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A. Alfiansyah |
Fig. 4.9 Role of the elastic term in geodesic active contour: In the upper row, the model is evolved without the elastic term; in the lower row, an elastic term is used
boundaries C. A reduced form of this problem consists simply in the restriction of EMS to piecewise constant functions and finding a partition of Ω such that f in Ω equals a constant.
To minimize the functional energy, Chan and Vese proposed a two-phase segmentation as follows:
C,Co,Cb |
μ δ (φ )| φ | |
+ ν |
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(φ ) |
+ λo |
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− |
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o| |
2H |
(φ ) |
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min |
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dxdy |
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H |
dxdy |
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f |
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c |
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dxdy |
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Ω |
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Ω |
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inside(C) |
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+ λo |
| f − co|2(1 − H(φ ))dxdy, |
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outside(C) |
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where φ is the level set function and H(φ ) is the Heaviside function.
Generally, the above deformable models implemented by means of the level set method suffer from a slower speed of convergence than parametric deformable models due to their computational complexity. Application of the Chan-Vese deformable model to segmentation on a low signal, noisy image is shown in Fig. 4.10.
4.3 Comparison of Deformable Models
Whereas geometric deformable models often have been presented as an improvement (and even superior) to classical snakes [23, 25, 30], one might argue that they actually are complementary to each other. In a wide range of image segmentation
4 Deformable Models and Level Sets in Image Segmentation |
77 |
Fig. 4.10 Image segmentation using a Chan–Vese deformable model: (a) The original image; note that it has a very weak border; (b) The degraded image with added salt-and-paper noises; (c) A segmentation result represented as a binary mask (the white area is the detected object)
applications there is a trade-off between desirable properties. Instances of such important trade-offs are the level of automation versus control, and topological flexibility versus a priori constraints. Other important properties are the role of the initialization, existence of a unique solution, dependence of the result on parameter choice and the robustness of the method with respect to noise, and imperfect image data.
Table 4.1 summarizes the main properties of a number of deformable model types which have been described in the previous section. We categorize the deformable models into three approaches following the previous presentation.
Classical and explicit parametric snakes are well-known in performing well if the initialization is close to the desired objects. Moreover, since a parameterization is