臺灣博碩士論文加值系統

本論文提出兩種對靜態模型迅速且有效的點簡化演算法；並且也針對動態模型提出一種使用主成分分析來壓縮資料量的資料簡化演算法。
針對靜態模型的簡化，本論文提出以八元樹為基礎之點的簡化及以離散外形運算子(DSO)為基礎之點的簡化方法。以八元樹為基礎之點的簡化演算法利用區域性的共平面分析來萃取代表點集合。區域性的共平面分析使用八元樹資料結構為基礎，利用八元樹節點中取樣點的散佈情形來判斷是否符合共平面的特性。此方法除了可以成功萃取出特徵區域，且可以依物體外觀變化情形與使用者的定義調整代表點集合的密度。代表點集合，又稱為基礎模型(base model)，將使用重建演算法重建出網狀模型。本論文除了可以成功重建外，對於簡化後模型也可保有不錯的誤差值，且相對於利用原始取樣點集合重建再進行簡化三角片的傳統方式，其所需花費之時間成本與計算代價均有良好的改善。本論文最終輸出為一階層式三角網格並保有多層次解析與自動多層次顯像的效果，並且可以產生點集合平均散佈或呈現出往特徵區域集中的兩種階層式成像法。
本論文並且提出以外形運算子為基礎之有效且低誤差的點簡化演算法用來保留物體的特徵。離散外形運算子(DSO)將用來針對點集合萃取特徵，並且所萃取出來的特徵點集合將會被延遲簡化。此簡化方法改善以二次誤差矩陣為基礎的網格點對簡化演算法，除了可以有效的在簡化物體時保留物體外型特徵，並且可以有效降低萃取特徵時的前處理時間。本論文並且提出一個得到單一簡化模型的方法。此單一簡化模型可以有效減少先對原始點集合進行重建的時間約百分之七十點六。換言之，此方法使用離散外形運算子可以獲得更多的幾何資訊，特別是在有高度變化的表面區域以及特徵點集合。並且離散外形運算子的特徵萃取不但有助於重建簡化後的點集合，而且在點簡化時可以適度地保留模型特徵，並且可以減少點簡化時所造成的誤差
針對動態模型的簡化，本論文使用仿射矩陣與主成份分析來壓縮3D動畫。使用主成份分析對一般的3D動畫模型均可以獲得不錯的簡化結果，因為主成份分析能將多個有相關性的變數簡化成少數幾個沒有相關的主成份，而且經由線性組合可還原出近似原始動畫。要選擇多少個主成分因子一直是一個值得研究的問題。選擇較多的主成分因子將可以獲得較精細的動畫模型並且降低誤差。因此，如何獲得主成分因子的公式是必需要的。本論文提出一個自動化選取主成分基底的方法，包含選擇最適當基底外並且分別應用主成分分析於x, y, z三軸上。
主成分分析適合用於原地運動的動態模型。若將原始動畫加入剛體運動，諸如:平移、旋轉、縮放，則對使用相同基底的主成份分析將會造成龐大的失真情形。本論文首先使用仿射矩陣來萃取模型運動特徵，並且將位移變化利用4�e4矩陣來記錄。轉換過後的模型可去除幾何轉換的影響，並將動畫模型正規化至侷限空間內；之後使用主成份分析將可以更正確的計算出最適當的基底(變異數)。

This investigation presents two novel rapid and effective point simplification algorithms for static model simplification utilizing point cloud without normals. And this thesis also presents a simplification algorithm for dynamic model, also called animation model, using principal component analysis.
For static model simplification, octree based point cloud simplification and DSO feature based point cloud simplification are proposed. The octree based point cloud simplification adopted local coplanar analysis to obtain the relevant points from a point set sampled from 3D objects. The local coplanar analysis, on the basis of an oc-tree data structure with an inner point distribution of a cube, can determine whether these points are coplanar. The proposed approach successfully extracts the feature area from point set. According to the object’s surface variation and user definition, the scheme also modulates the density of these relevant points. The relevant points, called the base model, were reconstructed to triangular mesh. In addition to the successful reconstruction, the error rate of the base model within a specific tolerance level. Compared with the traditional methods in which surface reconstruction is completed prior to mesh simplification, this approach applies point cloud simplification prior to surface reconstruction to improved time expense and calculation cost. By using the octree data structure, this thesis proposes some hierarchical rendering for the recon-structed model to suit user demand and produce a uniform or feature-sensitive simpli-fied model that facilitates rapid further mesh based applications. Finally, output of the proposed method is a hierarchical triangular mesh that inherently supports generation of multi-resolution representations for the applications of level of detail.
