El conflicte en el sistema sanitar

This model proposed byCaselles et al. (1997b) is an extension of GAC (Caselles et al.,

1997a) into 3-D. GAC mathematically connects two edge detection approaches, classical

snakes (Kass et al.,1988) and geometric curve propagations (Caselles et al.,1993;Malladi

2.2.1 Classical Snakes in 3-D

Let S(p, q) : [0, 1] × [0, 1] → R3 _{represent a parametric 3-D surface and I : Ω → R an}

image, where S = (x(p, q), y(p, q), z(p, q))T_{and Ω is bounded open set of R}3_{denoting the}

image domain. Let ˆΩ = [0, 1] × [0, 1], the energy functional introduced by Terzopoulos

et al. (1988) is given as Esnk(S) = Z ˆ Ω  µ1|∂pS|2+ µ2|∂qS|2+ µ3|∂ppS|2+ µ4|∂pqS|2+ µ5|∂qqS|2+ λ|∇I|  dp dq (2.1) where µi and λ are positive constants. The first and second order derivatives of S are to

regularize the curve (internal energy). The first order derivatives control elasticity and the second bending. The deformation of S are also affected by gradients of I (external energy), which attracts S to image boundaries. Given a set of constant parameters, S evolves to minimize Esnk. Trade-off between surface smoothness and proximity to object

edges is manually adjusted and topology is dealt with using specific schemes.

2.2.2 Implicit Surface Propagation

Malladi et al.(1995) proposed an implicit 3-D surface propagation for segmentation. S in

this model is embedded into an one-dimension-higher volume and propagates according to the mean curvature and geometric features of I. The advantage of the embedment is that it uses value changes of voxels to represent deformation of S (see figure 2.1), and thus vastly facilitates the numerical implementation.

Figure 2.1: A 1-D demonstration of the motion of S when embedded into Φ. A point on surface S (the circle in dashes) is implicitly represented by an one- dimension-higher line between two grid points on volume Φ (the square with solid lines). The value change of a point on Φ gives rise to the motion of the point on S. The motion of the point set that constitute S is deformation.

Let Φ(x) : Ω → R be a volumetric level-set function, where x = (x, y, z)T _{and in this}

explicit surfaces propagate according to the following motion equation:

∂tS = ~nV (2.2)

where V is the propagation velocity and ~n is the unit inward normal of S. Replacing ~n by ∇Φ/|∇Φ|, the implicit surface propagation equation proposed by Osher and Sethian

(1988) is dtΦ = ∂tΦ + ∂xΦ∂tx + ∂yΦ∂ty + ∂zΦ∂tz = ∂tΦ + ∇Φ · ∂tS = ∂tΦ + ∇Φ · (V~n) = ∂tΦ + V(∇Φ · ∇Φ |∇Φ|) = ∂tΦ + V|∇Φ| = 0 (2.3)

Therefore, the motion equation of Φ is ∂tΦ = −|∇Φ|V. In the work of Malladi et al.

(1995), Φ evolves according to ∂tΦ = |∇Φ|  g(I)(∇ · ∇Φ |∇Φ|+ ν)  = |∇Φ|g(I)(KΦ+ ν) (2.4)

Here KΦ= ∇ · (∇Φ/|∇Φ|) is twice the mean curvature (sum of the two principal curva-

tures) of the zero level set of Φ and g(I) = 1/(1 + (|G ∗ ∇I|/ξ)2_{), where G is the Gaussian}

filter that smooths the image and ξ is used to normalize the smoothed image gradients. ξ is necessary due to the fact that HRCT data have high dynamic range (0-3000) which may cause numerical instability of GAS (g(I) that corresponds to different values of ξ is given in figure 2.2). Therefore ξ is chosen so that g(I) is smooth enough to let GAS perform well. At ideal edges in the image, where ∇I = ∞, and therefore g(I) = 0, Φ stops evolving. This can also be considered as manually intervened mean curvature motion equation that stops at edges of image during its evolution that minimizes the Euclidean area of zero level surface S. ν here serves to minimize the volume of inside region of S when ν < 0 and when ν > 0, it acts as balloon force that dilates S.

2.2.3 Geodesic Active Surfaces

Caselles et al.(1997a) generalized classical snakes and geometric curve propagation into

finding a minimal path in a given Riemannian space. This model is extended to 3-D in the work of Caselles et al. (1997b). In 3-D, while surface-propagation-based model minimizes the Euclidean area A = R

Figure 2.2: (a) One slice of HRCT scans. (b) Smoothed gradient image of the slice. (c) g(I) when ξ = 1 and max(g(I)) = 0.01. (d) g(I) when ξ = 20 and max(g(I)) = 0.82. (e) g(I) when ξ = 50 and max(g(I)) = 0.97.

element of S, the GAS minimizes a weighted area: AGAS =

Z

S

g(S) da (2.5)

Image gradients are naturally embedded into the above functional. The corresponding Euler-Lagrange equation is given as

∂tS = ~n



g(I)K − ∇g(I) · ~n 

(2.6) where K = ∇ · ~n is twice the mean curvature (sum of two principal curvatures) of the explicit surface S. Motion of S naturally stops at image edges. Bring ~n = ∇Φ/|∇Φ| into the above equation, level-set formulation of GAS is

∂tΦ = |∇Φ|  g(I)KΦ+ ∇g(I) · ∇Φ |∇Φ|  = |∇Φ|g(I)∇ · ( ∇Φ |∇Φ|) + ∇g(I) · ∇Φ = |∇Φ|∇ · (g(I) ∇Φ |∇Φ|) (2.7)

GAS is closely related to surface propagation model ofMalladi et al.(1995) and a general GAS model includes the constant speed term:

∂tΦ = |∇Φ|  ∇ · (g(I) ∇Φ |∇Φ|) + νg(I)  (2.8) The constant speed term is optional, however, in practice, properly choosing the value of ν and the stopping term g(I) gives rise to a much faster convergence. Superior to its predecessors, the GAS model is capable of detecting thinner and sharper structures while keeping its smoothness, since it provides an optimal balance between object boundary presentation and surface regularity. Furthermore, in the level-set framework, surfaces naturally split and merge, therefore multiple objects could be detected simultaneously without specially designed topology handling schemes.