Segmentation of Stochastic Images using ... - Jacobs University

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Chapter 5 Stochastic Images Figure 5.5: Expected value (left) and variance (right) of a stochastic US-image. The expected value looks like a deterministic image and in the variance, regions with a high gray value uncertainty are visible as white dots. Figure 5.6: Two samples drawn from a stochastic image. The images differ due to realizations of the noise. In a printed version, these images look nearly the same. random variables it is also possible to generate selected samples from stochastic images by prescribing the values for every random variable. Then, we evaluate the basis functions from the polynomial chaos at these points and generate the image as the sum of the deterministic images, one for every basis function from the polynomial chaos. This can be automated by generating a dynamic image, which automatically loops over all possible realizations of the stochastic image [61]. In the chapter dealing with stochastic level sets it is necessary to visualize stochastic contours, i.e. contours whose position and shape are dependent on random variables. The easiest possibility is to visualize realizations of the stochastic contour (see Fig. 5.7). Using this approach, we visually detect regions with a high uncertainty of the contour position, i.e. regions where the distance between realizations of the contour is greater than in other regions. For 3D stochastic surfaces, the visualization is even harder, because a slicing through 2D-images is cumbersome. Thus, a technique for the visualization of 3D stochastic surfaces is required. One possibility is to visualize the expected value and to color-code them by the variance [125]. Fig. 5.8 shows such a visualization. The result is an image, which is comparable to the 2D result from Fig. 5.5, but combines the information into one image. Furthermore, Djurcilov [43] presented ideas for the volume rendering of stochastic images. 54
5.4 Visualization of Stochastic Images Figure 5.7: Visualization of realizations of a stochastic 2D contour. Every yellow line corresponds to a MC realization of the stochastic contour encoded in the stochastic image. Conclusion In this chapter, we presented the concept of stochastic images and introduced the polynomial chaos approximation of stochastic images. With the projection method from Section 5.2, we are able to construct stochastic images from samples. This is a crucial task, because without this projection method, stochastic images are a theoretical construct only, but applications cannot use them. Furthermore, we presented visualization techniques for stochastic images. The visualization is important to bring stochastic images into applications. Without an intuitive visualization of the additional stochastic content, it might be difficult to bring the concept of stochastic images into applications. Having the concept of stochastic images at hand, we investigate in the next chapters how segmentation methods can be extended to be able to accept stochastic images as input. Figure 5.8: Visualization of a 3D contour encoded in a 3D stochastic image. The expected value of the 3D stochastic contour is color-coded by the variance. Regions with a high variance are red and regions with a low variance green. 55
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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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Chapter 8 Segmentation of Classical
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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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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 [147] J. Tryoen, O. Le
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