臺灣博碩士論文加值系統

科技的演進帶動時代的進步，就在人們變得更強調視覺享受的推波助瀾下，電腦圖學的相關研究在近幾年更是蓬勃發展。電腦圖學中絕大部份的研究主題都必須使用或分析虛擬3D模型，而3D模型網格結構的規則與否，左右了這些研究的成效。不幸的是，現今被廣泛採用的3D模型，是透過雷射掃瞄設備自動建構的，其網格結構往往散亂不佳。為此，本論文提出了三種可保住外形尖銳特徵的網格重製演算法。這三種演算法，雖是採用不同的演算技術，但都保證可以在不會破壞3D模型原有外表所含之尖銳特徵的原則下，有效提升其網格結構的規則度。
為了成功避免尖銳特徵在網格重製過程中遭受破壞，本論文採用了以法向量方向為基礎的表面分割技術來自動萃取尖銳特徵的所在之地。事先被萃取出來的特徵，在後續的網格重製步驟中將會被演算法視為神聖不可侵犯之處，藉此降低網格重製模型與原始模型之間的外形差異程度。
本論文提出的第一個網格重製演算法，結合了區域性的幾何編輯運算與全域性的頂點調整概念來實踐。透過簡單的網格編輯技術，演算法可以有效提升頂點之間的相連關係規則性；此外，藉助架構在全域性最佳化概念上的頂點調整法，以線性系統求解的概念，讓模型頂點分佈的均勻程度可以一次推演到理想的層級。
本論文所提出的第二個半規則網格重製演算法，可以讓重製後的模型具備階層式顯示的架構。過往的半規則網格重製演算法，大多選用計算較為耗時的網格參數化技術來完成。此外，欲在重製模型上成功保留原始外形尖銳特徵，對於現今的半規則網格重製技術仍然是一大挑戰。為此，本論文提出的演算法首先沿用先前自動產生的表面分割結果，建立保有原始尖銳特徵的簡化基底模型。之後，演算法會透過網格細分與網格鬆弛的概念，在基底模型上重新建置出半規則網格結構。此時，重新取樣後的模型因與原始模型外觀之間仍具有差距，必須藉助由表面分割法衍生出之表面對應關係，迅速推算新取樣點在原始模型表面上的正確投影位置，藉此完成取樣點投影與網格重製模型的重建步驟。
本論文的第三個網格重製演算法，打破了過去單純依靠網格鬆弛技術來提升網格規則度的想法。此演算法使用了一個迅速且有效的重新取樣技術，讓分割後的每個表面區塊內部，在短時間內產生大量的正三角形。藉此讓後續網格鬆弛的執行次數大幅縮短，將網格規則度提升至理想層級。
為了證明本論文提出之重製演算法的成效，在介紹各個方法的演算技術之後。將會與過往之相關技術進行比較，透過視覺結果與統計數據，讓奪讀者了解本論文所提出之演算法，皆可以重新製造出擁有高網格規則度與低外形誤差的新3D模型。

