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42
Multilevel Partition of Unity Implicits
 ACM Transactions on Graphics
, 2003
"... We present a shape representation, the multilevel partition of unity implicit surface, that allows us to construct surface models from very large sets of points. There are three key ingredients to our approach: 1) piecewise quadratic functions that capture the local shape of the surface, 2) weighti ..."
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Cited by 157 (6 self)
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We present a shape representation, the multilevel partition of unity implicit surface, that allows us to construct surface models from very large sets of points. There are three key ingredients to our approach: 1) piecewise quadratic functions that capture the local shape of the surface, 2) weighting functions (the partitions of unity) that blend together these local shape functions, and 3) an octree subdivision method that adapts to variations in the complexity of the local shape.
Variational Implicit Surfaces
, 1999
"... We introduce a new method of creating smooth implicit surfaces of arbitrary manifold topology. These surfaces are described by specifying locations in 3D through which the surface should pass, and also identifying locations that are interior or exterior to the surface. A 3D implicit function is crea ..."
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Cited by 52 (2 self)
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We introduce a new method of creating smooth implicit surfaces of arbitrary manifold topology. These surfaces are described by specifying locations in 3D through which the surface should pass, and also identifying locations that are interior or exterior to the surface. A 3D implicit function is created from these constraints using a variational scattered data interpolation approach. We call the isosurface of this function a variational implicit surface. Like other implicit surface descriptions, these surfaces can be used for CSG and interference detection, may be interactively manipulated, are readily approximated by polygonal tilings, and are easy to ray trace. A key strength is that variational implicit surfaces allow the direct specification of both the location of points on the surface and surface normals. These are two important manipulation techniques that are difficult to achieve using other implicit surface representations such as sums of spherical or ellipsoidal Gaussian functions ("blobbies"). We show that these properties make variational implicit surfaces particularly attractive for interactive sculpting using the particle sampling technique introduced by Witkin and Heckbert in [30]. Our formulation also yields a simple method for converting a polygonal model to a smooth implicit model.
Morse Theory for Implicit Surface Modeling
 Mathematical Visualization
, 1997
"... . Morse theory describes the relationship between a function's critical points and the homotopy type of the function's domain. The theorems of Morse theory were developed specifically for functions on a manifold. This work adapts these theorems for use with parameterized families of implicit surface ..."
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Cited by 23 (4 self)
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. Morse theory describes the relationship between a function's critical points and the homotopy type of the function's domain. The theorems of Morse theory were developed specifically for functions on a manifold. This work adapts these theorems for use with parameterized families of implicit surfaces in computer graphics. The result is a theoretical basis for the determination of the global topology of an implicit surface, and supports the interactive modeling of implicit surfaces by direct manipulation of a topologicallycorrect triangulated representation. 1 Introduction Implicit surfaces provide a powerful and versatile shape model in computer graphics by representing geometry as the zeroset of a function over threespace, although displaying such surfaces requires a search through space. The display of an implicit surface is hastened by maintaining a triangulation that can be quickly rendered on modern graphics workstations. However, when the implicit surface changes topological t...
PerPixel Displacement Mapping with Distance Functions
 In GPU Gems 2
, 2005
"... In this chapter, we present distance mapping, a technique for adding smallscale displacement mapping to objects in a pixel shader. We treat displacement mapping as a raytracing problem, beginning with texture coordinates on the base surface and calculating texture coordinates where the viewing ray ..."
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Cited by 23 (0 self)
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In this chapter, we present distance mapping, a technique for adding smallscale displacement mapping to objects in a pixel shader. We treat displacement mapping as a raytracing problem, beginning with texture coordinates on the base surface and calculating texture coordinates where the viewing ray intersects the displaced surface. For this purpose, we precompute a threedimensional distance map, which gives a measure of the distance between points in space and the displaced surface. This distance map gives us all the information necessary to quickly intersect a ray with the surface. Our algorithm significantly increases the perceived geometric complexity of a scene while maintaining realtime performance. Cook (1984) introduced displacement mapping as a method for adding smallscale detail to surfaces. Unlike bump mapping, which affects only the shading of surfaces, displacement mapping adjusts the positions of surface elements. This leads to effects not possible with bump mapping, such as surface features that occlude each other and
Interactive ray tracing of arbitrary implicits with simd interval arithmetic
 In Proceedings of the 2nd IEEE/EG Symposium on Interactive Ray Tracing
, 2007
"... We present a practical and efficient algorithm for interactively ray tracing arbitrary implicit surfaces. We use interval arithmetic (IA) both for robust root computation and guaranteed detection of topological features. In conjunction with ray tracing, this allows for rendering literally any progra ..."
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Cited by 21 (7 self)
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We present a practical and efficient algorithm for interactively ray tracing arbitrary implicit surfaces. We use interval arithmetic (IA) both for robust root computation and guaranteed detection of topological features. In conjunction with ray tracing, this allows for rendering literally any programmable implicit function simply from its definition. Our method requires neither special hardware, nor preprocessing or storage of any data structure. Efficiency is achieved through SIMD optimization of both the interval arithmetic computation and coherent ray traversal algorithm, delivering interactive results even for complex implicit functions.
