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Reconstruction and Representation of 3D Objects with Radial Basis Functions
 Computer Graphics (SIGGRAPH ’01 Conf. Proc.), pages 67–76. ACM SIGGRAPH
, 2001
"... We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from pointcloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs al ..."
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Cited by 505 (1 self)
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We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from pointcloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs allow us to model large data sets, consisting of millions of surface points, by a single RBFpreviously an impossible task. A greedy algorithm in the fitting process reduces the number of RBF centers required to represent a surface and results in significant compression and further computational advantages. The energyminimisation characterisation of polyharmonic splines result in a "smoothest" interpolant. This scaleindependent characterisation is wellsuited to reconstructing surfaces from nonuniformly sampled data. Holes are smoothly filled and surfaces smoothly extrapolated. We use a noninterpolating approximation when the data is noisy. The functional representation is in effect a solid model, which means that gradients and surface normals can be determined analytically. This helps generate uniform meshes and we show that the RBF representation has advantages for mesh simplification and remeshing applications. Results are presented for realworld rangefinder data.
A survey of freeform object representation and recognition techniques
 Computer Vision and Image Understanding
, 2001
"... Advances in computer speed, memory capacity, and hardware graphics acceleration have made the interactive manipulation and visualization of complex, detailed (and therefore large) threedimensional models feasible. These models are either painstakingly designed through an elaborate CAD process or re ..."
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Cited by 200 (1 self)
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Advances in computer speed, memory capacity, and hardware graphics acceleration have made the interactive manipulation and visualization of complex, detailed (and therefore large) threedimensional models feasible. These models are either painstakingly designed through an elaborate CAD process or reverse engineered from sculpted prototypes using modern scanning technologies and integration methods. The availability of detailed data describing the shape of an object offers the computer vision practitioner new ways to recognize and localize freeform objects. This survey reviews recent literature on both the 3D model building process and techniques used to match and identify freeform objects from imagery. c ○ 2001 Academic Press 1.
PatientSpecific Vascular NURBS Modeling for Isogeometric Analysis of Blood Flow
 IN PROCEEDINGS OF THE 15TH INTERNATIONAL MESHING ROUNDTABLE
, 2006
"... We describe an approach to construct hexahedral solid NURBS (NonUniform Rational BSplines) meshes for patientspecific vascular geometric models from imaging data for use in isogeometric analysis. First, image processing techniques, such as contrast enhancement, filtering, classification, and segm ..."
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Cited by 46 (9 self)
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We describe an approach to construct hexahedral solid NURBS (NonUniform Rational BSplines) meshes for patientspecific vascular geometric models from imaging data for use in isogeometric analysis. First, image processing techniques, such as contrast enhancement, filtering, classification, and segmentation, are used to improve the quality of the input imaging data. Then, luminal surfaces are extracted by isocontouring the preprocessed data, followed by the extraction of vascular skeleton via Voronoi and Delaunay diagrams. Next, the skeletonbased sweeping method is used to construct hexahedral control meshes. Templates are designed for various branching configurations to decompose the geometry into mapped meshable patches. Each patch is then meshed using onetoone sweeping techniques, and boundary vertices are projected to the luminal surface. Finally, hexahedral solid NURBS are constructed and used in isogeometric analysis of blood flow. Piecewise linear hexahedral meshes can also be obtained using this approach. Examples of patientspecific arterial models are presented.
Automatic Reconstruction of 3D CAD Models from Digital Scans
 International Journal of Computational Geometry and Applications
, 1999
"... We present an approach for the reconstruction and approximation of 3D CAD models from an unorganized collection of points. Applications include rapid reverse engineering of existing objects for use in a synthetic computer environment, including computer aided design and manufacturing. Our reconstruc ..."
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Cited by 40 (10 self)
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We present an approach for the reconstruction and approximation of 3D CAD models from an unorganized collection of points. Applications include rapid reverse engineering of existing objects for use in a synthetic computer environment, including computer aided design and manufacturing. Our reconstruction approach is flexible enough to permit interpolation of both smooth surfaces and sharp features, while placing few restrictions on the geometry or topology of the object. Our algorithm is based on alphashapes to compute an initial triangle mesh approximating the object's surface. A mesh reduction technique is applied to the dense triangle mesh to build a simplified approximation, while retaining important topological and geometric characteristics of the model. The reduced mesh is interpolated with piecewise algebraic surface patches which approximate the original points. The process is fully automatic, and the reconstruction is guaranteed to be homeomorphic and error bounded with respec...
Highquality extraction of isosurfaces from regular and irregular grids
 IEEE Transactions on Visualization and Computer Graphics
"... Abstract—Isosurfaces are ubiquitous in many fields, including visualization, graphics, and vision. They are often the main computational component of important processing pipelines (e.g., surface reconstruction), and are heavily used in practice. The classical approach to compute isosurfaces is to a ..."
