## Variational Implicit Surfaces (1999)

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Citations: | 52 - 2 self |

### BibTeX

@MISC{Turk99variationalimplicit,

author = {Greg Turk and James F. O'Brien},

title = {Variational Implicit Surfaces},

year = {1999}

}

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### Abstract

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 iso-surface 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.

### Citations

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Citation Context ...cell. At such cells, a small set of polygons can be created that approximates the behavior of the surface within the cell. Perhaps the best known approach of this type is the Marching Cubes algorithm =-=[17]-=-. When an implicit function is to be extracted from a measured dataset such as from medical CT or MRI, an isosurface extraction algorithm typically examines each voxel of the given volume. For an anal... |

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Citation Context ...kes on positive values interior to the sphere and is negative outside the surface, as will be the convention in this paper. An important class of implicit surfaces are the blobby or metaball surfaces =-=[3, 19]-=-. The implicit functions of these surfaces are the sum of radially symmetric functions that have a Gaussian profile. Here is the general form of such an implicit function: f (x) =-t+ n i=1 h i (x) (1)... |

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Citation Context ...ed in turn, and the process eventually crawls over the entire surface defined by the implicit function. We use the implementation of this method from [5] that is described in detail by Bloomenthal in =-=[4]-=-. Figure 7: Ray tracing of variational implicit surfaces. The left image shows reflection and shadows of two implicit surfaces, and the right image illustrates constructive solid geometry. 7.2 Ray Tra... |

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Citation Context ...en by c i . Evaluating an exponential function is computationally expensive, so some authors have used piecewise polynomial expressions instead of exponentials to define these blobby sphere functions =-=[19, 31]-=-. A greater variety of shapes can be created with the blobby approach by using ellipsoidal rather than spherical atomic functions. Another important class of implicit surfaces are the algebraic surfac... |

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Citation Context ...monly know as the thin-plate radial basis function. For 3D interpolation, the appropriate radial basis function to use is f(x) = |x| 3 . Duchon did much of the early work on variational interpolation =-=[8]-=-, and the report by Girosi, Jones and Poggio is a good entry point into the mathematics of variational interpolation [11]. Using the appropriate radial basis functions, we can then write the interpola... |

224 | S.: Using particles to sample and control implicit surfaces
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Citation Context ...obbies"). We show that these properties make variational implicit surfaces particularly attractive for interactive sculpting using the particle sampling technique introduced by Witkin and Heckber=-=t in [30]-=-. Our formulation also yields a simple method for converting a polygonal model to a smooth implicit model. 1 Introduction The computer graphics, computer-aided design and computer vision literatures a... |

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Citation Context ...ated, by determining suitable weights for each of the basis functions. This approach has been widely used for height-field interpolation and deformable models, and examples of its use can be found in =-=[28, 26, 7, 29]-=-. While finite elements and finite differencing techniques have proven useful for many problems, the discretization that they require can lead to artifacts. An alternate approach is to express the sol... |

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Citation Context ...between the poles. This springy resistance is mimicked by the energy function E. Thin-plate interpolation is often used in the computer vision domain, where there are often sparse surface constraints =-=[13, 28]-=-. The above curvature minimization process is sometimes referred to as regularization, and can be thought of as an additional constraint that selects a unique surface out of an infinite number of surf... |

132 | Interval analysis for computer graphics
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Citation Context ...ed to guarantee topologically correct tessellation of such surfaces [24]. Interval techniques, explored by Duff, Snyder and others, might be applied to tiling and ray tracing of variational implicits =-=[9, 23]-=-. The interactive texture placement methods of Pedersen should be directly applicable to variational implicit surfaces [20, 21]. Finally, many marvelous animations have been produced using blobby impl... |

120 | An implicit surface polygonizer
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Citation Context ...The neighboring vertices of these cubes are examined in turn, and the process eventually crawls over the entire surface defined by the implicit function. We use the implementation of this method from =-=[5]-=- that is described in detail by Bloomenthal in [4]. Figure 7: Ray tracing of variational implicit surfaces. The left image shows reflection and shadows of two implicit surfaces, and the right image il... |