Anisotropic point set surfaces
 IN AFRIGRAPH ’06: PROCEEDINGS OF THE 4TH INTERNATIONAL CONFERENCE ON COMPUTER GRAPHICS, VIRTUAL REALITY, VISUALISATION AND INTERACTION IN AFRICA (2006), ACM
"... Point Set Surfaces define smooth surfaces from regular samples based on weighted averaging of the points. Because weighting is done based on a spatial scale parameter, point set surfaces apply basically only to regular samples. We suggest to attach individual weight functions to each sample rather t ..."
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Cited by 13 (0 self)
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Point Set Surfaces define smooth surfaces from regular samples based on weighted averaging of the points. Because weighting is done based on a spatial scale parameter, point set surfaces apply basically only to regular samples. We suggest to attach individual weight functions to each sample rather than to the location in space. This extends Point Set Surfaces to irregular settings, including anisotropic sampling adjusting to the principal curvatures of the surface. In particular, we describe how to represent surfaces with ellipsoidal weight functions per sample. Details of deriving such a representation from typical inputs and computing points on the surface are discussed.
Combining CSG modeling with soft blending using Lipschitzbased implicit surfaces
, 1996
"... In this paper a general method is given for combining CSG modeling with soft blending using implicit surfaces. A class of various blending functions sharing some desirable properties like differentiability and intuitive blend control are given. The functions defining the CSG objects satisfy the Lips ..."
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Cited by 11 (0 self)
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In this paper a general method is given for combining CSG modeling with soft blending using implicit surfaces. A class of various blending functions sharing some desirable properties like differentiability and intuitive blend control are given. The functions defining the CSG objects satisfy the Lipschitz condition which gives the possibility of fast rootfinding, but can also prove useful in the field of collision detection and adaptive triangulation. Introduction Methods for defining smooth surfaces can be divided into two categories: ffl Parametric functions Parametric functions are functions of the form f(u; v) = (f x (u; v); f y (u; v); f z (u; v)). Typical examples are Bezier or Bspline patches (See [Bohm84] for an overview). The surface is defined by control points and the surface can be adjusted by moving the control points. The surface can be rendered by evaluating the function for different wellchosen values of u and v. ffl Implicit functions Implicit functions have th...
Iterative methods for visualization of implicit surfaces on gpu
 In ISVC, International Symposium on Visual Computing, Lecture Notes in Computer Science, Lake Tahoe, Nevada/California, November 2007. SBC  Sociedade Brasileira de Computacao
, 2007
"... Abstract. The raycasting of implicit surfaces on GPU has been explored in the last few years. However, until recently, they were restricted to second degree (quadrics). We present an iterative solution to ray cast cubics and quartics on GPU. Our solution targets efficient implementation, obtaining ..."
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Cited by 11 (2 self)
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Abstract. The raycasting of implicit surfaces on GPU has been explored in the last few years. However, until recently, they were restricted to second degree (quadrics). We present an iterative solution to ray cast cubics and quartics on GPU. Our solution targets efficient implementation, obtaining interactive rendering for thousands of surfaces per frame. We have given special attention to torus rendering since it is a useful shape for multiple CAD models. We have tested four different iterative methods, including a novel one, comparing them with classical tessellation solution. Fig. 1. The faces of two bounding boxes are used to trigger the fragment shader responsible for rendering the tori. 1
Fast Ray Tracing of Arbitrary Implicit Surfaces with Interval and Affine Arithmetic
"... Existing techniques for rendering arbitraryform implicit surfaces are limited, either in performance, correctness or flexibility. Ray tracing algorithms employing interval arithmetic (IA) or affine arithmetic (AA) for rootfinding are robust and general in the class of surfaces they support, but tr ..."
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Cited by 9 (4 self)
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Existing techniques for rendering arbitraryform implicit surfaces are limited, either in performance, correctness or flexibility. Ray tracing algorithms employing interval arithmetic (IA) or affine arithmetic (AA) for rootfinding are robust and general in the class of surfaces they support, but traditionally slow. Nonetheless, implemented efficiently using a stackdriven iterative algorithm and SIMD vector instructions, these methods can achieve interactive performance for common algebraic surfaces on the CPU. A similar algorithm can also be implemented stacklessly, allowing for efficient ray tracing on the GPU. This paper presents these algorithms, as well as an inclusionpreserving reduced affine arithmetic (RAA) for faster raysurface intersection. Shader metaprogramming allows for immediate and automatic generation of symbolic expressions and their interval or affine extensions. Moreover, we are able to render even complex forms robustly, in realtime at high resolution.
PointSampled Cell Complexes
"... A piecewise smooth surface, possibly with boundaries, sharp edges, corners, or other features is defined by a set of samples. The basic idea is to model surface patches, curve segments and points explicitly, and then to glue them together based on explicit connectivity information. The geometry is d ..."
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Cited by 9 (1 self)
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A piecewise smooth surface, possibly with boundaries, sharp edges, corners, or other features is defined by a set of samples. The basic idea is to model surface patches, curve segments and points explicitly, and then to glue them together based on explicit connectivity information. The geometry is defined as the set of stationary points of a projection operator, which is generalized to allow modeling curves with samples, and extended to account for the connectivity information. Additional tangent constraints can be used to model shapes with continuous tangents across edges and corners.