Georg Umlauf's Publications
               


        
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2010


Adaptive and freature-preserving subdivision for h
igh-quality tetrahedral meshes.
Burkhart, Hamann, Umlauf
To appear: Computer Graphics Forum, 2010.The tetrahedral subdivision scheme

We present an adaptive subdivision scheme for unstructured tetrahedral meshes inspired by the sqrt(3)-subdivision scheme for triangular meshes. Existing  tetrahedral subdivision schemes do not support adaptive refinement and have traditionally been driven by the need to generate smooth three-dimensional deformations of solids. These schemes use edge bisections to subdivide tetrahedra, which generates octahedra in addition to tetrahedra. To split octahedra into tetrahedra one routinely chooses a direction for the diagonals for the subdivision step. We propose a new topology-based refinement operator that generates only tetrahedra and supports adaptive refinement. Our tetrahedral subdivision algorithm is motivated by the need to have one representation for the modeling, the  simulation and the visualization and so to bridge the gap between CAD and CAE. Our subdivision algorithm design emphasizes on geometric quality of the tetrahedral meshes, local and adaptive refinement operations, and preservation of sharp geometric features on the boundary and in the interior of the physical domain.

Available formats: pdf
Citation: BibTeX





Real-time triangulation of point streams.

Denker, Lehner, Umlauf
To appear: Engineering with Computers, 2010.

Hand-held laser scanners are commonly used in industry for reverse engineering and quality measurements. In this process, it is difficult for the human operator to scan the target object completely and uniformly. Therefore, an interactive triangulation of the scanned points can assist the operator in this task.

In this paper we describe the technical and implementational details of our real-time triangulation approach for point streams, presented at the 17th International Meshing Roundtable. Our method computes a triangulation of the point stream generated by the laser scanner online, i.e., the data points are added to the triangulation as they are received from the scanner. Multiple scanned areas and areas with a higher point density result in a finer mesh and a higher accuracy. On the other hand, the vertex density adapts to the estimated surface curvature. To guide the operator the resulting triangulation is rendered with a visualization of its uncertainty and the display of an optimal scanning direction.

Available formats: pdf
Citation: BibTeX






Geometric properties of the adaptive Delauney triangulation.
Bobach, Constantiniu, Steinmann, Umlauf
To appear: Mathematical Methods for Curves and Surfaces, 2010.

Recently, the Adaptive Delaunay Tessellation (ADT) was introduced in the context of computational mechanics as a tool to support Voronoi-based nodal integration schemes in the finite element method. While focusing on applications in mechanical engineering, the former presentation lacked rigorous proofs for the claimed geometric properties of the Adt necessary for the computation of the nodal integration scheme. This paper gives the missing proofs for the three main claims which are uniqueness of the ADT, connectedness of the ADT, and coverage of the Voronoi tiles by adjacent ADT tiles. Furthermore, this paper provides a critical assessment of the ADT for arbitrary point sets.

Available formats: pdf
Citation: BibTeX






Generalized swap operation for tetrahedrizations.
Lehner, Hamann, Umlauf
To appear in: H.Hagen, A.Knoll (eds.): Proceedings of NN, 2010.

Mesh optimization of 2D and 3D triangulations is used in multiple applications extensively. For example, mesh optimization is crucial in the context of adaptively discretizing geometry, typically representing the geometrical boundary conditions of a numerical simulation, or adaptively discretizing the entire space over which various dependent variables of a numerical simulation must be approximated. Together with operations applied to the vertices the so-called edge or face swap operations are the building block of all optimization approaches. To speed up the optimization or to avoid local minima of the function measuring overall mesh quality these swaps are combined to generalized swap operations with a less local impact on the triangulation. Despite the fact that these swap operations change only the connectivity of a triangulation, it depends on the geometry of the triangulation whether the generalized swap will generate inconsistently oriented or degenerate simplices. Because these are undesirable for numerical reasons, this paper is concerned with geometric criteria that guarantee the generalized swaps for a 3D triangulation to yield only valid, non-degenerate triangulations.

Available formats: pdf
Citation: BibTeX


2009


Natural neighbor extrapolation using ghost points.