102 |
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Citation Context ...ated, by determining suitable weights for each of the basis functions. This approach has been widely used for height-field interpolation and deformable models, and examples of its use can be found in =-=[28, 26, 7, 29]-=-. While finite elements and finite differencing techniques have proven useful for many problems, the discretization that they require can lead to artifacts. An alternate approach is to express the sol... |

99 | Guaranteeing the topology of an implicit surface polygonization for interactive modeling
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Citation Context ...to variational implicits, indicating several directions for future work. The critical point analysis of Stander and Hart could be used to guarantee topologically correct tessellation of such surfaces =-=[24]-=-. Interval techniques, explored by Duff, Snyder and others, might be applied to tiling and ray tracing of variational implicits [9, 23]. The interactive texture placement methods of Pedersen should be... |

96 |
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Citation Context ...ed to guarantee topologically correct tessellation of such surfaces [24]. Interval techniques, explored by Duff, Snyder and others, might be applied to tiling and ray tracing of variational implicits =-=[9, 23]-=-. The interactive texture placement methods of Pedersen should be directly applicable to variational implicit surfaces [20, 21]. Finally, many marvelous animations have been produced using blobby impl... |

95 | Guaranteed ray intersections with implicit surfaces
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Citation Context ...implicit function is a good area for future work. It is also likely that other techniques can successfully be applied to ray tracing these surfaces, such as the LG-surfaces approach of Kalra and Barr =-=[16]-=-. Figures 7 (left) is an image of two variational implicit surfaces that were ray traced using sphere tracing. Note that this figure includes shadows and reflections. Figure 7 (right) illustrates cons... |

78 | Stabilizers and Basis Functions: from regularization to radial, tensor and additive splines
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Citation Context ...s f(x) = |x| 3 . Duchon did much of the early work on variational interpolation [8], and the report by Girosi, Jones and Poggio is a good entry point into the mathematics of variational interpolation =-=[11]-=-. Using the appropriate radial basis functions, we can then write the interpolation function in the following form: f (x) = n j=1 d j f(x- c j ) +P(x) (5) In the above equation, c j are the locations ... |

77 |
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(Show Context)
Citation Context ...kes on positive values interior to the sphere and is negative outside the surface, as will be the convention in this paper. An important class of implicit surfaces are the blobby or metaball surfaces =-=[3, 19]-=-. The implicit functions of these surfaces are the sum of radially symmetric functions that have a Gaussian profile. Here is the general form of such an implicit function: f (x) =-t+ n i=1 h i (x) (1)... |

69 |
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Citation Context ...ed to solve the system grows as O(k 3 ). Work by Beatson and by Suter describe fast evaluation techniques that can be used to evaluate equation 5 in O(1) rather than the O(k) implied by the summation =-=[25, 1]-=-. The cost of solving the system can be reduced to approximately O(k 2 ) by using the a fast evaluation technique with an iterative solution method such as biconjugate gradient, and much larger system... |

65 |
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(Show Context)
Citation Context ...ated, by determining suitable weights for each of the basis functions. This approach has been widely used for height-field interpolation and deformable models, and examples of its use can be found in =-=[28, 26, 7, 29]-=-. While finite elements and finite differencing techniques have proven useful for many problems, the discretization that they require can lead to artifacts. An alternate approach is to express the sol... |

61 | Sphere tracing: A geometric method for the antialiased ray tracing of implicit surfaces
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(Show Context)
Citation Context ... a review of these techniques can be found in [14]. We have produced ray traced images of variational implicit surfaces using a particular technique introduced by Hart that is known as sphere tracing =-=[15]-=-. The basis for this method is that some implicit functions (including those that we are interested in) have what is called the Lipschitz property. A function f is said to have the Lipschitz property ... |

53 |
K.: Decorating implicit surfaces
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(Show Context)
Citation Context ...hers, might be applied to tiling and ray tracing of variational implicits [9, 23]. The interactive texture placement methods of Pedersen should be directly applicable to variational implicit surfaces =-=[20, 21]-=-. Finally, many marvelous animations have been produced using blobby implicit surfaces [3, 31]. We anticipate that the interpolating properties of variational implicit surfaces may provide animators w... |