This thesis also proposes a Discrete Shape operator (DSO) feature based method for effective low-error point cloud simplification method to retain the physi-cal features of models. The DSO value is adopted to extract the features of the point cloud models, and the feature vertices are postponed to simplify. The proposed method improves the quadric error metric of the vertex pair contraction; it not only effectively simplifies the model while retaining the features of the object model but also decreases the pre-processing time cost for feature analysis. This algorithm pro-poses a method to obtain unique simplified model for each model. The unique sim-plified model obtained can significantly reduce the computation cost about 70.6% than mesh simplification which reconstruct original points first. In other words, the proposed method using DSO can adaptively collect more geometric information, par-ticularly on the high variation surfaces and the feature points. The DSO extraction can help to successfully reconstruct the simplified point cloud, preserve the features of simplified models, and reduce the errors caused by point cloud simplification.
For dynamic model (animation model) simplification, this thesis investigates the use of the affine transformation matrix when employing Principal Component Analy-sis (PCA) to simplify the data of 3D animation models. Satisfactory results were achieved for the common 3D models by using PCA because it can simplify several related variables to a few independent main factors, in addition to making the anima-tion identical to the original by using linear combinations. The selection of the prin-cipal component factor (also known as the base) is still a subject for further research. Selecting a large number of bases could improve the precision of the animation and reduce distortion for a large data volume. Hence, a formula is required for base selec-tion. This thesis develops an automatic PCA selection method, which includes the se-lection of suitable bases and a PCA separately on the three axes to select the number of suitable bases for each axis.
PCA is more suitable for animation models for apparent stationary movement. If the original animation model is integrated with transformation movements such as translation, rotation, and scaling, the resulting animation model will have a greater distortion in the case of the same base vector with regard to apparent stationary movement. This thesis is the first to extract the model movement characteristics using the affine transformation matrix and then to compress 3D animation using PCA. The affine transformation matrix can record the changes in the geometric transformation by using 4 × 4 matrices. The transformed model can eliminate the influences of geo-metric transformations with the animation model normalized to a limited space. Sub-sequently, by using PCA, the most suitable base vector (variance) can be selected more precisely.

Contents
中文摘要 I
Abstract III
Acknowledgements VI
Contents VII
List of Figures X
List of Table XIII
Chapter 1 Introduction 1
1.1 Surface Reconstruction and Mesh Simplification 3
1.2 Dynamic Model (Animation model) Simplification 5
1.3 Outlines 7
Chapter 2 Related Works 8
2.1 Surface Reconstruction 8
2.1.1 Spatial Subdivision 8
2.1.2 Distance Functions 8
2.1.3 Implicit Functions 9
2.2 Mesh Simplification 11
2.3 Feature Extraction 16
2.3.1 Correlation Ellipsoid 16
2.3.2 Multi-scale Feature Extraction 16
2.3.3 View-Independent Computational Procedure 17
2.4 Point Cloud Simplification 18
2.5 Dynamic Model Simplification 21
2.5.1 Vertex Based Predictive Techniques 21
2.5.2 Wavelet-Based Approaches 22
2.5.3 Time-Varying Geometries 22
2.5.4 PCA- Based Approaches 24
Chapter 3 Related Theory 26
3.1 Finding Neighbor Points of Each Point 27
3.1.1 K-Nearest Neighbor Points 27
3.1.2 Neighbor Points in Adaptive Sphere of Influence 28
3.1.3 Neighbor Points in Uniform Subdivision 31
3.2 Normal Estimation by Principal Component Analysis 33
3.3 Discrete Shape Operator base on Point Cloud 35
3.3.1 Curvature Calculation 36
3.3.2 Torsion Calculation 37
3.3.3 Shape Operator 38
3.3.4 The Estimation of DSO of 3D Models and Feature Extraction 39
3.4 Surface Reconstruction Algorithm using Discrete Shape Operator 43
3.5 Mesh Simplification using Discrete Shape Operator 48
3.6 Error Measurement for Compression Model 50
3.7 PCA and its Properties 51
Chapter 4 Point Cloud Simplification Algorithm 54
4.1 Octree Subdivision Using Coplanar Criterion for Hierarchical Point Simplification 57
4.1.1 Iteratively and Spatially Subdivide All Sampled Points 57
4.1.2 Choose the Coplanar Variable 58
4.1.3 Choose the Relevant Point of Each Cluster 59
4.1.4 Identify the Near Surface and Adjust the Appropriate Relevant Points 60
4.1.5 Adaptively Add Dummy Vertices to Avoid Undersampling 62
4.1.6 Merge Dummy Vertex 64
4.1.7 Reconstruction Algorithm 67
