Error analysis for local coarsening in univariate spline spaces

In this article we analyze the error produced by the removal of an arbitrary knot from a spline function. When a knot has multiplicity greater than one, this implies a reduction of its multiplicity by one unit. In particular, we deduce a very simple formula to compute the error in terms of some neighboring knots and a few control points of the considered spline. Furthermore, we show precisely how this error is related to the jump of a derivative of the spline at the knot. We then use the developed theory to propose efficient and very low-cost local error indicators and adaptive coarsening algorithms. Finally, we present some numerical experiments to illustrate their performance and show some applications

Evalutionary algorithms for ship hull skinning approximation

Traditionally, the design process of a hull involves simulation using clay models. This must be done cautiously, accurately and efficiently in order to sustain the performance of ship. Presently, the current technology of Computer Aided Design, Manufacturing, Engineering and Computational Fluid Dynamic has enabled a 3D design and simulation of a hull be done at a lower cost and within a shorter period of time. Besides that, automated design tools allow the transformation of offset data in designing the hull be done automatically. One of the most common methods in constructing a hull from the offset data is the skinning method. Generally, the skinning method comprised of skinning interpolation and skinning approximation. Skinning interpolation constructs the surface perfectly but improper selection of parameterization methods may cause bumps, wiggles, or uneven surfaces on the generated surface. On the other hand, using the skinning surface approximation would mean that the surface can only be constructed closer to data points. Thus, the error between the generated surface and the data points must be minimized to increase the accuracy. Therefore, this study aims to solve the error minimization problem in order to produce a smoother and fairer surface by proposing Non Uniform Rational B-Spline surface using various evolutionary optimization algorithms, namely, Gravitational Search Algorithm, Particle Swarm Optimization and Genetic Algorithm. The proposed methods involve four procedures: extraction of offset data from line drawing plan; generation of control points; optimization of a surface; and validations of hull surfaces. Validation is done by analyzing the surface curvature and errors between the generated surface and the given data points. The experiments were implemented on both ship hull and free form models. The findings from the experiments are compared with interpolated skinning surface and conventional skinning surface approximation. The results show that the optimized skinning surfaces using the proposed methods yield a smaller error, less control points generation and feasible surfaces while maintaining the shape of the hull

A Universal Parametrization in B-Spline Curve and Surface Interpolation and Its Performance Evaluation.

The choice of a proper parametrization method is critical in curve and surface fitting using parametric B-splines. Conventional parametrization methods do not work well partly because they are based only on the geometric properties of given data points such as the distances between consecutive data points and the angles between consecutive line segments. The resulting interpolation curves don\u27t look natural and they are often not affine invariant. The conventional parametrization methods don\u27t work well for odd orders k. If a data point is altered, the effect is not limited locally at all with these methods. The localness property with respect to data points is critical in interactive modeling. We present a new parametrization based on the nature of the basis functions called B-splines. It assigns to each data point the parameter value at which the corresponding B-spline N\sb{ik}(t) is maximum. The new method overcomes all four problems mentioned above; (1) It works well for all orders k, (2) it generates affine invariant curves, (3) the resulting curves look more natural, in general, and (4) it has the semi-localness property with respect to data points. The new method is also computationally more efficient and the resulting curve has more regular behavior of the curvature. Fairness evaluation and knot removal are performed on curves obtained from various parametrizations. The results also show that the new parametrization is superior. Fairness is evaluated in terms of total curvature, total length, and curvature plot. The curvature plots are looking natural for the curves obtained from the new parametrization. For the curves obtained from the new method, knot removal is able to provide with the curves which are very close to the original curves. A more efficient and effective method is also presented for knot removal in B-spline curve. A global norm is utilized for approximation unlike other methods which are using some local norms. A geometrical view makes the computation more efficient

Optimale Box-Einschließungen von NURBS Kurven

In dieser Arbeit werden die Splines in den ersten beiden Kapiteln schrittweise eingeführt. Im Mittelpunkt der Aufmerksamkeit liegen aber sogenannte NURBS-Kurven. Nach einer Kurzen Einführung in die Intervallanalysis, wird schließlich eine Methode entwickelt um eine Box um beliebige NURBS Kurven legen zu können. Im Anschluss sind noch die Implementationen angeführt

Analysis and design of developable surfaces for shipbuilding

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Ocean Engineering, 1997, and Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 1997.Includes bibliographical references (p. 97-99).by Julie Steele Chalfant.M.S

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