|
|
2008 | 2007 |
2006 |
2005 |
2004 | 2004 > |
Natural neighbor extrapolation using ghost points.
Bobach, Farin, Hansford, Umlauf
Accepted: Computer Aided-Design.
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
Available formats:pdf
Citation: BibTeX
Online triangulation of laser-scan data.
Denker, Lehner, Umlauf
To appear: 17th International Meshing Roundtable 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.
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
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.Citation: BibTeX
Natural neighbor concepts in scattered data interpolation and discrete function approximation.
Bobach, Umlauf
To appear in: H. Hagen, C. Garth, M. Bertram (eds.), GI Lecture Notes in Informatics, Visualization of Large and Unstructured Data Sets, 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.
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.
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.
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.
Available formats: 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.
Available formats: 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.
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
To appear in: H. Hagen, C. Garth, M. Bertram (eds.), GI Lecture Notes in Informatics, Visualization of Large and Unstructured Data Sets, 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.
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
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.
Available formats: 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.
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
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.
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
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.
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
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.
Available formats: pdf
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.
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