State of the Art of Level Set Methods in Segmentation and Registration of Medical Imaging Modalities - Biomedical Image Analysis - page 66

In this chapter we selected to focus on recent works applied to segmentation
and registration of medical images as this application typically involves tuning
of a general framework to the specificity of the task at hand. We describe in
details two different approaches in the next sections.

2.3.2 Shape Priors into a Variational

Segmentation Framework

Several applications in medical imaging can benefit from the introduction of
shape priors in the segmentation process using deformable models [49; 63-65].
Only few works on segmentation of medical imaging with level set framework
attempted to perform simultaneous registration and segmentation into a single
energy functional and we review three of them in this section.
We first review the work of Chen et al. [60, 66, 67] that proposes a Mumford-
Shah type energy functional plus a parameterized registration term embedded in
a level set formulation for segmentation of brain MRI. Their approach consists
of constraining the segmentation process with a level set framework by incorpo-
rating an explicit registration term between the detected shape and a prior shape
model. They proposed two approaches either with a geodesic, gradient-based
active contour or with a Mumford-Shah region-based functional.
The geodesic active contour minimizes the following functional:

1 g ( |∇ I | ) C ( p ) + 2 d 2 ( sRC ( p ) + T )

E ( C , s , R , T ) = | C ( p ) | dp (2.31)

0

with C ( p ) a differentiable curve parameterized with ( p ∈ [0 , 1] ) defined on im-
age I , g a positive decreasing function, ( s , R , T ) are rigid transformation pa-
rameters for scale, rotation and translation and d ( C ( p )) = d ( C ∗ , C ( p )) is the
distance between a point C ( p ) on the curve C and the curve C ∗ representing
the shape prior for the segmentation task. A level set formulation is derived
by embedding the curve C into a level set function φ positive inside the curve.
Let's introduce the Heaviside function H ( z ) = 1if z ≥ 0 , H ( z ) = 0 otherwise ,
and the Dirac measure δ ( z ) = H ( z ) (with derivative in the distribution sense),
the energy functional in Eq. (2.31) is reformulated as:

δ ( φ ) g ( |∇ I | ) + 2 d 2 ( µ Rx + T ) E ( φ,µ, R , T ) = (2.32) |∇ φ | .