Segmentation of 3D Tubular Tree Structures in Medical Images ...

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20 Chapter 2. Extraction of Tubular Structures (a) Original image showing 2D cross sections of 3D tubular structures. Left: Cross section orthogonal to tubes tangent direction. Right: Cross section along the tubes tangent direction. (b) Filtered image and corresponding gradient vectors (vector: direction, grayvalue: magnitude) on a small scale. (c) Filtered image and corresponding gradient vectors (vector: direction, grayvalue: magnitude) on a large scale. Figure 2.1: Tubular structures and derived gradient vectors in Gaussian scale space.
2.2. Tube Detection Filters in Gaussian Scale Space 21 and eigenvectors v 1 , v 2 , and v 3 , encode information about the basic shape of image structures (blob, tube, or plane) as well as the polarity of the structures (brighter or darker than the surrounding). Based on this information, bright tubular structures can be identified as they require their two larger eigenvalues to be negative e 1 < 0 and e 2 < 0 and the third eigenvalue e 3 being close to zero. For those structures, the tangent direction is then indicated by v 3 . Frangi et al. [44]: One method based on the eigenvalues of the Hessian matrix that has been utilized extensively in practice due to it’s simple geometric interpretation is the method of Frangi et al. [44]: { 0 if e1 > 0 or e 2 > 0 T (x) = ( ( R 1 − exp − A 2 )) ( R 2α exp − 2 )( ( )) (2.2) 2 B 2β 1 − exp − S 2 2 2c otherwise 2 with R A = |e 3 |/ √ |e 1 ||e 2 | indicating blob-like structures, R B = |e 2 |/|e 1 | to distinguish between plate-like and line-like structures, and S = √ e 2 1 + e2 2 + e2 3 for suppression of random noise effects. The parameters α, β, and c allow controlling the weighting of the measures R A , R B , and S, respectively. Frangi’s suggested default values are α = 0.5 and β = 0.5, while c depends on the expected contrast and setting it to half the value of the maximum Hessian norm in the image is suggested. 2.2.2 Offset Medialness Functions Offset medialness functions incorporate information away from the tubes center at a distance that is related to the tubes size. It was shown by Krissian et al. [70] that such offset medialness functions allow for a greater robustness compared to central medialness functions. Krissian’s approach utilizes the Hessian matrix solely for pre-selection of candidate tube points and to determine the tubes cross-section plane, while in a second step boundariness information b i along a circle in this plane is sampled and combined into the actual tube-likeliness response as illustrated in Fig. 2.2. Considering the Hessian matrix, the eigenvectors associated with the two smaller eigenvalues provide estimates about the tubes cross-section plane spanned by the eigenvectors v 1 and v 2 as well as it’s tangent direction given by v 3 . In the following, we present two approaches based on this idea of estimating the tubes cross section plane using the Hessian matrix and afterwards sampling along the expected surface in this plane; the TDFs of Krissian et al. [70] and Pock et al. [9, 114].
Page 1: Graz University of Technology Insti
Page 5 and 6: Abstract The segmentation of tubula
Page 7 and 8: Kurzfassung Die Segmentierung tubul
Page 9: Acknowledgements I am deeply gratef
Page 13 and 14: Contents 1 Introduction 1 1.0.1 Req
Page 15: CONTENTS xv 8 Conclusion and Outloo
Page 18 and 19: xviii LIST OF FIGURES 4.2 Shape pri
Page 21: List of Tables 3.1 Average centerli
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70 Chapter 4. Tube Segmentation ass
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72 Chapter 4. Tube Segmentation (a)
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74 Chapter 4. Tube Segmentation Tab
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Chapter 5 Liver Vascular Tree Segme
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5.2. Method 79 (a) MIP of the datas
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5.3. Evaluation and Results 81 5.3.
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5.3. Evaluation and Results 83 cent
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5.3. Evaluation and Results 85 (a)
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5.4. Discussion 93 results. Selle
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5.5. Conclusion 95 a prior to const
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98 Chapter 6. Coronary Artery Tree
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Chapter 7 Airway Tree Segmentation
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7.2. Methods 111 shown in Fig. 7.1.
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7.2. Methods 113 Additionally, the
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7.3. Evaluation and Results 115 Gro
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7.3. Evaluation and Results 117 Tab
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7.4. Discussion 119 Figure 7.5: “
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7.5. Conclusion 121 Comparison to o
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7.5. Conclusion 123 Figure 7.6: Exa
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Chapter 8 Conclusion and Outlook Co
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8.1. Conclusion 127 arteries - info
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8.2. Directions for Future Work 129
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132 Chapter A. List of Publications
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BIBLIOGRAPHY 135 Bibliography [1] A
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BIBLIOGRAPHY 137 [21] Choyke, P., Y
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BIBLIOGRAPHY 139 [41] Florin, C., P
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BIBLIOGRAPHY 141 [63] Kitslaar, P.
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BIBLIOGRAPHY 143 [85] Lindeberg, T.
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BIBLIOGRAPHY 145 [108] O’Donnell,
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BIBLIOGRAPHY 147 [129] Schmugge, S.
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BIBLIOGRAPHY 149 [150] van Bemmel,
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BIBLIOGRAPHY 151 [171] Yi, J. and R
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