Segmentation of Stochastic Images using ... - Jacobs University

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List of Figures 6.16 Comparison of the stochastic Ambrosio-Tortorelli model (left column) with the extended model using the edge linking procedure described in Section 2.3.3 (middle column) and a combination of the edge linking and adaptive grid approach (right column). Note that these results are computed with the same parameter set. The differences in the results are due to the additional edge linking parameter c only. . . . 75 6.17 Comparison of the full grid and adaptive grid solution. The full grid and adaptive grid solution are visually identical, but the computation of the adaptive grid solution needs significantly less DOFs. Thus, it can be applied on high-resolution images. . . 77 7.1 Stochastic level sets do not have a fixed position where φ(x) = 0. Instead, there is a band with positive probability that the level set is equal to zero, i.e. the position of the zero level set is random and it is possible to estimate the PDF of the interface location in the normal direction of the expected value of the interface (lower right corner). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7.2 Comparison of expected value and variance of the resulting phase field for the cosine test of (7.18) using the polynomial chaos (PC), stochastic collocation (SC), Monte Carlo simulation (MC), and Monte Carlo simulation of the original level set equation (MCL). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.3 Comparison of the expected value and variance of the resulting phase field for the rarefaction fan and the shock, two classical tests for level set propagation. The figure shows the comparison of the four discretizations of the stochastic phase field equation. 86 7.4 Expected value color-coded by the variance for the Stanford bunny after shrinkage under an uncertain speed in the normal direction. Red indicates regions with a high variance and green regions with low variance. In addition, we show one slice of the variance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.5 Mean of the CT data set (left) and the liver data set (right) for the segmentation test. . 88 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.7 Resulting image with the expected value of the contour (red) of the segmented object and the phase field variance with the expected value of the contour for gradientbased segmentation of a stochastic CT image. The variance is constant in the normal direction of the expected value of zero level set. . . . . . . . . . . . . . . . . . . . . 90 7.8 Mean and variance of the stochastic geodesic active contour segmentation of the stochastic CT data set. The variance is constant in the normal direction of the zero level set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.9 Left: Evolution of the expected value contour of the stochastic geodesic active contour method. The shown variance corresponds to the contour after 240 iterations (the magenta contour). Right: Mean value of the stochastic image to be segmented and the contours at time points of the level set evolution. The final contour matches the object boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 7.10 Mean (left) of the stochastic CT and the variance (right) of the stochastic Chan-Vese solution. Additionally, we show the expected value contour at different time steps. . . 93 7.11 Left: MC-realizations of the stochastic contour from the stochastic Chan-Vese segmentation applied on the CT data set. Right: Realizations of the stochastic contour from the stochastic geodesic active contour approach applied on the CT data set. . . . 94 114
List of Figures 7.12 Mean (left) of the stochastic liver image and the variance of the stochastic Chan-Vese solution. In addition, we show the expected value contour at different time steps. . . 95 7.13 Variance of the stochastic image to segment (left), the expected value is not depicted, because the expected value is an image with the same gray value at every pixel. On the right, the segmentation result is depicted on one realization (one sample) of the stochastic image to segment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.1 Left: Realizations of the stochastic contour obtained from the random walker segmentation with stochastic parameter. Right: Mean and variance of the stochastic contour obtained from the random walker segmentation with stochastic parameter. . . 99 8.2 Left: Realizations of the stochastic contour obtained from the random walker segmentation with stochastic parameter. Right: Mean and variance of the stochastic contour obtained from the random walker segmentation with stochastic parameter. . . 100 8.3 Volume of the stochastic contour obtained from the random walker segmentation with stochastic parameter. The left curve shows the PDF of the object in the US image, the right curve the PDF of the liver in the liver data set. The PDFs are obtained using (6.12). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 8.4 Ambrosio-Tortorelli model applied on the expected value of the liver data set using a stochastic parameter µ. The upper row shows the expected value (left) and the variance (right) of the smoothed image, the lower row the expected value (left) and the variance (right) of the phase field. . . . . . . . . . . . . . . . . . . . . . . . . . 102 8.5 Ambrosio-Tortorelli model applied on the expected value of the liver data set using a stochastic parameter ε. The upper row shows the expected value (left) and the variance (right) of the smoothed image and the lower row the expected value (left) and the variance (right) of the phase field. . . . . . . . . . . . . . . . . . . . . . . . 103 8.6 Result of the gradient-based segmentation with stochastic parameter b, i.e. with a stochastic curvature smoothing. The upper row shows the results for the CT data set, expected value of the image and contour realizations (left) and variance of the level set with contour realizations (right). The lower row shows the same results for the liver data set. In all figures we add Monte Carlo realizations of the stochastic object boundary. The red contour corresponds to a b = 0.75, yellow to b = 1.0 and blue to b = 1.25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 8.7 Result of the geodesic active contour segmentation with stochastic parameters for the CT data set. On the left the expected value of the image and contour realizations and on the right the variance of the level set with contour realizations are shown. . . 105 8.8 Result of the geodesic active contour segmentation with stochastic parameters for the liver data set. On the left the expected value of a detail of the image and contour realizations and on the right the variance of the level set with contour realizations are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 9.1 Visualization of the PDF of the object boundary in the case of the segmentation of the stochastic US image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 115
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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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Figure 1.3: This thesis combines fi
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the presentation of the stochastic
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Chapter 2 Image Segmentation and Li
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Chapter 3 SPDEs and Polynomial Chao
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Chapter 4 Discretization of SPDEs F
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Chapter 4 Discretization of SPDEs 4
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Chapter 4 Discretization of SPDEs a
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Chapter 5 Stochastic Images
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Chapter 5 Stochastic Images Figure
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Chapter 5 Stochastic Images like qu
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Chapter 5 Stochastic Images Figure
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Chapter 6 Segmentation of Stochasti
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6.1 Random Walker Segmentation on S
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6.1 Random Walker Segmentation on S
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