Results 1  10
of
31
FreeForm Shape Design Using Triangulated Surfaces
, 1994
"... We present an approach to modeling with truly mutable yet completely controllable freeform surfaces of arbitrary topology. Surfaces may be pinned down at points and along curves, cut up and smoothly welded back together, and faired and reshaped in the large. This style of control is formulated as a ..."
Abstract

Cited by 153 (0 self)
 Add to MetaCart
We present an approach to modeling with truly mutable yet completely controllable freeform surfaces of arbitrary topology. Surfaces may be pinned down at points and along curves, cut up and smoothly welded back together, and faired and reshaped in the large. This style of control is formulated as a constrained shape optimization, with minimization of squared principal curvatures yielding graceful shapes that are free of the parameterization worries accompanying many patchbased approaches. Triangulated point sets are used to approximate these smooth variational surfaces, bridging the gap between patchbased and particlebased representations. Automatic refinement, mesh smoothing, and retriangulation maintain a good computational mesh as the surface shape evolves, and give sample points and surface features much of the freedom to slide around in the surface that oriented particles enjoy. The resulting surface triangulations are constructed and maintained in real time. 1 Introduction ...
Scattered Data Interpolation with Multilevel Splines
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
, 1997
"... This paper describes a fast algorithm for scattered data interpolation and approximation. Multilevel Bsplines are introduced to compute a C²continuous surface through a set of irregularly spaced points. The algorithm makes use of a coarsetofine hierarchy of control lattices to generate a sequen ..."
Abstract

Cited by 106 (9 self)
 Add to MetaCart
This paper describes a fast algorithm for scattered data interpolation and approximation. Multilevel Bsplines are introduced to compute a C²continuous surface through a set of irregularly spaced points. The algorithm makes use of a coarsetofine hierarchy of control lattices to generate a sequence of bicubic Bspline functions whose sum approaches the desired interpolation function. Large performance gains are realized by using Bspline refinement to reduce the sum of these functions into one equivalent Bspline function. Experimental results demonstrate that highfidelity reconstruction is possible from a selected set of sparse and irregular samples.
Fitting Scattered Data on SphereLike Surfaces Using Spherical Splines
 J. Comput. Appl. Math
"... . Spaces of polynomial splines defined on planar triangulations are very useful tools for fitting scattered data in the plane. Recently, [4, 5], using homogeneous polynomials, we have developed analogous spline spaces defined on triangulations on the sphere and on spherelike surfaces. Using these s ..."
Abstract

Cited by 48 (11 self)
 Add to MetaCart
. Spaces of polynomial splines defined on planar triangulations are very useful tools for fitting scattered data in the plane. Recently, [4, 5], using homogeneous polynomials, we have developed analogous spline spaces defined on triangulations on the sphere and on spherelike surfaces. Using these spaces, it is possible to construct analogs of many of the classical interpolation and fitting methods. Here we examine some of the more interesting ones in detail. For interpolation, we discuss macroelement methods and minimal energy splines, and for fitting, we consider discrete least squares and penalized least squares. 1. Introduction Let S be the unit sphere or a spherelike surface (see Sect. 2 below) in IR 3 . In addition, suppose that we are given a set of scattered points located on S, along with real numbers associated with each of these points. The problem of interest in this paper is to find a function defined on S which either interpolates or approximates these data. This pr...
Scattered Data Interpolation Methods for Electronic Imaging Systems: A Survey
, 2002
"... Numerous problems in electronic imaging systems involve the need to interpolate from irregularly spaced data. One example is the calibration of color input/output devices with respect to a common intermediate objective color space, such as XYZ or L*a*b*. In the present report we survey some of the m ..."
Abstract

