Capacity-constrained point distributions

Abstract

Figure 1: (Left) 1024 points with constant density in a toroidal square and its spectral analysis to the right; (Center) 2048 points with the density function ρ = e (−20x 2 −20y 2 ) + 0.2 sin 2 (πx) sin 2 (πy); (Right) 4096 points with a density function extracted from a grayscale image. AbstractWe present a new general-purpose method for optimizing existing point sets. The resulting distributions possess high-quality blue noise characteristics and adapt precisely to given density functions. Our method is similar to the commonly used Lloyd's method while avoiding its drawbacks. We achieve our results by utilizing the concept of capacity, which for each point is determined by the area of its Voronoi region weighted with an underlying density function. We demand that each point has the same capacity. In combination with a dedicated optimization algorithm, this capacity constraint enforces that each point obtains equal importance in the distribution. Our method can be used as a drop-in replacement for Lloyd's method, and combines enhancement of blue noise characteristics and density function adaptation in one operation.

Variance analysis for Monte Carlo integration

“…For example, the tiled version of Poisson Disk sampling in Wachtel et al [2014] has power spectrum converging to zero towards zero frequency. Similarly, all methods based on the capacity constraint [Balzer et al 2009] seem to guarantee a power spectrum that converges to zero. However, these methods still have residual energy spread along low frequencies which can be improved.…”

“…Mitchell [1991] has also pointed out that energy in low-frequency part of the Fourier spectrum of the sampling pattern should be avoided. Since then, different algorithms [Balzer et al 2009;Schlömer et al 2011;de Goes et al 2012] and tile-based Spherical harmonic function of degree l and order m whose inner product over the sphere is 4π P m l (cos θ) associated Legendre polynomial of degree l and order m. Θ the Fourier domain without the DC peak. Φ the Fourier domain with the DC peak.…”

“…Similar observations are made in the spherical domain for the jittered sampling patterns. CCVT [Balzer et al 2009], jittered, and regular sampling patterns have asymptotically similar behaviors in the spherical domain, though jittered sampling patterns show higher variance.…”

Parallel Blue‐noise Sampling by Constrained Farthest Point Optimization

“…Unfortunately, CVT usually closely resembles a regular hexagonal grid, exhibiting high regularity with no blue noise characteristics, so the process must be stopped before it converges. Balzer et al [BSD09] proposed the Capacity-Constrained Voronoi Tessellation (CCVT), a modification of the CVT, by imposing an extra capacity constraint -that all generalized Voronoi cells should have the same area. They demonstrate that the resulting point distributions possess high-quality blue noise spectra and the distribution does not converge to the hexagonal grid due to the random initialization in their pixel-based algorithm.…”

Weighted linde-buzo-gray stippling

“…First, there is no clear mechanism to stop the iteration-it will finally end up in hexagonal subsets (cf. Balzer et al [2009]) which are distracting for visual applications. Second, a good initial point distribution has to be given.…”