4.1.8 Hierarchical Representation and Rendering 67
4.1.9 Experimental Results 70
4.1.10 Conclusions 76
4.2 Point Cloud Simplification Algorithm using DSO 77
4.2.1 Determination of Localized Neighbour 77
4.2.2 Determination of the Sequence of Decimation 79
4.2.3 Determination of the Point Simplification Algorithm 81
4.2.4 Reconstruction Algorithm 84
4.2.5 Experimental Results 85
4.2.6 Conclusion 89
Chapter 5 Dynamic Model Simplification Algorithm 90
5.1 Suitable Bases Selection for PCA 92
5.2 Exploration of Storage Requirements using PCA 99
5.3 Animation Model with Geometric Transformation 101
5.4 Affine Transformation Matrix 103
5.5 Comparison of Representation Methods and Distortion for 3D Animation 108
5.6 Conclusions 115
Chapter 6 Conclusions and Future Works 116
References 118
作者簡介 125

List of Table
Table 3.1 Analysis of the usage of adaptive r = ms on different types of point clouds. 29
Table 4.1 The size generated by different models and simplified error measurement by Metro tool [48] and the flatness is using 0.005. 71
Table 4.2 The unique simplified results. 82
Table 4.3 The unique simplified results. 87
Table 4.4 The execution time (secs.) of experimental results measured on a Pentium 4 (3.0GHz) with 1 Gb of main memory. Comparison between reconstruction first method and point cloud simplification first method. The attributes of models are shown in Table 4.3. 88
Table 5.1 Properties of the animated mesh sequence considered in the evaluation. 92
Table 5.2 Compression ratios for principal component representation. 95
Table 5.3 Comparison between the proposed method and the methods in which Ek = 90% and 95%. The error evaluated is represented by Df. 98
Table 5.4 Corresponding proportion of selected bases. 98
Table 5.5 Storage requirements for the Chicken model. With close distortion, our method reduces the memory storage relative to the original method of PCA. 100
Table 5.6 The normalization of the 192 frames of the original horse animation model maintains a stable error rate. 106
Table 5.7 PCA applied to the Chicken and Cow animation using normalization. 106
Table 5.8 Distortion for 3 types of animation compression using the same bases that were selected for the Horse model. 110
Table 5.9 Comparison between the data requirement for application of normalization and PCA for similar distortion using the horse animation. 112
Table 5.10 The comparison measured on a Pentium 4 (3.0 GHz) with 2GB of main memory and implemented with Matlab 6.5. 114

List of Figures
Figure 2.1 Vertex pair contraction 13
Figure 2.2 Simplification results of vertex pair contraction at low resolution 15
Figure 3.1 Locally non-uniform surface. 28
Figure 3.2. Gopi selected ms to be the adjacent point range, but generated undersampled. 29
Figure 3.3. Jong adopted a fixed size to choose the adjacent point range, and generated oversampled and some big triangles on original holes. 30
Figure 3.4 The dotted circle indicates the sampling region of sampled point p using the fourth-closest point 31
Figure 3.5 The results of adopted the 4-closest point. 31
Figure 3.6 Diagram illustrating geodesic distance. 32
Figure 3.7 Diagram of the normal vector 33
Figure 3.8 (a) Unit normal vector field of surface M; (b) Shape Operator for tangent vector v at p. 35
Figure 3.9 Diagram of curvature (the curvature is in reverse proportion to radius) and curvature calculation, . 37
Figure 3.10.Diagram of torsion calculation, . 38
Figure 3.11 Shape Operator of saddle surface 39
Figure 3.12. Estimate the DSO in sampling region. 40
Figure 3.13 Surface variation detection. 41
Figure 3.14: Reconstruction results. 45
Figure 3.15 Comparison of the simplification results between vertex pair contraction and Jong’s method 49
Figure 4.1 Sequence of processing 54
Figure 4.2 Sequence of proposed algorithm using space subdivision. 55
Figure 4.3 Sequence of proposed algorithm using DSO. 56
Figure 4.4 The Bunny model (34838 points) as an example using different thresholds. (a) When the coefficient f = 0.005, the number of relevant points is 5616; (b) When f = 0.001, the number of relevant points is 12307. 59
Figure 4.5 (a) The gravitational center of cluster c is chosen as the relevant point; (b) P is chosen as the closest point to the gravitational center c, and set to the relevant point. 60
Figure 4.6 (a) The near surface may lead to incorrect judgment; (b) Incorrect judgment for a near surface may cause non-manifold occurrence; (c) Reduction of the near surface to p causes inconsistent curvature and topology errors. 60
Figure 4.7 Auxiliary points at the cell center and corners are adopted to detect a near surface. 61
Figure 4.8 The inner product of the normal vector and adjacent points is adopted to determine whether a near surface exists. 62
Figure 4.9. (a) The cluster containing a possible near surface is subdivided to avoid non-manifold occurrence; (b)Produced correct surface. 62