Cited by 47 (0 self)
 Add to MetaCart
Numerous problems in electronic imaging systems involve the need to interpolate from irregularly spaced data. One example is the calibration of color input/output devices with respect to a common intermediate objective color space, such as XYZ or L*a*b*. In the present report we survey some of the most important methods of scattered data interpolation in twodimensional and in threedimensional spaces. We review both singlevalued cases, where the underlying function has the form f:R #R, and multivalued cases, where the underlying function is f:R . The main methods we review include linear triangular (or tetrahedral) interpolation, cubic triangular (CloughTocher) interpolation, triangle based blending interpolation, inverse distance weighted methods, radial basis function methods, and natural neighbor interpolation methods. We also review one method of scattered data fitting, as an illustration to the basic differences between scattered data interpolation and scattered data fitting.
Construction of Vector Field Hierarchies
, 1999
"... We present a method for the hierarchical representation of vector fields. Our approach is based on iterative refinement using clustering and principal component analysis. The input to our algorithm is a discrete set of points with associated vectors. The algorithm generates a topdown segmentation o ..."
Abstract

Cited by 41 (4 self)
 Add to MetaCart
We present a method for the hierarchical representation of vector fields. Our approach is based on iterative refinement using clustering and principal component analysis. The input to our algorithm is a discrete set of points with associated vectors. The algorithm generates a topdown segmentation of the discrete field by splitting clusters of points. We measure the error of the various approximation levels by measuring the discrepancy between streamlines generated by the original discrete field and its approximations based on much smaller discrete data sets. Our method assumes no particular structure of the field, nor does it require any topological connectivity information. It is possible to generate multiresolution representations of vector fields using this approach. Keywords: vector field visualization; Hardy's multiquadric method; binaryspace partitioning; data simplification. 1 Introduction The rapid increase in the power of computer systems coupled with the improving precis...
Scattered Data Fitting on the Sphere
 in Mathematical Methods for Curves and Surfaces II
, 1998
"... . We discuss several approaches to the problem of interpolating or approximating data given at scattered points lying on the surface of the sphere. These include methods based on spherical harmonics, tensorproduct spaces on a rectangular map of the sphere, functions defined over spherical triangulat ..."
Abstract

Cited by 34 (5 self)
 Add to MetaCart
. We discuss several approaches to the problem of interpolating or approximating data given at scattered points lying on the surface of the sphere. These include methods based on spherical harmonics, tensorproduct spaces on a rectangular map of the sphere, functions defined over spherical triangulations, spherical splines, spherical radial basis functions, and some associated multiresolution methods. In addition, we briefly discuss spherelike surfaces, visualization, and methods for more general surfaces. The paper includes a total of 206 references. x1. Introduction Let S be the unit sphere in IR 3 , and suppose that fv i g n i=1 is a set of scattered points lying on S. In this paper we are interested in the following problem: Problem 1. Given real numbers fr i g n i=1 , find a (smooth) function s defined on S which interpolates the data in the sense that s(v i ) = r i ; i = 1; : : : ; n; (1) or approximates it in the sense that s(v i ) ß r i ; i = 1; : : : ; n: (2) Data f...
Leastsquares meshes
 In Shape Modeling International (SMI
, 2004
"... Figure 1: LSmesh: a mesh constructed from a given connectivity graph and a sparse set of control points with geometry. In this example the connectivity is taken from the camel mesh. In (a) the LSmesh is constructed with 100 control points and in (c) with 2000 control points. The connectivity graph ..."
Abstract

Cited by 30 (4 self)
 Add to MetaCart
Figure 1: LSmesh: a mesh constructed from a given connectivity graph and a sparse set of control points with geometry. In this example the connectivity is taken from the camel mesh. In (a) the LSmesh is constructed with 100 control points and in (c) with 2000 control points. The connectivity graph contains 39074 vertices (without any geometric information). (b) and (d) show closeups on the head; the control points are marked by red balls. In this paper we introduce Leastsquares Meshes: meshes with a prescribed connectivity that approximate a set of control points in a leastsquares sense. The given mesh consists of a planar graph with arbitrary connectivity and a sparse set of control points with geometry. The geometry of the mesh is reconstructed by solving a sparse linear system. The linear system not only defines a surface that approximates the given control points, but it also distributes the vertices over the surface in a fair way. That is, each vertex lies as close as possible to the center of gravity of its immediate neighbors. The Leastsquares Meshes (LSmeshes) are a visually smooth and fair approximation of the given control points. We show that the connectivity of the mesh contains geometric information that affects the shape of the reconstructed surface. Finally, we discuss the applicability of LSmeshes to approximation of given surfaces, smooth completion, mesh editing and progressive transmission.
Topological segmentation in threedimensional vector fields
 IEEE Transactions on Visualization and Computer Graphics
, 2004
"... We present a new method for topological segmentation in steady threedimensional vector fields. Depending on desired properties, the algorithm replaces the original vector field by a derived segmented data set, which is utilized to produce separating surfaces in the vector field. We define the conce ..."
Abstract

