Sampling implicit objects with physically-based particle systems

doi:10.1016/0097-8493(96)00005-2

Computers & Graphics

Volume 20, Issue 3, May–June 1996, Pages 365-375

https://doi.org/10.1016/0097-8493(96)00005-2 Get rights and content

Abstract

After reviewing three classical sampling methods for implicit objects, we describe a new sampling method that is not based on scanning the ambient space. In this method, samples are “randomly” generated using physically-based particle systems.

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Cited by (21)

Point-based rendering of implicit surfaces in <sup>R4</sup>
2013, Computers and Graphics (Pergamon)
Citation Excerpt :
It is well known that computing a polygonal approximation of an implicit object is a challenging problem [12] for two main reasons: it is difficult to find points on the object [13] and it is difficult to connect isolated points into a mesh [14].
Stochastic visualization of intersection curves of implicit surfaces
2007, Computers and Graphics (Pergamon)
Citation Excerpt :
2) The method also offers the capability to perform an efficient ‘intensive search’ that enables quick investigation of only a part of interest on an implicit surface. ( 3) The method generates high-density points on intersection curves, and these points enable precise visualization of the curves with point-based graphics (see [8–10] and references in [10]). Thus, the main contribution of this paper is introducing the ideas of ‘stochastic search’ and ‘point-based graphics’ in finding and precisely visualizing intersection curves of implicit surfaces.
We propose a stochastic method to visualize intersection curves of complex implicit surfaces. Our method performs a stochastic ‘uniform search’ on an implicit surface to visualize its intersection curves, which scatter over the whole surface disconnectedly. The method also offers the capability to perform an efficient ‘intensive search’ that enables quick investigation of only a part of interest on an implicit surface. The method generates high-density points on intersection curves. This realizes precise visualization of the curves with point-based graphics. The visualization precision realized by our method is higher than that of the conventional intersection-finding/visualizing methods, e.g., the marching methods. Our method is also applicable to precise visualization of contours based on general curvilinear coordinates. This can help our visual understanding of rough/fine structures of complex implicit surfaces. To demonstrate effectiveness of the method, we apply it to complex implicit surfaces that approximate data sets of $10^{5}$ or $10^{6}$ 3D points.
Robust adaptive polygonal approximation of implicit curves
2002, Computers and Graphics (Pergamon)
Citation Excerpt :
Applications usually need a geometric model of the implicit object, typically a polygonal approximation. While it is easy to compute polygonal approximations for parametric objects, computing polygonal approximations for implicit objects is a challenging problem for two main reasons: first, it is difficult to find points on the implicit object [3]; second, it is difficult to connect isolated points into a mesh [4]. An example of what our algorithm does is shown in Fig. 1 for the ellipse given implicitly by the equation x2/6+y2=1.
Application of the stochastic sampling method to various implicit surfaces
2001, Computers and Graphics (Pergamon)
Recently, we proposed the stochastic sampling method (SSM) to numerically generate sample points on implicit surfaces quickly and uniformly. In this paper, we apply the SSM to various complicated implicit surfaces to demonstrate its usefulness. Namely, we demonstrate that the SSM works well for surfaces with much ruggedness and/or many holes, blobbies, and open surfaces. We also show that the SSM enables effective techniques for quick rendering and visualizing outlines of implicit surfaces. Many other new information/hints, which are useful in applying the SSM to complicated surfaces effectively and easily, are also presented.
Stochastic algorithm for detecting intersection of implicit surfaces
2000, Computers and Graphics (Pergamon)
Recently, we proposed the stochastic sampling method (SSM), which can sample implicit surfaces with Monte Carlo simulation based on a stochastic differential equation. In this paper, we demonstrate that the SSM realizes an excellent intersection-detection algorithm for implicit surfaces. The algorithm is fast enough for practical use, reliable, and widely applicable to complicated surfaces topologically, mathematically, and/or in shape. Moreover, the algorithm is suitable to parallel calculation using multiple CPU powers, and it can accelerate the intersection detection to a great extent.
Sampling implicit surfaces based on stochastic differential equations with converging constraint
2000, Computers and Graphics (Pergamon)
We propose a new stochastic sampling method for implicit surfaces based on an Ito-type stochastic differential equation with the converging constraint. The equation can realize a stationary state in which the probability distribution becomes uniform over the whole area of a constraint surface. A hypothetical stochastic particle described by the equation performs Brownian motion, i.e. random walk, being confined on a constraint surface, i.e. an implicit surface. Therefore its trajectory gives us sample points on it. The stochastic sampling method is fast and widely applicable to implicit surfaces defined with twice differentiable constraint functions. It is applicable to non-polynomial surfaces and surfaces with branches, too. It does not require an initial sample point on a sampled surface to start sampling. It works well with a single-particle system without necessity of introducing interaction between particles. It also has advantageous features at the rendering stage.

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Sampling implicit objects with physically-based particle systems

Abstract

Computer Aided Geometric Design

Contour tracing by piecewise linear approximations

ACM Transactions on Graphics

Adaptive polygonization of implicitly defined surfaces

IEEE Computer Graphics & Applications

A simple and relatively efficient triangulation of the n-cube

Discrete and Computational Geometry

Affine arithmetic and its applications to computer graphics