Biomedical Engineering Reference
In-Depth Information
I ( x, y ) | 2 , where γ controls the magnitude of the potential,
defined as
γ
|∇
G σ
is the gradient operator, and G σ
I ( x, y ) denotes the image intensity I ( x, y )
convolved with a (Gaussian) smoothing filter whose characteristic width σ controls
the spatial extent of the local minima of the convolution kernel. Note here that
the expression for the edge operator has a negative sign associated with it. The
reason for this is that the local minima of the contour need to coincide with the
maxima of the gradient functional. Also, note that the squared magnitude has
been used for the edge functional computation. A different approach is also used,
where the vector form of the gradient is used instead of the scalar information.
However, in the Lagrangian setting an energy expression is required for solving
the minimization problem. Thus, for the vector-based approach the dot product
of the contour normal with the gradient is used for defining the energy functional
and is expressed as (
G σ
I ( x, y )) · N ( τ ( x ( s ) ,y ( s ))), where
N ( τ ( x ( s ) ,y ( s )))
is the normal to the contour
τ ( s ) at location x ( s ) ,y ( s ). Similarly, image intensity
has been widely used along with edge information to formulate the external energy
functional. The total external energy of the contour can be defined as
1
E ext ( τ )=
E ext ( τ ( s )) ds,
(3)
0
where
E ext ( τ ( s )) denotes the energy functional given by the image properties at
the point
τ ( s ).
In summary, the basic definition of deformable parametric curve contains two
terms: (a) internal energy, which defines the geometric properties of the curve;
and (b) external energy, which combines all other forces that guide the curve to
delineate the desired structure. Once the basic energy formulation is done, the idea
is to find a methodology for energy minimization. A number of approaches have
been proposed so far for energy minimization of the contour. The most well known
is by solving the partial differential equation (PDE) for force (defined through an
Euler-Lagrangian) using a finite-difference [10] or finite-element method [13]. A
dynamic programming-based approach [14] and greedy snakes [15] are also used in
many applications. The next subsection will briefly touch upon these approaches.
Since these are quite standard ways of solving minimization problems, this chapter
gives only the basic idea behind each of the methodologies. The pseudocode for
the Euler-Lagrangian and greedy snakes are provided in Appendix 1.
2.3. Energy Minimization
According to the calculus of variations, the contour that minimizes the energy
E snake ( τ ) must satisfy the Euler-Lagrange equation [10]
α
β 2 τ ( s )
∂s 2
∂s
τ ( s )
∂s
2
∂s 2
−∇ E ext ( τ ( s ))=0 .
(4)
Search WWH ::




Custom Search