Variable Radii Poisson Disk Sampling

We introduce three natural and well-defined generalizations of maximal Poisson-disk sampling. The first is to decouple the disk-free (inhibition) radius from the maximality (coverage) radius. Selecting a smaller inhibition radius than the coverage radius yields samples which mix advantages of Poisson-disk and uniform-random samplings. The second generalization yields hierarchical samplings, by scaling inhibition and coverage radii by an abstract parameter, e.g. time. The third generalization is to allow the radii to vary spatially, according to a formally characterized sizing function. We state bounds on edge lengths and angles in a Delaunay triangulation of the points, dependent on the ratio of inhibition to coverage radii, or the sizing function’s Lipschitz constant. Hierarchical samplings have distributions similar to those created directly. 1 Maximal Poisson-disk Sampling A sampling is a set of ordered points taken from a domain at random. Each point is the center of a disk that precludes additional points inside it, but points are otherwise chosen uniformly. The sampling is maximal if the entire domain is covered by disks. Together these define maximal Poisson-disk sampling (MPS). More formally, a sampling X = (xi) n i=1, xi ∈ Ω satisfies the inhibition or empty disk property if ∀i < j ≤ n, |xi − xj | ≥ r. (1) The set of uncovered points is defined to be S(X) = {y ∈ Ω : |y− xi| ≥ r, i = 1..n}. (2) A sampling X is maximal if S(X) is empty: S(X) = ∅. (3) Given a non-maximal sampling, the next sample is biasfree if the probability of selecting it from any uncovered subregion is proportional to the subregion’s area, i.e., ∀A ⊂ S(X) : P (xn+1 ∈ A |X) = |A| |S(X)| . (4) ∗Sandia National Laboratories, [email protected] †Institute for Computational Engineering and Sciences, The University of Texas at Austin ‡Sandia National Laboratories §Dept. of Computer Science and Institute for Computational Engineering and Sciences, The University of Texas at Austin We generalize these equations: decoupling the radii in the empty disk and uncovered equations; scaling the radii for a hierarchy of denser samplings; and varying the radii spatially by a sizing function. The purpose of this short paper is to introduce these generalizations in a mathematically consistent way. Examples illustrate the properties of the resulting output distributions. For simplicity our language is twodimensional, e.g. “disks” instead of “spheres,” but the definitions are general dimensional. Also for simplicity, we consider only periodic (or free-boundary) domains. These domains are used in some applications: computer graphics texture synthesis and mesh generation of material grains. 2 Motivation and Previous Work An MPS sampling is a separated-yet-dense point set: points are not too close together and lie throughout the entire domain. This is an efficient way to distribute a fixed budget of points. In mesh generation, separated-yet-dense points yield Delaunay triangulations (DT) with provable quality bounds [4, 9, 19]. Delaunay Refinement (DR) [20] introduces points to improve DT triangle quality and a separated-yet-dense point set follows. Variations of DR provide adaptivity and sizing control [16]. DR is usually deterministic; although regions of acceptable points have been characterized [12, 13], and one may select from regions randomly to improve tetrahedron quality [5], randomized point positions are not a traditional requirement. However, random meshes are of independent interest for certain applications; e.g. in some fracture mechanics methods, cracks propagate only along mesh edges. Meshes from MPS produce more physically realistic cracks [1, 2, 8, 7]. Ensembles of MPS meshes can model natural material strength variations. In a sphere packing no two disks overlap. If the disk radii satisfy a Lipschitz condition then a quality mesh results [19]. As in MPS and in reverse to DR, algorithms add disks until the packing is (nearly) maximal, and a good-quality DT follows. A fixed-r MPS sampling is a sphere packing: halve the disk radius r so no disks overlap. We define four new spatial variations for MPS, however none are equivalent to maximal sphere packings. Conflicts are defined by disks containing each other’s centers; for unequal radii this is not equivalent to non-overlapping 1/2-radii disks. Also, we achieve a 24 Canadian Conference on Computational Geometry, 2012 maximal distribution following a characterized statistical process. MPS is popular for computer graphics [15] for texture synthesis because the distribution avoids repeating patterns of distances between points which produce visible artifacts. Fixed radius disks are traditional, but not suitable in all situations. In real-time games and data exploration [17] with level-of-detail adaptivity, renderings use a finer sampling as the camera zooms in. Switching between discrete sets of samples is common, but has the potential to introduce visible artifacts or scene jumps [23]. Our definitions enable smoothly increasing density in time. Spatially varying samplings are useful for objects with varying curvature and lighting [3, 14]. Curvature and solution gradients motivate spatially-varying finite element meshes, and incremental adaptivity is preferred over mesh replacement. Varying density sampling is popular in Graphics but often the algorithms are heuristic, and the requirements not well understood. This paper seeks to provide some formal guidance. For example, the spatially-varying sampling algorithm of Bowers et al. [3] uses a datastructure that holds all the nearby points whose Poissondisks might conflict with a new point. This datastructure sometimes overflows in practice. We show that this is the fault of the input and not their algorithm: the bigger-disks criteria in Section 5 shows that a sizing function with Lipschitz constant L < 1/2 is necessary to bound the number of nearby points. Classic dart throwing [6] generates samples and rejects those inside prior disks. The probability of generating an acceptable sample becomes vanishingly small, so maximality is not reached. After many rejected samples McCool and Flume [18] reduce the radii of disks, either locally or globally, to make room for more samples. An adaptive MPS variation [23] for deforming point clouds coarsens to remove points that are too close together, and refines to re-achieve maximality. For coarsening the disk-free and maximal criteria hold approximately, subject to a tolerance band. In Section 3 we effectively tune this tolerance band by the ratio of the two radii, and scale the radii continuously in Section 4. 3 Different inhibition and coverage radii Here we relax the condition that the coverage and inhibition radii are equal. We focus on a particular relaxation that proves useful for generating hierarchical point sets, and flatter FFT radial power spectra. Let Rf ≤ Rc denote the inhibition and coverage radii, respectively. The empty disk property is ∀i < j ≤ n, |xi − xj | ≥ Rf . (5) The set of free points is defined to be S(X) = {y ∈ Ω : |y− xi| ≥ Rf , i = 1..n}. (6) The set of uncovered points is defined to be U(X) = {y ∈ Ω : |y− xi| ≥ Rc, i = 1..n}. (7) The sampling is maximal if U(X) is empty,