Bobach, Farin, Hansford, Umlauf
Computer Aided-Design, 41(5): 350-365, 2009.

Among locally supported scattered data schemes, natural neighbor interpolation has some unique features that makes it interesting for a range of applications. However, its restriction to the convex hull of the data sites is a limitation that has not yet been satisfyingly overcome. We use this setting to discuss some aspects of scattered data extrapolation in general, compare existing methods, and propose a framework for the extrapolation of natural neighbor interpolants on the basis of dynamic ghost points.

Available formats: pdf
Citation: BibTeX
2008


Online triangulation of laser-scan data.
Denker, Lehner, Umlauf
In: R. Garimella (ed.), Proceedings of the 17th International Meshing Roundtable, Springer-Verlag, 415-432, 2008.

Hand-held laser scanners are used massively in industry for reverse engineering and quality measurements. In this process, it is difficult for the human operator to cover the scanned object completely and uniformly. Therefore, an interactive triangulation of the scanned surface points can assist the human operator in this task.

Our method computes a triangulation of the point stream generated by the laser scanner online, i.e., the data points are added to the triangulation as they are received from the scanner. Multiple scanned areas and areas with a higher point density result in a finer mesh and a higher accuracy. On the other hand, the vertex density adapts to the estimated surface curvature. To assist the human operator the resulting triangulation is rendered with a visualization of its faithfulness. Additionally, our triangulation method allows for a level-of-detail representation to reduce the mesh complexity for fast rendering on low-cost graphics hardware.


Available formats: pdf
Citation: BibTeX
Videos: avi





Video compression using data-dependent triangulations.

Lehner, Umlauf, Hamann
In: Y. Xiao, E. ten Thij (eds.), Computer Graphics and Visualization '08, 244-248, 2008
.

Best short-paper award.

We present a method for compression of video clips using data-dependent triangulations. This technique utilizes the time coherence of a video to transfer information from one frame to the next, reducing the computation time for the compression. The results of this method are compared to those obtained with MJPEG and MPEG-2.



Available formats:
pdf
Citation: BibTeX
Videos: avi





The adaptive Delaunay tessellation: A neighborhood covering meshing technique.
Constantiniu, Steinmann, Bobach, Farin, Umlauf
Computational Mechanics, 42(5), 655-669, 2008.


In this paper we propose an unstructured hybrid tessellation of a scattered point set that minimally covers the proximal space around each point. The mesh is automatically obtained in a bounded period of time by transforming an initial Delaunay tessellation. Novel types of polygonal interpolants are used for interpolation applications and geometric quantities of the elements make them also useful for discretization schemes. The approach proves to be superior to classical Delaunay one in a finite element context.

Available formats:
pdf
Citation: BibTeX



 

Local energy-optimizing subdivision algorithms.
Ginkel, Umlauf
Computer Aided Geometric Design, 25(3): 137-147, 2008.


In this paper a method is presented to fair the limit surface of a subdivision algorithm locally around an extraordinary point. The dominant six eigenvalues of the subdivision matrix have to satisfy linear and quadratic equality- and inequality-constraints in order to guarantee normal-continuity and bounded curvature at the extraordinary point. All other eigenvalues can be chosen arbitrarily within certain intervals and therefore can be used to optimize the shape of the subdivision surface by minimizing quadratic energy functionals. Additionally, if the sub- and subsub-dominant eigenvalues vary within predefined intervals, C1-regularity of the surface and locality of the stencils can be guaranteed, although eigenvectors are changed.

Available formats:
pdf
Citation: BibTeX






Symmetry of shape charts.
Ginkel, Umlauf
Computer Aided Geometric Design, 24(3): 131-136, 2008.

For subdivision surfaces, the so-called shape chart can be used to characterize the curvature behavior at an extraordinary point a priori from the initial control net. Of late, it has been used in different approaches to tune subdivision algorithms to handle the so-called hybrid shapes. For this the shape charts are computed numerically. In this paper, symmetries of shape charts are analyzed that can be used to simplify the computations and to reduce the computation costs significantly.

Available formats:
pdf
Citation: BibTeX

2007


Discrete harmonic functions from local coordinates.

