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.

Variational Blue Noise Sampling

“…Recently, Balzer et al [5] presented a variant of Lloyd's method for generating point distribution by introducing the capacity-constrained Voronoi tessellation (CapVT), in which each point obtains equal capacity (i.e., the mass of its Voronoi cell). By requiring also that each point coincides with the centroid of its Voronoi cell in the CapVT, the resulting point distributions were shown to possess the blue noise properties.…”

“…Inspired by [5] and [4], we propose a variational framework based on a new energy function combining the CVT energy and the CapVT energy. Both the capacity constraint and the centroid constraint are considered as soft constraints in our framework.…”

“…Our method is implemented in the continuous setting and achieves significantly faster performance (generally two orders of magnitude faster for a large number of points) over the method of [5] without sacrificing blue noise proper-ties. We also study the accommodation of the density functions and reveal the relationship between the density functions used in the CVT and CapVT energy functions.…”

“…Based on the accurate evaluation of gradient of the energy function, we present an efficient optimization approach to minimizing the energy function and gain significantly faster performance than the method of [5].…”

An adaptive point sampler on a regular lattice

“…There are many algorithms for this purpose, e.g. Balzer et al 2009;Cook 1986;de Goes et al 2012;Fattal 2011;Jiang et al 2015;McCool and Fiume 1992;Schlömer et al 2011]. In all these algorithms the production cost (time and memory) is high, which leads to the idea of tabulating the blue-noise sets for subsequent reuse, replacing on-the-fly generation of sample points by a lookup framework.…”