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37
A Signal Processing Approach To Fair Surface Design
, 1995
"... In this paper we describe a new tool for interactive freeform fair surface design. By generalizing classical discrete Fourier analysis to twodimensional discrete surface signals  functions defined on polyhedral surfaces of arbitrary topology , we reduce the problem of surface smoothing, or fai ..."
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Cited by 654 (15 self)
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In this paper we describe a new tool for interactive freeform fair surface design. By generalizing classical discrete Fourier analysis to twodimensional discrete surface signals  functions defined on polyhedral surfaces of arbitrary topology , we reduce the problem of surface smoothing, or fairing, to lowpass filtering. We describe a very simple surface signal lowpass filter algorithm that applies to surfaces of arbitrary topology. As opposed to other existing optimizationbased fairing methods, which are computationally more expensive, this is a linear time and space complexity algorithm. With this algorithm, fairing very large surfaces, such as those obtained from volumetric medical data, becomes affordable. By combining this algorithm with surface subdivision methods we obtain a very effective fair surface design technique. We then extend the analysis, and modify the algorithm accordingly, to accommodate different types of constraints. Some constraints can be imposed without any modification of the algorithm, while others require the solution of a small associated linear system of equations. In particular, vertex location constraints, vertex normal constraints, and surface normal discontinuities across curves embedded in the surface, can be imposed with this technique.
Piecewise smooth surface reconstruction
, 1994
"... We present a general method for automatic reconstruction of accurate, concise, piecewise smooth surface models from scattered range data. The method can be used in a variety of applications such as reverse engineering — the automatic generation of CAD models from physical objects. Novel aspects of t ..."
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Cited by 303 (13 self)
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We present a general method for automatic reconstruction of accurate, concise, piecewise smooth surface models from scattered range data. The method can be used in a variety of applications such as reverse engineering — the automatic generation of CAD models from physical objects. Novel aspects of the method are its ability to model surfaces of arbitrary topological type and to recover sharp features such as creases and corners. The method has proven to be effective, as demonstrated by a number of examples using both simulated and real data. A key ingredient in the method, and a principal contribution of this paper, is the introduction of a new class of piecewise smooth surface representations based on subdivision. These surfaces have a number of properties that make them ideal for use in surface reconstruction: they are simple to implement, they can model sharp features concisely, and they can be fit to scattered range data using an unconstrained optimization procedure.
Efficient, Fair Interpolation using CatmullClark Surfaces
, 1993
"... We describe an efficient method for constructing a smooth surface that interpolates the vertices of a mesh of arbitrary topological type. Normal vectors can also be interpolated at an arbitrary subset of the vertices. The method improves on existing interpolation techniques in that it is fast, robus ..."
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Cited by 205 (9 self)
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We describe an efficient method for constructing a smooth surface that interpolates the vertices of a mesh of arbitrary topological type. Normal vectors can also be interpolated at an arbitrary subset of the vertices. The method improves on existing interpolation techniques in that it is fast, robust and general. Our approach is to compute a control mesh whose CatmullClark subdivision surface interpolates the given data and minimizes a smoothness or "fairness" measure of the surface. Following Celniker and Gossard, the norm we use is based on a linear combination of thinplate and membrane energies. Even though CatmullClark surfaces do not possess closedform parametrizations, we show that the relevant properties of the surfaces can be computed efficiently and without approximation. In particular, we show that (1) simple, exact interpolation conditions can be derived, and (2) the fairness norm and its derivatives can be computed exactly, without resort to numerical integration.
Geometric signal processing on polygonal meshes”. Eurographics State of the Art Report.
, 2000
"... ..."
Curved PN Triangles
 In Symposium on Interactive 3D Graphics
, 2001
"... quadratically varying normals). To improve the visual quality of existing trianglebased art in realtime entertainment, such as computer games, we propose replacing flat triangles with curved patches and higherorder normal variation. At the hardware level, based only on the three vertices and three ..."
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Cited by 75 (3 self)
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quadratically varying normals). To improve the visual quality of existing trianglebased art in realtime entertainment, such as computer games, we propose replacing flat triangles with curved patches and higherorder normal variation. At the hardware level, based only on the three vertices and three vertex normals of a given flat triangle, we substitute the geometry of a threesided cubic Bézier patch for the triangle’s flat geometry, and a quadratically varying normal for Gouraud shading. These curved pointnormal triangles, or PN triangles, require minimal or no change to existing authoring tools and hardware designs while providing a smoother, though not necessarily everywhere tangent continuous, silhouette and more organic shapes. CR Categories: I.3.5 [surface representation, splines]: I.3.6— graphics data structures
On surface normal and gaussian curvature approximations given data sampled from a smooth surface. Computer Aided Geometric design
, 2000
"... smooth surface ..."
