臺灣博碩士論文加值系統

於本論文中，我們提出了一個快速、精確、對稱且微分同構的核磁共振影像(MRI)對位演算法。我們利用一個對數－歐幾里德的架構以建立微分同構的模型，在此模型中，微分同構的李代數是由不隨時間變化的速度場表示，此速度場的模型是由多個具緊支撐的Wendland徑向基底函數的線性組合構成。最佳化所用的目標函數是由一有對稱性質的相關比及經過權重後的拉普拉斯算子模型組成。我們使用一具區域性及貪婪演算法性質的最佳化架構，藉以增進演算法的速度。在此架構中，我們使用一具對稱性的單純形演算法，來分別且逐一地找出各個徑向基函數的係數。為了在與仿射對位的結果合併時仍能保有整體對稱性，我們設計出一個應用“中途空間”概念的架構。藉由此架構，如果使用具對稱性的仿射對位演算法，則能確保整體對位流程也具對稱性。我們使用一個階層式的架構以增加演算法的速度及準確度，在此架構中，徑向基底函數是以由粗略至精細的順序逐一地被部署及估計。我們利用LPBA40數據集的40個T1-權重MRI影像的共1560對影像對的對位以驗證本論文提出的演算法。經由驗證可得知此演算法完全滿足微分同構的性質，且對稱性的誤差也小於體素寬度。為了驗證準確度，我們利用Klein等人於2010年提出的驗證架構評估本演算法，並與其他14種對位演算法進行比較。驗證結果顯示本演算法的中位目標重疊值高於全部14個演算法。另外，在使用5層規模級別時，本演算法較14種演算法中所有具微分同構性質者快速。

Abstract

A fast symmetric and diffeomorphic non-rigid registration algorithm for magnetic resonance images (MRIs) is proposed in this work. A log-Euclidean framework is used to model diffeomorphisms, in which the Lie Algebra of the diffeomorphism is modeled by time-invariant velocity fields. The velocity fields are modeled using linear combinations of compactly-supported Wendland radial basis functions. A symmetric correlation ratio combined with a weighted Laplacian model is used as the objective function for optimization. We used a greedy local optimization scheme to increase the speed of the algorithm. In this setup, a symmetric downhill simplex method is used to estimate the coefficient of each radial basis function separately and consecutively. To incorporate the result of initial affine registration while maintaining overall symmetry, a framework utilizing the concept of “halfway space” is devised. This framework can ensure overall symmetry if the affine registration algorithm is symmetric. To increase the speed and accuracy, we used a hierarchical framework in which the RBFs are deployed and estimated in a coarse-to-fine manner. The proposed algorithm was evaluated using the results of 1560 pairwise registrations of 40 T1-weighted MRIs in LPBA40 dataset. According to the evaluation results, the proposed algorithm is completely diffeomorphic and has sub-voxel accuracy in terms of symmetry. The accuracy of the proposed algorithm was evaluated and compared with 14 registration methods using the evaluation framework by Klein et al., 2010. The median target overlap of the proposed algorithm using LPBA40 dataset is higher than all 14 registration methods. In addition, the proposed algorithm is faster than all diffeomorphic registration methods in the comparison when using 5 scale levels.

Chapter 1 Introduction 1
1.1 Backgrounds 2
1.2 Small-Deformation Registration Frameworks 3
1.3 Diffeomorphic Registration frameworks 4
1.4 Symmetric Registration 4
1.5 Motivation 5
1.6 Thesis Overview 6
2 Background Knowledge and Related Works 7
2.1 Diffeomorphisms and Lie Algebra 8
2.2 Models of Diffeomorphisms 10
2.3 Basis Functions 12
Chapter 3 Methods 13
3.1 Model of Diffeomorphism 14
3.2 Radial Basis Functions 14
3.3 The Objective Function 16
3.3.1 The Likelihood Term 17
3.3.2 The Prior Term 23
3.4 Optimization 24
3.4.1 Local Optimization Scheme 24
3.4.2 Optimization Algorithm 25
3.5 Incorporation of Affine Registrations 26
3.6 A Hierarchical Framework 29
Chapter 4 Implementation Issues 35
4.1 Solving Partial Differential Equations 36
4.2 Re-sampling Algorithms 37
4.2.1 Nearest Neighbor Interpolation 38
4.2.2 Trilinear Interpolation 38
4.2.3 Tricubic Interpolation 41
4.2.4 Sinc Interpolation 42
4.3 The Evaluation Point Set 44
4.4 The Selection of Bin Width 45
Chapter 5 Results 49
5.1 Data and Registration 50
5.2 Evaluation of Diffeomorphism 50
5.3 Evaluation of Symmetry 53
5.4 Evaluation of Accuracy 57
5.5 Evaluation of Speed 58
Chapter 6 Discussion 65
6.1 Validity of Evaluation Results 66
6.2 Robustness of Symmetry 67
6.3 Effect of Basis Functions on Accuracy 69
6.4 Similarity Measure 70
Chapter 7 Conclusion 75
Bibliography 77

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