Bobach, Farin, Hansford, Umlauf
In: R. Martin, M. Sabin, J. Winkler (eds.), Mathematics of Surfaces XII, Springer-Verlag, 93-103, 2007.

In this work we focus on approximations of continuous harmonic functions by discrete harmonic functions based on the discrete Laplacian in a triangulation of a point set. We show how the choice of edge weights based on generalized barycentric coordinates influences the approximation quality of discrete harmonic functions. Furthermore, we consider a varying point set to demonstrate that generalized barycentric coordinates based on natural neighbors admit discrete harmonic functions that continuously depend on the point set.

Available formats: pdf
Citation: BibTeX





Natural neighbor concepts in scattered data interpolation and discrete function approximation.

Bobach, Umlauf
In: H. Hagen, M. Hering-Bertram, C. Garth (eds.), GI Lecture Notes in Informatics, Visualization of Large and Unstructured Data Sets, 23-35, 2007.

The concept of natural neighbors employs the notion of distance to define local neighborhoods in discrete data. Especially when querying and accessing large scale data, it is important to limit the amount of data that has to be processed for an answer. Because of its implicit definition on distances, the natural neighbor concept is extremely well suited to provide meaningful neighborhoods in spatial data with a scattered, inhomogeneous distribution.

This paper revisits some unique properties of natural neighbor based methods and summarizes important findings for their successful application to scattered data interpolation, and the computation of discrete harmonic functions.


Available formats: pdf
Citation: BibTeX





Analyzing a generalized Loop subdivision scheme.

Ginkel, Umlauf
Computing, 79(2-4), 353-363, 2007.

In this paper a class subdivision schemes generalizing the algorithm of Loop is presented. The stencils have same support as those from the algorithm of Loop, but allow a variety of weights. By varying the weights a class of C1 regular subdivision schemes is obtained. This class includes the algorithm of Loop and the midpoint schemes of order one and two for triangular nets. The proof of C1 regularity of the limit surface for arbitrary triangular nets is provided for any choice of feasible weights.

The purpose of this generalization of the subdivision algorithm of Loop is to demonstrate the capabilities of the applied analysis technique. Since this class includes schemes that do not generalize box spline subdivision, the analysis of the characteristic map is done with a technique that does not need an explicit piecewise polynomial representation. This technique is computationally simple and can be used to analyze classes of subdivision schemes. It extends previously presented techniques based on geometric criteria.

Available formats:
pdf
Citation: BibTeX





Tuning subdivision algorithms using constrained energy minimization.
Ginkel, Umlauf
In: R. Martin, M. Sabin, J. Winkler (eds.), Mathematics of Surfaces XII, Springer-Verlag, 166-176, 2007.

In this paper a method is presented to fair the limit surface of a subdivision algorithm around an extraordinary point. The eigenvalues and eigenvectors of the subdivision matrix determine the continuity and shape of the limit surface. The dominant, sub-dominant and subsub-dominant eigenvalues should satisfy linear and quadratic equality- and inequality-constraints to guarantee continuous normal and bounded curvature globally. The remaining eigenvalues need only satisfy linear inequality-constraints. In general, except for the dominant eigenvalue, all eigenvalues can be used to optimize the shape of the limit surface with our method.

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pdf
Citation: BibTeX





Normals of subdivision surfaces and their control polyhedra.

Ginkel, Peters, Umlauf
Computer Aided Geometric Design, 24(2): 112-116, 2007.

For planar spline curves and bivariate box-spline functions, the cone of normals of a polynomial spline piece is enclosed by the cone of normals of its spline control polyhedron. This note collects some concrete examples to show that this is not true for subdivision surfaces, both at extraordinary points and in the regular, box-spline setting. A larger set, the cross products of families of control net edges, has to be considered.

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pdf
Citation: BibTeX





Image compression using data-dependent triangulations.
Lehner, Umlauf, Hamann
In:
G. Bebis et al. (eds.), Advances in Visual Computing, Part I, Springer-Verlag, 352-362, 2007.

We present a method to speed up the computation of a high-quality data-dependent triangulation approximating an image using simulated annealing by probability distributions guided by local approximation error and its variance. The triangulation encodes the image, yielding compression rates comparable to or even superior to JPEG and JPEG2000 compression.

