Point-Normal Subdivision Curves and Surfaces

This paper proposes to generalize linear subdivision schemes to nonlinear subdivision schemes for curve and surface modeling by refining vertex positions together with refinement of unit control normals at the vertices. For each round of subdivision, new control normals are obtained by projections of linearly subdivided normals onto unit circle or sphere while new vertex positions are obtained by updating linearly subdivided vertices along the directions of the newly subdivided normals. Particularly, the new position of each linearly subdivided vertex is computed by weighted averages of end points of circular or helical arcs that interpolate the positions and normals at the old vertices at one ends and the newly subdivided normal at the other ends. The main features of the proposed subdivision schemes are three folds: (1) The point-normal (PN) subdivision schemes can reproduce circles, circular cylinders and spheres using control points and control normals; (2) PN subdivision schemes generalized from convergent linear subdivision schemes converge and can have the same smoothness orders as the linear schemes; (3) PN $C^2$ subdivision schemes generalizing linear subdivision schemes that generate $C^2$ subdivision surfaces with flat extraordinary points can generate visually $C^2$ subdivision surfaces with non-flat extraordinary points. Experimental examples have been given to show the effectiveness of the proposed techniques for curve and surface modeling.Comment: 30 pages, 17 figures, 22.5M

Multiresolution editing for B-spline curves and surfaces

Since 1980 surface modeling has been used in industrial design, CAD and entertainment to create and represent complex forms. Even with this comparatively long history of development, challenges remain in free-form surface modeling. One such challenge is building surface creation and editing techniques that effectively balance the need for local control with the need to control the overall global shape, or sweep of the surface. This dissertation presents a multiresolution approach to the creation of surfaces that allows a designer to more easily manage this balance between local and global control. The techniques presented in this dissertation utilize a wavelet decomposition of B-spline curves and surfaces to allow a designer to easily develop the basic shape using lower level representations, and then seamlessly switch to higher level representations to achieve fine control over local features. The algorithms described in the dissertation are implemented in an interactive software system that is used to demonstrate their effectiveness in comparison to existing methods

Non-linear subdivision of univariate signals and discrete surfaces

During the last 20 years, the joint expansion of computing power, computer graphics, networking capabilities and multiresolution analysis have stimulated several research domains, and developed the need for new types of data such as 3D models, i.e. discrete surfaces. In the intersection between multiresolution analysis and computer graphics, subdivision methods, i.e. iterative refinement procedures of curves or surfaces, have a non-negligible place, since they are a basic component needed to adapt existing multiresolution techniques dedicated to signals and images to more complicated data such as discrete surfaces represented by polygonal meshes. Such representations are of great interest since they make polygonal meshes nearly as exible as higher level 3D model representations, such as piecewise polynomial based surfaces (e.g. NURBS, B-splines...). The generalization of subdivision methods from univariate data to polygonal meshes is relatively simple in case of a regular mesh but becomes less straightforward when handling irregularities. Moreover, in the linear univariate case, obtaining a smoother limit curve is achieved by increasing the size of the support of the subdivision scheme, which is not a trivial operation in the case of a surface subdivision scheme without a priori assumptions on the mesh. While many linear subdivision methods are available, the studies concerning more general non-linear methods are relatively sparse, whereas such techniques could be used to achieve better results without increasing the size support. The goal of this study is to propose and to analyze a binary non-linear interpolatory subdivision method. The proposed technique uses local polar coordinates to compute the positions of the newly inserted points. It is shown that the method converges toward continuous limit functions. The proposed univariate scheme is extended to triangular meshes, possibly with boundaries. In order to evaluate characteristics of the proposed scheme which are not proved analytically, numerical estimates to study convergence, regularity of the limit function and approximation order are studied and validated using known linear schemes of identical support. The convergence criterion is adapted to surface subdivision via a Hausdorff distance-based metric. The evolution of Gaussian and mean curvature of limit surfaces is also studied and compared against theoretical values when available. An application of surface subdivision to build a multiresolution representation of 3D models is also studied. In particular, the efficiency of such a representation for compression and in terms of rate-distortion of such a representation is shown. An alternate to the initial SPIHT-based encoding, based on the JPEG 2000 image compression standard method. This method makes possible partial decoding of the compressed model in both SNR-progressive and level-progressive ways, while adding only a minimal overhead when compared to SPIHT

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