48 |
An improved algorithm for algebraic curve and surface fitting
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(Show Context)
Citation Context ...h, it may be described by a single polynomial expression. A good deal of attention has been devoted to this approach, and a good entry point into these kinds of surfaces is the work of Gabriel Taubin =-=[27]-=-. Much of the work in this area has been devoted to fitting an algebraic surfaces to a given collection of points. Usually it is not possible to interpolation all of the data points, so error minimizi... |

11 |
A Framework for Interactive Texturing on Curved Surfaces
- Pedersen
- 1996
(Show Context)
Citation Context ...hers, might be applied to tiling and ray tracing of variational implicits [9, 23]. The interactive texture placement methods of Pedersen should be directly applicable to variational implicit surfaces =-=[20, 21]-=-. Finally, many marvelous animations have been produced using blobby implicit surfaces [3, 31]. We anticipate that the interpolating properties of variational implicit surfaces may provide animators w... |

10 |
Surface consistency constraints in vision
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- 1983
(Show Context)
Citation Context ...between the poles. This springy resistance is mimicked by the energy function E. Thin-plate interpolation is often used in the computer vision domain, where there are often sparse surface constraints =-=[13, 28]-=-. The above curvature minimization process is sometimes referred to as regularization, and can be thought of as an additional constraint that selects a unique surface out of an infinite number of surf... |

7 |
Marie-Paule Gascuel. Automatic reconstruction of unstructured 3d data: Combining medial axis and implicit surfaces
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(Show Context)
Citation Context ... 243 blobby spheres "took a few days on a UNIX workstation (Stardent TITAN3000 2 CPU)." Similar blobby sphere data approximation by Eric Bittar and co-workers was limited to roughly 50 blobb=-=y spheres [2]-=-. In contrast to these methods, the bunny in Figure 5 (right) is a variational implicit surface with 800 boundary and 800 normal constraints. It required 1 minute 43 seconds to solve the matrix equati... |

5 | Fast evaluation of splines using poisson formula
- Suter
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(Show Context)
Citation Context ...ed to solve the system grows as O(k 3 ). Work by Beatson and by Suter describe fast evaluation techniques that can be used to evaluate equation 5 in O(1) rather than the O(k) implied by the summation =-=[25, 1]-=-. The cost of solving the system can be reduced to approximately O(k 2 ) by using the a fast evaluation technique with an iterative solution method such as biconjugate gradient, and much larger system... |

3 |
Volumetric shape description of range data using ’blobby model
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(Show Context)
Citation Context ... many beautiful objects have been sculpted in this manner. Can this process be entirely automated? Shigeru Muraki demonstrated a way in which a given range image may be approximated by blobby spheres =-=[18]-=-. The method begins with a single blobby sphere that is positioned to match the data. Then the method repeatedly selects one blobby sphere and splits it into two new spheres, invoking an optimization ... |

3 |
casting as a method for solid modeling
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(Show Context)
Citation Context ...ional implicit surfaces. The figure shows (from left to right) intersection and subtraction of two implicit surfaces. This figure was created using standard ray tracing CSG techniques as described in =-=[22]-=-. The rendering techniques of this section highlight a key point-- variational implicit surfaces may be used in almost all of the contexts in which other implicit formulations have been used. This new... |

2 |
Interpolation of scattered data by radial basis functions
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(Show Context)
Citation Context ...arise from such ill-conditioning. Even if such problems do arise, variational interpolation is such a well-studied problem that methods exist for improving the conditioning of the system of equations =-=[10]-=-. 4 Creating Variational Implicit Surfaces With tools for solving the scattered data interpolation problem in hand, we now turn our attention to creating implicit functions. In this section, we will e... |

2 |
tracing implicit surfaces
- Ray
- 1993
(Show Context)
Citation Context ...right image illustrates constructive solid geometry. 7.2 Ray Tracing There are a number of techniques that may be used to ray trace implicit surfaces, and a review of these techniques can be found in =-=[14]-=-. We have produced ray traced images of variational implicit surfaces using a particular technique introduced by Hart that is known as sphere tracing [15]. The basis for this method is that some impli... |