Compound Biorthogonal Wavelets on Quadrilaterals and Polar Structures

In geometric models with high-valence vertices, current subdivision wavelets may not deal with the special cases well for good visual effect of multiresolution surfaces. In this paper, we present the novel biorthogonal polar subdivision wavelets, which can efficiently perform wavelet analysis to the control nets with polar structures. The polar subdivision can generate more natural subdivision surfaces around the high-valence vertices and avoid the ripples and saddle points where Catmull-Clark subdivision may produce. Based on polar subdivision, our wavelet scheme supports special operations on the polar structures, especially suitable to models with many facets joining. For seamless fusing with Catmull-Clark subdivision wavelet, we construct the wavelets in circular and radial layers of polar structures, so can combine the subdivision wavelets smoothly for composite models formed by quadrilaterals and polar structures. The computations of wavelet analysis and synthesis are highly efficient and fully in-place. The experimental results have confirmed the stability of our proposed approach

Compound Biorthogonal Wavelets on Quadrilaterals and Polar Structures

In geometric models with high-valence vertices, current subdivision wavelets may not deal with the special cases well for good visual effect of multiresolution surfaces. In this paper, we present the novel biorthogonal polar subdivision wavelets, which can efficiently perform wavelet analysis to the control nets with polar structures. The polar subdivision can generate more natural subdivision surfaces around the high-valence vertices and avoid the ripples and saddle points where Catmull-Clark subdivision may produce. Based on polar subdivision, our wavelet scheme supports special operations on the polar structures, especially suitable to models with many facets joining. For seamless fusing with Catmull-Clark subdivision wavelet, we construct the wavelets in circular and radial layers of polar structures, so can combine the subdivision wavelets smoothly for composite models formed by quadrilaterals and polar structures. The computations of wavelet analysis and synthesis are highly efficient and fully in-place. The experimental results have confirmed the stability of our proposed approach

