Variational Blue Noise Sampling

Abstract

Abstract-Blue noise point sampling is one of the core algorithms in computer graphics. In this paper we present a new and versatile variational framework for generating point distributions with high-quality blue noise characteristics while precisely adapting to given density functions. Different from previous approaches based on discrete settings of capacity-constrained Voronoi tessellation, we cast the blue noise sampling generation as a variational problem with continuous settings. Based on an accurate evaluation of the gradient of an energy function, an efficient optimization is developed which delivers significantly faster performance than the previous optimization-based methods. Our framework can easily be extended to generating blue noise point samples on manifold surfaces and for multi-class sampling. The optimization formulation also allows us to naturally deal with dynamic domains, such as deformable surfaces, and to yield blue noise samplings with temporal coherence. We present experimental results to validate the efficacy of our variational framework. Finally, we show a variety of applications of the proposed methods, including non-photorealistic image stippling, color stippling, and blue noise sampling on deformable surfaces.

Weighted linde-buzo-gray stippling

“…More recently, Xin et al [2016] improve the speed of this method by an order of magnitude by restricting kernel functions to squared Euclidean distances. Chen et al [2012] propose a variational framework for producing Blue Noise samples in 2D and on geometry with a variant of CCVT. Their combined energy term allows sets to adapt very fast to different density levels.…”

Low-discrepancy blue noise sampling

“…Recently, a number of new optimization-based algorithms have been proposed for blue noise [Balzer et al 2009;Chen et al 2012;Schmaltz et al 2010;Schlömer et al 2011;Fattal 2011;de Goes et al 2012;Jiang et al 2015], as well as general noise [Zhou et al 2012;Öztireli and Gross 2012;Heck et al 2013]. Since our method relies on an offline optimization (Section 4), many of these algorithms can be used.…”

“…The first obstacle is that a regular grid is a stable minimum for the associated energy of many optimizers (Lloyd's algorithm [McCool and Fiume 1992] and kins [Balzer et al 2009;de Goes et al 2012;Chen et al 2012;Xu et al 2011]); hence, such optimizers tend to restore the grid structure of the jittered grid. Rather than employing direct optimization algorithms, we overcome this problem by employing a target-matching algorithm, such as Heck et al [2013].…”

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]. …”