Biomedical Engineering Reference
In-Depth Information
discrete representation of the contour. Minimum energy estimation is equivalent to
setting the gradient of Eq. (10) to 0, which results in the following linear equation:
A
τ
=
F
,
(10)
where
A
is the penta-diagonal stiffness matrix, and
and
F
represent the position
τ
vectors
F
(
τ
i
), respectively.
As the energy-space cannot be ascertained to be convex, so there is a high
probability of getting local minima in the energy surface. In fact, finding the
global minimum of the energy is not necessarily meaningful. Indeed, the main
interest is finding a good contour that optimally fits to delineate the anatomic
structure of interest in the best possible manner.
A neighborhood around each control point is considered and the total energy
of the contour is computed for each neighborhood. Energyminimization continues
until the energy between two consecutive iterations changes. This dynamic pro-
gramming approach [14] searches for global minima in the image space. Greedy
snakes [15], on the other hand, searches for local minima for each of the control
points. The local motion of the points is considered in the neighborhood for en-
ergy minimization. In contrast to dynamic programming [14], greedy snakes [15]
minimizes the energy of each local control point. Figure 4 illustrates the neigh-
borhood search around a control point for dynamic programming and the greedy
snakes algorithm.
τ
i
=
τ
(
iδ
) and the corresponding force at these points
3.
ACTIVE CONTOUR EVOLUTION
The image processing task can be broadly classified into two categories:
region-based and boundary based operations. Image processing techniques like
mathematical morphological operations, region growing, and other region-based
operations use regional homogeneity statistics to drive the task of image process-
ing. Boundary-based operations (e.g., edge detection, gradient computation) use
the statistics of variation in a local neighborhood. Low-level image processing
techniques, if used independently for the purpose of segmentation, require a high
level of manual intervention, rendering the result prone to inter- and intra-operator
variability.
This chapter will focus on gradient-based approaches that rely mostly on
image edges for convergence. Subsequent modifications for the gradient-based
approaches and challenges faced at various levels of medical image segmenta-
tion will be discussed. Eventually, incorporation of region-based forces within
the snake model help in providing a more compact model for the segmentation
process. Region-based information can be incorporated in many ways into the
energy minimization equation.
The active contour model allows user interaction at various stages. The main
intention of the active contour is to reduce the amount of user intervention in the