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

CUHK electronic theses & dissertations collection

随着3D 图形学技术的快速发展，基于细分小波的多分辨率方法受到了越来越多的关注。为了提高运算效率， 一些细分小波采用了厅局部提升用的方法以避免解全局方程组的庞大开销。这种方法虽然极大地提高了小波分解的速度，但也使得这些小波较之一些经典的细分小波在生成曲面的质量上有所不如。在本篇论文里，我们提出了一组新型细分小波。这些小波变换不但保留了"局部提升"波运算速度快，节省内存的优点，在生成模型的质量上也大大提高，接近了经典的全局优化小波。我们构造了极细分小波用于极结构快速简化和重构。极细分小波变换有效地避免Catmull -Clark 细分小波用于极结构时所造成的"皱裙"和鞍点，可以在高度数的异常点区域生成非常自然的二次连续曲面。为了更好的应用于普通的四边形网格曲面，我们还改进了极细分小波使之生成的曲面可以在边界处与Catmull-Clark 细分小波曲面光滑地融合。实验表明我们构造的混合极细分小波不但运算效率高，节省内存，还具有良好的稳定性，生成的曲面质量良好。基于矩阵值细分，我们还构造了一组近似和插值类型的矩阵值小波。由于矩阵值小波变换直接作用于向量，我们可以利用向量中额外的项作为参数以控制生成的多分辨率由面的形状。通过优化这些形状控制参数，我们在保持高效低内存消耗的同时，还可以进一步提高"局部提升"小波曲面的质量。我们还将矩阵值小波应用于3D 模型的几何压缩。为了避免存储形状控制参数所带来的额外消耗，我们采用固定的形状控制参数从而将矩阵值小波简化为一种特殊的标量值小波。实验表明采用我们的小波的压缩方法，其压缩率接近于经典的全局优化小波，远高于"局部提升"小波。其压缩速度则接近于"局部提升"小波，远高于经典的采用全局优化小波。在未来的研究工作中，我们会进一步优化形状控制参数的选择策略，并尝试将其应用范围从目前三角形网格由面扩展到四边形曲面， T 样条曲面以及混合曲面。我们还将研究如何应用己有的小波变换提高多分辨率编辑与动画技术。During the rapid development of 3D graphics applications, the wavelet-based multiresolution approaches have attracted more attention because they can effectively reduce the process/storage costs of high-detailed models. For the efficiency, many wavelets are constructed by using local lifting, which makes the fitting quality of results are not good as the usual wavelets with global optimization. On the other hand, once the wavelet transforms were constructed, the multiresolution meshes got by them cannot be adjusted any more. It is important to develop the new adaptive wavelets with better fitting quality, while keeping the high efficiency. In this dissertation, we provide several secondgeneration wavelets with improved fitting qualities, which include the compound biorthogonal wavelets for the hybrid quadrilateral meshes, and the efficient matrix-valued wavelets for complex triangular meshes.We propose the novel polar subdivision wavelet, which efficiently generate multiresolution the polar structures. Polar structures are the natural representations of the self-revolution structures or high-valence regions of quadrilateral grids. The traditional multiresolution methods for the polar structures often generate deficits caused by high valence vertices. By adopting the polar subdivision and the special lifting operations on the polar structures, our wavelet transforms can generate smooth multiresolution surfaces without ripples and saddle points. To process the hybrid meshes made of quadrilaterals and polar structures, we extend the polar wavelet to the vertices in the circular layers, which makes it possible to fuse the surfaces generated by different wavelet transforms seamlessly. To improve the fitting quality of local lifting wavelets, we extend wavelet constructions from the scalar-valued scheme to the matrix-valued scheme, and propose a family of novel approximate and interpolatory matrix-valued subdivision wavelets. The matrix-valued wavelets are constructed from the refinable basis function vectors, which deal with the additional parameters to the geometric position of vertices. Since the final results of wavelet transforms are sensitive to the parameters, these parameters can be used to adjust the shape of multiresolution surfaces. By applying the lifting scheme, the computations of wavelet transform are local and in-placed. We also discuss the strategy of better shape controls for improving the fitting quality of simplified surfaces. The experiments showed that these novel wavelet transforms were sufficiently stable and performed well for noise reduction. With the suitable shape control parameters, the fitting quality of multiresolution surfaces can be further improved.We study how to apply the efficient compression approach to the real applications, such as the compression of meshes. Since the shape control parameters need the additional storage, they will decrease the compression ratio if we apply the original versions of matrix-valued wavelets. We revise the construction of the matrix-valued wavelet transform and proposed the parameterized scalar-valued wavelet transform. With the special optimization of wavelet construction