Cited by 27 (5 self)
 Add to MetaCart
We present a new method for topological segmentation in steady threedimensional vector fields. Depending on desired properties, the algorithm replaces the original vector field by a derived segmented data set, which is utilized to produce separating surfaces in the vector field. We define the concept of a segmented data set, develop methods that produce the segmented data by sampling the vector field with streamlines, and describe algorithms that generate the separating surfaces. This method is applied to generate local separatrices in the field, defined by a movable boundary region placed in the field. The resulting partitions can be visualized using standard techniques for a visualization of a vector field at a higher level of abstraction. 1.
ERTL T.: Interactively Visualizing Procedurally Encoded Scalar Fields
 In Proceedings of EG/IEEE TCVG Symposium on Visualization VisSym ’04 (2004), Deussen O., Hansen C., Keim D.„ Saupe D., (Eds
"... Figure 1: RBF reconstruction of unstructured CFD data. (a) Volume rendering of 1,943,383 tetrahedral shock data set using 2,932 RBF functions. (b) Volume rendering of a 156,642 tetrahedral oil reservoir data set using 222 RBF functions organized in a hierarchy of 49 cells. While interactive visualiz ..."
Abstract

Cited by 20 (3 self)
 Add to MetaCart
Figure 1: RBF reconstruction of unstructured CFD data. (a) Volume rendering of 1,943,383 tetrahedral shock data set using 2,932 RBF functions. (b) Volume rendering of a 156,642 tetrahedral oil reservoir data set using 222 RBF functions organized in a hierarchy of 49 cells. While interactive visualization of rectilinear gridded volume data sets can now be accomplished using texture mapping hardware on commodity PCs, interactive rendering and exploration of large scattered or unstructured data sets is still a challenging problem. We have developed a new approach that allows the interactive rendering and navigation of procedurallyencoded 3D scalar fields by reconstructing these fields on PC class graphics processing units. Since the radial basis functions (RBFs) we use for encoding can provide a compact representation of volumetric scalar fields, the large grid/mesh traditionally needed for rendering is no longer required and ceases to be a data transfer and computational bottleneck during rendering. Our new approach will interactively render RBF encoded data obtained from arbitrary volume data sets, including both structured volume models and unstructured scattered volume models. This procedural reconstruction of large data sets is flexible, extensible, and can take advantage of the Moore’s Law cubed increase in performance of graphics hardware.
Reconstructing surfaces and functions on surfaces from unorganized 3d data
 Algorithmica
, 1997
"... Creating a computer model from an existing part is a common problem in Reverse Engineering. The part might be scanned with a device like the laser range scanner, or points might be measured on its surface with a mechanical probe. Sometimes, not only the spatial location of points, but also some asso ..."
Abstract

Cited by 17 (0 self)
 Add to MetaCart
Creating a computer model from an existing part is a common problem in Reverse Engineering. The part might be scanned with a device like the laser range scanner, or points might be measured on its surface with a mechanical probe. Sometimes, not only the spatial location of points, but also some associated physical property can be measured. The problem of automatically reconstructing from this data a topologically consistent and geometrically accurate model of the object and of the sampled scalar field is the subject of this paper. The algorithm proposed in this paper can deal with connected,orientable manifolds of unrestricted topological type, given a sufficiently dense and uniform sampling of the object’s surface. It is capable of automatically reconstructing both the model and a scalar field over its surface. It uses Delaunay triangulations, Voronoi diagrams and alphashapes for efficiency of computation and theoretical soundness. It generates a representation of the surface and the field based on BernsteinBézier polynomial implicit patches (Apatches), that are guaranteed to be smooth and singlesheeted.