Lifting-based subdivision wavelets with geometric constraints.

Qin, Guiming."August 2010."Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.Includes bibliographical references (p. 72-74).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.5Chapter 1.1 --- B splines and B-splines surfaces --- p.5Chapter 1. 2 --- Box spline --- p.6Chapter 1. 3 --- Biorthogonal subdivision wavelets based on the lifting scheme --- p.7Chapter 1.4 --- Geometrically-constrained subdivision wavelets --- p.9Chapter 1.5 --- Contributions --- p.9Chapter 2 --- Explicit symbol formulae for B-splines --- p.11Chapter 2. 1 --- Explicit formula for a general recursion scheme --- p.11Chapter 2. 2 --- Explicit formulae for de Boor algorithms of B-spline curves and their derivatives --- p.14Chapter 2.2.1 --- Explicit computation of de Boor Algorithm for Computing B-Spline Curves --- p.14Chapter 2.2.2 --- Explicit computation of Derivatives of B-Spline Curves --- p.15Chapter 2. 3 --- Explicit power-basis matrix fomula for non-uniform B-spline curves --- p.17Chapter 3 --- Biorthogonal subdivision wavelets with geometric constraints --- p.23Chapter 3. 1 --- Primal subdivision and dual subdivision --- p.23Chapter 3. 2 --- Biorthogonal Loop-subdivision-based wavelets with geometric constraints for triangular meshes --- p.24Chapter 3.2.1 --- Loop subdivision surfaces and exact evaluation --- p.24Chapter 3.2.2 --- Lifting-based Loop subdivision wavelets --- p.24Chapter 3.2.3 --- Biorthogonal Loop-subdivision wavelets with geometric constraints --- p.26Chapter 3. 3 --- Biorthogonal subdivision wavelets with geometric constraints for quadrilateral meshes --- p.35Chapter 3.3.1 --- Catmull-Clark subdivision and Doo-Sabin subdivision surfaces --- p.35Chapter 3.3.1.1 --- Catmull-Clark subdivision --- p.36Chapter 3.3.1.2 --- Doo-Sabin subdivision --- p.37Chapter 3.3.2 --- Biorthogonal subdivision wavelets with geometric constraints for quadrilateral meshes --- p.38Chapter 3.3.2.1 --- Biorthogonal Doo-Sabin subdivision wavelets with geometric constraints --- p.38Chapter 3.3.2.2 --- Biorthogonal Catmull-Clark subdivision wavelets with geometric constraints --- p.44Chapter 4 --- Experiments and results --- p.49Chapter 5 --- Conclusions and future work --- p.60Appendix A --- p.62Appendix B --- p.67Appendix C --- p.69Appendix D --- p.71References --- p.7

Wavelet representation of contour sets

Journal ArticleWe present a new wavelet compression and multiresolution modeling approach for sets of contours (level sets). In contrast to previous wavelet schemes, our algorithm creates a parametrization of a scalar field induced by its contours and compactly stores this parametrization rather than function values sampled on a regular grid. Our representation is based on hierarchical polygon meshes with subdivision connectivity whose vertices are transformed into wavelet coefficients. From this sparse set of coefficients, every set of contours can be efficiently reconstructed at multiple levels of resolution. When applying lossy compression, introducing high quantization errors, our method preserves contour topology, in contrast to compression methods applied to the corresponding field function. We provide numerical results for scalar fields defined on planar domains. Our approach generalizes to volumetric domains, time-varying contours, and level sets of vector fields

Wavelet-based multiresolution data representations for scalable distributed GIS services

