Blue noise through optimal transport

Abstract

We present a fast, scalable algorithm to generate high-quality blue noise point distributions of arbitrary density functions. At its core is a novel formulation of the recently-introduced concept of capacityconstrained Voronoi tessellation as an optimal transport problem. This insight leads to a continuous formulation able to enforce the capacity constraints exactly, unlike previous work. We exploit the variational nature of this formulation to design an efficient optimization technique of point distributions via constrained minimization in the space of power diagrams. Our mathematical, algorithmic, and practical contributions lead to high-quality blue noise point sets with improved spectral and spatial properties.

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.…”

Numerical resolution of Euler equations through semi-discrete optimal transport

“…Coordinate-wise increments were the first introduced optimization method [37], and have cubic complexity O(N 3 (ln N )/ε) where N := #(Y ) and ε is the desired precision. Gradient based methods such as LBFGS are much more efficient in practice [32,29], and are themselves outperformed by Newton methods [21]. Their implementation requires to identify the first and second derivatives of the Kantorovich functional.…”

“…The expressions of the second derivatives, first identified in [21], involve integrating the density ρ over Laguerre cell boundaries. Assuming a quadratic cost c(x, y) = 1 2 |x − y| 2 , the Kantorovich functional is twice differentiable, and for all distinct y, z ∈ Y …”

Earth mover's distances on discrete surfaces

“…More recently, [Bonneel et al 2011] applies approximations of optimal transportation to interpolate between BRDFs, intensity histograms, and other simple distributions; similar problems are considered in [Bonneel et al 2013] after defining the barycenter of a set of distributions with respect to approximated transportation distances. de Goes et al 2012] compute transportation distances from two-dimensional point sets for application in shape processing and blue noise generation, while [Mullen et al 2011] employs a similar formulation to triangulation problems. These distances also have been applied to geometry analysis [Lipman and Daubechies 2011;Lipman et al 2013], spherical parameterization [Dominitz and Tannenbaum 2010], and matching [Mémoli 2011;Solomon et al 2012].…”