C¹ surface splines
 SIAM Journal of Numerical Analysis
, 1995
"... The construction of quadratic C¹
surfaces from Bspline control points is generalized to a wider class of control meshes capable of outlining arbitrary freeform surfaces in space. Irregular meshes with non quadrilateral cells and more or fewer than four cells meeting at a point are allowed so that ..."
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Cited by 34 (12 self)
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The construction of quadratic C¹
surfaces from Bspline control points is generalized to a wider class of control meshes capable of outlining arbitrary freeform surfaces in space. Irregular meshes with non quadrilateral cells and more or fewer than four cells meeting at a point are allowed so that arbitrary freeform surfaces with or without boundary can be modeled in the same conceptual frame work as tensorproduct Bsplines. That is, the mesh points serve as control points of a smooth piecewise polynomial surface representation that is local, evaluates by averaging and obeys the convex hull property. For a regular region of the input mesh, the representation reduces to the standard quadratic spline. In general, any surface spline can be represented by BernsteinBezier patches of degree two and three. According to the user's choice, these patches can be polynomial or rational, threesided, foursided or a combination thereof.
A Survey of Parametric Scattered Data Fitting Using Triangular Interpolants
, 1992
"... This paper has been published as a chapter in "Curve and Surface Design", H. Hagen, (ed), SIAM, 1992 Some of the figures from that paper are missing from this version, as are all of the blackandwhite photographs. There are currently a number of methods for solving variants of the followi ..."
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Cited by 23 (5 self)
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This paper has been published as a chapter in "Curve and Surface Design", H. Hagen, (ed), SIAM, 1992 Some of the figures from that paper are missing from this version, as are all of the blackandwhite photographs. There are currently a number of methods for solving variants of the following problem: Given a triangulated polyhedron P in three space with or without boundary, construct a smooth surface that interpolates the vertices of P. In general, while the methods satisfy the continuity and interpolation requirements of the problem, they often fail to produce pleasing shapes. The purpose of this paper is to present a unifying survey of the published methods, to identify causes of shape defects, and to offer suggestions for improving the aesthetic quality of the interpolants. 8.1 Introduction.
Polar Forms and Triangular BSpline Surfaces
 In Blossoming: The New PolarForm Approach to Spline Curves and Surfaces, SIGGRAPH '91 Course Notes #26
, 1992
"... This paper presents a new triangular Bspline scheme that allows to construct piecewise polynomial surfaces over arbitrary triangulations of the parameter plane. The development of this scheme is based on the study of polar forms [79]. Polar forms have originally been a tool from classical mathemati ..."
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Cited by 23 (5 self)
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This paper presents a new triangular Bspline scheme that allows to construct piecewise polynomial surfaces over arbitrary triangulations of the parameter plane. The development of this scheme is based on the study of polar forms [79]. Polar forms have originally been a tool from classical mathematics [88]. They have first been introduced to CAGD by P. de Faget de Casteljau [24, 26] and by L. Ramshaw [65, 66, 67]. The author has subsequently extended this theory to more general surface representations and has used polar forms for the development of Bpatches [77, 76, 84]. Further extensions to simplex splines have finally led to the new triangular Bspline scheme described in this paper [18, 38, 39, 49, 78, 81]. While previous approaches to the construction of Bspline like surfaces over irregular domains have been based on subdivision, interpolation, and on the use of multisided patches the new scheme is based on blending functions and control points. The resulting surfaces are defined as linear combinations of the blending functions and are parametric piecewise polynomials over an arbitrary triangulation of the parameter plane, whose shape is determined by their control points. The paper is organized as follows: After a brief introduction to bivariate polynomials and polar forms (Section 2) we discuss triangular B'ezier patches (Section 3) and introduce a new surface representation for bivariate polynomials, the Bpatch (Section 4). In connection with simplex splines (Section 5) this finally leads to the construction of the new triangular Bspline scheme (Section 6). We hope that our presentation will provide a thorough unterstanding of the polar form of a polynomial surface, of triangular B'ezier patches, and of some of the main issues that are involved in the constr...