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2002.Includes bibliographical references (p. 155-160).Demand for providing scalable distributed GIS services has been growing greatly as the Internet continues to boom. However, currently available data representations for these services are limited by a deficiency of scalability in data formats. In this research, four types of multiresolution data representations based on wavelet theories have been put forward. The designed Wavelet Image (WImg) data format helps us to achieve dynamic zooming and panning of compressed image maps in a prototype GIS viewer. The Wavelet Digital Elevation Model (WDEM) format is developed to deal with cell-based surface data. A WDEM is better than a raster pyramid in that a WDEM provides a non-redundant multiresolution representation. The Wavelet Arc (WArc) format is developed for decomposing curves into a multiresolution format through the lifting scheme. The Wavelet Triangulated Irregular Network (WTIN) format is developed to process general terrain surfaces based on the second generation wavelet theory. By designing a strategy to resample a terrain surface at subdivision points through the modified Butterfly scheme, we achieve the result: only one wavelet coefficient needs to be stored for each point in the final representation. In contrast to this result, three wavelet coefficients need to be stored for each point in a general 3D object wavelet-based representation. Our scheme is an interpolation scheme and has much better performance than the Hat wavelet filter on a surface. Boundary filters are designed to make the representation consistent with the rectangular boundary constraint.(cont.) We use a multi-linked list and a quadtree array as the data structures for computing. A method to convert a high resolution DEM to a WTIN is also provided. These four wavelet-based representations provide consistent and efficient multiresolution formats for online GIS. This makes scalable distributed GIS services more efficient and implementable.by Jingsong Wu.Ph.D

Operator-adapted wavelets for finite-element differential forms

We introduce in this paper an operator-adapted multiresolution analysis for finite-element differential forms. From a given continuous, linear, bijective, and self-adjoint positive-definite operator L, a hierarchy of basis functions and associated wavelets for discrete differential forms is constructed in a fine-to-coarse fashion and in quasilinear time. The resulting wavelets are L-orthogonal across all scales, and can be used to derive a Galerkin discretization of the operator such that its stiffness matrix becomes block-diagonal, with uniformly well-conditioned and sparse blocks. Because our approach applies to arbitrary differential p-forms, we can derive both scalar-valued and vector-valued wavelets block-diagonalizing a prescribed operator. We also discuss the generality of the construction by pointing out that it applies to various types of computational grids, offers arbitrary smoothness orders of basis functions and wavelets, and can accommodate linear differential constraints such as divergence-freeness. Finally, we demonstrate the benefits of the corresponding operator-adapted multiresolution decomposition for coarse-graining and model reduction of linear and non-linear partial differential equations

Operator-adapted wavelets for finite-element differential forms

We introduce in this paper an operator-adapted multiresolution analysis for finite-element differential forms. From a given continuous, linear, bijective, and self-adjoint positive-definite operator L, a hierarchy of basis functions and associated wavelets for discrete differential forms is constructed in a fine-to-coarse fashion and in quasilinear time. The resulting wavelets are L-orthogonal across all scales, and can be used to derive a Galerkin discretization of the operator such that its stiffness matrix becomes block-diagonal, with uniformly well-conditioned and sparse blocks. Because our approach applies to arbitrary differential p-forms, we can derive both scalar-valued and vector-valued wavelets block-diagonalizing a prescribed operator. We also discuss the generality of the construction by pointing out that it applies to various types of computational grids, offers arbitrary smoothness orders of basis functions and wavelets, and can accommodate linear differential constraints such as divergence-freeness. Finally, we demonstrate the benefits of the corresponding operator-adapted multiresolution decomposition for coarse-graining and model reduction of linear and non-linear partial differential equations

