State of the Art of Level Set Methods in Segmentation and Registration of Medical Imaging Modalities - Biomedical Image Analysis - page 61

The authors provide a very nice application for segmentation of cortical
gray matter surfaces from MRIs derived from the initial work of Zeng et al.
[33]. With this method, if the curve evolution in Eq. (2.24) is implemented with
model in Eq. (2.5), the magnitude of the gradients ( |∇ φ in | , |∇ φ out | ) will increase
and the estimation of the distance between the zero-levels of the two func-
tions will be overestimated, leading ( φ in ,φ out ) to get closer as they evolve and
eventually collide until the level set functions are reinitialized. Results are illus-
trated on three regions of interest from three MRI slices and show very inter-
esting results but no quantitative evaluation of the accuracy of the method was
performed.

2.2.8 Region-based Level Set Active Contours

Region-based active contour were derived from the Mumford-Shah segmenta-
tion framework initially proposed in [36]. In their initial work, Mumford and Shah
defined a new segmentation framework performing segmentation of a given im-
age I into a set of contours S and a smooth approximation f of the image via
minimization of the following framework:

( f − I ) 2 dx + β

\ S |∇ f | dx + H n − 1 ( S ) ,

E ( S , f ) = α (2.27)

where H n − 1 ( S ) is the ( n − 1) dimensional Hausdorff measure, and ( α,β ) are
positive real parameters. In this functional, the first term ensures that f is a good
approximation of the original image I , the second term ensures that f is smooth
and the last term minimizes the length of the set of contours of the segmentation.
This type of region-based segmentation method relies on the homogeneity of the
object to segment. This assumption is often violated with medical images due
to motion of the organ, presence of corrupting machine noise or acquisition
artifacts that introduce flat field inhomogeneities.
Based on the Mumford-Shah segmentation framework, Chan and Vese intro-
duced in a series of papers a new type of active contour models without gradient
information [37-41]. In the simplest case, assume that an image I defined on is
composed of two regions (e.g., an object and a background) with homogeneous
intensities around values c 0 and c 1 . Given a curve C that defines the boundary
of a region inside the image I , they introduce the following homogeneity-based