The specific contributions of our paper are a speed-up of the simulated annealing optimization and a comparison of our approach to other image approximation and compression methods. Furthermore, we propose an adaptive vertex insertion/removal strategy and termination criteria for the simulated annealing to achieve specified approximation error bounds.

Available formats: pdf
Citation: BibTeX





Survey of techniques for data-dependent triangulations.

Lehner, Umlauf, Hamann
In: H. Hagen
, M. Hering-Bertram, C. Garth (eds.), GI Lecture Notes in Informatics, Visualization of Large and Unstructured Data Sets, 178-187, 2007.

We present a survey of different techniques to approximate a color image using a piecewise linear interpolation induced by a triangulation of the image domain. We also include a detailed description of a method we designed. We give a short overview of possible applications and extensions.

Available formats: pdf
Citation: BibTeX





SoPC for 3D point rendering.
Middendorf, Mühlbauer, Bobda, Umlauf
To appear: DASIP, 2007.


Embedded vertex shader in FPGA.
Middendorf, Mühlbauer, Umlauf, Bobda
In: A. Rettberg et al. (eds.), Embedded System Design: Topics, Techniques and Trendes, Springer, 155-164, 2007.



Real-time 3D visualization of objects or information becomes increasingly important in everyday life e.g. in cellular phones or mobile systems. Care should be taken in the design and implementation of 3D rendering in such embedded devices like handhelds devices in order to meet the performance requirement, while maintaining power consumption low. In this work, the design and implementation of a vertex shader on a reconfigurable hardware is presented. The main focus is placed on the efficient hardware/software partitioning of the vertex shader computation, in order to maximize the performance while maintaining a high flexibility. The resulting solution must be compatible to existing vertex shaders in oder to allow the large amount of existing program to be easily ported to our platform. A prototype consisting of a PowerPC, peripherals and some custom hardware modules is realized a on an FPGA-board. The implementation of a point rendering shows considerable speed up compared to a pure software solution.

Available formats: pdf
Citation: BibTeX, BibTeX

2006


Comparison of Voronoi based scattered data interpolation schemes.
Bobach, Bertram, Umlauf
In: J.J. Villanueva (ed.), Proceedings of
International Conference on Visualization, Imaging and Image Processing, 342-349, 2006.


Voronoi based interpolation employs the concept of natural neighbors to define an interpolating function over discrete data known at scattered sample points. In this work we review the two main concepts for improving interpolation continuity inside the convex hull of the sample domain and compare four natural neighbor interpolants of C1 and C2 continuity. We give a visual presentation of all interpolants to provide insight into their overall behavior in addition to a comparison of their analytical and practical properties.

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pdf
Citation: BibTeX





Issues and implementation of C1 and C2 natural neighbor interpolation.
Bobach, Bertram, Umlauf
In: G. Bebis et al. (eds.), Advances in Visual Computing, Part II, Springer-Verlag, 186-195, 2006.

Smooth local coordinates have been proposed by Hiyoshi and Sugihara 2000 to improve the classical Sibson’s and Laplace coordinates. These smooth local coordinates are computed by integrating geometric quantities over weights in the power diagram. In this paper we describe how to efficiently& implement the Voronoi based C2 local coordinates. The globally C2 interpolant that Hiyoshi and Sugihara presented in 2004 is then compared to Sibson’s and Farin’s C1 interpolants when applied toscattered data interpolation.

Available formats: pdf
Citation: BibTeX





Natural neighbor interpolation and order of continuity.
Bobach, Umlauf
In: H. Hagen, A. Kerren, P. Dannenmann (eds.), GI Lecture Notes in Informatics, Visualization of Large and Unstructured Data Sets, 69-86, 2006.

In this paper we give a survey on natural neighbor based interpolation, a class of local scattered data interpolation schemes that define their support on natural neighbors in the Voronoi diagram of the input data sites. We discuss the existing work with respect to common aspects of scattered data interpolation and focus on smoothness of the interpolant.