Figure 4.10 (a) Unexpected holes during reconstruction resulting from undersampling. (b) Reconstructed correct base model. 64
Figure 4.11 Diagram of merge dummy vertex. Where the dummy vertices are marked as green color. 65
Figure 4.12 The experimental results obtained by (a) QEM, (b) Uniform subdivision and (c) The proposed method to simplify the same models with similar numbers of reconstructed model points. The QEM method achieves the best error rate, since it has mesh information; the proposed method adopts a coplanar restriction to simplify the original point cloud without additional information and to perform reconstruction, therefore maintaining a good error rate. Uniform subdivision has a regular distribution, but causes undersampling in the feature area. 66
Figure 4.13 Depth First reduction reduces the deepest level each time. 68
Figure 4.14 Reduction by one ring neighbor coplanar measurement reduces the points according to the variation error of each relevant point pair Q*pq=Qp+Qq (Qp = epq1 + epq2 + epq3 + epq4 + epq5 + epq6 ). 69
Figure 4.15 The number of relevant points is slowly reduced when f exceeds 0.005. 70
Figure 4.16 The reconstructed results of the Dragon model on various levels after adopting the Depth First reduction. 72
Figure 4.17 The reconstructed results of the Dragon model on various levels after using the one ring neighbor coplanar measurement. 73
Figure 4.18 The reconstructed results of the Armadillo model (172974 points) on various levels after adopting the Depth First reduction. From left to right, base model (45077 points), reconstructed model (43981 points), 43978 points, 11154 points, 3051 points (1.76% of original). 74
Figure 4.19 The reconstructed results of the Armadillo model on various levels after using the one ring neighbor coplanar measurement. From left to right, base model (45077 points), reconstructed model (43981 points), 30007 points, 10014 points, 5018 points. 74
Figure 4.20 The reconstructed results of the Budda model on various levels after adopting the Depth First reduction. From left to right, original model, base model (135205 points), reconstructed model (96967 points), 96301 points, 85300 points, 47519 points, 18665 points; and 5278 points (0.97% of original). 74
Figure 4.21 The reconstructed results of the Budda model on various levels after using the one ring neighbor coplanar measurement. From left to right, original model, base model (135205 points), reconstructed model (96967 points), 70204 points, 50178 points, 40155 points, 20066 points; and 4963 points (0.9% of original). 75
Figure 4.22 The different point distribution of hierarchical rendering. (a) The depth first reduction obtains uniform distribution (11154 points) and (b) The one ring neighbor coplanar measurement product the feature-sensitive result (10014 points). 75
Figure 4.23 Oversampling in the flat region needlessly increases the number of calculations. The simplified model produces the same effect of solid representation. 76
Figure 4.24 Diagram illustrating geodesic distance. 78
Figure 4.25 The point distribution of the unique simplified models. 83
Figure 4.26 The mesh display of the unique simplified models. For each model pair, left side is the original model, and right side is the unique simplified model. 87
Figure 5.1 Principal component analysis applied to the Chicken animation. The relationship between base selection and error is depicted, showing that the error decreases as the number of bases increases. The animation appears reasonable when at least ten bases are selected. 93
Figure 5.2 Principal component analysis applied to the Cow animation. Decreasing the number of bases will decrease memory storage, but increase more distortion. 94
Figure 5.3. The relationship between base selection and reconstructed error (Df). The red points indicate the automatically selected base number, and in practice, the improvement of the error is insignificant with the increase in base number. 96
Figure 5.4 Our method automatically calculates the optimal number of bases and limits the error to a certain range. 97
Figure 5.5 Relation between the animation of the Horse model and that of the model with geometric transformation- Rotation, Translation, and Scaling (RTS). 102
Figure 5.6 PCA applied to the Horse animation by using 3 bases. 102
Figure 5.7 Production methods for 3D animation. 108
Figure 5.8 Depiction of the Horse animation frames 42, 64, 73, 90, and 125 by using the 3 methods mentioned in the text. Method 3 obtains the best effect owing to the affine matrix. 111
Figure 5.9 Improve ratio curve for animation models using our method for various bases with close distortion. The Horse animation can reach approximately 3.27 times that of PCA used alone and 1.6 and 2.24 times the Chicken and Cow animation, respectively. 113

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