image information in a way that equilibrium is accomplished when
it balances with the physical constraints on the contour. Thus,
F
ext
is expressed
as
F
ext
=
−∇
E
ext
(
τ
(
s
))
,
(7)
which pulls the contour toward the salient image features of interest. Other forces
can be added to impose constraints defined by the user. We will make use of
additional forces.
To solve the energy minimization problem, it is customary to construct the
snake as a dynamical system that is governed by the functional to evolve the system
toward equilibrium. The snake is made dynamic by treating the evolving contour
τ
as a function of both time
t
and arc-length
s
. This unifies the description of
shape and motion within the same framework of Lagrangian mechanics. Thus,
this formulation not only captures the shape of the contour but also quantifies its
evolution over time. The Lagrange equations of motion for a snake is given by
α
∂
β
∂
2
τ
(
s
)
∂s
2
µ
∂
2
τ
(
s
)
∂t
2
∂
2
∂s
2
+
ν
∂
τ
(
s
)
∂t
∂
∂s
τ
(
s
)
∂s
+
−
=
∇
E
ext
(
τ
(
s
))
,
(8)
where
µ
is themass constant and
ν
the damping density followingNewton's laws of
motion. The system achieves equilibriumwhen the internal stretching and bending
forces balance with the external forces and the contour ceases to move, i.e., both
the acceleration and velocity terms vanish; in other words,
∂
2
τ
∂t
=0.
For numerical solution of the equation, discretization of the equation is re-
quired. This is in general accomplished using a finite difference for solving the
partial differential equation. In the discrete domain the energy equation can be
expressed as
∂
∂t
2
=
E
(
v
)=
1
2
vAv
+
E
ext
(
v
)
,
(9)
where
is the stiffness
matrix. For all practical purposes, in this text we will use the symbol
v
is the discretized version of the contour
τ
(
s
), and
A
τ
(
δ
) for