Available formats: pdf
Citation: BibTeX






Controlling a subdivision tuning method.
Ginkel, Umlauf
In: A. Cohen, J.-L. Merrien, L.L. Schumaker (eds.), Curve and Surface Fitting, 170-179, 2006.

In this paper the problem of curvature behavior around extraordinary points of a Loop subdivision surface is addressed. For subdivision surfaces, configurations of the initial control net exist in which neither the elliptic nor the hyperbolic component of the initial control net becomes dominant. This leads to so-called hybrid shapes which often cause visible artifacts. A solution to this problem is to split it into two parts: first an eigenvalue tuning to allow for bounded Gauss curvature with arbitrary sign and, second, an eigencoefficient tuning to avoid hybrid shapes. The techniques for eigencoefficient tuning will now be analyzed in detail. The analysis allows to quantify the difference between the original and the modified surfaces. Additionally, extensions to the eigencoefficient tuning techniques are given to solve various problems that might be imposed by the topology of the initial control net.

Available formats: pdf
Citation: BibTeX






Loop subdivision with curvature control.
Ginkel, Umlauf
In: K. Polthier, A. Sheffer (eds.), Eurographics Symposium on Geometry Processing, Eurographics Association, 163-171, 2006.

In this paper the problem of curvature behavior around extraordinary points of a Loop subdivision surface is addressed. A variant of Loop’s algorithm with small stencils is used that generates surfaces with bounded curvature and prescribed elliptic or hyperbolic behavior. We present two different techniques that avoid the occurrence of hybrid configurations, so that an elliptic or hyperbolic shape can be guaranteed. The first technique uses a symmetric modification of the initial control-net to avoid hybrid shapes in the vicinity of an extraordinary point. To keep the difference between the original and the modified mesh as small as possible the changes are formulated as correction stencils and spread to a finite number of subdivision steps. The second technique is based on local optimization in the frequency domain. It provides more degrees of freedom and so more control over the global shape.

Available formats: pdf
Citation: BibTeX





Topographic distance functions for interpolation of meteorological data.
Lehner, Umlauf, Hamann, Ustin
In: H. Hagen, A. Kerren, P. Dannenmann (eds.), GI Lecture Notes in Informatics, Visualization of Large and Unstructured Data Sets, 119-131, 2006.

The reference evapotranspiration ET0 is an important meteorological quantity in agriculture and water resource management. It is usually estimated from other meteorological quantities measured at weather stations. To estimate ET0 at an arbitrary geographical position these quantities must be interpolated. The Center for Spatial Technologies And Remote Sensing (CSTARS) at UC Davis uses the DayMet approach for this task.

We discuss some inconsistencies within the DayMet approach and suggest improvements. One significant problem of DayMet is the lack of consideration of terrain topography. We define new distance functions that are elevation-dependent and show preliminary results of the comparison of the classic and the improved DayMet approach.

Available formats: pdf
Citation: BibTeX





Parametrizations for triangular
Gk spline surfaces of low degree.
Prautzsch, Umlauf
ACM Transactions on Graphics, 25(4), 1281-1293, 2006.

In this paper, we present regularly parametrized Gk free form spline surfaces that extend box and half-box splines over regular triangular grids. The polynomial degree of these splines is max{4k+1; floor(3k/2 +1)}, where r \in N can be chosen arbitrarily and determines the flexibility at extraordinary points. The Gk splines presented in this paper depend crucially on low-degree (re-)parametrizations of piecewise polynomial hole fillings. The explicit construction of such parametrizations forms the core of this paper and we present two classes of singular and regular parametrizations. Also we show how to build box and half-box spline surfaces of arbitrarily high smoothness with holes bounded by only n patches in principle.

Available formats: pdf
Citation: BibTeX

2005


On normals and control nets.
Ginkel, Peters, Umlauf
In: R. Martin, H. Bez, M. Sabin (eds.): The Mathematics of Surfaces XI, Springer-Verlag, 2005, 233-239.

This paper characterizes when the normals of a spline curve or spline surface lie in the more easily computed cone of the normals of the segments of the spline control net.

