Farthest-point optimized point sets with maximized minimum distance

Abstract

δ X =0.009,δ X =0.469 input δ X =0.049,δ X =0.645 1/4 iteration δ X =0.079,δ X =0.756 1/2 iteration δ X =0.772,δ X =0.865 1 iteration δ X =0.814,δ X =0.905 2 iterations δ X =0.925,δ X =0.932 63 iterationsFigure 1: Farthest-point optimization of a random point set with 1024 points. Both the global minimum distance δX and the average minimum distanceδX increase rapidly using our optimization technique. After one iteration the point set is already well-distributed. AbstractEfficient sampling often relies on irregular point sets that uniformly cover the sample space. We present a flexible and simple optimization strategy for such point sets. It is based on the idea of increasing the mutual distances by successively moving each point to the "farthest point," i.e., the location that has the maximum distance from the rest of the point set. We present two iterative algorithms based on this strategy. The first is our main algorithm which distributes points in the plane. Our experimental results show that the resulting distributions have almost optimal blue noise properties and are highly suitable for image plane sampling. The second is a variant of the main algorithm that partitions any point set into equally sized subsets, each with large mutual distances; the resulting partitionings yield improved results in more general integration problems such as those occurring in physically based rendering.

Variational Blue Noise Sampling

“…To avoid the regularity artifacts, a statistical model is defined to allow solutions that are less energetically-favorable. By iteratively enlarging the minimum distance between points, Schlömer et al [22] construct irregular distributions with a significantly higher minimum distance than previous methods.…”

“…However, Schlömer (e) CapCVT result with ρ(x) = ̺ 2 (x) and ̺(x) = ϕ(x) (λ = 30) et al [22] show that Poisson disk distributions can have radii up to 0.93 or higher. Considering also the spectrum properties, we found that our method with λ ∈ [20,100] gives a point distribution with good blue noise characteristics.…”

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.…”

“…Typical optimization algorithms include Lloyd's algorithm [Lloyd 1982;McCool and Fiume 1992], a localized version of Farthest Point Optimization (FPO) [Schlömer et al 2011], and the more recent Push-Pull Optimization (PPO) . Since all the mentioned algorithms are based on a Delaunay triangulation of the point set, they can easily be combined, leading to a sequence of optimizations applied to each visited point.…”

AA patterns for point sets with controlled spectral properties

“…Direct production algorithms to generate such point sets include stratified jittering, dart throwing [Dippé and Wold 1985;Cook 1986;Mitchell 1987], and their variants. There are also iterative optimization techniques to modify a given point set, including Lloyd's relaxation algorithm [McCool and Fiume 1992] and its variants [Balzer et al 2009;Xu et al 2011;Chen et al 2012;de Goes et al 2012], other iterative methods [Schmaltz et al 2010;Fattal 2011;Schlömer et al 2011], and the recently invented target-matching algorithms [Zhou et al 2012;Öztireli and Gross 2012;Heck et al 2013]. …”