Biomedical Engineering Reference
In-Depth Information
in which stained pathology specimens are to be segmented. With a limited set
of mathematical jargon and symbols, the emphasis is weighted to inspire interest
for the problems at hand. This has been done by selecting those methods that are
best suited for conveying the fundamental characteristics of deformable models.
Of course, our selection of the techniques and numerical examples are limited by
the usual constraints: author bias and author limitations. Our goal is to provide a
general framework for deformable models while providing references which will
enable readers to pursue any remaining details.
In this chapter, we will review two deformable models: the parametric de-
formable model and the geodesic or level set-based deformable model. The chap-
ter is organized as follows: the next section provides the essential mathematical
background related to deformable models. It discusses some selected numeri-
cal methods for solving discretized optimization problems in deformable models.
Emphasis is given to the well developed Finite Difference Method (FDM). Section
3 serves as a review of different types of parametric deformable models. We will
also provide the numerical solution to the optimization problem. Stained pathol-
ogy specimens will be used as the test set for evaluating the performance of the
algorithms. Section 4 is devoted to illustrating how to use level set techniques to
solve topological changes that include a description of the geodesic deformable
models. This section also provides their applications in pathology image segmen-
tation and the actual experimental results.
2.
MATHEMATICALBACKGROUNDOFDEFORMABLEMODELS
In the present section we provide a brief overview of some of the requisite
mathematics that are needed to understand deformable models. While detailed
proofs are not included, we have provided citations for proof and additional detailed
descriptions.
2.1. Calculus ofVariations andEuler-LagrangeDifferential Equation
. We claim that
a necessary condition for
y
to have an extreme points is
f
(
x
0
)=0. Furthermore,
if
f
J
(
x
0
)
=0, we can conclude that
x
0
Let's consider a real function
y
=
f
(
x
) with both
x
and
y
∈
R
is the extreme point of
y
. Similarly
a functional is defined as
f
(
G
(
x
)) =
y
with
G
(
x
) denoting a set of functions
defined in
. Therefore,
the fundamental problem of calculus of
variations
is: how to find the extreme function
G
0
(
x
) that will minimize the value
of
y
. This is also called a variational problem.
and
y
∈
R
R
Theorem 1 (Euler-Lagrange principle)
For a functional defined as
b
g
(
x, f
(
x
)
,f
(
x
))
dx
I
(
f
(
x
)) =
(6)
a