Available formats: pdf
Citation: BibTeX






A
nalysis and tuning of subdivision schemes.
Umlauf
In: B. Jüttler (ed.),  Proceedings of Spring Conference on Computer Graphics SCCG 2005, ACM Press, 33-40, 2005.

This paper surveys the current state in analyzing and tuning of subdivision algorithms. These two aspects of subdivision algorithms are very much intertwined with the differential geometry of the subdivision surface. This paper deals with the interconnection of these different aspects of subdivision algorithms and surfaces. The principal idea for the analysis of a subdivision algorithm dates back to the late 70s although the overall technique is only well understood since the early 90s. Most subdivision algorithms are analyzed today but the proofs involve time consuming computations. Only recently simple proofs for a certain class of subdivision algorithms were developed that are based on geometric reasoning. This allows for easier smoothness proofs for new developed or tuned subdivision algorithms.

The analysis of the classical algorithms, such Catmull-Clark, Loop, etc., shows that the subdivision surfaces at the extraordinary points are not as smooth as the rest of the surface. It was also shown that the subdivision surfaces of these classical algorithms cannot model certain basic shapes. One way to tune a stationary subdivision algorithms to overcome this problem is to drop the stationarity while at the same time using the smoothness proof of the stationary algorithms.


Available formats: pdf
Citation: BibTeX

2004


Post-processing polygonal voxel data from numerical simulation.
Bobach, Umlauf
Interner Bericht 332/04, Fachbereich Informatik, TU Kaiserslautern, 2004.

Many applications dealing with geometry acquisition and processing produce polygonal meshes that carry artifacts like discretization noise. While there are many approaches to remove the artifacts by smoothing or filtering the mesh, they are not tailored to any specific application demanding certain restrictive constraints. We show how to incorporate smoothing schemes based on the general Laplacian approximation to process the results of flow simulation in car manufacturing.

In the presented application setting the major restrictions come from the bounding volume of the flow simulation, the so-called installation space. In particular, clean mesh regions (without noise) should not be smoothed while at the same time the installation space must not be violated by the smoothing of the noisy mesh regions. Additionally, aliasing effects at the boundary between clean and noisy mesh regions must be prevented. To address the fact that the meshes come from flow simulation, the presented method is versatile enough to preserve their exact volume and to apply anisotropic filters using the flow information.


Available formats: pdf
Citation: BibTeX





A technique for verifying the smoothness of subdivision schemes.

Umlauf
In:  M.L. Lucian, M. Neamtu (eds.), Geometric Modeling and Computing: Seattle 2003, Nashboro Press, 513-521, 2004.

In recent years, subdivision schemes for surfaces of arbitrary topology were developed that do not generalize box-splines subdivision schemes. Examples for this kind of subdivision schemes are the sqrt(3)-scheme and certain averaging schemes for triangular and hexahedral nets. In order to analyze the smoothness of the limit surface, it is necessary to know if its characteristic map is regular and injective. For box-spline based schemes this can be done based on the explicit piecewise polynomial representation. In this paper, a general approach is introduced that allows analyzing the characteristic map even if no explicit representation is available. The proposed technique requires that the first divided differences schemes are scalar and use convex combinations. Then simple geometric properties of the sub-dominant eigenvectors of the subdivision matrix can be used to prove regularity and injectivity of the characteristic map for any valence. This is demonstrated for a midpoint scheme for triangular nets.

Available formats: pdf
Citation: BibTeX

before 2003


Computing curvature bounds for bounded curvature subdivision.

Peters, Umlauf
Computer Aided Geometric Design, 18 (5): 455-462, 2001.

For a stationary, affine invariant, symmetric, linear and local subdivision scheme that is C2 apart from the extraordinary point the curvature is bounded if the square of the subdominant eigenvalue equals the subsubdominant eigenvalue. This paper gives a technique for quickly establishing an interval to which the curvature is confined at the extraordinary point. The approach factors the work into precomputed intervals that depend only on the scheme and a mesh-specific component. In many cases the intervals are tight enough to be used as a test of shape-faithfulness of the given subdivision scheme; i.e. if the limit surface in the neighborhood of the extraordinary point of the subdivision scheme is consistent with the geometry implied by the input mesh.

