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is indeed very helpful when the contour is far away from the real contour to be detected. For this purpose we assume two regions in the images (which can be expanded two more times) with different probability distributions in which each of these regions has different means and variances. Staib’s [24] formulation to determine the region likelihood function can be used for this case:

Eregion = − log(P(f(s)|s R))dS −	log(P(f(s)|s R ))dS.	(4.18)
S	S

Where R and R denote the different regions of the curve and S and S indicate the position inside or outside the region, respectively. The energy will be maximum when R = S and R = S , and the regional based energy can be reformulated as:

E	=	−	log	P(f(s)|s R)		dS.	(4.19)
region				P(f(s) s	)
		S		| R

4.2.2.3 External Constraint Energy Definition

External constraint, proposed by Kass, can be integrated in such a way that the user might specify a few points which should lie on the contour to be detected. It can be performed by adding an energy term which is the distance between these given points and the corresponding closest points on the curve.

As mentioned previously, image segmentation finally yields the totality of the energy minimization process that will place a regular contour at the edge of the object which we want to detect. Sometimes, we do not require the global optimum solution, since the initial contour can be provided interactively by the user to obtain a rough initial contour near the edge, even though a robust optimization scheme that converges to the minimum solution in an acceptable number of iterations is strongly desired. Some examples of image segmentation results using a Bspline snake which integrate gradient and region based energy are demonstrated in Fig. 4.5.

4.2.3 Geometric Deformable Models (Active Contours)

These models were introduced independently by Caselles [23] and Malladi [25] to propose an efficient solution addressing the primary limitations of the parametric deformable model using a geometric deformable model (level set). Advantages of the implicit contour formulation of the geometric deformable model over parametric formulation include: (1) no parameterization of the contour, (2) topological flexibility, (3) good numerical stability, and (4) straightforward extension of the 2D formulation to higher dimensions.

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Fig. 4.5 B-Spline snake performance with respect to noisy image (i.e. echo cardiogram): (a) Initial contour by means of manual contouring; (b) Segmentation using only gradient-based energy; anatomical details of the upper part of the heart can be captured since this region has a bright area, but not in the lower part due to the noises; (c) Region-based energy, noisy problems can be solved but not anatomical details in the upper part; (d) Result using combination of 75% regionbased and 25% edge-based energies; the advantage of both energies can be taken and limitations can be solved, thus enabling a better result

These models are based on curve evolution theory and the level set method [23, 26] proposed by Sethian and Osher [27] to track the surface interface and shape evolution in physical situations. Using this approach, curves and surfaces are evolved using only geometric measures, resulting in an evolution that is independent of parameterization. As in the other types of deformable models, the evolution is coupled with the image data in such a way that the process recovers object boundaries. The evolving curves and surfaces can be represented implicitly as a level set of a higher-dimensional function, so the evolution is independent of parameterization. As a result, topological changes can be handled automatically.

4.2.3.1 Curve Evolution

Let C(t, p) be a kind of closed curves where t parameterizes the family and p the given curve, where 0 ≤ p ≤ 1. As a closed curve, we assume that C(0, t) = C(1, t) and similarly for the first derivatives for closed curves. Using this curve definition, the curve shortening flow, in the sense that the Euclidean curve length shrinks as

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quickly as possible when the curve evolves, can be obtained from the first variation of the length functional [28]:

∂ C
∂ p	= κ N,	(4.20)

κ

where is the local mean curvature of the contour at a point, and N is the unit inward normal. For the intrinsic property of being closed, the curve under the evolution of the curve shortening flow will continue to shrink until it vanishes. By adding a constant ν , which we will refer to as the “inflation term,” the curve tends to grow and counteracts the effect of the curvature term when κ is negative [29]:

∂ C
∂ p	= (ν + κ )N.	(4.21)

A stopping evolution term can be introduced into the above framework by changing the ordinary Euclidean arc-length function along the curve C to a geodesic arc length, by multiplying with a conformal factor g, where g = g(x, y) is a positive differentiable function that is defined based on the given image I (x, y). From the first variation of the geodesic curve length function, we obtain a new evolution equation by combining both the internal property of the contour and the external image force:

∂ C
∂ p	= g(ν + κ )N − g.	(4.22)

Equation (4.22) gives us an elegant curve evolution definition for deformable model formulation applicable for curve evolution analysis. The main problem arises then in relation to how to represent the contour efficiently in terms of geometric and topologic stability as well as numerical implementation. One of the most common methods for representing the contour in using this scheme is the level set concept.

For a more detailed explanation of curve evolution in terms of image segmentation, interested readers are suggested to refer to the works of Casseles [23, 30], Kichenassamy [29], and Yezzi [31].

4.2.3.2 Level Set Concept

Curve evolution analysis using level set method views a curve as the zero level set of a higher-dimensional function φ (x, t). Generally, the level set function satisfies

φ (x, t) < 0 inside Ω(t)
φ (x, t) φ (x, t) = 0 C(t)	,
φ (x, t) > 0 outsideΩ(t)

where the artificial time t denotes the evolution process, C(t) is the evolving curve, and Ω(t) represents the region enclosed by C(t).

Figure 4.6 illustrates an important property of the level set method in handling the topological change in the object of interest. The first image shows a closed curve

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Fig. 4.6 Level set visualization

Fig. 4.7 Level set definition in three-dimensional objects: (a) An object represented implicitly using a 0th level set; (b) An example of a plane in the distance map representing the level set of the object

shape with a well-behaved boundary. The red surface below the object is a graph of a level set function φ determined by the shape, and the flat blue region represents the x − y plane. The boundary of the shape is then the zero-th level set of φ , while the shape itself is the set of points in the plane for which φ has positive values (interior of the shape) or zero (at the boundary).

Extension of the level set method to higher dimensions is also possible and straightforward as shown in Fig. 4.7. Topological change in three dimensional space in this kind of deformable model can be observed in Fig. 4.8, where an initial surface defined as a sphere evolves to become a two-connected torus.

Mathematically speaking, instead of explicitly calculating the curve propagation directly on C, we can transform the contour implicitly using level set representation.