The efficiency of various computer graphics algorithms is generally depended on the mesh regularity of input 3D models. Unfortunately, the mesh regularity on most of 3D models acquired from general laser scanners is unsatisfactory. Thus, this thesis proposes three feature preservation remeshing algorithms. All of the proposed remeshing algorithms are focused on the same goal, improving the mesh regularity as well as preserving shape features on the input irregular meshes, by using different methods.
For the three remeshing algorithms proposed in this thesis, we adopt a normal-based surface segmentation to automatically extract the surface features on the input 3D models. These extracted features will be considered as geometric constraints to prevent the approximation errors caused by the further remeshing processes.
In our first mesh optimization algorithm, the mesh regularity of input mesh is improved by using the combination of geometric operations and a global vertex reposition technique. The proposed geometric operations are easy-to-implement and have the capability of efficiently improving the regularity of mesh connectivity. The global vertex reposition process is used to optimize the vertex distribution on the input irregular mesh, which is achieved by solving a global linear system. The experimental results demonstrate the robustness of the proposed algorithm.
In our second remeshing algorithm, we consider that benefited from the hierarchical representations, 3D models generated by semi-regular remeshing algorithms based on either global parameterization or normal displacement have more advantages for digital geometry processing applications than the ones produced from traditional isotropic remeshing algorithms. Nevertheless, while original models have sharp features or multiple self-intersecting surfaces, it is still a challenge for previous algorithms to produce a semi-regular mesh with sharp features preservation as well as high mesh regularity. Therefore, we proposes a robust semi-regular remeshing algorithm that uses a two-step surface segmentation scheme to build the high quality base mesh, as well as the regional relationship between the original surface and subdivision domain surface. Using the regional relationship, the proposed algorithm substantially enhances the accuracy and robustness of the backward projection process of subdivision vertices based on normal displacement. Furthermore, the mesh regularity of remeshed models is improved by the quadric mesh relaxation scheme. The experimental results demonstrate the capabilities of the proposed semi-regular remeshing algorithm to preserve geometric features and have good triangle aspect ratio.
In the third remeshing algorithm, instead of the global mesh relaxation method proposed in the previous study conducted on remeshing, we proposes an equilateral triangle grid-resampling scheme for achieving mesh optimization more efficiently. In order to improve the feasibility of resampling by directly using an equilateral triangle grid, the surface structure of the original model is correctly extracted by the automatic surface segmentation technique before the resampling step is executed. Results of this study show that the proposed remeshing algorithm can automatically and substantially improve the quality of triangulation, as well as automatically preserve shape features under an acceptable level of measurement error in the shape approximation, which is suitable for a mesh with a specific topology.

Contents
Chapter 1
Introduction 1
1.1 Motivation 1
1.2 Surface Remeshing 2
1.3 Feature Preservation Remeshing Algorithms 4
Chapter 2 Related Works 8
2.1 Isotropic Surface Remeshing 8
2.2 Global Mesh Optimization 9
2.3 Semi-regular Remeshing 9
2.4 Subdivision Surface Fitting 11
2.5 Feature Preservation for Remeshing 11
Chapter 3 Features Extraction with Normal-based Surface Segmentation Algorithm 13
3.1 Variational Shape Approximation (VSA) 14
3.2 Automatic Determination of Initial Triangles 15
3.3 Comparison 20
Chapter 4 Global Mesh Optimization with Automatic Surface Structure Preservation 23
4.1 Algorithm Overview 23
4.2 Structure Extraction by Surface Segmentation 24
4.3 Global Mesh Optimization 24
4.3.1 Mesh Connectivity Optimization 25
4.3.2 Vertex Distribution Optimization 27
4.4 Results 30
Chapter 5 A Robust Feature- Preserving Semi-regular Remeshing Method for Triangular Meshes 34
5.1 Algorithm Overview 34
5.2 Construction of Regular Base Mesh 37
5.3 Semi-regular Mesh Reconstruction 41
5.3.1 Construction of Subdivision Domain Mesh 41
5.3.2 Regional Projection of Subdivision Vertices 41
5.3.3 Mesh Relaxation 45
5.4 Results 50
Chapter 6 High Quality Surface Remeshing with Equilateral Triangle Grid 59
6.1 Algorithm Overview 59
6.2 Smoothing Boundary of Surface Patches 60
6.3 Mesh optimization 60
6.3.1 Equilateral triangle grid-resampling scheme 62
6.4 Results 67
Chapter 7 Conclusions 78
Reference 80

