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21
Fair Surface Reconstruction Using Quadratic Functionals
, 1995
"... An algorithm for surface reconstruction from a polyhedron with arbitrary topology consisting of triangular faces is presented. The first variant of the algorithm constructs a curve network consisting of cubic B'ezier curves meeting with tangent plane continuity at the vertices. This curve netwo ..."
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An algorithm for surface reconstruction from a polyhedron with arbitrary topology consisting of triangular faces is presented. The first variant of the algorithm constructs a curve network consisting of cubic B'ezier curves meeting with tangent plane continuity at the vertices. This curve network is extended to a smooth surface by replacing each of the networks facets with a split patch consisting of three triangular B'ezier patches. The remaining degrees of freedom of the curve network and the split patches are determined by minimizing a quadratic functional. This optimization process works either for the curve network and the split patches separately or in one simultaneous step. The second variant of our algorithm is based on the construction of an optimized curve network with higher continuity. Examples demonstrate the quality of the different methods. 1 Introduction The reconstruction of a surface from a set of (a priori unorganized) points as well as the design of surfaces with a...
Physics based geometric design
 International J. of Shape Modeling
, 1996
"... Geometric modeling has proved to be crucial to computer graphics and computer aided geometric design (CAGD). During the past several decades, numerous geometric formulations have been proposed for a large variety of geometric modeling applications. In this paper, we survey the diversity of shape rep ..."
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Geometric modeling has proved to be crucial to computer graphics and computer aided geometric design (CAGD). During the past several decades, numerous geometric formulations have been proposed for a large variety of geometric modeling applications. In this paper, we survey the diversity of shape representations ranging from the primitive polynomial to the sophisticated NonUniform Rational BSpline (NURBS). We demonstrate that, among various geometric representations, NURBS have become an industrial standard primarily because of their many superior properties. By reviewing commonly used design paradigms such asinterpolation, approximation, interactive modi cation and variational optimization, we can also show that these conventional geometric design techniques are generally awkward when designers are confronted by complex, realworld objects. This is primarily because they only allow freeform primitives such as NURBS to be indirectly manipulated through numerous degrees of freedom (DOFs). To overcome the disadvantages of this indirect process, we summarize the prior work of physicsbased modeling and review dynamic NURBS (DNURBS) as a physicsbased generalization of geometric NURBS for shape design. DNURBS can unify the features of the industrystandard NURBS geometry with the many demonstrated conveniences of interaction within the new physicsbased design framework. We demonstrate that DNURBS can not only serve as a basis for the future research of physicsbased geometric design but also become readily appropriate for a large variety of important applications in graphics, vision, and scienti c visualization.
HALF DAY, 1:30 5 PM Course Speakers:
, 2001
"... Very large polyhedral models, which are used in more and more graphics applications today, are routinely generated by a variety of methods such as surface reconstruction algorithms from 3D scanned data, isosurface construction algorithms from volumetric data, and photogrametric methods from aerial p ..."
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Very large polyhedral models, which are used in more and more graphics applications today, are routinely generated by a variety of methods such as surface reconstruction algorithms from 3D scanned data, isosurface construction algorithms from volumetric data, and photogrametric methods from aerial photography. The course will provide an overview of several closely related methods designed to smooth, denoise, edit, compress, transmit, and animate very large polygonal models, based on signal processing techniques, constrained energy minimization, and the solution of diffusion differential equations. SPEAKERS
FITTING SMOOTH SURFACES TO DENSE POLYGON MESHES
, 2000
"... that I have read this dissertation and that in my opinion it is fully adequate, ..."
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that I have read this dissertation and that in my opinion it is fully adequate,
Surface Reconstruction Based Upon
"... In this paper we present a method for surface reconstruction using an extension of the Minimum Norm Network (MNN) designed for interpolating (functional) scattered data to the parametric case. Given a polyhedron with triangular faces, we obtain a smooth surface interpolating the vertices of the ..."
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In this paper we present a method for surface reconstruction using an extension of the Minimum Norm Network (MNN) designed for interpolating (functional) scattered data to the parametric case. Given a polyhedron with triangular faces, we obtain a smooth surface interpolating the vertices of the polyhedron and preserving its topology. As for functional MNN, a curve network is constructed satisfying certain conditions at the vertices from where the curves emanate, and having minimal norm with respect to a certain functional. The G MNN can then be extended to a smooth surface using, for example, methods described by Shirman/Sequin, Peters and Mann. Additionally we give a construction for a G MNN and describe its extension to a smooth surface.
Chapter 5
"... Introduction 97 The goal of this chapter is to develop techniques for constraining the volume bounded by a surface. We discuss the motivation for developing such a constraint, derive expressions for the constraint function, and describe techniques for generating fair, volume constrained surfaces. T ..."
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Introduction 97 The goal of this chapter is to develop techniques for constraining the volume bounded by a surface. We discuss the motivation for developing such a constraint, derive expressions for the constraint function, and describe techniques for generating fair, volume constrained surfaces. To measure fairness we use both the linearized thin plate and the exact thin plate formulations. We use Loop's Bezier patch scheme and the CatmullClark subdivision scheme as surface representations, and draw comparisons between the two. In Section 2.3 and the previous chapter, we surveyed techniques for designing surfaces using constrained optimization. The physical situation simulated by most of these techniques is a pliable surface being coerced into shape by userdefined constraints. In some applications, or in certain phases of design, a designer may prefer to construct an object from a lump of clay than from an elastic membrane. This suggests that we should extend existing surfa
An ObjectOriented Approach to Curves and Surfaces
 ObjectOriented and Mixed Programming Paradigms, chapter I/3
, 1996
"... This paper presents a top down approach to the design of an objectoriented framework for curves and surfaces together with its C++ implementation. We start from an abstract class of general differentiable curves and surfaces and in turn refine this design to various parametric representations of cu ..."
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This paper presents a top down approach to the design of an objectoriented framework for curves and surfaces together with its C++ implementation. We start from an abstract class of general differentiable curves and surfaces and in turn refine this design to various parametric representations of curves and surfaces. This design includes all of the standard curve and surface types and provides a powerful and uniform interface for applications. Examples from differential geometry, blending, and scattered data interpolation illustrate the approach.
Interpolating Scattered Data With C² Surfaces
 ComputerAided Design
, 1995
"... A method is presented for interpolating bivariate scattered data based upon a minimum norm network. This new method is related to G. M. Nielson's minimum norm network and to H. Pottmann's generalization of that method. The shape of the resulting C 2 interpolant is controlled parametri ..."
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A method is presented for interpolating bivariate scattered data based upon a minimum norm network. This new method is related to G. M. Nielson's minimum norm network and to H. Pottmann's generalization of that method. The shape of the resulting C 2 interpolant is controlled parametrically. Examples are given showing how the surface responds to changes in the control parameter. To illustrate the results, curvature plots as well as shaded images of the interpolants are given. Introduction The problem of interpolating scattered data has been investigated by many authors in recent years (for an overview: cf. [1], [3], [5] and [8]). In 1983, Nielson introduced a method for solving this problem based on variational principles, which leads to the minimum norm network (MNN) (cf. [14]). This approach was later extended using splines under tension (instead of cubic splines) as basis functions on the edges of a given domain triangulation (cf. [15]). Both methods yield a C 1 interpolan...