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A. Alfiansyah |
The equation can be solved by changing C over time t, then C is a function of the contour s and time t → C(s, t). When the snake reaches a stable state Ct (s, t) = 0, the solution of (4.12) is obtained. Thus, insertion of this term gives:
∂ Eext |
− (w1(s)C ) + (w2(s)C ) + Eext(C(s)) = 0. |
(4.13) |
∂ s |
If the sum of all external forces and image energy is Eext(s), the equation can be solved numerically by using a finite difference approach and represented implicitly in matrix multiplication form such as:
F = A.V, |
(4.14) |
where A is a penta-diagonal banded matrix. This expression is correct for
snakes with fixed-point positions at their extremities or closed snakes (i.e.
C(0) =C(N − 1)).
Generally, we assume that elasticity w1 and rigidity w2 are constant for both
discretized space and time during curve evolution, thus: |
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a1 |
= e1 |
= |
w2 |
b1 = d1 = |
w1 |
+ 4 |
w2 |
c = 2 |
w1 |
+ 6 |
w2 |
. |
(4.15) |
h4 |
h4 |
h4 |
h4 |
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h4 |
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To solve (4.14) iteratively, the successive over-relaxation method [21] can be applied. In the two-dimensional image case, the resulting equations for evaluating time t from time t − 1 can be solved iteratively after matrix inversion using:
Vt = τ(A + I)−1 · (Vt−1 + τF(xt−1, yt−1), |
(4.16) |
I is the identity matrix, and τ(A + I)−1 is also penta-diagonal.
This optimization approach does not guarantee a global minimum solution, and requires estimates of high-order derivatives on the discrete data. Moreover, hard constraints cannot be directly enforced. A desired constraint term like mean or minimum snaxel spacing can only be enforced by increasing the associated weighting term, which will force more effect on this constraint, but at the cost of other terms.
4.2.2 Parametric Deformable Models
These deformable models represent the curve or surface in an explicit parametric form during the model deformation. This representation allows for direct interaction and gives a compact representation for real-time implementation. Parametric
4 Deformable Models and Level Sets in Image Segmentation |
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deformable models are usually too sensitive to their initial conditions because of the non-convexity of the energy functional and the contraction force which arises from the internal energy term.
B-Spline is often used as a representation of parametric deformable models [42]. In this case, the deformable model is split into segments by knot points. Each curve segment C(t) = {x(t), y(t)} is approximated by a piecewise polynomial function, which is obtained by a linear combination of basis functions βi and a set of control points ν = {xi, y}. In general, however, representations using smooth basis functions require fewer parameters than point-based approaches and thus result in faster optimization algorithms [22]. Moreover, such curve models have inherent regularity and hence do not require extra constraints to ensure smoothness [22, 23].
Both point-based and parametric snakes represent the model in an explicit way, hence it is easier to integrate an a priori shape constraint into the deformable model. Moreover, the user’s interaction can be accommodated in a straightforward manner by allowing the user to specify some points through the desired contour evolution. The inconvenience of this model lies in reduced flexibility in accounting for topological changes during the evolution, although much effort has been spent to overcome this limitation.
4.2.2.1 Internal Energy Definition
Similar to the discrete point-based snake, internal energy is responsible for ensuring the smoothness of the contour. Actually, Kass proposed a linear combination of the length of the contour and the integral of the square of the curvature along the contour. Thus, in explicit contour representation, this energy can be defined as:
M |
M |
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2 |
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Econtour = w1 (x (t)2 + y (t)2) 21 dt + w2 |
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x (t)y (t) − x (t)y (t) |
dt. (4.17) |
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0 |
0 |
(x (t)2 |
+ y (t)2) 23 |
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where the second term represents the curvature at point r(t).
4.2.2.2 Image Energy Definition
The most common image energy applied for the snake is defined as the integral of the square of the gradient magnitude along the curve. The main drawback widely known in using this energy is the lack of gradient direction. This information can be used to detect edges, since at the boundary image gradient is usually perpendicular to the curve. This direction should be incorporated into the image energy to make the snake more robust for image segmentation.
Region-based energy. This region-based energy represents the statistical characteristics in a region in the contour and provides snake boundary information. This