List of Figures
Fig. 1.1. The workflow of generating a desirable triangular mesh for further applications. 2
Fig. 1.2. The remeshing without the consideration of sharp feature preservation. 3
Fig. 1.3. The semi-regular remeshing method can generate remeshed models with miltiresolution representation. 4
Fig. 3.1. Surface segmentation process of the VSA approach [CAD04]. 15
Fig. 3.2. Remeshing result without sharp features preservation caused by using undesirable surface segmentation result. 16
Fig. 3.3. Remeshing result with unsatisfactory mesh regularity caused by using undesirable surface segmentation result. 16
Fig. 3.4. The experimental result of L2,1 error distribution using 50 randomly generated seed triangle sets. 17
Fig. 3.5. The steps of the automatic surface segmentation. 19
Fig. 3.6. Segmentation result for model after rotation. 20
Fig. 3.7. Comparison of L2,1 error between the VSA approach [CAD04] and our proposed method. 21
Fig. 4.1. Work flow of the proposed mesh optimization algorithm 24
Fig. 4.2. Automatic feature points extraction. 25
Fig. 4.3. Comparison of mesh optimization results between the original mesh with irregular vertex degree and the one with regular vertex degree. 26
Fig. 4.4. Edge-flip operation. 27
Fig. 4.5. Comparison of optimized results of various models between [NISA06] and our method. 32
Fig. 5.1. Semi-regular remeshing results of a mechanical model with sharp features produced by our method and previous semi-regular remeshing algorithms. 35
Fig. 5.2. Overview of the proposed semi-regular remeshing algorithm. 37
Fig. 5.3. Resampling and relaxation by using the second segmentation step. 40
Fig. 5.4. The effect of proposed regional projection method on a model with sharp features. 43
Fig. 5.5. Regional projection scheme. 45
Fig. 5.6. Visual comparison of mesh relaxation method. 48
Fig. 5.7. Mesh relaxation results using different relaxation weights of vertices. 50
Fig. 5.8. Semi-regular remeshing results. 53
Fig. 5.9. Comparison of the influence of base mesh quality on corresponding semi-regular remeshing result. 54
Fig. 5.10. Comparison of remeshing results of mechanical models between our algorithm and subdivision surface fitting methods proposed by [LWY08] and [LD09]. 56
Fig. 5.11. Comparison of remeshing results of Kitten model between our method and isotropic surface remeshing approach [YLSW09]. 57
Fig. 5.12. Comparison of remeshing results of Rabbit model between various semi-regular remeshing schemes. 58
Fig. 6.1. Work flow of the proposed high quality surface remeshing algorithm. 60
Fig. 6.2. Smoothing boundary of surface patch by removing ear triangle (blue triangle in (a)). 61
Fig. 6.3. Surface segmentation result for model with high genus and the corresponding segmented patches flatten results. 62
Fig. 6.4. The steps of the resampling scheme. 63
Fig. 6.5. Local mesh relaxation. 66
Fig. 6.6. Remeshed results for the experiment models with different geometric characteristics. 69
Fig. 6.7. Comparison of sharp feature preservation. 70
Fig. 6.8. Result of experiment with high genus model. 71
Fig. 6.9. Comparison of efficiency and quality of mesh optimization process between different states of initial distribution of resample points. 74
Fig. 6.10. Comparison of remeshing result of car-door model between our method and other approaches. 75
Fig. 6.11. Comparison of remeshing results for models of different types between our method and the approach in Valette et al. [VCP08]. 77

List of Tables
Table 4.1. Comparisons of mesh qualities for models shown in Fig. 4.5. 33
Table 5.1. Experimental parameters for remeshed model generated by the proposed algorithm. 51
Table 5.2. Mesh quality and computing time of remeshed models shown in Fig. 5.8. 52
Table 5.3. Mesh quality of simplified meshes and corresponding semi-regular remeshed models shown in Fig. 5.9. 54
Table 5.4. Mesh quality of remeshed mechanical models shown in Fig. 5.10. 55
Table 5.5. Mesh quality of remeshing results shown in Fig. 5.11 and Fig. 5.12. 57
Table 6.1. Mesh quality, approximation error, and connectivity compression rate for the experiment models. 68
Table 6.2. Computing times for the major stages of our proposed algorithm for the experiment models. 68
Table 6.3. Comparison of remeshing aspect ratio of car-door model between our method and other approaches. 72
Table 6.4. Comparison of remeshing approximation error of car-door model between our method and other approaches. 73
Table 6.5. Comparison of remeshing aspect ratio between our method and the approach in Valette et al. [VCP08]. 73
Table 6.6. Comparison of remeshing approximation error between our method and the approach in Valette et al. [VCP08]. 73

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