and suitable parameters, our compression approach has the high compression ratio close to the well-known approaches using the global orthogonal wavelets, and much higher compression ratio than the compression using the local lifting wavelets, while keeping the good efficiency. In the future work, we plan to extend the matrix-valued wavelets from triangular meshes to quadrilateral, normal and hybrid meshes. We will study how to apply the matrix-valued wavelets to the applications, such as multiresolution editing and animations. Further optimization of the shape control parameters for mobile and online applications is also an important issue.Detailed summary in vernacular field only.Detailed summary in vernacular field only.Detailed summary in vernacular field only.Zhao, Chong.Thesis (Ph.D.)--Chinese University of Hong Kong, 2012.Includes bibliographical references (leaves 136-149).Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.Abstract also in Chinese.Abstract --- p.iAcknowledgement --- p.vChapter 1 --- Introduction --- p.1Chapter 1.1 --- Introduction --- p.1Chapter 1.2 --- Wavelet Transform --- p.4Chapter 1.2.1 --- Wavelets and Multiresolution Analysis --- p.4Chapter 1.2.2 --- Continuous Wavelet Transforms --- p.7Chapter 1.2.3 --- Discrete Wavelet Transform --- p.8Chapter 1.3 --- Second Generation Wavelets --- p.9Chapter 1.3.1 --- Lifting Scheme --- p.9Chapter 1.3.2 --- Subdivision --- p.11Chapter 1.3.3 --- Subdivision Wavelets --- p.13Chapter 1.4 --- Main Contributions --- p.15Chapter 1.5 --- Chapter Summary --- p.16Chapter 2 --- Compound Wavelets on Quadrilaterals --- p.18Chapter 2.1 --- Introduction --- p.18Chapter 2.2 --- Related Work --- p.19Chapter 2.3 --- Polar Subdivision --- p.20Chapter 2.3.1 --- Subdivision in radial layers --- p.21Chapter 2.3.2 --- Subdivision in circular layers --- p.24Chapter 2.4 --- Subdivision Wavelets Using Lifting --- p.25Chapter 2.4.1 --- Lifting wavelets --- p.25Chapter 2.4.2 --- Wavelet transforms --- p.29Chapter 2.5 --- Compound Subdivision Wavelets --- p.32Chapter 2.6 --- Experimental Results --- p.34Chapter 2.7 --- Chapter Summary --- p.38Chapter 3 --- Matrix-valued Loop Wavelets --- p.40Chapter 3.1 --- Introduction --- p.40Chapter 3.2 --- Related Work --- p.41Chapter 3.3 --- Matrix-valued Loop Subdivision --- p.43Chapter 3.4 --- Matrix-valued Loop Subdivision Wavelet --- p.46Chapter 3.4.1 --- Lazy Wavelet --- p.46Chapter 3.4.2 --- Inner Product --- p.49Chapter 3.4.3 --- Wavelet Transform --- p.54Chapter 3.4.4 --- Shape Control Parameters --- p.55Chapter 3.5 --- Experiments and Discussion --- p.57Chapter 3.6 --- Chapter Summary --- p.61Chapter 4 --- Matrix-valued Interpolatory Wavelets --- p.63Chapter 4.1 --- Introduction --- p.63Chapter 4.2 --- Matrix-valued Interpolatory Subdivision --- p.64Chapter 4.3 --- Matrix-valued 1-ring Wavelets --- p.67Chapter 4.3.1 --- Biorthogonal Wavelet Transform --- p.67Chapter 4.3.2 --- Extraordinary Points Treatment --- p.74Chapter 4.3.3 --- Shape Control Parameters --- p.75Chapter 4.4 --- Matrix-valued 2-ring Wavelets --- p.81Chapter 4.5 --- Experiments and Discussion --- p.86Chapter 4.5.1 --- 1-ring Wavelet Transform --- p.86Chapter 4.5.2 --- 2-ring Wavelet Transform --- p.92Chapter 4.6 --- Chapter Summary --- p.98Chapter 5 --- Geometry compression using wavelets. --- p.100Chapter 5.1 --- Introduction --- p.100Chapter 5.2 --- Related Work --- p.101Chapter 5.3 --- Matrix-valued Wavelet Transform --- p.105Chapter 5.3.1 --- Matrix-valued Loop Subdivision --- p.105Chapter 5.3.2 --- Lazy Wavelet --- p.108Chapter 5.3.3 --- Inner Product --- p.109Chapter 5.3.4 --- Wavelet Transform --- p.112Chapter 5.3.5 --- Coding --- p.113Chapter 5.4 --- Experiments and discussion --- p.114Chapter 5.4.1 --- Stability --- p.117Chapter 5.4.2 --- Efficiency --- p.118Chapter 5.4.3 --- Compressions --- p.120Chapter 5.5 --- Chapter Summary --- p.124Chapter 6 --- Conclusions --- p.125Chapter 6.1 --- Research Summary --- p.125Chapter 6.2 --- Future Work --- p.127Chapter A --- Inner Products of Wavelets in Radial Layers --- p.130Chapter B --- Publication List --- p.133Bibliography --- p.13