Compression of 3D models with NURBS

With recent progress in computing, algorithmics and telecommunications, 3D models are increasingly used in various multimedia applications. Examples include visualization, gaming, entertainment and virtual reality. In the multimedia domain 3D models have been traditionally represented as polygonal meshes. This piecewise planar representation can be thought of as the analogy of bitmap images for 3D surfaces. As bitmap images, they enjoy great flexibility and are particularly well suited to describing information captured from the real world, through, for instance, scanning processes. They suffer, however, from the same shortcomings, namely limited resolution and large storage size. The compression of polygonal meshes has been a very active field of research in the last decade and rather efficient compression algorithms have been proposed in the literature that greatly mitigate the high storage costs. However, such a low level description of a 3D shape has a bounded performance. More efficient compression should be reachable through the use of higher level primitives. This idea has been explored to a great extent in the context of model based coding of visual information. In such an approach, when compressing the visual information a higher level representation (e.g., 3D model of a talking head) is obtained through analysis methods. This can be seen as an inverse projection problem. Once this task is fullled, the resulting parameters of the model are coded instead of the original information. It is believed that if the analysis module is efficient enough, the total cost of coding (in a rate distortion sense) will be greatly reduced. The relatively poor performance and high complexity of currently available analysis methods (except for specific cases where a priori knowledge about the nature of the objects is available), has refrained a large deployment of coding techniques based on such an approach. Progress in computer graphics has however changed this situation. In fact, nowadays, an increasing number of pictures, video and 3D content are generated by synthesis processing rather than coming from a capture device such as a camera or a scanner. This means that the underlying model in the synthesis stage can be used for their efficient coding without the need for a complex analysis module. In other words it would be a mistake to attempt to compress a low level description (e.g., a polygonal mesh) when a higher level one is available from the synthesis process (e.g., a parametric surface). This is, however, what is usually done in the multimedia domain, where higher level 3D model descriptions are converted to polygonal meshes, if anything by the lack of standard coded formats for the former. On a parallel but related path, the way we consume audio-visual information is changing. As opposed to recent past and a large part of today's applications, interactivity is becoming a key element in the way we consume information. In the context of interest in this dissertation, this means that when coding visual information (an image or a video for instance), previously obvious considerations such as decision on sampling parameters are not so obvious anymore. In fact, as in an interactive environment the effective display resolution can be controlled by the user through zooming, there is no clear optimal setting for the sampling period. This means that because of interactivity, the representation used to code the scene should allow the display of objects in a variety of resolutions, and ideally up to infinity. One way to resolve this problem would be by extensive over-sampling. But this approach is unrealistic and too expensive to implement in many situations. The alternative would be to use a resolution independent representation. In the realm of 3D modeling, such representations are usually available when the models are created by an artist on a computer. The scope of this dissertation is precisely the compression of 3D models in higher level forms. The direct coding in such a form should yield improved rate-distortion performance while providing a large degree of resolution independence. There has not been, so far, any major attempt to efficiently compress these representations, such as parametric surfaces. This thesis proposes a solution to overcome this gap. A variety of higher level 3D representations exist, of which parametric surfaces are a popular choice among designers. Within parametric surfaces, Non-Uniform Rational B-Splines (NURBS) enjoy great popularity as a wide range of NURBS based modeling tools are readily available. Recently, NURBS has been included in the Virtual Reality Modeling Language (VRML) and its next generation descendant eXtensible 3D (X3D). The nice properties of NURBS and their widespread use has lead us to choose them as the form we use for the coded representation. The primary goal of this dissertation is the definition of a system for coding 3D NURBS models with guaranteed distortion. The basis of the system is entropy coded differential pulse coded modulation (DPCM). In the case of NURBS, guaranteeing the distortion is not trivial, as some of its parameters (e.g., knots) have a complicated influence on the overall surface distortion. To this end, a detailed distortion analysis is performed. In particular, previously unknown relations between the distortion of knots and the resulting surface distortion are demonstrated. Compression efficiency is pursued at every stage and simple yet efficient entropy coder realizations are defined. The special case of degenerate and closed surfaces with duplicate control points is addressed and an efficient yet simple coding is proposed to compress the duplicate relationships. Encoder aspects are also analyzed. Optimal predictors are found that perform well across a wide class of models. Simplification techniques are also considered for improved compression efficiency at negligible distortion cost. Transmission over error prone channels is also considered and an error resilient extension defined. The data stream is partitioned by independently coding small groups of surfaces and inserting the necessary resynchronization markers. Simple strategies for achieving the desired level of protection are proposed. The same extension also serves the purpose of random access and on-the-fly reordering of the data stream