Available formats: pdf
Citation: BibTeX





Triangular
Gk splines.
Prautzsch, Umlauf
Interner Bericht Nr. 2001-8, Fakultät für Informatik, Universität Karlsruhe, 2001.

In this paper a new approach is presented to construct piecewise polynomial Gk-surfaces of arbitrary topology and smoothness order k>0 of degree O(k). This approach generalizes some results presented in 1997 in CAGD and in 1999 at the Saint-Malo conference, respectively. In our construction only 4n polynomial patches are needed to fill an n-sided hole in a generalized Ck-(half-)box spline surface. This is achieved by coalescing certain control points while at the same time maintaining a regular parametrization.





Gaussian and mean curvature of subdivision surfaces.

Peters, Umlauf
In: R. Cipolla, R. Martin (eds.), The Mathematics of Surfaces IX, Springer-Verlag, 59-69, 2000.

By explicitly deriving the curvature of subdivision surfaces in the extraordinary points, we give an alternative, more direct account of the criteria necessary and sufficient for achieving curvature continuity than earlier approaches that locally parametrize the surface by eigenfunctions. The approach allows us to rederive and thus survey the important lower bound results on piecewise polynomial subdivision surfaces by Prautzsch, Reif, Sabin and Zorin, as well as explain the beauty of curvature continuous constructions like Prautzsch’s. The parametrization neutral perspective gives also additional insights into the inherent constraints and stiffness of subdivision surfaces.

Available formats: pdf
Citation: BibTeX





A
G1 and a G2 subdivision scheme for triangular nets.
Prautzsch, Umlauf
International Journal on Shape Modelling, 6(1): 21-35, 2000.

In this article we improve the butterfly and Loop's algorithm. As a result we obtain subdivision algorithms for triangular nets which can be used to generate G1- and G2-surfaces, respectively.

Available formats: pdf
Citation: BibTeX





Analyzing the characteristic map of triangular subdivision schemes.

Umlauf
Constructive Approximation, 16(1): 145-155, 2000.

Tools for the analysis of generalized triangular box spline subdivision schemes are developed. For the first time the full analysis of Loop's algorithm can be carried out with these tools.

Available formats: pdf
Citation: BibTeX





Glatte Freiformflächen und optimierte Unterteilungsalgorithmen.

Umlauf
April 1999. Verlag Shaker, Aachen, 1999.
In German.





Triangular
G2- splines.
Prautzsch ,Umlauf
In: P.-L. Laurent, P. Sablonniere, L.L. Schumaker (eds.), Curve and Surface Design, Vanderbilt University Press, 335-342, 1999.

We introduce curvature continuous regular free-form surfaces with triangular control nets. These surfaces are composed of quartic box spline surfaces and are piecewise polynomial multisided patches of total degree 8 which minimize some energy integral. The Bézier nets can be computed efficiently form the spline control net by some fixed masks, i.e. matrix multiplications.





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Citation: BibTeX





Improved triangular subdivision schemes.

Prautzsch, Umlauf
In: F.-E. Wolter and N.M. Patrikalakis (eds.), Proceedings of the CGI '98, Hannover, 626-632, 1998. 

In this article we improve the butterfly and Loop’s algorithm. As a result we obtain subdivision algorithms for triangular nets which can be used to generate G1- and G2-surfaces, respectively.

Available formats: pdf
Citation: BibTeX





Konstruktion von
Gk-Flächen.
Umlauf
In: T. Ertl (ed.), "Trends und Höhepunkte der Graphischen Datenverarbeitung", Gesellschaft für Informatik, FA 4.1, Bonn, 1998.
In German.





A
G2-subdivision algorithm.
Prautzsch, Umlauf
Computing, 13: 217-224, 1998.

In this paper we present a method to optimize the smoothness order of subdivision algorithms generating surfaces of arbitrary topology. In particular we construct a subdivision algorithm with some negative weights producing G2-surfaces. These surfaces are piecewise bicubic and are flat at their extraordinary points. The underlying ideas can also be used to improve the smoothness order of subdivision algorithms for surfaces of higher degree or triangular nets.

Available formats: pdf
Citation: BibTeX


© Georg Umlauf Last modified: 26.01.2010