74	A. Alfiansyah

Fig. 4.8 Level set representation advantage in handling the topological change during segmentation. Initialized using a sphere the method can detect the connected chain as the final result

Thus, contour evolution, which is	∂C
	∂ p	= FN,
representation:

∂ φ = F| φ |. ∂ t

can be transformed in level set

(4.23)

It would be very hard to elaborate upon this topology transformation using parameterized curve representation. One would need to develop an algorithm able to detect the moment the shape split (or merged), and then construct parameterizations for the newly obtained curves.

4.2.3.3 Geodesic Active Contour

Slightly different from the other models previously explained, this model does not impose any rigidity constraints (i.e. w2 = 0); hence, the minimized energy is formulated as:

		1						∂C
	E(C) = g(| I(C(s))|)ds							∂ p		dp.	(4.24)
							regularity term
	0
			attraction term

Where g is a monotonically decreasing function, ds is the Euclidian arc-length element and L the Euclidian length of C(t, p).

4 Deformable Models and Level Sets in Image Segmentation

75

Using this formulation, we aim to detect an object in the image by finding the minimal-length geodesic curve that best takes into account the desired image characteristics. The stated energy in (4.24) can be minimized locally using the steepest descent optimization method, as demonstrated in [30]. It shows that in order to deform the contour towards the minimum local solution with respect to the geodesic curve length in Riemannian space, the curve evolves according to the following equation:

	∂ C
	∂ t	= gκ N − ( g · N)N.		(4.25)
The segmentation result can be achieved in the equilibrium state where				∂C	= 0.
				∂t

Following the level set method for the curve evolution in (4.25), we obtain the curve evolution using the geodesic active contour in terms of the level set:

∂ φ	= gκ | φ | + g φ .	(4.26)
∂ t

Finally, in order to accelerate the convergence and place the model in the correct boundary, we integrate an elastic term in the curve evolution that will pull the model towards the desired object [30] and writing the curvature κ explicitly:

∂ φ

= g · div

φ

| φ | +

g φ +ν g| φ |.

(4.27)

∂

t

φ |

|

elastic term

The role of this elastic term can be observed in Fig. 4.9 in which the upper-row active contour evolves without any elastic term, and hence the converged contour does not match the desired contour.

4.2.3.4 Chan–Vese Deformable Model

One limitation of the geodesic active contour lies in its dependence on image energy represented by the gradient. In order to stop the curve evolution, we need to define g(| I|) which defines the edges of the object. In practice, discrete gradients are bounded, and so the stopping function is never zero on the edges, and the curve may pass through the boundary. Moreover, if the image is very noisy, the isotropic smoothing Gaussian has to be strong, which will smooth the edges as well.

To overcome these problems, Chan and Vese [32–34] proposed stopping process based on the general Mumford–Shah formulation of image segmentation [15], by minimizing the functional:

EMS( f ,C) = μ .Length(C) + λ	| f − f0|2dxdy + | f |2dxdy,	(4.28)
Ω	C
	Ω\

where f0 : Ω → R is a given image to be segmented, and μ and λ are positive parameters. The solution image f is formed in smooth regions Ri and sharp