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Chapter 7 Stochastic Level Sets Figure 7.5: Mean of the CT data set (left) and the liver data set (right) for the segmentation test. 7.5 Segmentation of Stochastic Images Using Stochastic Level Sets Level set evolution with an uncertain speed can be useful in applications, e.g. in physical applications where level sets track the interface between materials. Often, the material parameters are not known exactly and the interface speed can be modeled dependent on the random material parameters. The focus of this thesis is on the application of segmentation methods. This is why we use this new concept for stochastic level sets at the moment for segmentation only. The author used level sets for the modeling of physical effects, the evaporation of water during radiofrequency ablation [10,11,13]. It is possible to use stochastic level sets in this context, because the material parameters can be modeled as random variables [87, 128] which leads to an uncertain interface speed. For segmentation, we investigate three segmentation methods based on level sets for stochastic extensions: gradient-based segmentation, geodesic active contours, and Chan-Vese segmentation. Other segmentation methods based on level sets can also be suitable for stochastic extensions. 7.5.1 Gradient-Based Segmentation of Stochastic Images Gradient-based segmentation of an image u : D → IR is introduced in Section 2.4.2 and given by φ t + v(1 − bκ)|∇φ| = 0 , (7.26) where the function v is called stopping function, because this function controls the stopping of the level set evolution at the desired boundaries. Often, the function v is given by v = 1/(1 + |∇u|) , (7.27) where u is the image to segment. Typically, the level set is initialized as the signed distance function of a small circle inside the object to segment. There is no theoretical justification of this method besides the observation that the level set speed is close to zero at sharp edges due to the reciprocal dependence between image gradient and speed. We replace the classical image u(x) by a stochastic image u(x,ω). The equation for stochastic gradient-based segmentation is and the speed is a stochastic quantity, too: φ t (t,x,ω) + v(t,x,ω)(1 − bκ(t,x,ω))|∇φ(t,x,ω)| = 0 (7.28) v(t,x,ω) = 1 1 + |∇u(t,x,ω)| . (7.29) 88
7.5 Segmentation of Stochastic Images Using Stochastic Level Sets Figure 7.6: Left: Mean contour during the evolution of the stochastic level set. The iso-contours are drawn on the variance image of the final, magenta contour. The contour detection is influenced by the image noise on the bottom and the right of the object (high variance). Right: Contour realizations of the stochastic gradient-based segmentation of the CT data. This method can be implemented by using the stochastic preconditioned phase field implementation introduced in the last section by rearranging the equation to where the stochastic speed ṽ is φ t (t,x,ω) + ṽ(t,x,ω)|∇φ(t,x,ω)| = 0 , (7.30) ṽ(t,x,ω) = 1 − bκ(t,x,ω) 1 + |∇φ(t,x,ω)| . (7.31) Using the decomposition into curvature dependent and independent parts, we end up with Numerical Results φ t + 1 1 + |∇φ| |∇φ| − b κ|∇φ| = 0 . (7.32) 1 + |∇φ| For the presentation of the results of the gradient-based segmentation of stochastic images we use two data sets. The first data set consists of 289 reconstructions of a CT data set with 100 × 100 pixels. Section 5.2.2 gives details about the generation of the reconstructions. These reconstructions are treated as independent realizations of a stochastic image and the polynomial chaos expansion of the stochastic image is calculated using the methods from Section 5.2. The second data set consists of a liver mask embedded into a 129 × 129 pixel image with a varying gradient strength to the background. This image is corrupted by uniform noise and 25 samples, i.e. noise realizations, of this image are treated as input for the generation of the stochastic image. In both data sets, the generated stochastic image contains two random variables, and we use a polynomial degree of two, i.e. n = 2 and p = 2. Fig. 7.5 shows the expected value of the stochastic image of both data sets. The parameter b is set to the grid spacing h and the level set is initialized as a circle centered in the center of the image with a radius of 0.15 of the image width. Fig. 7.6 shows the expected value during the level set evolution (the colored contours) and the variance after 280 iterations of the gradient-based segmentation with time step τ = 0.2h of the second data set. It shows the typical behavior of a rapid propagation of the level set towards the object boundary and the influence of the stopping function that tries to stop the evolution at the boundary. 89
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Segmentation of Stochastic Images u
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Abstract The task of segmentation,
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7.5 Segmentation of Stochastic Imag
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Chapter 1 Introduction The developm
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