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Cited by 31 (7 self)
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Abstract—Isosurfaces are ubiquitous in many fields, including visualization, graphics, and vision. They are often the main computational component of important processing pipelines (e.g., surface reconstruction), and are heavily used in practice. The classical approach to compute isosurfaces is to apply the Marching Cubes algorithm, which although robust and simple to implement, generates surfaces that require additional processing steps to improve triangle quality and mesh size. An important issue is that in some cases, the surfaces generated by Marching Cubes are irreparably damaged, and important details are lost which can not be recovered by subsequent processing. The main motivation of this work is to develop a technique capable of constructing highquality and highfidelity isosurfaces. We propose a new advancing front technique that is capable of creating highquality isosurfaces from regular and irregular volumetric datasets. Our work extends the guidance field framework of Schreiner et al. to implicit surfaces, and improves it in significant ways. In particular, we describe a set of sampling conditions that guarantee that surface features will be captured by the algorithm. We also describe an efficient technique to compute a minimal guidance field, which greatly improves performance. Our experimental results show that our technique can generate highquality meshes from complex datasets.
Surface Reconstruction From Nonparallel Curve Networks
"... Building surfaces from crosssection curves has wide applications including biomedical modeling. Previous work in this area has mostly focused on connecting simple closed curves on parallel crosssections. Here we consider the more general problem where input data may lie on nonparallel crosssect ..."
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Cited by 23 (4 self)
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Building surfaces from crosssection curves has wide applications including biomedical modeling. Previous work in this area has mostly focused on connecting simple closed curves on parallel crosssections. Here we consider the more general problem where input data may lie on nonparallel crosssections and consist of curve networks that represent the segmentation of the underlying object by different material or tissue types (e.g., skin, muscle, bone, etc.) on each crosssection. The desired output is a surface network that models both the exterior surface and the internal partitioning of the object. We introduce an algorithm that is capable of handling curve networks of arbitrary shape and topology on crosssection planes with arbitrary orientations. Our algorithm is simple to implement and is guaranteed to produce a closed surface network that interpolates the curve network on each crosssection. Our method is demonstrated on both synthetic and biomedical examples. 1.
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 ..."
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Cited by 19 (0 self)
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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.
Rational parametrizations of nonsingular real cubic surfaces
 ACM Trans. on Graphics
, 1998
"... Real cubic algebraic surfaces may be described by either implicit or parametric equations. One particularly useful representation is the rational parametrization, where the three spatial coordinates are given by rational functions of two parameters. These parametrizations take on different forms for ..."
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Cited by 13 (0 self)
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Real cubic algebraic surfaces may be described by either implicit or parametric equations. One particularly useful representation is the rational parametrization, where the three spatial coordinates are given by rational functions of two parameters. These parametrizations take on different forms for different classes of cubic surfaces. Classification of real cubic algebraic surfaces into five families for the nonsingular case is based on the configuration of 27 lines on them. We provide a method of extracting all these lines by constructing and solving a polynomial of degree 27. Simple roots of this polynomial correspond to real lines on the surface, and real skew lines are used to form rational parametrizations for three of these families. Complex conjugate skew lines are used to parametrize surfaces from the fourth family. The parametrizations for these four families involve quotients of polynomials of degree no higher than four. Each of these parametrizations covers the whole surface except for a few points, lines, or conic sections. The parametrization for the fifth family, as noted previously in the literature, requires a square root. We also analyze the image of the derived rational parametrization for both real and complex parameter values, together with “base ” points where the parametrizations are illdefined.
Algebraic Methods for Computer Aided Geometric Design
"... The concepts and methods of algebra and algebraic geometry have found significant applications in many disciplines. This chapter presents a collection of gleanings from algebra or algebraic geometry that hold practical value for the field of computer aided geometric design. We focus on the insights, ..."
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Cited by 13 (0 self)
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The concepts and methods of algebra and algebraic geometry have found significant applications in many disciplines. This chapter presents a collection of gleanings from algebra or algebraic geometry that hold practical value for the field of computer aided geometric design. We focus on the insights, algorithm enhancements and practical capabilities that algebraic methods have contributed to CAGD. Specifically, we examine resultants and Gröbner basis, and discuss their applications in implicitization, inversion, parametrization and intersection algorithms. Other topics of CAGD research work using algebraic methods are also outlined.
A TriangulationBased Object Reconstruction Method
"... Reconstructing the shape of a 3D object from a digital scan of its surface has a range of applications, such asreverse engineering, authoring 3D synthetic worlds, shape analysis, 3D faxing and tailorfit modeling. Input data ..."
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Cited by 11 (0 self)
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Reconstructing the shape of a 3D object from a digital scan of its surface has a range of applications, such asreverse engineering, authoring 3D synthetic worlds, shape analysis, 3D faxing and tailorfit modeling. Input data