Table 4.1 Comparison of different types of deformable models. We indicate whether the approach satisfies some properties (+) or not (−) or it is partially fulfilled ( ). These properties are topological flexibility (a); steady final state (b); possibility of segmenting multiple objects (c); support to a priori knowledge (d); independence from initialization (e); absence of ad-hoc parameters (f)

available, morphological properties can directly be derived from the representation. A drawback is that they are easily trapped by spurious edges representing local minimum solution when the initialization is poor. This problem has been tackled by the balloon approach, where an additional inflation force pushes the level sets over insignificant edges. This introduces an additional arbitrary parameter on which the result strongly depends.

Interestingly, the same difference is present between the classical parametric snake model and its geodesic version. In the latter approach, one parameter less is required, but as a consequence the result depends more strongly on the initialization.

4.4 Applications

4.4.1 Bone Surface Extraction from Ultrasound

In order to extract bone surfaces from ultrasound images by following the expert reasoning from clinicians, bones can be characterized by a strong intensity change from a bright pixel group to a global dark area below since the high absorption rate of bones generates an acoustic shadow behind them. Discontinuities are normally not present on the bone surface; hence the segmentation method should produce a smooth contour.

Considering all of these, an open active contour initialized below the edge in the acoustic shadow was proposed, going from one side of the image to another, and evolving vertically towards the top of the image until it meets the bone surface. A posterior treatment is then necessary to choose only real bone surface points in the image. A deformable model with an external balloon force is applied for this segmentation, but we only take into account the vertical component of the normal

Fig. 4.11 Snake evolution, initialized with a simple contour at the bottom of image (a) and then evolved until it reaches convergence

to the curve and the horizontal point positions of the snake are fixed. This approach allows the snake to move in the vertical direction. We also require an open contour with all points able to freely move vertically, as illustrated in Fig. 4.11.

Then the discretization of the energy minimization scheme gives a stiffness matrix A similar to (4.15), except for the elements related to the two-model extremities. The elements of the matrix A become:

A new local energy definition [35] is proposed to accommodate the image intensity changes and reduce the noise effect during the segmentation process. This regional energy can be defined as a difference between mean intensities in the regions above, below, and in the considered location. When this difference is negative, then a penalization value is applied, and the final energy term is defined as the product of this region-based measurement and the gradient term, and Fig. 4.12 illustrates the role of this energy in deformable model evolution.

Since the active contour initialization goes from one side to another in the image, and considering that the real bone contour does not always behave in a similar fashion, we need to perform a posterior point selection in which only points having a high enough intensity are retained.

A method has been proposed to improve its performance [36] by applying a Gradient Vector Flow in the narrow band around the evolved contour. Furthermore, addressing serial ultrasound images, snake initialization for the next slice can be obtained from the retrospective results of previous slices. This Gradient Vector Flow imparts bi-directional force to the evolved contour; in consequence, the model is capable of moving back when placed above the desired contour. To reduce