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Chapter 2 Image Segmentation and Limitations Phase fields are frequently used for interface tracking (see [23, 143] and the references therein), also the image processing community uses phase fields for segmentation purposes [15, 123]. In contrast to the phase field used in the Ambrosio-Tortorelli approach, the phase fields needed in this context differentiate between object and background, i.e. they describe closed contours. They have a value of −1 in the object, 0 at the interface, +1 on the background and vary smoothly between these values. Typically, the phase field is ±1 away from the interface and changes smoothly inside a small layer with thickness ε around the interface in a tangential profile. A phase field equation [143] like ( φ t + F|∇φ| + u e · ∇φ = b ∆φ + φ(1 − φ 2 ) ) ε 2 (2.31) controls the evolution of the diffusive phase field interface. In this equation, φ is the phase field, u e an external advection, b the interface speed depending on the curvature, ε the thickness of the diffusive interface, and F again the speed in the normal direction. In contrast to the level set approach (2.30), this is a parabolic equation, which avoids the numerical difficulties arising in the discretization of the level set equation. When the curvature-depending interface speed vanishes, i.e. when b = 0, this equation is kept parabolic by adding a counter term introduced by Folch [51]. Following [143] this counter term leads to the parabolic equation ( φ t + F|∇φ| + u e · ∇φ = b ∆φ + φ(1 − φ 2 ( )) ) ∇φ ε 2 − |∇φ|∇ · , (2.32) |∇φ| where b is a purely numerical parameter, because the curvature at the end cancels out the Laplacian and the term φ(1−φ 2 ) . It is easier to discretize (2.32) than (2.30) using central differences. Sun et ε 2 al. [143] developed a phase field equation based on nonlinear preconditioning [56] ( φ t + a|∇φ| + u e · ∇φ = b ∆φ + 1 ε (1 − |∇φ|2 ) √ ( ) ( )) φ ∇φ 2tanh √ − |∇φ|∇ · , (2.33) 2ε |∇φ| which is an integrated reinitialization scheme for the phase field. The phase field φ in this equation becomes a signed distance function. Thus, this equation is a parabolic level set equation with integrated reinitialization. Again, it is possible to discretize (2.33) using simple difference schemes. This connection between phase fields and level sets allows us to use known segmentation algorithms from the level set context and embed them into this nonlinear preconditioned phase field equation. The discretization of the parabolic phase field equations is easier in the stochastic context, cf. Chapter 7. 2.4.2 Gradient-Based Segmentation The idea of the level set propagation is useful to segment objects inside an image. The simplest approach for segmentation based on level sets is to use a speed F in the level set equation that depends on characteristics of the image. Popular is F = F(|∇u|), i.e. to stop the evolution on edges inside the image (see [29, 96, 138] and the references therein). Caselles et al. [29] proposed to use with g u = φ t + g u |∇φ| = 0 (2.34) 1 (1 − εκ) , (2.35) 1 + |∇G σ ∗ u| where u is the image, G σ a Gaussian smoothing filter with width σ, κ the curvature of the level set function and ε a small scale parameter that controls the influence of the curvature smoothing term. Although this idea sounds simple, the method achieves good results when a high gradient separates the objects from the background (see Fig. 2.7). One major drawback of this method is the need for 18
2.4 Level Sets for Image Segmentation Figure 2.7: Segmentation of a medical image based on a level set propagation with gradient-based speed function. The time increases from left to right and the zero level set (red line) approximates the boundary of the object (a liver mask) at the end. finding a stopping criterion. The evolution speed g u is always positive, even close to edges. Thus, it is possible that the zero level set passes the edge. A typical stopping criterion is to stop the evolution when the difference between the level sets of subsequent time steps is small. This occurs when the level set reached the boundary of the object and the speed dropped down. Using methods that are more sophisticated, it is possible to stop the zero level set at the edge. Thus, these methods have a convergent solution. The next section presents one of these methods, geodesic active contours. Remark 5. It is also possible to formulate the gradient-based segmentation based on the phase field model presented in the last section. This yields the equation ( φ t + g u |∇φ| = ε ∆φ + φ(1 − φ 2 ) ) ε 2 . (2.36) 2.4.3 Geodesic Active Contours Caselles et al. [30] and simultaneously Kichenassamy et al. [82] developed geodesic, or minimal distance, active contours. They minimize an energy B that depends on the curve C and on the parametrization of the curve C(q) : [0,1] → IR 2 : ∫ 1 B(C) = α |C ′ (q)| 2 dq + β 0 ∫ 1 0 g u (|∇u(C(q))|) 2 dq , (2.37) where g u is the edge indicator from the last section. They computed a minimizer of this energy by using a level set representation of the curve and computing the Euler-Lagrange equations of the resulting energy. This leads to a level set equation with an additional advection term that forces the zero level set to stay in regions with high gradient: φ t = −α∇g u · ∇φ − βg u |∇φ| + εκ|∇φ| . (2.38) The user chooses the parameters α,β and ε. For given parameters and an initial level set we solve to steady state. Fig. 2.8 shows a typical geodesic active contours segmentation result. 2.4.4 Chan-Vese Segmentation The segmentation methods presented so far are based on a high gradient that separates the object from the background. When such a gradient is not present, the methods fail to segment the object. 19
Page 1: Segmentation of Stochastic Images u
Page 5: Abstract The task of segmentation,
Page 8 and 9: 7.5 Segmentation of Stochastic Imag
Page 11 and 12: Chapter 1 Introduction The developm
Page 13 and 14: Figure 1.3: This thesis combines fi
Page 15: the presentation of the stochastic
Page 18 and 19: Chapter 2 Image Segmentation and Li
Page 36 and 37: Chapter 3 SPDEs and Polynomial Chao
Page 48 and 49: Chapter 4 Discretization of SPDEs F
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Chapter 9 Summary, Discussion, and
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List of Figures 1.1 Left: CT image
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List of Figures 6.2 Mean and varian
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List of Figures 7.12 Mean (left) of
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Appendix A Publications Written Dur
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Bibliography [14] L. Ambrosio and M
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Bibliography [46] B. Echebarria, R.
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Bibliography [81] B. N. Khoromskij
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Bibliography [113] A. Nouy. A gener
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Bibliography [147] J. Tryoen, O. Le
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