Segmentation of Stochastic Images using ... - Jacobs University

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Chapter 2 Image Segmentation and Limitations ✈ ✈ ✈ ✈ ✈ ✈ ✈✘ ✘ ✘ v k ✎☞ w 7 ✍✌ jk ✘✘✘ ✈ v j ✘ ✘ ✘ e i j ✘✈ ✘ ✘ v i Figure 2.2: The graph generated from a 3 × 3 image contains 9 nodes and 12 edges. The edges e mn connect the nodes (the black dots) v l . Every edge e mn has a weight w mn describing the costs for traveling along this edge. 2.2.1 Relation to the Dirichlet Problem The simulation of an infinite number of random walks is equivalent to solving a combinatorial Dirichlet problem [45,59]. Therefore, we review the Dirichlet problem in this section and show its relation to random walks. The Dirichlet integral is R[u] = 1 ∫ |∇u| 2 dx . (2.7) 2 D A minimizer of the Dirichlet integral is a harmonic function, i.e. a function satisfying ∆u = 0 and the prescribed boundary conditions. The Dirichlet integral presented above is only useful for graphs with equal weights, for different weights we have to use a Dirichlet integral that respects these weights: R w [u] = 1 ∫ w(x)|∇u| 2 dx . (2.8) 2 D A minimizer of (2.8) is a function that satisfies ∇ · (w∇u) = 0. To compute the minimizer of the discrete Dirichlet problem, i.e. to find a minimizer of the discrete version of (2.8), we introduce the combinatorial Laplacian matrix: ⎧ ⎨ d i if i = j L i j = −w i j if v i and v j are adjacent nodes (2.9) ⎩ 0 otherwise . Using the combinatorial Laplacian matrix, the graph discrete version of (2.8) is R[x] = 1 2 xT Lx , (2.10) where x is a vector containing all nodes/pixels of the graph resp. the image, i.e. x = (v 1 ,...,v n ). The user prescribes seed points V M = V O ∪V B for the object V O and background V B , see Fig. 2.3. These points act as boundary conditions for the Dirichlet problem, because the probability that a random walk starting at a seed point reaches it is one. The unseeded points V U are the degrees of freedom. Reordering the nodes according to the set they belong to, (2.10) is written in block form R[x U ] = 1 [ x T 2 M xU] [ ][ ] T L M B xM B T L U x U = 1 (2.11) ( x T 2 M L M x M + 2xUB T T x M + xUL T ) U x U . In (2.11) L M is the part of the matrix L describing the dependencies between the seed points V M . L U is the part with the dependencies between the unseeded pixels. Finally, B, respectively B T , is the 10
2.2 Random Walker Segmentation Figure 2.3: Left: Definition of the seed regions for the object (yellow) and the background (red). Middle: The probability that a random walker reaches an object seed. Black denotes probability zero, white probability one. Right: Random walker segmentation result of the ultrasound image. As input we used the seed regions from the left image and β = 200. part of the matrix describing the coupling between the seeded and unseeded pixels. Differentiation of (2.11) yields a minimizer of (2.11) given by the solution of L U x U = −B T x M . (2.12) Remark 2. For 2D-images, the matrices L U and B are band matrices with five bands used only. The numerical solution of the system benefits from the use of numerical methods that make use of this special matrix structure, e.g. , it is necessary to store five bands as single vectors only. Furthermore, arithmetic operations for matrices with band structure can be implemented efficiently [57]. As already mentioned, the random walker segmentation is an interactive segmentation method. Due to the fast calculation of the random walker result, the user interactively defines new seed regions or eliminates unwanted seed regions to get an optimal segmentation result. Fig. 2.4 shows three steps of the computer/user interaction for the refinement of a segmentation result. Another, more mathematical, motivation for the derivation of (2.12) is that we have to solve −∇ · (w∇u) = 0 in D u = 1 on V O u = 0 on V B . (2.13) Transforming this PDE to homogeneous boundary conditions [124], applying the reordering of the nodes, and using the combinatorial Laplacian, we end up with (2.12). Eq. (2.12) is a system of linear equations solvable by using iterative methods, e.g. the method of conjugate gradients. Remark 3. The presented segmentation method, the random walker segmentation, sounds like a stochastic method for segmentation that includes randomness and uncertainty, but this is false. Using the equivalence to the Dirichlet problem presented above, the method computes deterministic weights for an elliptic PDE on a graph. Thus, all “randomness” is lost. The “randomness” comes from the interpretation of the result: Every pixel gets a value between zero and one, and we interpret these values as probabilities for reaching the seed region from this specific pixel. The random walker segmentation result is a probability for every pixel for belonging to the object (see middle of Fig. 2.3). Typically, the threshold 0.5 distinguishes between the object and the background. A pixel having a probability above 50% is assigned to the object and a pixel with a probability below 50% to the background. Fig. 2.3 shows a random walker segmentation result. 11
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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Page 58 and 59: Chapter 5 Stochastic Images
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Page 62 and 63: Chapter 5 Stochastic Images like qu
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6.1 Random Walker Segmentation on S
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6.2 Ambrosio-Tortorelli Segmentatio
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Chapter 7 Stochastic Level Sets whe
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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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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 [147] J. Tryoen, O. Le
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